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1. 15 i i 1750 1800 1850 i 1900 i 1950 2000 The residuals are more random than for the simple polynomial fit As might be expected the residuals tend to get larger in magnitude as the population increases But overall the logarithmic model provides a more accuratefit tothe population data 6 27 6 Data Analysis and Statistics 6 28 Error Bounds Error bounds are useful for determining if your data is reasonably modeled by the fit An optional second output parameter can be obtained frompol yfit and passed as an input parameter topol yval in order to obtain the error bounds For example the code below usespol yf it andpol yval toproduceerror bounds for a second order polynomial model Year values are normalized This code uses an interval of 2A corresponding to a 95 confidence interval p2 52 polyfit sdate pop 2 pop2 del2 polyval p2 sdate 2 plot cdate pop cdate pop2 g cdate pop2 2 del2 r cdate pop2 2 del2 r grid on 300 25075 200 150 100 505 L L L L 1750 1800 1850 1900 1950 2000 Difference Equations and Filtering Difference Equations and Filtering MATLAB has functions for working with difference equations and filters These functions operate primarily on vectors Vectors are used to hold sampled data signals or sequences for signal processing and data analysis For multi input systems each row of a
2. 0 0 ees 2 26 Memory Utilization 0 00 0 2 33 Microsoft Windows Handbook 2 35 UNIX Handbook 000000 c eee eee 2 54 3 MATLAB Debugger 0000e cece eee eae 3 2 M File Profiler 0 0 0c eee eee eens 3 17 Contents Contents Matrices and Linear Algebra 4 Matrices and Linear Algebra l una auaa 4 2 Matrices in MATLAB 0 00 0c ee eee eee 4 4 Solving Linear Equations 0 0000 4 13 Inverses and Determinants 2 0055 4 20 LU QR and Cholesky Factorizations 4 24 Matrix Powers and Exponentials 4 31 Eigenvalues 20c 0c 0 eens web de oe ee een 4 34 Singular Value Decomposition 4 38 Polynomials and Interpolation 5 Polynomials auaa 5 2 Interpolation uua 5 9 Data Analysis and Statistics 6 Column Oriented Data Sets uaaa aaaea 6 3 Basic Data Analysis Functions 000000s 6 7 Data Pre Processing 0 0 cece eee eee eee 6 12 Regression and Curve Fitting 6 15 Case Study Curve Fitting 0 0 0 ea eee 6 20 Difference Equations and Filtering 6 29 Fourier Analysis and the Fast Fourier Transform FFT 6 31 Function Functions 7 Representing Functionsin MAT
3. matches findstr line literal num length matches if num gt 0 y y num fprintf 1 d s n num line end end fclose fid Given the following input datafile called badpoem Oranges and emons Pineapples and tea Orangutans and monkeys Dragonflys or fleas Calling the i t count function with the string an produces the output litcount badpoem an 2 Oranges and lemons 1 Pineapples and tea 3 Orangutans and monkeys Reading Formatted Text Thefscanf function is likethef scanf function in standard C Both functions operate in a similar manner reading a line of data from a file and assigning it to one or more variables Both functions use the same set of conversion specifiers to control the interpretation of the input data The conversion specifiers for f scanf begin with a character common conversion specifiers include 15 14 Formatted Files e s to match a string e d to match an integer in base 10 format e g to match a double precision floating point value Despite all the similarities between the MATLAB and C versions off scanf there are some significant differences For example consider a file named moon dat for which the contents are as follows 3 654234533 2 71343142314 5 34134135678 The following code reads all three elements of this file into a matrix named MyData fid fopen moon dat r MyData fscanf fid q status fclose f
4. 1 12 17 12 Multidimensional Arrays To access a subimage use subi mage RGB 20 40 50 85 The RGB image is a good example of data that needs to be accessed in planes for operations like display or filtering In other instances however the data itself might be multidimensional F or example consider a set of temperature measurements taken at equally spaced points in a room Here the location of each value is an integral part of the data set the physical placement in three space of each element is an aspect of the information Such data also lends itself to representation as a multidimensional array Array TEMP es 68 0 68 0 67 8 7 5 OPO 67 89 67 69 77777 67 89 67 6 67 6 Now to find the average of all the measurements use mean mean mean TEMP To obtain a vector of the middle values element 2 2 in the room on each page use B TEMP 2 2 12 18 M ultidimensional Cell Arrays Multidimensional Cell Arrays Like numeric arrays the framework for multidimensional cell arrays in MATLAB is an extension of the two dimensional cell array model You can use thecat function to build multidimensional cell arrays just as you use it for numeric arrays For example create a simple three di mensional cell array c A l 1 1 2 4 5 A 1 2 Name A 2 1 2 41 A 2 2 7 B 1 1 Name2 B 1 2 3 B 2 1 0 1 3 B 2 2 4 5 C cat 3 A
5. Structure struct data repmat struct x 1 3 array repmat Yolo OLN 5 E E Preallocation also helps reduce memory fragmentation if you work with large matrices In the course of a MATLAB session memory can become fragmented due to dynamic memory allocation and deallocation This can result in plenty of free memory but not enough contiguous space to hold a large variable Preallocation helps prevent this by allowing MATLAB to grab sufficient space for large data constructs at the beginning of a computation 10 65 10 M File Programming 10 66 Notes on Memory Use This section discusses ways to conserve memory and improve memory use Memory Management Functions MATLAB has five functions to improve how memory is handled e clear removes variables from memory e pack Saves existing variables to disk then reloads them contiguously Because of time considerations you should not usepack within loops or M file functions e quit exits MATLAB and returns all allocated memory to the system e save selectively stores variables to disk e oad reloads a data file saved with thes ave command Note save and oad are faster than MATLAB low level file 1 O routines save and oad have been optimized to run faster and reduce memory fragmentation See Chapter 2 MATLAB Working Environment for details on these functions On some systems the whos command displays the amount of free memory remaining However
6. To convert between the two representations let speye n bean identity matrix of the appropriate size Then 1 pyr I1 p 1in P P 1 n o gt UU TheinverseofP issimplyR P You can compute the inverse of p with r p lin r p 1 5 r Reordering for Sparsity Reordering the columns of a matrix can often make its Cholesky LU or QR factors sparser The simplest such reordering is to sort the columns by nonzero count This is sometimes a good reordering for matrices with very irregular structures especially if there is great variation in the nonzero counts of rows or columns The function p col perm computes this column count permutation The col permM file has only a single line ignore p sort full sum spones S This line performs these steps 1 Theinner call tos pones creates a sparse matrix with ones at the location of every nonzero element ins 2 Thesum function sums down the columns of the matrix producing a vector that contains the count of nonzeros in each column Sparse M atrix O perations 3 full converts this vector to full storage format 4 sort sorts the values in ascending order The second output argument from sort is the permutation that sorts this vector Reordering to Reduce Bandwidth The reverse Cuthill M cK ee ordering is intended to reduce the profile or bandwidth of the matrix It is not guaranteed to find the smallest possible bandwidth
7. Compare the original data and the smoothed data with an overlaid plot of the two curves t l length x plot t x t y grid on legend Original Data Smoothed Data 2 120 i r iah Original Data i Smoothed Data 100 1i 80F 60 F 40 207 The filtered data represented by the solid line is the 4 hour moving average of the observed traffic count data represented by the dashed line For practical filtering applications the Signal Processing Toolbox includes numerous functions for designing and analyzing filters Fourier Analysis and the Fast Fourier Transform FFT Fourier Analysis and the Fast Fourier Transform FFT Fourier analysis is extremely useful for data analysis as it breaks down a signal into constituent sinusoids of different frequencies For sampled vector data Fourier analysis is performed using the discrete Fourier transform DFT The fast Fourier transform FFT is an efficient algorithm for computing the DFT of a sequence it is not a separate transform It is particularly useful in areas such as signal and image processing where its uses range from filtering convolution and frequency analysis to power spectrum estimation MATLAB provides a collection of functions for computing and working with Fourier transforms Function Description fft fft2 fftn ifft ifft2 ifftn abs angle unwrap fftshift cpl xpair next pow2 Discrete
8. X jordan A X 1 7500 1 5000 2 7500 3 0000 3 0000 3 0000 1 2500 1 5000 1 2500 1 0 0 0 1 1 0 0 1 Thej ordan Canonical Form is an important theoretical concept but it is not a reliable computational tool for larger matrices or for matrices whose elements are subject to roundoff errors and other uncertainties 4 36 Eigenvalues MATLAB s advanced matrix computations do not require eigenvalue decompositions They are based instead on the Schur decomposition A USU where U is an orthogonal matrix and S is a block upper triangular matrix with 1 by 1 and 2 by 2 blocks on the diagonal The eigenvalues are revealed by the diagonal elements and blocks of S whilethe columns of U providea basis with much better numerical properties than a set of eigenvectors The Schur decomposition of our defective example is U S schur A U 0 4741 0 6571 0 5861 0 8127 0 0706 0 5783 0 3386 0 7505 0 5675 1 0000 21 3737 44 4161 0 1 0081 0 6095 0 0 0001 0 9919 The double eigenvalue is contained in the lower 2 by 2 block of S 4 37 4 Matrices and Linear Algebra 4 38 Singular Value Decomposition A singular value and corresponding singular vectors of a rectangular matrix A are a scalar o and a pair of vectors u and v that satisfy AV ou Alu ov With the singular values on the diagonal of a diagonal matrix and the corresponding singular vectors forming the columns of two ort
9. additional solutions at equally spaced points within each time step or view statistics about the computations Property Value Description OutputFcn String The name of an output function Out put Sel Vector of indices Indices of solver output components to pass to an output function Refine Positive integer Produces smoother output increasing the number of output points by a factor of Refine If Refine isl the solver returns solutions only at the end of each time step If Refine isn gt 1 the solver uses continuous extension to subdivide each time step inton smaller intervals and returns solutions at each time point Refi ne is1 by default in all solvers except ode45 whereit is 4 because of the solver s large step sizes Refine does not apply when length tspan gt 2 Stats on off Specifies whether statistics about the solver s computations should be displayed OutputFcn The Out put Fen property lets you define your own output function and pass the name of this function tothe ODE solvers If no output arguments are specified the solvers call this function after each successful time step You can use this feature for example to plot results as they are computed You must code your output function in a specific way for it to interact properly with the ODE solvers When the name of an executable M file function e g myf un iS passed to an ODE solver as the Out put Fcn property options odeset OutputFc
10. b 2 4 6 Calculate the derivative of the product a b by calling pol yder with a single output argument c polyder a b 8 30 56 38 5 5 5 Polynomials and Interpolation Calculate the derivative of the quotient a b by callingpol yder with two output arguments q d polyder a b q 4 16 40 48 36 q d is the result of the operation Polynomial Curve Fitting polyfit finds the coefficients of a polynomial that fits a set of dataina least squares sense p polyfit x y n x andy are vectors containing the x and y data to be fitted and n is the order of the polynomial to return For example consider the x y test data x 12 3 4 5 y 5 5 43 1 128 290 7 498 4 A third order polynomial that approximately fits the data is p polyfit x y 3 p 0 1917 31 5821 60 3262 35 3400 5 6 Polynomials Compute the values of the pol yf it estimate over a finer range and plot the estimate over the real data values for comparison x2 1 1 5 y2 polyval p x2 plot x y 0 x2 y2 grid on To use these functions in an application example see Chapter 6 Partial Fraction Expansion residue finds the partial fraction expansion of the ratio of two polynomials This is particularly useful for applications that represent systems in transfer function form For polynomials b and a if there are no multiple roots b s ri r3 Fh gt t k a S S P S P S Ph a
11. information Use thes ave command and then write a program in Fortran or C totranslate the MAT file into the desired format See the MATLAB Application Program Interface Guide for more information Delimiter Separated Text Files The functions d mr ead and dl mwr i te let you read and write delimiter separated values from an ASCII data file A ddimiter is any character that separates the file s values These functions are also useful for reading or writing into a specific MATLAB variable name For example consider a file named ph dat whose contents are separated by semicolons 12 8 5 6 2 6 6 5 4 9 2 8 1 7 2 To read the entire contents of this file into an array named A use A dil mread ph dat 2 29 2 MATLAB W orking Environment The second argument tod mr ead specifies the delimiter which in the previous example is a semicolon In addition to the delimiter you specify d mr ead also interprets all whitespace characters as delimiters So the preceding dl mr ead command works even if the contents of ph dat are Ta 2s 8 5 6 2 6 6 TEF 9 2 8 1 7 2 Note Thefirst argument tod mr ead is a filename not a file identifier Do not open the file with f open before usingdl mr ead ord mwrite Similarly di mwr i te writes delimiter separated text to an external file 1 2 3 4 5 6 dl mwrite myfile A myfile now contains PAoa 4 5 6 Reading Files that Have a Uniform Format Thef
12. that contains these statistics Jacobian Matrix Properties Thestiff ODE solvers often execute faster if you provide additional information about the J acobian matrix oF dy a matrix of partial derivatives of the function defining the differential equation OF oF OX Xp OF OF gt OX 8 25 8 O rdinary Differential Equations There are two aspects to providing information about the J acobian e You can set up your ODE file to calculate and return the value of the J acobian matrix for the problem In this case you must also useodeset to set the acobi an property e If you donot calculate thejJ acobianin the ODE file ode15s andode23s call the helper function numj ac to approximate J acobians numerically by finite differences In this case you may be abletousethe Constant Vectorized or J Pattern properties TheJ acobian matrix properties pertain only to the stiff solversode15s and ode23s for which the acobian matrix oF oy is critical to reliability and efficiency Property Value Description Constant on off Set on if the acobian matrix dF oy is constant does not depend ont ory Jacobian on Off Set on toinform the solver that the ODE file is coded such that F t y Jacobian returns oF oy J Pattern on ff Set on if dF oy is a sparse matrix and the ODE file is coded so that F J Pattern returns a sparsity pattern matrix Vectorized on off Set on toin
13. 12 6 M ultidimensional Arrays B 2 5 5 5 5 5 5 5 5 5 5 5 5 Note Any dimension of an array can have size zero making it a form of empty array For example 10 by 0 by 20 is a valid size for a multidimensional array Using the cat Function to Build Multidimensional Arrays Thecat function is a simple way to build multidimensional arrays it concatenates a list of arrays along a specified dimension B cat dim Al A2 whereAl A2 andsoon arethe arrays to concatenate and di mis the dimension along which to concatenate the arrays F or example to create a new array with cat B cat 3 2 8 0 5 1 3 7 9 Bie te ay ee 2 8 0 5 B 2 1 3 7 9 12 7 12 Multidimensional Arrays 12 8 Thecat function accepts any combination of existing and new data In addition you can nest calls tocat The lines below for example create a four dimensional array A cat 3 9 2 6 5 7 1 8 4 B cat 3 3 5 0 1 5 6 2 1 D cat 4 A B cat 3 1 2 3 4 4 3 2 1 cat automatically adds subscripts of 1 between dimensions if necessary F or example to create a 2 by 2 by 1 by 2 array enter C cat 4 1 2 4 5 7 8 3 2 In the previous case cat inserts as many singleton dimensions as needed to create a four dimensional array whose last dimension is not a singleton dimension If thedi m argument had been 5 the previous statement would have produced a 2 by 2 by 1 by 1 by 2 array This adds ad
14. 6 19 6 Data Analysis and Statistics 6 20 Case Study Curve Fitting This section provides an overview of some of MATLAB s basic data analysis capabilities in the form of a case study The examples that follow work with a collection of census data using MATLAB functions to experiment with fitting curves to the data Thefilecensus mat contains U S population data for the years 1790 through 1990 Load it into MATLAB load census Your workspace now contains two new variables cdate andpop e cdate isa column vector containing the years from 1790 to 1990 in increments of 10 e pop isa column vector with theU S population figures that correspond tothe years incdate Polynomial Fit A first try in fitting the census data might be a simple polynomial fit Two MATLAB functions help with this process Function Description polyfit Polynomial curve fit pol yval Evaluation of polynomial fit MATLAB s pol yfit function generates a best fit polynomial in the least squares sense of a specified order for a given set of data For a polynomial fit of the fourth order p polyfit cdate pop 4 Warning Matrix is close to singular or badly scaled Results may be inaccurate RCOND 5 429790e 20 1 0e 05 0 0000 0 0000 0 0000 0 0126 6 0020 Case Study Curve Fitting The warning arises because thepol yf it function uses thec date values as the basis for a matrix with very large values it creates a Va
15. 9 W day of week single letter 10 1996 year four digits 11 96 year two digits 12 Mar 96 month year 13 15 45 17 hour minute second 14 03 45 17 PM hour minute second AM or PM 15 15 45 hour minute 16 03 45 PM hour minute AM or PM 17 Q1 96 calendar quarter year 18 Ql calendar quarter Times and Dates Here are some examples of converting the date March 1 1996 to various forms using thedat estr function d 01 Mar 1996 d 01 Mar 1996 datestr d ans 01 Mar 1996 datestr d 2 ans 03 01 96 datestr d 17 ans Q1 96 10 59 10 M File Programming Current Date and Time The function date returns a string for today s date date ans 02 Oct 1996 The function now returns the serial date number for the current date and time now ans 729300 71 datestr now ans 02 Oct 1996 16 56 16 datestr floor now ans 02 Oct 1996 10 60 O btaining User Input Obtaining User Input To obtain input from a user during M file execution you can e Display a prompt and obtain keyboard input e Pause until the user presses a key e Build a complete graphical user interface This section covers the first two topics The third topic is discussed in Using MATLAB Graphics and Building GUIs with MATLAB Prompting for Keyboard Input Thei nput function displays a prompt and waits for a user response Its syntax is n input prompt_string The function di
16. In MATLAB there is no equivalent to an abstract class In MATLAB there is no equivalent to a J ava interface In MATLAB there is no equivalent to the C scoping operator In MATLAB there is no virtual inheritance or virtual base classes In MATLAB there is no equivalent to C templates References for Object Oriented Design For more detailed information about object oriented design we recommend these references Object Oriented Software Construction Bertrand Meyer Object Oriented Analysis and Design with Applications Grady Booch Designing User Classes in MATLAB Designing User Classes in MATLAB This section discusses how to approach the design of a class and describes the basic set of methods that should be included in a class The MATLAB Canonical Class When you design a MATLAB class you should include a standard set of methods that enable the class to behave in a consistent and logical way within the MATLAB environment Depending on the nature of the class you are defining you may not need to include all of these methods and you may include a number of other methods to realize the class s design goals This table lists the basic methods included in MATLAB classes Class Method Description class constructor display set and get subsref andsubsasgn end subsindex converters likedouble and char Creates an object of the class Called whenever MATLAB displays the contents of an o
17. In addition to the simple syntax all of the ODE solvers accept a fourth input argument opti ons which can be used to change the default integration parameters t y solver F tspan y0 options Theoptions argument is created with theodeset function see Creating an Options Structure The odeset Function on page 8 20 Any input parameters after theopt ions argument are passed to the ODE file every time it is called For example T Y solver F tspan y0 options pl p2 calls F t y flag pl p2 Obtaining Statistics About Solver Performance Use an additional output argument to obtain statistics about the ODE solver s computations T Y solver F tspan y0 options S isasix element column vector Element 1 is the number of successful steps Element 2 is the number of failed attempts Element 3 is the number of times the ODE file was called to evaluate F t y Element 4 is the number of times that the partial derivatives matrix dF dy was formed e Element 5 is the number of LU decompositions e Element 6 is the number of solutions of linear systems The last three elements of the list apply to the stiff solvers only The solver automatically displays these statistics if theSt ats property see 8 25 is set in theopt ions argument 8 13 8 O rdinary Differential Equations 8 14 Creating ODE Files Thevan der Pol examples in the previous sections show some simple ODE file
18. Q uestions and Answers Error Tolerance and Other Options Question Answer How dol tell the solver that don t care about getting an accurate answer for one of the solution components You can increase the absolute error tolerance corresponding to this solution component If the tolerance is bigger than the component this specifies no correct digits for the component The solver may have to get some correct digits in this component to compute other components accurately but it generally handles this automatically Solving Different Kinds of Systems Question Answer Can the solvers handle PDEs that have been discretized by the Method of Lines Can I solve differential algebraic equation DAE systems Yes What you obtain is a system of ODEs Depending on the discretization you might have a form involving mass matrices odel5s ode23s ode23t andode23tb provide for this Often the system is stiff This is to be expected when the PDE is parabolic and when there are phenomena that happen on very different time scales such as a chemical reaction in a fluid flow In such cases use one of the four solvers mentioned above If as usual there are many equations set the Patt ern property This is easy and might make the difference between success and failure due to the computation being too expensive When the system is not stiff or not very stiff ode23 or ode45 will be more efficient than od
19. asset a varargin l else error Wrong argument type end case 2 create object using specified values a descriptor varargin 1 a date date a current_value varargin 2 a class a asset otherwise error Wrong number of input arguments end The function uses as wi t ch statement to accommodate three possible scenarios e Called with no arguments the constructor returns a default asset object e Called with one argument that is an asset object the object is simply returned e Called with two arguments subclass descriptor and current value the constructor returns a new asset object The asset constructor method is not intended to be called directly it is called from the child constructors since its purpose is to provide storage for common data 14 39 14 MATLAB Classes and O bjects The Asset get Method Theasset class needs methods to access the data contained in asset objects The following function implements a get method for the class It uses capitalized property names rather than literal field names to provide an interface similar to other MATLAB objects function val get a prop_name GET Get asset properties fromthe specified object and return the value switch prop_name case Descriptor val a descriptor case Date val a date case Current Value val a current_value otherwise error prop_name Is not a valid asset property en
20. get p ind_assets i C elseif isa p ind_assets i sa Savings amt Savings amt get p ind_assets i get p elseif isa p c end end i 1 if stock amt 0 abel i Stocks pie_vector i stock_amt i 1 end if bond amt 0 abel i Bonds pie_vector i bond_amt i 1 end if savings_amt 0 abel i Savings pie_vector i Savings amt end pie3 pie_vector abel urrentVa nd urrentVa vings Poata urrentVa set gcf Renderer zbuffer set findobj gca Type Text FontSize 14 cm gray 64 colormap cm 48 end stg 1 Portfolio Composition stg 2 Total Value of Assets title stg FontSize 12 for p name num2str p total value Example The Portfolio C ontainer There are three parts in the overloaded pi e3 method e The first uses the asset subclass get methods to access the Current Val ue property of each contained object The total value of each class is summed e Thesecond part creates the pie chart labels and builds a vector of graph data depending on which objects are present e The third part calls the MATLAB pi e3 function makes some font and colormap adjustments and adds a title Creating a Portfolio Suppose you have implemented a collection of asset subclasses in a manner similar to the stock class You can then use a portfolio object to present the
21. MATLAB zaf On Windows platforms MATLAB provides a special window with si Windows only features X On UNIX systems the Command Window is the terminal window from which you start MATLAB The MATLAB interpreter displays a prompt gt gt indicating that it is ready to accept commands from you For example to enter a 3 by 3 matrix you can type A 12 3 4 5 6 7 8 10 When you press the Enter or Return key MATLAB responds with Toinvert this matrix enter B inv A MATLAB responds with the result Command Line Editing Arrow and control keys on your keyboard allow you to recall edit and reuse commands you have typed earlier For example suppose you mistakenly enter rho 1 sqt 5 2 You have misspelled sqrt MATLAB responds with Undefined function or variable sqt Instead of retyping the entire line simply press the M key The misspelled command is redisplayed Use the lt key to move the cursor over and insert the missingr Repeated use of the M key recalls earlier lines 2 MATLAB W orking Environment The commands you enter during a MATLAB session are stored in a buffer Y ou can use smart recall to recall a previous command whose first few characters you specify F or example typing the letters p o and pressing the N key recalls the last command that started with pI o as in the most recent pI ot command The complete list of arrow and control keys provides additional control M any of
22. Memory Use and Computation Speed Question Answer How large a problem can solve with the ODE suite The primary constraints are memory and time At each time step the nonstiff solvers allocate vectors of length n wheren is the number of equations in the system The stiff solvers allocate vectors of length n but also an n by n J acobian matrix F or these solvers it may be advantageous to use the sparse option If the problem is nonstiff or if you are using the sparse option it may be possible to solve a problem with thousands of unknowns In this case however storage of the result can be problematic Q uestions and Answers Problem Size Memory Use and Computation Speed Question Answer I m solving a very large system but only care about a couple of the components of y ls there any way to avoid storing all of the elements How many time steps is too many What is the startup cost of the integration and how can reduce it Yes The user installable output function capability is designed specifically for this purpose When an output function is installed and the solver call does not include output arguments the solver does not allocate storage to hold the entire solution history Instead the solver calls Out put Fcen t y at each time step To keep the history of specific elements write an output function that stores or plots only the elements you care about If your integrati
23. S S You can use the met hods command to produce a list of all of the methods that are defined for a class Private Methods Private methods can be called only by other methods of their class You define private methods by placing the associated M files in apri vate subdirectory of the cl ass_ name directory In the example class_name private update obj m the method update obj has scope only within theclass_ name class This means that updat e_obj can be called by any method that is defined in the class_name directory but it cannot be called from the MATLAB command line or by methods outside of the class directory including parent methods Private methods and private functions differ in that private methods in fact all methods have an object as one of their input arguments and private functions do not You can use private functions as helper functions such as described in the next section Helper Functions In designing a class you may discover the need for functions that perform support tasks for the class but do not directly operate on an object These functions are called hd per functions A helper function can be a subfunction in a class method file or a private function When determining which version of a particular function to call MATLAB looks for these functions in the order listed above For more information about the order in which MATLAB calls functions and methods see H ow MATLAB Determines Which Method to Ca
24. Sample 1 10 30 95 fi ndstr returns the starting position of a substring within a longer string To find all occurrences of the string amp inside abe position findstr amp abel position 2 The position within abel where the only occurrence of amp begins is the second character Thestrtok function returns the characters before the first occurrence of a delimiting character in an input string The default delimiting characters are the set of whitespace characters You can usethestrtok function to parse a sentence into words for example function all_words words input_string remainder input string all_words while any remainder chopped remainder strtok remainder all_words strvcat all_ words chopped end 11 12 String N umeric C onversion String Numeric Conversion MATLAB s string numeric conversion functions change numeric values into character strings You can store numeric values as digit by digit string representations or convert a value into a hexadecimal or binary string Consider a the scalar x 5317 By default MATLAB stores the number x as a 1 by 1 double array containing the value 5317 Thei nt 2str integer to string function breaks this scalar into a 1 by 4 vector containing the string 5317 y int2str x size y ans 1 4 A related function num2str provides more control over the format of the output string An optional second argume
25. Solution of van der Pol Equation mu 1 xlabel time t ylabel solution y legend yl y2 Solution of van der Pol Equation mu 1 yi i amp solution y 3 fi L fi fi fi fi 1 1 f 0 2 4 6 8 10 12 14 16 18 20 8 7 8 O rdinary Differential Equations 8 8 Example The van der Pol Equation u 1000 Stiff Stiff ODE Problems This section presents a stiff problem For a stiff problem solutions can change on a time scale that is very short compared to the interval of integration but the solution of interest changes on a much longer time scale Methods not designed for stiff problems are ineffective on intervals where the solution changes slowly because they use time steps small enough to resolve the fastest possible change When u is increased to 1000 the solution to the van der Pol equation changes dramatically and exhibits oscillation on a much longer time scale Approximating the solution of the initial value problem becomes a more difficult task Because this particular problem is stiff a nonstiff solver such as ode45 issoinefficient that it is impractical The stiff solver ode15s is intended for such problems This code shows how to represent the van der Pol system in an ODE file with u 1000 function dy vdpl000 t y dy y 2 1000 1 y 1 2 y 2 y 1 1 Now usetheodel5s function tosolvevdp1000 Retain the initial condition vector of 2 0 but useatimeint
26. The portfolio constructor method needs to be modified to create a new field account _number which is initialized to the empty string when an object is created saveobj A new portfolio method designed to add an account number toa portfolio object during the save operation only if the object does not already have one oadobj A new portfolio method designed to update older versions of portfolio objects that were saved before the account number structure field was added subsref A new portfolio method that enables subscripted reference to portfolio objects outside of a portfolio method get Account Number a MATLAB function that returns an account number that consists of the first three letters of the client s name The saveobj Method MATLAB looks for the portfolios aveobj method whenever thes ave command is passed a portfolio object If portfolio saveobj exists MATLAB passes the portfolio object tos aveobj which must then return the modified object as an output argument The following implementation of saveobj determines if the object has already been assigned an account number from a previous save operation If not saveobj callsgetAccount Number to obtain the number and assigns it totheaccount_number field function b saveobj a if isempty a account_ number a account_ number getAccount Number a end b a The loadobj Method MATLAB looks for the portfolio 0adobj method whenever the oad command detects portfo
27. case Type val get s asset Type case Current Value val get s asset CurrentValue otherwise error prop_name ls not a valid stock property end Note that the asset object is accessed via the stock object s asset field s asset MATLAB automatically creates this field when thec ass function is called with the parent argument The Stock subsref Method Thesubsref method defines subscripted indexing for the stock class In this example subsref is implemented to enable numeric and structure field name indexing of stock objects Example Assets and Asset Subclasses function b subsref s index SUBSREF Define field name indexing for stock objects fc fieldcount s asset switch index type case if index subs lt fc b subsref s asset index else switch index subs fc case 1 b s num shares case 2 b s share_ price otherwise error Index must be in the range 1 to mum2str fc 2 end end case switch index subs case num shares b s num shares case share price b s share_ price otherwise b subsref s asset index end end The outer s wi t ch statement determines if the index is anumeric or field name syntax Thefieldcount asset method determines how many fields there arein the asset structure and thei f statement calls the asset s ubsr ef method for indices 1 tofiel dcount See The Asset fieldcount Method on page 14 4
28. dbcont ans 2 5000 Ending A Debugging Session Usedbqui t to end the debugging session and return to the base workspace Editsqsum msothat itsf or statement runs from1 tol engt h x rather than 1 tol ength mu for i l length x 3 15 3 Debugger and Profiler Repeating the original trial run we now get the expected results variance v ans 2 5000 vari ance w ans 468 3876 3 16 M File Profiler M File Profiler One way to improve the performance of your M files is to profile them MATLAB provides an M file profiler that lets you see how much computation time each line of an M file uses Profiling An Overview Profilingis a way to measure where a program spends its time Measuringisa much better method than guessing where the most execution timeis spent Y ou probably deal with obvious speed issues at design time and can then discover unanticipated effects through measurement One key to effective coding is to create an original implementation that is as simple as possible and then use a profiler to identify bottlenecks if speed is an issue Premature optimization often increases code complexity unnecessarily without providing a real gain in performance Use a profiler to identify functions that are consuming the most time then determine why you are calling them and look for ways to minimize their use It is often helpful to decide whether the number of times a particular function is called i
29. grid on 10 fi fi L fi fi fi fi ji L 5 4 3 2 1 0 1 2 3 4 5 You can also pass an expression for f pl ot to graph asin fplot 2 sin x 3 1 1 You can plot more than one function on the same graph with one call tof plot If you use this with a M file function then the M file must take a column vector x and return a matrix where each column corresponds to each function evaluated at each value of x If you pass an expression of several functions tof pl ot it also must return a matrix where each column corresponds to each function evaluated at each value of x as in fplot 2 sin x 3 humps x J 1 1 7 Function Functions which plots the first and second expressions on the same graph 100 T 80F lt 60F 4 40 d 207 A 7 IV co OF ee ee ere Wee 20 L L L L id id j ji L 5 4 3 2 1 0 1 2 3 4 5 Note that the expression 2 sin x 3 humps x evaluates toa matrix of two columns one for each function when x isa column vector Minimizing Functions and Finding Zeros Minimizing Functions and Finding Zeros MATLAB provides a number of high level function functions that perform optimization related tasks This section describes e Minimizing a function of one variable e Minimizing a function of several variables e Setting minimization options e Finding a zero of a function of one variable For more sophisticated optimization capabilities
30. observation points Basic Data Analysis Functions Y ou can subtract the mean from each column of the data using an outer product involving a vector of n ones n p size count e ones n 1 X count e mu Rearranging the data may help you evaluate a vector function over an entire data set For example to find the smallest value in the entire data set use mi n count which produces ans 7 The syntax count rearranges the 24 by 3 matrix into a 72 by 1 column vector Covariance and Correlation Coefficients MATLAB s statistical capabilities include two functions for the computation of correlation coefficients and covariance Function Description cov Variance of vector measure of spread or dispersion of sample variable Covariance of matrix measure of strength of linear relationships between variables corrcoef Correlation coefficient normalized measure of linear relationship strength between variables cov returns the variance for a vector of data The variance of the data in the first column of count is cov count 1 ans 643 6522 6 9 6 Data Analysis and Statistics 6 10 For an array of data cov calculates the covariance matrix The variance values for the array columns arearranged along thediagonal of the covariance matrix The remaining entries reflect the covariance between the columns of the original array For an m by n matrix the covariance matrix has s
31. or one of them ke a scalar If the dimensions are incompatible an error results X A C Error using gt Matrix dimensions must agree w v s 4 6 Matrices in MATLAB Vector Products and Transpose A row vector anda column vector of the same length can be multiplied in either order The result is either a scalar the inner product or a matrix the outer product x vu X 2 X u y X 6 0 33 2 0 1 8 0 4 For real matrices the transpose operation interchanges aj and aji MATLAB uses the apostrophe or single quote to denote transpose Our example matrix A issymmetric soA is equal toA But B is not symmetric 4 7 4 Matrices and Linear Algebra 4 8 Ifx andy are both real column vectors the product x y is not defined but the two products x Fy and y x are the same scalar This quantity is used so frequently it has three different names inner product scalar product or dot product For a complex vector or matrix z the quantity z denotes the complex conj ugatetranspose The unconjugated complex transpose is denoted byz in analogy with the other array operations So if z 142i 344i then z is 1 2 3 4i whilez is 1 2i 3 4i For complex vectors the two scalar products x y andy x are complex conjugates of each other and the scalar product x x of a complex vector with itself is real Matrix Multiplication Multiplication of matrices is defined
32. sin x stores integers in the range from 0 to 255 in 1 8 the memory required for a double precision array No mathematical operations are defined for ui nt 8 arrays User defined data type O perators Operators MATLAB s operators fall into three categories e Arithmetic operators that perform numeric computations for example adding two numbers or raising the elements of an array toa given power e Relational operators that compare operands quantitatively using operators like less than and not equal to e Logical operators that use the logical operators AND OR and NOT Arithmetic Operators MATLAB provides these arithmetic operators Addition Subtraction as Multiplication aa Right division iN Left division Unary plus Unary minus Colon operator Power 10 21 10 M File Programming 10 22 Arithmetic O perators and Arrays Except for some matrix operators MATLAB s arithmetic operators work on corresponding elements of arrays with equal dimensions F or vectors and rectangular arrays both operands must be the same size unless oneis a scalar If one operand is a scalar and the other is not MATLAB applies the scalar to every element of the other operand this property is known as scalar expansion This example uses scalar expansion to compute the product of a scalar operand Transpose Complex conjugate transpose Matrix multiplication Matrix right division Matrix le
33. sqsum breakpoints Run variance again still using the vector v as input vari ance v 4 tot tot x i mu 2 Usedbstack to view the function call stack verifying that vari ance did call sqsum K gt gt dbstack In Pat Applications V5 sqsumm at line 4 In Pat Applications V5 variance m at line 3 Check the value of the loop index i then the value of t ot After checkingt ot dbstep again to check the next value ofi K gt gt i K gt gt dbstep K gt gt tot tot 4 K gt gt dbstep End of M file function Pat Applications V5 sqsum m The function only goes through the loop once ending after the first iteration From thef or statement for i 1 length mu MATLAB Debugger you can see that there is a mistake in the line The loop only iterates until the length of mu a scalar rather than the length of x the input vector Toseeif changingt ot to its expected value produces the correct answer type K gt gt tot 10 tot 10 Use thedbup command to move up one workspace context into thevari ance function workspace It s clear that thevari ance function was taking the returnedt ot value of 4 dividing by engt h x 1 also4 in this case and coming up with the incorrect answer of 1 Verify that it comes up with the correct answer now that tot has the correct value usingdbcont to continue execution K gt gt dbup In workspace belonging to Pat Applications V5 variance m K gt gt
34. startup 2 2 startup m 2 2 temporary 15 2 15 6 text 15 13 writing with C or Fortran programs 2 29 fill in of sparse matrix 9 20 filtering 6 29 fi nd function and sparse matrices 9 14 and subscripting 10 27 finding nonzero elements 6 13 substring within a string 11 12 finish m 2 4 finishdi g m 2 4 finishsav m2 4 finite differences 6 11 finite element discretization ODE example 8 40 first order differential equations representation for ODE solvers 8 5 flag argument ODE 8 18 FLEXIm license manager 2 59 float 15 8 floating point number largest 10 18 smallest 10 18 floating point operations count 10 18 floating point precision 15 8 floating point relative accuracy 10 18 flops 10 18 flow control 10 30 catch 10 36 else 10 30 elseif 10 30 for 10 35 if 10 30 return 10 36 switch 10 32 try 10 36 while 10 34 Index fmi nbnd 7 7 7 9 fminsearch 7 8 font in Command Window 2 39 in Editor 2 50 fopen 2 27 15 2 15 3 failing 15 4 for 10 35 13 32 example 10 35 loop indexing 10 35 matrices as indices 10 35 nested 10 35 syntax 10 35 vectorization 10 30 format date 10 54 numeric 2 38 Seealsos printf format 2 7 formatted files 15 13 formatting of syntax in Editor 2 49 Fortran programs for reading files 2 28 for writing files 2 29 Fourier analysis 6 31 Fourier transform fast FF T based interpolation 5 11 specifying length 6 38 fplot 7 4 fprintf 15 2 15 13 fragmentation reducing 10 65
35. temporary 15 6 type for input 15 8 data analysis finite differences 6 11 triangulation 5 18 data class hierarchy 14 3 data fitting 6 20 confidence intervals 6 28 error bounds 6 28 exponential fit 6 25 exponential fits 6 25 polynomial fits 6 20 data gridding multidimensional 5 17 data normalization 6 21 data organization cell arrays 13 28 multidimensional arrays 12 17 structure arrays 13 11 data types 10 19 cell 10 20 char 10 20 classes 14 3 double 10 20 for image processing 10 19 for toolbox creation 10 19 numeric 10 20 precision 15 8 char 15 8 double 15 8 float 15 8 long 15 8 short 15 8 uchar 15 8 sparse 10 20 sparse matrices 10 19 storage 10 20 struct 10 20 ui nt 10 20 user defined 14 3 User Obj ect 10 20 datatips 2 48 date 10 60 Index dat enum 10 55 10 56 dates base 10 54 conversions 10 55 formats 10 54 handling and converting 10 54 number 10 54 string vector of input 10 57 datestr 10 55 10 57 datevec 10 55 dbcl ear 3 10 dbcont 3 10 dbdown 3 10 dbquit 3 10 dbstack 3 10 dbstatus 3 10 dbstep 3 10 dbstop 3 10 dbtype 3 10 dbup 3 10 debl ank 11 6 Debugger 3 2 buttons 3 5 changing workspace context 3 8 continue 3 5 example 3 3 highlighting 3 7 pause 3 7 red stop sign 3 6 stack 3 2 3 8 step 3 5 stepping through code 3 12 techniques 3 2 values viewing 3 2 yellow arrow 3 7 debugging 2 38 2 39 debugging class methods 14 6 debugging commands 2 54 3 10 decimal representation to bin
36. theoretical 9 15 graphical debugging 2 38 graphics 2 18 2 19 annotations 2 18 greater than operator 10 23 greater than or equal to operator 10 23 griddata 5 21 H H1 line 2 22 10 7 and hel p command 10 3 and ookf or command 10 3 defined 10 3 Handle Graphics 1 4 handling and converting dates 10 54 Harwell Boeing data format 9 10 hecurve 7 15 help at command line 2 20 commands by subject 2 21 directory 2 38 for MATLAB 2 20 hel p 2 20 10 8 and H1 line 10 3 Help Desk 2 23 help text defined 10 3 Help Window 2 21 hel pdesk 2 23 hel pwin 2 21 hexadecimal converting from decimal 11 13 hierarchy of data classes 14 3 highlighting in Editor Debugger 2 39 humps 7 3 l I O S input output icons in command window 2 36 identity matrix 4 10 i f 10 30 and empty arrays 10 32 example 10 31 Index nested 10 31 image files reading 2 27 writing 2 29 image processing data type 10 19 imaginary unit 10 18 importing data 2 26 sparse matrix 9 10 improving solver performance 8 17 imread 2 27 imwrite 2 29 incomplete factorization 9 32 indenting in Editor Debugger 2 39 indexed reference 14 14 indexing 10 40 advanced 10 42 cell array 13 20 content 13 20 for loops 10 35 multidimensional arrays 12 10 nested cell arrays 13 30 nested structure arrays 13 17 structures within cell arrays 13 33 within structure fields 13 7 indices how MATLAB calculates 10 45 nf 10 18 inferiorto function 14 66 infinity represented in MATL
37. 0 pDevtools mat lab5 ni ght Functions Because rE detail builtin XI 0 03 15 0 34 0 0009 0 00 0 0 builtin Option was Used for 0 02 10 0 11 0 0018 0 01 5 0 Devtools mat labS night profile on diag oo 5 0 34 0 0003 0 01 5 0 biitin x 4 gt Document Done ge SP ye Z Time Per Function Time Per Function Induding Time Spent in Not Including Time Spent in Child Functions Called Child Functions Called 3 22 M File Profiler Function Details Profile Report The function details report provides statistics for the parent and child functions of a function and reports the line numbers on which the most time was spent Below is the detail report for the ot ka function which is one of the functions called in An Example Using the Profiler MATLAB Profiler Report Netscape OF x File Edit View Go Communicator Help Summary Function Details Function Call History lotka Devtools matlab5 nightly toolbox mat lab demos lotka mr Time 0 03s 15 0 Calls 34 Selftime 0 02s 15 0 Function Time Calls lotka 0 03 34 Parent functions oo feval 34 Child functions diag 0 01 333 66 of the total time in this function was spent on the following lines p diagihl 0tes 2 e ee Osh ta Time in Seconds Percentage of the Line Number Function s Time Spent on that Line 3 23 3 Debugger and Profiler Function Call History
38. 10 ode23s 8 11 8 41 8 53 8 55 8 56 description 8 11 ode45 8 6 8 8 8 10 8 23 8 24 8 53 8 55 description 8 10 odeget 8 21 odephas2 8 24 odephas3 8 24 odeplot 8 24 odeprint 8 24 odeset 8 20 offsets for indexing 10 45 one dimensional interpolation 5 9 5 10 cubic spline 5 10 linear 5 10 nearest neighbor 5 10 ones 9 23 12 6 one step solver ODE 8 10 online help 2 22 2 23 10 8 open 2 16 opening files 2 16 15 2 15 3 failing 15 4 operating system command 2 25 operator 10 21 applying to cell arrays 13 27 applying to structure fields 13 9 arithmetic 10 21 colon 8 28 10 21 12 6 12 10 12 16 13 24 complex conjugate 10 22 deletion 10 42 equal to 10 23 greater than 10 23 greater than or equal to 10 23 less than 10 23 less than or equal to 10 23 matrix power 10 22 not equal to 10 23 power 10 21 precedence 10 29 relational 10 23 second difference 9 7 subtraction 10 21 unary minus 10 21 operators amp 10 24 10 24 10 24 colon 4 8 logical 10 24 overloading 14 2 14 20 semicolon 2 9 table of 14 21 optimization 10 63 practicalities 7 12 preallocation array 10 65 troubleshooting 7 13 vectorization 10 63 optimizing performance of M files 3 17 options for Editor 2 55 for startup 2 3 Index local file 2 60 minimization 7 9 OR operator rules for evaluating 10 25 orbitode 8 46 order of function arguments 10 13 10 15 Ordinary Differential Equation solvers SeeODE solvers
39. 1001 values ranging from 0 to 10 i 0 fort 0 01 10 i i 41 y i sin t end A vectorized version of the same code is t 0 01 10 y sin t The second example executes much faster than the first and is the way MATLAB is meant to be used Test this on your system by creating M file scripts that contain the code shown then using thet i c and toc commands to time the M files An Advanced Example repmat is an example of a function that takes advantage of vectorization It accepts three input arguments an array A a row dimension M and a column dimension N 10 63 10 M File Programming repmat creates an output array that contains the elements of array A replicated and tiled in an M by N arrangement A 12 3 45 6 B repmat A 2 3 B 1 2 3 1 2 3 1 2 3 4 5 6 4 5 6 4 5 6 1 2 3 1 2 3 1 2 3 4 5 6 4 5 6 4 5 6 repmat uses vectorization to create the indices that place elements in the output array function B repmat A M N if nargin lt 2 error Requires at least 2 inputs elseif nargin N M end 1 Get row and column sizes m n size A 2 Generate vedtorsof indices Mmi nd 1 m from 1 to row column size mind 1 n mind mind ones 1 M Lnind nind ones 1 N B A mind nind 3 Create index matrices from vectors above 1 above obtains the row and column sizes of the input array 2 abovecreates twocolumn vectors m
40. 2 van der Pol Equation vdpode iS amore general version of the van der Pol example that has been used in various forms throughout this chapter For illustrative purposes it is coded for both fast numerical J acobian computation Vect ori zed property and for analytical J acobian evaluation Jacobi an property In practice you would supply only one or the other of these options It is not necessary to supply either The van der Pol equation is written as a system of two equations y d 2 y2 u 1 y1 Y2 Y1 vdpode t y orvdpode t y mu returns the derivatives vector for the van der Pol equation By default mu is 1 and the problem is not stiff Optionally pass in the mu parameter as an additional input argument to an ODE solver The problem becomes more stiff as mu is increased and the period of oscillation becomes larger 8 35 8 O rdinary Differential Equations 8 36 When mu is 1000 the equation is in relaxation oscillation and the problem is very stiff The limit cycle has portions where the solution components change slowly and the problem is quite stiff alternating with regions of very sharp change where it is not stiff quasi discontinuities This examplesetsVect ori zed on withodeset becausevdpode is coded so that vdpode t yl y2 returns vdpode t yl vdpode t y2 forscalar timet and vectors y1 y2 The stiff ODE solvers take advantage of this feature only when approximating the columns of the J acobian
41. 4 3 1 Input Dimension 1 2 3 4 The order of dimensions in array A Size 5 l4 3 per mut e s argument list deter mines the size and shape of the output array In this example the second dimension moves to the first position Because the second ae dimension of the original array had Output Dimension 2 3 f4 size four the output array s first array B E L 3 E dimension also has size four Size You can think of per mut e s operation as an extension of thet rans pose function which switches the row and column dimensions of a matrix F or per mut e the order of the input dimension list determines the reordering of the subscripts In the example above element 4 2 1 2 of A becomes element 2 2 1 4 of B element 5 4 3 2 of A becomes element 4 2 3 5 of B and so on Inverse Permutation Thei per mut e function is the inverse of per mute Given an input array A anda vector of dimensionsv i per mut e produces an array B such that per mut e B v returns A For example these statements create an array E that is equal to the input array C D ipermute C 1 4 2 3 E permute D 1 4 2 3 You can obtain the original array after permuting it by callingi per mut e with the same vector of dimensions Computation with M ultidimensional Arrays Computation with Multidimensional Arrays Many of MATLAB s computational and mathematical functions accept multidimensional arrays as argument
42. 9 27 9 Sparse Matrices 9 28 LU Factorization IfS is a Sparse matrix the statement below returns three sparse matrices L U and P such that P S L U L U P I u S u obtains the factors by Gaussian elimination with partial pivoting The permutation matrixP has onlyn nonzero elements As with dense matrices the statement L U u returns a permuted unit lower triangular matrix and an upper triangular matrix whose product iss By itself u S returnsL and U in a single matrix without the pivot information The sparse LU factorization does not pivot for sparsity but it does pivot for numerical stability In fact both the sparse factorization line 1 and the full factorization line 2 below producethesameL and U even though thetimeand storage requirements might differ greatly L U lu S sparse factorization L U sparse lu full full factorization MATLAB automatically allocates the memory necessary to hold the sparse L and U factors during the factorization MATLAB does not use any symbolic LU prefactorization to determine the memory requirements and set up the data structures in advance Reordering and factorization f you obtain a good column permutation p that reduces fill in perhaps from sy mr cm or col mmd then computing u p will take less time and storage than computing u S Two permutations are the symmetric reverse Cuthill McK ee ordering and the symmetric minimum degree orde
43. Asset Subclasses 14 37 Example The Portfolio Container 14 54 Saving and Loading Objects 5 14 61 Object Precedence 0 00 cc ccc eee eens 14 66 15 How MATLAB Determines Which Method to Call 14 68 Filel O Opening and Closing Files 00 ce ueues 15 3 Temporary Files and Directories 5 15 6 Binary Filesi aaea wae fees peed ees a ees 15 7 Controlling Position inaFile 15 10 Formatted Files 15 13 vii viii Contents Introduction WhatISMATLAB 0 0 0 004 1 3 The MATLAB System 2 ee ee es 1 4 How to Use the Documentation Set 1 6 About Simulink aoaaa a a a a a a 1 8 About Toolboxes 2 a a a e aD a a a a a a 1 8 1 Introduction About the Cover The cover of this guide depicts a solution toa problem that has played a small but interesting role in the history of numerical methods during the last 30 years The problem involves finding the modes of vibration of a membrane supported by an L shaped domain consisting of three unit squares The nonconvex corner in the domain generates singularities in the solutions thereby providing challenges for both the underlying mathematical theory and the computational algorithms There are important applications including wave guides structures and semiconductors Two of the founders of modern numeric
44. B The subscripts for the cells of look like cell 1 1 2 cell 1 2 2 Name2 Pes cell 2 1 2 cell 2 2 2 cell 1 1 clli2i i 3 a 5 1 2 Name 4 5 cell 2 1 1 cell 2 2 1 2 4i 7 12 19 12 Multidimensional Arrays Multidimensional Structure Arrays Multidimensional structure arrays are extensions of rectangular structure arrays Like other types of multidimensional arrays you can build them using direct assignment or the cat function 12 20 patient 1 1 name John Doe patient 1 1 1 billing 127 00 patient 1 1 test 79 75 73 180 178 177 5 220 210 205 patient 2 1 name Ann Lane patient 1 2 1 billing 28 50 patient 2 1 test 68 70 68 118 118 119 172 170 169 patient 1 2 name Al Smith patient 1 1 2 billing 504 70 patient 1 2 test 80 80 80 153 153 154 181 190 182 patient 2 2 name Dora Jones patient 1 2 2 billing 1173 90 patient 1 2 2 test 73 73 75 103 103 102 201 198 200 patient 1 1 2 patient 1 2 2 name AL Smith name Dora Jones billing 504 70 L billing 1173 99 test 80 80 80 test 775 735 75 153 153 154 103 103 102 181 190 182 201 198 200 patient 1 1 1 patient 1 2 1 name John Doe name Ann Lane L billing
45. Discretization on page 8 40 Event Location Property In some ODE problems the times of specific events are important such as the time at which a ball hits the ground or the time at which a spaceship returns to the earth or the times at which the ODE solution reaches certain values Improving Solver Performance While solving a problem the MATLAB ODE solvers can locate transitions to from or through zeros of a vector of user defined functions String Value Description Events on off Set this on if the ODE file evaluates and returns the event functions and returns information about the events Events Set this parameter on to inform the solver that the ODE file is coded so that F t y events returns appropriate event function information By default events iSoff For example the statement T Y TE YE E solver F tspan y0 options with theEvents property in options seton solves an ODE problem while also locating zero crossings of an events function defined in the ODE file In this case the solver returns three additional outputs e TE isa column vector of times at which events occur e Rows of YE are solutions corresponding to times in TE e Indices in vector E specify which event occurred at thetimein TE The ODE file must be coded to return three values in response tothe event s flag value isterminal direction F t y events The first output argument val ue is
46. M t varargout l mass t y N case demo Run a demo demo otherwise error Unknown flag flag end function dydt f t y N e N 1 pi zeros N 1 h pi Nt1 e 1 h zeros N 1 R spdiags e 2 e e 1 1 N N dydt R y Opes ied ochre She Att EE E BAGS NS a Mal ea De tot E E tas tee SN AAS Te hl function tspan y0 options init N tspan 0 pi yO sin pi N 1 1 N options odeset Mass M t Vectorized on 8 43 8 O rdinary Differential Equations 8 44 function M mass t y N e exp t pi 6 Nt 1 zeros N 1 h pi N 1 e exp t h 6 zeros M spdiags e 4 e e 1 1 N N function demo t y odel5s femlode surf 1 9 t y set gca ZLim 0 1 vi ew 142 5 30 title Finite element problem with time dependent mass matrix solved by ODE15S xlabel space ylabel time zl abel solution Example 5 Simple Event Location ball ode t y returns the derivatives vector for the equations of motion of a bouncing ball This ODE file illustrates the event location capabilities of the ODE solvers The equations for the bouncing ball are y y 9 8 ballode t y events returns a zero crossing vector val ue evaluated at t y aS well as two constant vectorSi ster mi nal and di rection By default the ODE solvers do not locate zero crossings However if theEvents prop
47. MATLAB Classes and O bjects Overloading the Operator You can implement the overloaded minus operator using the same approach as the plus operator MATLAB calls pol ynom mi nus m to compute p q function r minus p q POLYNOM MINUS Implement p q for polynoms p polynom p q polynom q k length q c length p c r polynom zeros 1 k p c zeros 1 k q c Overloading the Operator MATLAB calls the method po ynom mt i mes m to compute the product p q The letter m at the beginning of the function name comes from the fact that it is overloading MATLAB s matrix multiplication Multiplication of two polynomials is simply the convolution of their coefficient vectors function r mtimes p q POLYNOM MTI MES Implement p q for polynoms p polynom p q polynom q r polynom conv p c q c Using the Overloaded Operators Given the polynom object p polynom 1 0 2 5 MATLAB calls these two functions po l ynom pl us mand pol ynom mt i mes m when you issue the statements q ptl r p q to produce q x 3 2 x 4 r X 6 4 x 4 9 X 3 4 x 2 18 x 20 Example A Polynomial C lass Overloading Functions for the Polynom Class MATLAB already has several functions for working with polynomials represented by coefficient vectors They should be overloaded to also work with the new polynom object In many cases the overloading methods can simp
48. MATLAB maintains a current directory for working with M files and MAT files zen On Windows platforms the initial current directory is specified in the 44 shortcut file you use to start MATLAB Right click on the shortcut file and select Properties to change the default On UNIX systems the initial current directory is the directory you are in on your UNIX file system when you invoke MATLAB Todisplay your current directory usethecd command with no arguments F or example on UNIX cd home roger To change your current directory usecd with a path For example for Windows cd bigproj phasel changes the current directory tothephasel directory located in bi gproj Opening Files in MATLAB You can open files in MATLAB based on their extension using theopen function open is a user extensible function that provides an interface to file open operations Default behavior is provided for these standard MATLAB file The Command W indow types You can extend the interface to include other file types and to override the default behavior for the standard files Name Action Figure file fig M file name m Model name mdi P file name p Variable Other extensions name cust om Open figure in a figure window Open M filename in Editor Open model name in Simulink Open the corresponding M file name m if it exists in the Editor Open array name in theArray Editor the array must be
49. Matching E nables matching for quotes 2 49 2 MATLAB W orking Environment Search distance Controls the number of lines forward or backward the Editor searches for a match Show match for Controls duration of the match indicator e Tab key settings These settings control what happens when you type a tab character with Emacs style tab key auto indenting off and when the Editor encounters a pre existing tab in your file Tab key inserts Chooses whether the Tab key inserts a tab character or a number of spaces equivalent to a tab character Tab size Controls number of space equivalents between tab stops Note that the Auto indent size parameter is separately tunable from this setting Automatic indenting always inserts spaces Font Usethe Font submenu to change the font family style and size for the text in the Editor Debugger and the Path Browser Font 21x Font Font style Size R egular fi 0 Cancel Courier New Courier New Fixedsys T Lucida Console Terminal Sample AaBbYyZz Script Westem ad Path Browser The Path Browser lets you view and modify MATLAB s search path and see all of its files To open the Path Browser select Set Path from the File menu or click the Path Browser toolbar button You can also access the Path Browser Microsoft W indows Handbook using the pat htool command or from the E ditor Debugger select Path Br
50. Microsoft Windows Handbook for a discussion of this tool gt Path Browser olx File Edit View Path Tools Help Files in general Current Directory home 1lchuss Path fdevel natlabS Nightly toolbox natLab geng devel AnatlabS nightly toolbox Anatlab ops devel AnatlabS nightly toolb ox Anatlab lanc devel AnatlabS Avightly toolb ox Anatlab elme devel natlabS nightly toolb ox Anatlab elfu develmatlabS nightly toolb ox matlab spec 4 Bi Ready 5 03 PM Z computer m Ej Contents m copyfile m clear m UN IX Handbook Workspace Browser MATLAB provides a Workspace Browser that lets you view the contents of the current MATLAB workspace It provides a graphical representation of thewhos display To open the Workspace Browser typeworkspace at the command line This is essentially the same Workspace Browser as the one provided for Windows platforms See the section Microsoft Windows H andbook for a discussion of this tool 2 3 Bytes Class double array 24 double array 48 double array 24 char array 84 sparse array P sss sus suu C IEE AwA Ta O Oo OM 192 double array 130 struct array 200 cell array 824 inline object Ba Grand total is 93 elements using 1534 bytes Open Editing Arrays The MATLAB Editor Debugger provides a visual representation of two dimensional numeric arrays which allows for eas
51. Often y t is a vector having elements y1 y2 yn ODEs often involve a number of dependent variables as well as derivatives of order higher than one To use the MATLAB ODE solvers you must rewrite such equations as an equivalent system of first order differential equations in terms of a vector y and its first derivative y F t y Once you represent the equation in this way you can code it as an ODE M file that a MATLAB ODE solver can use Initial Value Problems and Initial Conditions Generally there are many functions y t that satisfy a given ODE and additional information is necessary to specify the solution of interest In an initial value problem the solution of interest has a specific initial condition that is y is equal to yg at a given initial time t An initial value problem for an ODE is then y F t y y to Yo Ifthefunction F t y is sufficiently smooth this problem has one and only one solution Generally there is no analytic expression for the solution so it is necessary to approximate y t by numerical means such as one of the solvers of the MATLAB ODE suite Example The van der Pol Equation An example of an ODE is the van der Pol equation 8 5 8 O rdinary Differential Equations 8 6 I 2 yy w 1 y y7 y 0 where u gt 0 is a scalar parameter Rewriting the System To express this equation as a system of first order differential equations for MATLAB introduce a
52. Pay special attention to the section about upgrading for guidance on converting your M files You should then refer to Using MATLAB and Using MATLAB Graphics for specific details about the new features What I Want What I Should Do want to know how to use a specific function want to find a function for a specific purpose but don t know its name want tolearn about a specific topic like sparse matrices ordinary differential equations or cell arrays want to know what functions are available in a general area havea problem want help with want to report a bug or make a suggestion want to contact The MathWorks Technical Support Use the online Help facility You can use the M file help window to get brief online help or access the MATLAB Function Reference via the Web based Help Desk These are available using the commands hel pwi n and hel pdesk or from the Help menu on the PC The F unction Reference is also available on the Help Desk in PDF format if you want to print out any of the function descriptions in high quality form There are three choices e Usel ookfor g 1 ookf or inverse from the command line e Use the online keyword search from the Help Desk e Visit The MathWorks Web site and see if thereisa user contributed file to solve your problem See the appropriate chapter in Using MATLAB Use the help window typehe pwi n or select from Help menu to see a tab
53. Profile Report The function call history displays the exact sequence of functions called To view this report you must have started the profiler using the hist ory option profile on history The profiler records up to 10 000 function entry and exit events F or more than 10 000 events the profiler continues to record other profile statistics but not the sequence of calls Below is the history report generated from An Example Using the Profiler 3 MATLAB Profiler Report Netscape of x File Edit View Go Communicator Help Summary Function Details Function Call Histo MATLAB Profile Report Function Call History display ode23 A Devtools matlab5 nightly toc nargin Exact Sequence of Calls isstr nargin nargin nargin isempty isempty 0 eee oa Document Done 36 ca gP Z Z 3 24 M File Profiler Profile Plot To view a bar graph for the functions using the most execution time type profile plot profile plot suspends the profiler The bar graph appears in a figure window This is the bar graph generated from An Example Using the Profiler Figure No 1 of x File Edit Tools Window Help osaakaan BER ode23 ode23 evalODEfile lotka feval odeget diag isempty abs callstats profile 0 0 2 0 4 0 6 0 8 1 Total time seconds Saving Profile Reports Type profile report basename The profiler saves the report tothe filebasename in thec
54. The examples in this section focus on two dimensional cell arrays F or examples of higher dimension cell arrays see Chapter 12 Creating Cell Arrays You can create cell arrays by e Using assignment statements e Preallocating thearray using thece s function then assigning data tocells 13 19 13 Structures and Cell Arrays 13 20 Using Assignment Statements You can build a cell array by assigning data to individual cells one cell at a time MATLAB automatically builds the array as you go along There are two ways to assign data to cells e Cell indexing Enclose the cell subscripts in parentheses using standard array notation Enclose the cell contents on the right side of the assignment statement in curly braces For example create a 2 by 2 cell array A ALLI Shh a 0 eee aT a y A 1 2 Anne Smith A 2 1 347i A 2 2 pi pi 10 pi Note The notation denotes the empty cell array just as denotes the empty matrix for numeric arrays You can use the empty cell array in any cell array assignments e Content indexing Enclose the cell subscripts in curly braces using standard array notation Specify the cell contents on the right side of the assignment statement A 1 1 1 4 3 05 8 7 2 9 A 1 2 Anne Smith A 2 1 347i A 2 2 pi pi 10 pi Cell Arrays The various examples in this guide do not use one syntax throughout but attempt t
55. The raw data is stored in thefile count dat 11 7 14 11 Use the oad command to import the data load count dat This creates a matrix count in the workspace 11 13 17 13 51 46 132 135 88 36 12 27 19 15 36 47 65 66 55 145 58 12 9 9 9 11 20 9 69 76 151 90 257 68 15 15 7 For this example there are 24 observations of three variables This is confirmed by n p n 24 p 3 size count 6 5 6 Data Analysis and Statistics Create a time vector t of integers from1 ton t lin Now plot the counts versus time and annotate the plot set 0 defaultaxeslinestyleorder set 0 defaultaxescolororder 0 0 0 plot t count legend Location 1 Location 2 Location 3 0 xlabel Time ylabel Vehicle Count grid on The plot shows the vehicle counts at three locations over a 24 hour period 300 T a Location 1 inaina Location 2 i 250i ae Location 3 fl i 150 F Vehicle Count 100 F 507 25 Time 6 6 Basic Data Analysis Functions Basic Data Analysis Functions A collection of functions provides basic column oriented data analysis capabilities Function Description cumprod cums um cumtrapz diff ma x mean medi an mi n prod sort sortrows std sum trapz Cumulative product of elements Cumulative sum of elements Cumulative trapezoidal numerical integration Differe
56. and Asset Subclasses function a subsasgn a index val SUBSASGN Define index assignment for asset objects switch index type case switch index subs case 1 a descriptor val case 2 a date val case 3 a current value val otherwise error Index out of range end case switch index subs case descriptor a descriptor val case date a date val case current value a current_ value val otherwise error Invalid field name end end Thesubsasgn method enables you to assign values to the asset object data structure using two techniques For example suppose you have a child stock object s s stock XYZ 100 25 Within stock class methods you could changethedescriptor field with either of the following statements s asset 1 ABC or S asset descriptor ABC 14 MATLAB Classes and O bjects See the The Stock subsasgn M ethod on page 14 51 for an example of how the childs ubsasgn method calls the parent s ubsas gn method The Asset display Method The asset di spl ay method is designed to be called from child class di spl ay methods Its purpose is to display the data it stores for the child object The method simply formats the data for display in a way that is consistent with the formatting of the child s display method function display a DISPLAY a Display an asset object stg sprintf Descriptor s nDate S nCurren
57. and that the elements of U are not much larger than those of A For example L U lu B L 1 0000 0 0 0 3750 0 5441 1 0000 0 5000 1 0000 0 8 0000 1 0000 6 0000 0 8 5000 1 0000 0 0 5 2941 LU QR and Cholesky Factorizations The LU factorization of A allows the linear system A x b to be solved quickly with x U L b Determinants and inverses are computed from the LU factorization using det A det L det U prod diag U and inv A inv U inv l QR Factorization An orthogonal matrix or a matrix with orthonormal columns is a real matrix whose columns all have unit length and are perpendicular to each other If Q is orthogonal then Q Q The simplest orthogonal matrices are two dimensional coordinate rotations cos 0 sin 0 sin 0 cos For complex matrices the corresponding term is unitary Orthogonal and unitary matrices are desirable for numerical computation because they preserve length preserve angles and do not magnify errors The orthogonal or QR factorization expresses any rectangular matrix as the product of an orthogonal or unitary matrix and an upper triangular matrix A column permutation may also be involved A QR or AP QR where Q is orthogonal or unitary R is upper triangular and P isa permutation 4 27 4 Matrices and Linear Algebra 4 28 There are four variants of the QR factorization full or economy size and with or without column permutation Over
58. as it is shown below complete with a planted bug function tot sqsum x mu tot 0 for i 1 length mu tot tot x i mu 2 end 3 3 3 Debugger and Profiler 3 4 Note The example above is coded for illustrative purposes only Whenever possible avoid f or loops and use vectorization for the most efficient execution Trial Run Try out the M files to see if they work correctly Use MATLAB sst d function to compute results Create a test vector at the command line v 123 45 Compute the variance usingstd varl std v 2 varli 2 5000 Now try thevar i ance function from above myvarl variance v myvarl 1 Theanswer is wrong Let s use the Debugger to isolate the error in the M files The following section shows a sample session using the Editor Debugger graphical user interface as well as one from the command line Debugging Using the Graphical User Interface Tostart debugging e f you havejust created the M files using the E ditor Debugger window you can continue from this point e f you ve created the M files using an external text editor start the Editor Debugger and then click the Open M file button on the toolbar MATLAB Debugger The Editor Debugger toolbar contains a series of debugging icons SA 8 2 42 BEE Toolbar Button Purpose Description Equivalent Command 2 mi y 2 Set Clear Breakpoint Clear All Breakpoin
59. but it usually does The function sy mrcm A actually operates on the nonzero structure of the symmetric matrix A A but the result is also useful for asymmetric matrices This ordering is useful for matrices that come from one dimensional problems or problems that are in some sense long and thin Minimum Degree Ordering The degree of a node in a graph is the number of connections to that node which is the same as the number of nonzero elements in the corresponding row of the adjacency matrix The minimum degree algorithm generates an ordering based on how these degrees are altered during Gaussian elimination It isa complicated and powerful algorithm that usually leads to sparser factors than most other orderings including column count and reverse Cuthill McK ee MATLAB has two versions s ymmd for symmetric matrices and col mmd for nonsymmetric matrices You can change various parameters associated with details of the algorithm using thes pparms function For more details on the algorithm and MATLAB s version of it see Gilbert J ohn R Cleve Moler and Robert Schreiber Sparse Matrices in MATLAB Design and Implementation SIAM J Matrix Anal Appl Vol 13 No 1 J anuary 1992 pp 333 356 Factorization This section discusses four important factorization techniques for sparse matrices e LU or triangular factorization e Cholesky factorization e QR or orthogonal factorization e Incomplete factorizations
60. can write a method to handle the assignment of an individual property The method should have the same name as the property name For example if you defined a class that creates objects representing employee data you might have a field in an employee object called sal ary You could then define a method called sal ary m that takes an employee object and a value as input arguments and returns the object with the specified value set Indexed Reference Using subsref and subsasgn User classes implement new data types in MATLAB It is useful to be able to access object data via an indexed reference as is possible with MATLAB s built in data types For example if A is an array of classdouble A i returns thei element of A As the class designer you can decide what an index reference to an object means F or example suppose you define a class that creates polynomial objects and these objects contain the coefficients of the polynomial An indexed reference to a polynomial object p 3 could return the value of the coefficient of x3 the value of the polynomial at x 3 or something different depending on the intended design You define the behavior of indexing for a particular class by creating two class methods subsref andsubsasgn MATLAB calls these methods whenever an subscripted reference or assignment is made on an object from the class If you do not define these methods for a class indexing is undefined for objects of this c
61. case jacobian Return Jacobian matrix df dy varargout 1 jacobian t y N otherwise error Unknown flag flag end Oy ie E A ENE Sell ie gtd iat GO afar Heit AE Ae J a E WN ie inten acta vei omens ieee i ee NEE vee dite function dydt f t y N c 0 02 N 1 2 dydt zeros 2 N size y 2 preallocate dy dt Evaluate the 2 components of the function at one edge of the grid with edge conditions i 1 dydt i 1 y i l y i 2 4 y i c 1 2 y i y i 2 dydt i 1 3 y i y i 1 y i 2 c 3 2 y i 1 y i 3 Evaluate the 2 components of the function at all interior grid points i 3 2 2 N 3 Cydia sol eyed a Py Gigs O20 Ay Cte Ae es ce y i 2 2 y i ty it2 dydt itl 3 y i y itl y i s 42 c y i 1 2 y i t1 ty i 3 Evaluate the 2 components of the function at the other edge of the grid with edge conditions ij 2 N 1 8 39 8 O rdinary Differential Equations 8 40 dydt i 1 y itl y i 2 4 y i ton c y i 2 2 y i 1 dydt i l 3 y i y itl yli 2 00 c y i 1 2 y i t1 3 unction tspan y0 options init N span 0 10 0 1l sin 2 pi N 1 1 N 3 zeros 1 N 0 yO ptions odeset Vectorized on unction dfdy jacobian t y N 0 02 N41 2 zeros 2 N 5 gt 2 N 1 1
62. continuous but the slope changes at the vertex points Cubic interpolation requires more memory and execution time than either the nearest neighbor or linear methods H owever both the interpolated data and its derivative are continuous Cubic spline interpolation has the longest relative execution time although it requires less memory than cubic interpolation It produces the smoothest results of all the interpolation methods You may obtain unexpected results however if your input data is non uniform and some points are much closer together than others The relative performance of each method holds true even for interpolation of two dimensional or multidimensional data For a graphical comparison of interpolation methods see the section Comparing Interpolation Methods on page 5 13 FFT Based Interpolation The function i nt er pft performs one dimensional interpolation using an FF T based method This method calculates the F ourier transform of a vector that contains the values of a periodic function It then calculates the inverse Fourier transform using more points Its form is y interpft x n x is a vector containing the values of a periodic function sampled at equally spaced points n is the number of equally spaced points to return 5 11 5 Polynomials and Interpolation 5 12 Two Dimensional Interpolation The function i nt erp2 performs two dimensional interpolation an important operation for image p
63. data constructs that are compatible with versions of MATLAB 4 therefore you cannot save structures cell arrays multidimensional arrays or objects In addition you must use filenames that are supported by MATLAB version 4 When you save workspace contents in ASCII format save only one variable at atime If you save morethan one variable MATLAB will createthe ASCII file but you will be unable to load it back into MATLAB later using oad Loading the Workspace The oad command loads a MAT file that you have previously created with save For example load juneld loadsj une10 mat into the workspace If the saved MAT filej une10 contains the variables A B and C then loadingj une10 places the variables A B and Cc back into the workspace If the variables already exist in the workspace they are overwritten If your MAT file has a filename extension other than mat you must use the mat switch or else MATLAB expects the file to be ASCII text format load filename mat en On Windows platforms the oad operation is also available by selecting i Load Workspace from the File menu The Command W indow Loading ASCII Data Files The oad command also imports ASCII data files It reads the contents of the file into a variable with the same name as the file without the extension F or example load tides dat creates a variable named ti des in the workspace If the ASCII data file has m lines with n values on each lin
64. dimensional interpolation nearest neighbor 5 15 tricubic 5 15 trilinear 5 15 time interval ODE 8 6 measured for M files 3 17 numbers 10 54 titlebar in Editor Debugger 2 40 token in string 11 12 tolerance 10 18 tool bar preferences 2 38 toolbar in command window 2 36 Index toolboxes 1 8 creation data type 10 19 tooltip 2 36 transformed data magnitude 6 37 phase 6 37 transforms 6 31 discrete F ourier 6 31 fast Fourier 6 31 fft 6 31 transpose 4 7 complex conjugate 4 8 unconjugated complex 4 8 transpose 12 14 triangular factorization for sparse matrices 9 28 triangular matrices 4 24 triangulation 5 18 closest point searches 5 22 Delaunay 5 19 Voronoi diagrams 5 22 tricubic interpolation 5 15 trigonometric functions 10 5 12 15 trilinear interpolation 5 15 try 10 36 tsearch 5 22 two dimensional interpolation 5 12 bicubic 5 12 bilinear 5 12 nearest neighbor 5 12 type 2 15 2 25 U uchar data type 15 8 ui nt datatype 10 20 unary minus operator 10 21 unconjugated complex transpose 4 8 UNIX environment 2 54 startup options 2 4 unix 2 25 unregserver startup option 2 4 unwrap 6 37 url for The MathWorks 2 23 user classes designing 14 9 user input obtaining interactively 10 61 User Obj ect data type 10 20 utilities running from MATLAB 2 25 V value largest system can represent 10 18 passing arguments by 10 11 values data type 15 8 van der Pol example 8 35 extra parameters 8 16 simpl
65. dispatching rules and is discussed in Chapter 14 Classes and Objects W hat Happens When You Call a Function When you call a function M file from either the command line or from within another M file MATLAB parses the function into pseudocode and stores it in memory This prevents MATLAB from having to reparse a function each time you call it during a session The pseudocode remains in memory until you clear Functions it using thec ear command or until you quit MATLAB Variants of thec ear command that you can use to clear functions from memory include clear function_name Remove specified function from workspace clear functions Remove all compiled M functions clear all Remove all variables and functions Creating P Code Files Y ou can save a preparsed version of a function or script called P code files for later MATLAB sessions using thepcode command For example pcode average parses average m and saves the resulting pseudocode to the file named average p This saves MATLAB from reparsing average m the first time you call it in each session MATLAB is very fast at parsing so thepcode command rarely makes much of a speed difference One situation wherepcode does provide a speed benefit is for large GUI applications In this case many M files must be parsed before the application becomes visible Another situation for pcode is when for proprietary reasons you want to hide algorithms you ve created in
66. dy dt F t y outl y 2 mu 1 y 1 2 y 2 y 1 elseif strcemp flag init Return tspan y0 options outl 0 20 tspan out2 2 0 initial conditions out3 odeset RelTol le 4 options end In this example the parameter mu is an optional argument specific to the van der Pol example MATLAB and the ODE solvers do not set a limit on the number of parameters you can pass to an ODE file Guidelines for Creating ODE Files e Theode file must have at least two input arguments t andy It is not necessary however for the function to use either t ory e The derivatives returned by F t y must be column vectors e Any additional parameters beyondt andy must appear at the end of the argument list and must begin at the fourth input parameter The third position is reserved for an optional flag as shown above in Coding the ODE Fileto Return Initial Values Thef ag argument is described in more detail in Special Purpose ODE Files and the flag Argument on 8 17 Improving Solver Performance Improving Solver Performance In some cases you can improve ODE solver performance by specially coding your ODE file For instance you might accelerate the solution of a stiff problem by coding the ODE file to compute the J acobian matrix analytically Another way to improve solver performance often used in conjunction with a specially coded ODE file is to tune solver parameters The default p
67. example 10 5 executing 10 5 script files 10 2 search path 2 50 10 10 and subfunctions 10 38 changing 2 14 for MATLAB files 2 14 searching for functions 2 22 in string 11 12 online help 2 22 2 23 second difference operator example 9 7 semicolon to suppress output 2 9 set method 14 13 setfield 13 8 shell escape 2 25 shell escape functions 10 62 shiftdi m 12 2 short 15 8 short integer 15 8 simple inheritance 14 34 Simulink 1 8 si n 10 5 12 15 single precision 15 8 singular value decomposition 4 38 size of structure arrays 13 9 of structure fields 13 9 si ze 9 23 12 9 13 9 smallest value system can represent 10 18 smoothing ODE solver output 8 24 solvers See ODE solvers solving linear systems of equations sparse 9 33 sort 9 27 sorting data 6 7 sound files reading 2 27 writing 2 29 sparse 9 6 9 23 sparse data type 10 20 sparse matrix 10 19 advantages 9 5 and complex values 9 5 Cholesky factorization 9 30 computational considerations 9 23 contents 9 11 conversion from full 9 2 9 6 creating 9 6 directly 9 7 from diagonal elements 9 8 defined 9 2 density 9 6 distance between nodes 9 21 eigenvalues 9 36 elementary 9 2 example 9 7 fill in 9 20 importing 9 10 linear algebra 9 3 linear equations 9 3 linear systems of equations 9 33 LU factorization 9 28 and reordering 9 28 mathematical operations 9 23 nonzero elements 9 11 maximum number 9 7 specifying when creating matrix 9 7 storage 9 5 9 11 1 25 Ind
68. feed characters the MATLAB interpreter tolerates all possible combinations H owever text editors and other tools may not work correctly with M files from other platforms e MAT files are binary and machine dependent but they can be transported between machines because they contain a machine signature in the file header MATLAB checks thesignature when it loads a file and if a signature indicates that a file is foreign performs the necessary conversion TouseMATLAB across different platforms you need a program for exchanging both binary and text data between the machines When using these programs be sure to transmit MAT files in binary file mode and M files in ASCII file mode Failure to set these modes correctly usually corrupts the data 2 31 2 MATLAB W orking Environment 2 32 The diary Command Thedi ar y command creates a verbatim copy of your MATLAB session in a disk file excluding graphics Y ou can view and edit the resulting text file using any word processor To create a file on your disk called sept 23 out that contains all the commands you enter as well as MATLAB s output enter diary sept23 out To stop recording the session use diary off Memory Utilization Memory Utilization MATLAB requires a contiguous area of memory to store each matrix In particular images and movies can consume large amounts of memory In addition to the storage required for the matrix the pixmap used to draw the im
69. forms The example above uses a form that sets the maximum number of nonzero elements in the matrix tol ength s If desired you can append a sixth argument that specifies a larger maximum allowing you to add nonzero elements later without changing storage requirements Example The Second Difference O perator The matrix representation of the second difference operator is a good example of a sparse matrix It is a tridiagonal matrix with 2s on the diagonal and 1s 9 7 9 Sparse Matrices on the super and subdiagonal There are many ways to generate it here s one possibility D sparse l n l n 2 0nes 1 n n n E sparse 2 n 1 n 1 o0nes 1 n 1 n n S E D E Forn 5 MATLAB responds with S 1 1 2 2 1 1 1 2 1 2 2 2 3 2 1 2 3 1 3 3 2 4 3 1 3 4 1 4 4 2 5 4 1 4 5 1 5 5 2 NowF full S displays the corresponding full matrix F full S F E 2 1 0 0 0 1 2 1 0 0 0 1 2 1 0 0 0 1 2 1 0 0 0 1 2 Creating Sparse Matrices from Their Diagonal Elements Creating sparse matrices based on their diagonal elements is a common operation so the function s pdi ags handles this task Its syntax is S spdiags B d m n 9 8 Introduction To create an output matrix S of size m by n with elements on p diagonals e B isa matrix of size mi n m n by p The columns of B are the values to populate the diagonals of e d is a vector of length p whose integer elements specify whi
70. i el d function Its most basic form is struc2 rmfield array field wherearray isa structure array and field is the name of afield to remove from it To remove the name field from the patient array for example enter patient rmfield patient name Applying Functions and Operators Operate on fields and field elements the same way you operate on any other MATLAB array Use indexing to access the data on which to operate F or example this statement finds the mean across the rows of thet est arrayin patient 2 mean patient 2 test 13 9 13 Structures and Cell Arrays 13 10 There are sometimes multiple ways to apply functions or operators across fields in astructurearray One way toaddall thebi ing fieldsinthepati ent array is total 0 for j l length patient total total patient j billing end To simplify operations like this MATLAB enables you to operate on all like named fields in a structure array Simply enclose thearray field expression in square brackets within the function call For example you can sum all thebil ling fields in thepatient array using total sum patient billing This is equivalent to using the comma separated list total sum patient 1 billing patient 2 billing This syntax is most useful in cases where the operand field is a scalar field Writing Functions to Operate on Structures You can write functions that work on structures with specific f
71. in as additional parameters If you know there is a sharp change at timet it might help to break thet s pan interval into two pieces t0 t and t tfinal and call the integrator twice If the differential equation has periodic coefficients or solution you might restrict the maximum step size to the length of the period so the integrator won t step over periods Sparse Matrices Introduction Sparse Matrix Storage Creating Sparse Matrices Importing Sparse Matrices from Outside MATLAB Viewing Sparse Matrices General Storage Information Information About Nonzero Elements Viewing Sparse Matrices Graphically The find Function and Sparse Matrices Example Adjacency Matrices and Graphs Graphing Using ee Matrices The Bucky Ball y An Airflow Model Sparse Matrix Operations Computational Considerations Standard Mathematical Operations Permutation and Reordering Factorization Simultaneous Linear Equations Eigenvalues and Singular Values 9 5 9 5 9 10 9 11 9 11 9 11 9 13 9 14 9 15 9 15 9 16 9 21 9 23 9 23 9 23 9 24 9 27 9 33 9 36 9 Sparse Matrices 9 2 MATLAB supports sparse matrices matrices that contain a small proportion of nonzero elements This characteristic provides advantages in both matrix storage space and computation time This chapter explains how to create sparse matrices in MATLAB and how to use them in bot
72. individual s financial portfolio For example given the following assets XYZStock stock XYZ 200 12 SaveAccount savings Acc 1234 2000 3 2 Bonds bond U S Treasury 1600 12 Now create a portfolio object p portfolio Gilbert Bates XYZStock SaveAccount Bonds The portfoliodi sp ay method summarizes the portfolio contents because this statement is not terminated by a semicolon 14 59 14 MATLAB Classes and O bjects Descriptor XYZ Date 24 Nov 1998 Type stock Current Value 2400 00 Number of shares 200 Share price 12 00 Descriptor Acc 1234 Date 24 Nov 1998 Type savings Current Value 2000 00 Interest Rate 3 2 Descriptor U S Treasury Date 24 Nov 1998 Type bond Current Value 1600 00 Interest Rate 12 Assets for Client Gilbert Bates Total Value 6000 00 The portfolio pi e3 method displays the relative mix of assets using a pie chart pi e3 p Portfolio Composition for Gilbert Bates Total Value of Assets 6000 Savings Saving and Loading O bjects Saving and Loading Objects You can usethe MATLAB save and oad commands to save and retrieve user defined objects to and from mat files just like any other variables When you load objects MATLAB calls the object s class constructor to register the object in the workspace The constructor function for the object class you are loading must be able to be called with no input argu
73. inheritance consider a general asset class that can be used torepresent any item that has monetary value Some examples of an asset are stocks bonds savings accounts and any other piece of property In designing this collection of classes theasset class holds the data that is common to all of the specialized asset subclasses The individual asset subclasses such as thest ock class inherit theass et properties and contribute additional properties The subclasses are kinds of assets Simple Inheritance An example of a simple inheritance relationship using an asset parent class is shown in this diagram Asset Class Structure Fields descriptor date current value Stock Class Bond Class Savings Class descriptor date current value Stock Fields num shares share_price asset Inherited Fields Inherited Fields descriptor date current value Bond Fields interest rate asset Inherited Fields descriptor date current value Savings Fields interest rate asset As shown in the diagram thest ock bond andsavings classes inherit structure fields from theasset class In this example theasset class is used to provide storage for data common to all subclasses and to share asset methods with these subclasses This example shows how to implement the 14 37 14 MATLAB Classes and O bjects 14 38 asset and stock classes Thebond andsavings classes can b
74. just as they do on full matrices In addition MATLAB provides a number of functions that perform operations specific to sparse matrices This section discusses e Computational Considerations e Standard Mathematical Operations e Permutation and Reordering e Factorization e Simultaneous Linear Equations e Eigenvalues and Singular Values Computational Considerations The computational complexity of sparse operations is proportional tonnz the number of nonzero elements in the matrix Computational complexity also depends linearly on the row sizem and column sizen of the matrix but is independent of the product m n the total number of zero and nonzero elements The complexity of fairly complicated operations such as the solution of sparse linear equations involves factors like ordering and fill in which are discussed in the previous section In general however the computer time required for a Sparse matrix operation is proportional to the number of arithmetic operations on nonzero quantities This is the time is proportional to flops rule Standard Mathematical Operations Sparse matrices propagate through computations according to these rules e Functions that accept a matrix and return a scalar or vector always produce output in full storage format For example thesi ze function always returns a full vector whether its input is full or sparse e Functions that accept scalars or vectors and return matrices such aszer
75. lines show the correct placement of varargin andvarargout function outl out2 examplel a b varargin function i varargout example2 xl yl x2 y2 flag 10 15 10 M File Programming Local and Global Variables 10 16 Thesame guidelines that apply to MATLAB variables at the command linealso apply to variables in M files e You donot need totype or declare variables Before assigning one variable to another however you must be sure that the variable on the right hand side of the assignment has a value e Any operation that assigns a value to a variable creates the variable if needed or overwrites its current value if it already exists e MATLAB variable names consist of a letter followed by any number of letters digits and underscores MATLAB distinguishes between uppercase and lowercase characters SoA anda are not the same variable e MATLAB uses only the first 31 characters of variable names Ordinarily each MATLAB function defined by an M file has its own local variables which are separate from those of other functions and from those of the base workspace However if several functions and possibly the base workspace all declare a particular name as global then they all share a single copy of that variable Any assignment to that variable in any function is available to all the other functions declaring it global Suppose you want to study the effect of the interaction coefficients a and 8 in th
76. numeric open callSopenvar Open name cust om by calling the helper function opencustom whereopencust omis a user defined function 2 17 2 MATLAB W orking Environment 2 18 The Figure Window MATLAB directs graphics output to a window that is separate from the Command Window In MATLAB this window is referred to as a figure Graphics functions automatically create new figure windows if none currently exist If a figure window already exists MATLAB uses that window If multiple figure windows exist one is designated as the current figure and is used by MATLAB this is generally the last figure used or the last figure you clicked the mouse in Thef i gure function creates figure windows For example figure creates a new window and makes it the current figure The plot function creates a plot in a figure window For example t 0 pi 100 2 pi y sin t plot t y draws a graph of thesine function from zero to 2z in the current figure window if one exists or in a new figure window if none exists Annotating Plots Using the Plot Editor After creating a plot you can make changes to it and annotate it with the Plot Editor which is an easy to use graphical interface The illustration below shows the plot in a figure window and labels the main features of the figure window and the Plot Editor The Figure W indow Use the Tools Click this Button Get Help for Annotate Zoom and Rotate the Menu to Access Plot
77. numerically vdpode init returns the default tspan y0 andoptions values for this problem The entire initial value problem is defined in this one file vdpode t y jacobian orvdpode t y jacobi an mu returns the J acobian matrix dF dy evaluated analytically at t y By default the stiff solvers of the ODE suite approximate J acobian matrices numerically However if acobi an isseton withodeset asolver calls the ODE file with the flag j acobian toobtain oF dy Providing the solvers with an analytic J acobian is not necessary but it can improve the reliability and efficiency of integration Examples Applying the O DE Solvers function varargout if nargin lt 4 mu 1 end switch flag case vdpode t y flag mu VDPODE Parameterizable van der Pol equation stiff for large mu isempt y mu Return dy dt f t y varargout 1l f t y mu case init varargout l 3 case jacobi an Return default tspan y0 options init mu Return Jacobian matrix df dy varargout 1 jacobian t y mu otherwise error Unknown flag flag end Oy ekai Give Sh a a a a o e EE Be aE tevin nah grin lat S ae EE a uct gute E OE ee he function dydt f t y mu dydt y 2 mu 1 y 1 2 y 2 y 1 Vectorized Oy ee nati ates Rea y aa E Ging he e Ea Ea EA bess Hp eke ein othe wie E hye ange ep erate eee a e E function
78. object data always accept an object as an input argument and return a new object with the data changed This is necessary because MATLAB does not support passing arguments by reference i e pointers Functions can change only their private temporary copy of an object Therefore to changean existing object you must create a new one and then replace the old one The following sections provide more detail about implementation techniques for theset get subsasgn andsubsref methods The set and get Methods Theset andget methods provide a convenient way to access object data in certain cases For example suppose you have created a class that defines an arrow object that MATLAB can display on graphs perhaps composed of existing MATLAB line and patch objects To produce a consistent interface you could defineset andget methods that operate on arrow objects the way the MATLAB set andget functions operate on built in graphics objects Theset andget verbs convey what operations they perform but insulate the user from the internals of the object Examples of set and get Methods See the following sections for examples of s et and get methods e The Asset get Method on page 14 40 and the The Asset set Method on page 14 41 e The Stock get Method on page 14 47 and the The Stock set Method on page 14 50 14 13 14 MATLAB Classes and O bjects Property Name Methods Asan alternative to a general set method you
79. of Strings Cell Arrays of Strings It s often convenient to store groups of strings in cell arrays instead of standard character arrays This prevents you from having to pad strings with blanks to create character arrays with rows of equal length A set of functions enables you to work with cell arrays of strings e You can convert between standard character arrays and cell arrays of strings e You can apply string comparison operations to cell arrays of strings For details on cell arrays see the Structures and Cal Arrays chapter Converting Between Character Arrays and Cell Arrays of Strings Thecel str function converts a character array into a cell array of strings Consider the character array data Allison Jones Development Phoenix Each row of the matrix is padded so that all have equal length in this case 13 characters Now usecel l str to create a column vector of cells each cell containing one of the strings from the data array celldata cellstr data celldata Allison Jones Development Phoenix Note that thecel Istr function strips off the blanks that pad the rows of the input string matrix length celldata 3 ans 11 7 11 Character Arrays Strings Usechar toconvert back to a standard padded character array strings char cell data strings Allison J ones Devel opment Phoeni x 11 8 String Comparisons String Comparisons There are
80. operand is a scalar and one an array the operator applies the scalar to each element of the array 12 15 12 Multidimensional Arrays 12 16 Functions that Operate on Planes and Matrices Functions that operate on planes or matrices such as the linear algebra and matrix functions in the mat f un directory do not accept multidimensional arrays as arguments That is you cannot use the functions in the mat f un directory or thearray operators or with multidimensional arguments Supplying multidimensional arguments or operands in these cases results in an error You can useindexing to apply a matrix function or operator to matrices within a multidimensional array For example create a three dimensional array A A cat 3 1 2 3 9 8 7 4 6 5 0 3 2 8 8 4 5 3 5 6 4 7 6 8 5 5 4 3 Applying theei g function to the entire multidimensional array results in an error ei g A Error using gt eig Input arguments must be 2 D You can however apply ei g to planes within thearray F or example use colon notation to index just one page in this case the second of the array ei g A 2 ans 2 6260 12 9129 2 4L 31 Note In the first subscripts are not colons you must uses queeze toavoid an error For example ei g A 2 results in an error because the size of the input is 1 3 3 Theexpression ei g squeeze A 2 however passes a valid two dimensional matrix toei g Organizing Dat
81. parent A child class can inherit from one parent singleinheritance or many parents multiple inheritance Inheritance can span one or more generations Inheritance enables sharing common parent functions and enforcing common behavior amongst all child classes e Aggregation You can create classes using aggregation in which an object contains other objects This is appropriate when an object type is part of another object type F or example a savings account object might bea part of a financial portfolio object MATLAB Data Class Hierarchy You can add new data types to MATLAB by extending the MATLAB data class hierarchy The MATLAB data class hierarchy is shown in this diagram array char numeric cel struct user class double int8 uint8 intl6 uintl6 sparse int32 uint32 single Thearray class shown at the root of the diagram is the fundamental data class in MATLAB upon which all other data classes are based The array class is a virtual class that you cannot directly instantiate This means that you cannot create a variable with thetypearray Thenumeric classis alsoa virtual class Theremainingcl asses char double sparse cell struct andthe 14 MATLAB Classes and O bjects individual storage types such as ui nt 8 are the classes that you use to work with data in MATLAB Thediagram shows a user class that inherits from thestruct class All classes that you create are structure based sincethis is the poin
82. reshape an array or if you perform a calculation that results in an array dimension of one B repmat 5 2 3 1 4 size B ans 2 3 1 4 Thes queeze function removes singleton dimensions from an array C squeeze B size C ans 12 12 Multidimensional Arrays Thesqueeze function does not affect two dimensional arrays row vectors remain rows Permuting Array Dimensions The per mute function reorders the dimensions of an array B permute A di ms dims iS a vector specifying the new order for the dimensions of A where 1 corresponds to the first dimension rows 2 corresponds to the second dimension columns 3 corresponds to pages and so on A B permute A 2 1 3 C permute A 3 2 1 A 1 Besni I Clee ES 1 4 7 Row and column 1 2 3 Row and page 5 8 subscripts are 0 5 4 subscripts are 9 reversed i J reversed As i 2 TEE Ue Na ransposition 4 5 5 0 2 9 2 7 6 5 7 3 4 6 1 caeai 7 8 9 9 3 1 For a more detailed look at the per mut e function consider a four dimensional array A of size 5 by 4 by 3 by 2 Rearrange the dimensions placing the column dimension first followed by the second page dimension the first page dimension then the row dimension The result is a 4 by 2 by 3 by 5 array 12 13 12 Multidimensional Arrays 12 14 Move dimension 2 of A to first subscript position of B dimension 4 to second sub script position and so on B permute A 2
83. scheme leads to the most fill in The fill in for the reverse Cuthill M cK ee ordering is concentrated within the band but it is almost as extensive as the first two orderings For the minimum degree ordering the relatively large blocks of zeros are preserved during the elimination and the 9 29 9 Sparse Matrices 9 30 amount of fill in is significantly less than that generated by the other orderings The spy plots below reflect the characteristics of each reordering Original Reverse Cuthill McKee 0 10 20 30 40 50 60 nz 1022 nz 968 Minimum Degree Cholesky Factorization IfS is asymmetric or Hermitian positive definite sparse matrix the statement below returns a sparse upper triangular matrix R sothatR R S R chol S chol does not automatically pivot for sparsity but you can compute minimum degree and profile limiting permutations for use with chol p p Since the Cholesky algorithm does not use pivoting for sparsity and does not require pivoting for numerical stability it is possible to do a quick calculation of the amount of memory required and allocate all the memory at the start of the factorization QR Factorization MATLAB will compute the complete QR factorization of a sparse matrix s with Q R qr S but this is usually impractical The orthogonal matrix Q often fails to havea high proportion of zero elements A more practical alternative someti
84. see the Optimization Toolbox Minimizing Functions of One Variable Given a mathematical function of a single variable coded in an M file you can use thef mi nbnd function to find a local minimizer of the function in a given interval For example to find a minimum of the humps function in the range 0 3 1 use x fminbnd humps 0 3 1 which returns X 0 6370 You can ask for a tabular display of output by passing a fourth argument created by theopti mset command tof mi nbnd x fminbnd humps 0 3 1 optimset Display iter 7 Function Functions which gives the output Func count X f x Procedure 1 0 567376 12 9098 initia 2 0 732624 13 7746 golden 3 0 465248 25 1714 golden 4 0 644416 11 2693 parabolic 5 0 6413 11 2583 parabolic 6 0 637618 11 2529 parabolic 7 0 636985 11 2528 parabolic 8 0 637019 11 2528 parabolic 9 0 637052 11 2528 parabolic X 0 6370 This shows the current value of x and the function value at f x each time a function evaluation occurs For f mi nbnd one function evaluation corresponds to one iteration of the algorithm The last column shows what procedure is being used at each iteration either a golden section search or a parabolic interpolation Minimizing Functions of Several Variables Thef mi nsearch function is similar tof mi nbnd except that it handles functions of many variables and you specify a starting vector Xo rather than a starting interval f mi nsea
85. sei f condition is false 0 The statements associated with it execute if its logical condition is true 1 You can have multipleel sei f s within ani f block if n lt 0 If n negative display error message disp Input must be positive elseif rem n 2 0 If n positive and even divide by 2 A n 2 else A n 1 2 If n positive and odd increment and divide end 10 31 10 M File Programming if Statements and Empty Arrays Ani f condition that reduces to an empty array represents a false condition That is if A 1 else 0 end will execute statement 0 when A is an empty array switch switch executes certain statements based on the value of a variable or expression Its basic form is switch expression scalar or string case valuel statements Executes if expression is value case val ue2 statements Executes if expression is value otherwise statements Executes if expression does not end match any case This block consists of e The word swi tch followed by an expression to evaluate e Any number of case groups These groups consist of the word case followed by a possible value for the expression all on a single line Subsequent lines contain the statements to execute for the given value of the expression These can be any valid MATLAB statement including another s wi t ch block Execution of acase group ends when MATLAB encounters the next case stateme
86. steps after the initial function evaluations would have included some search steps whilef zero searched for an interval containing a sign change You can specify a relative error tolerance using opti mset In the call above passing in the empty matrix causes the default relative error tolerance of eps to be used Tips Optimization problems may take many iterations to converge Most optimization problems benefit from good starting guesses Providing good starting guesses improves the execution efficiency and may help locate the global minimum instead of a local minimum Sophisticated problems are best solved by an evolutionary approach whereby a problem with a smaller number of independent variables is solved first Solutions from lower order problems can generally be used as starting points for higher order problems by using an appropriate mapping The use of simpler cost functions and less stringent termination criteria in the early stages of an optimization problem can also reduce computation time Such an approach often produces superior results by avoiding local minima Minimizing Functions and Finding Zeros Troubleshooting Below is alist of typical problems and recommendations for dealing with them Problem Recommendation The solution found by f mi nbnd or f minsearch does not appear to bea global minimum Sometimes an optimization problem has values of x for which it is impossible to evaluate f The min
87. subsets of the data as comma separated variable lists e You don t havea fixed set of field names e You routinely remove fields from the structure As an example of accessing multiple fields with one statement assume that your data consists of e A 3 by 4 array consisting of measurements taken for an experiment e A 15 character string containing a technician s name e A 3 by 4 by 5 array containing a record of measurements taken for the past five experiments For many applications the best data construct for this data is a structure However if you routinely access only the first two fields of information then a cell array might be more convenient for indexing purposes This example shows how to access the first and second elements of the cell array TEST newdata name deal TEST 1 2 This example shows how to access the first and second elements of the structureTESsT newdata TEST measure name TEST name Cell Arrays Thevarargin andvarargout arguments are examples of the utility of cell arrays as substitutes for comma separated lists Create a 3 by 3 numeric array A A 01 2 4 0 7 3 1 2 Now apply thenor mest 2 norm estimate function toA and assign the function output to individual cells of B B 1 2 normest A B 8 8826 4 All of the output values from the function are stored in separate cells ofB B 1 contains the norm estimate B 2 contains the iteration count Nesting Cell Arra
88. that contains a MATLAB expression statement or function call In its simplest form theeval syntax is eval string For example this code uses eval on an expression to generate a Hilbert matrix of order n ts l itj 1 fori 1 n forj lin a i j eval t end end Here s an example that uses eval on a statement eval t clock feval feval differsfromeval in that it executes a function whose name isin a string rather than an entire MATLAB expression You can usef eval and thei nput function to choose one of several tasks defined by M files In this example the functions have names likesin cos and soon fun sin cos log k input Choose function number x input Enter value feval fun k x String Evaluation Note Usef eval rather than eval whenever possible M files that usef eval execute faster and can be compiled with the MATLAB compiler Constructing Strings for Evaluation You can concatenate strings to create a complete expression for input toeval This code shows how eval can create 10 variables namedP1 P2 P10 and set each of them to a different value for i l 10 eval P int2str i i 2 end 10 47 10 M File Programming Command Function Duality 10 48 MATLAB commands are statements like load help Many commands accept modifiers that specify operands load August17 dat help magic type rank An al
89. the Harwell Boeing collection load west0479 whos Name Size Bytes Class west 0479 479x479 24576 sparse array This matrix models an eight stage chemical distillation column Try these commands nnz west 0479 ans 1887 format short e west 0479 west0479 25 1 1 0000e 00 FLA 3 7648e 02 87 1 3 4424e 01 26 2 1 0000e 00 3172 2 4523e 02 88 2 3 7371e 01 21 3 1 0000e 00 31 3 3 6613e 02 89 3 8 3694e 01 28 4 1 3000e 02 nonzeros west0479 9 12 Viewing Sparse M atrices 0e 8e 0 4 2 00e 23e 71e 01 0 1 9 0 0e 3e 1 w WWrnon FoF x4 Qe Note Use Ctrl C to stop thenonzeros listing at any time Note that initially nnz has the same value as nz max by default That is the number of nonzero elements is equivalent to the number of storage locations allocated for nonzeros However MATLAB does not dynamically release memory if you zero out additional array elements Changing the value of some matrix elements to zero changes the value of nnz but not that of nz max You can add as many nonzero elements to the matrix as desired however you are not constrained by the original value of nz max Viewing Sparse Matrices Graphically It is often useful to use a graphical format to view the distribution of the nonzero elements within a sparse matrix MATLAB ss py function produces a template view of the sparsity struct
90. the parameters 6 17 multiple 6 19 polynomial 6 15 regserver startup option 2 4 relational operators and empty arrays 10 24 and strings 11 11 relative accuracy ODE 8 21 remainder 10 13 removing cells from cell array 13 25 fields from structure arrays 13 9 singleton dimensions 12 12 reorderings 9 24 and LU factorization 9 28 for sparser factorizations 9 26 minimum degree ordering 9 27 to reduce bandwidth 9 27 replacing substring within string 11 12 repmap 12 6 reports function call history 3 24 function details 3 23 representing polynomial roots 5 3 polynomials 5 2 problems for ODE solvers 8 5 reshape 12 11 13 25 reshaping cell arrays 13 25 multidimensional arrays 12 11 residuals 6 22 for exponential data fit 6 27 residue 5 7 return 10 36 rigid body ODE example 8 34 rigidode 8 34 rmfield 13 9 rmpath 2 14 roots 5 3 roots of polynomial 5 3 round 5 10 row vector 4 5 for polynomial representation 5 2 rows deleting 10 42 running external program 10 62 script 10 5 running in Editor 2 41 runtime errors finding 3 2 S save 2 11 2 28 2 29 9 10 10 66 saveobj example 14 62 saving ASCII files 2 12 2 28 Editor options 2 55 files in Editor 2 40 objects 14 61 variables 2 11 workspace 2 11 scalar 4 5 and relational operators 11 11 expansion 10 22 string 11 11 schur 4 37 Schur decomposition 4 37 Index script 10 5 and creation of new data 10 5 and data in workspace 10 5 characteristics 10 3 defined 10 3
91. the appropriate subscript and assign the desired values e Assign thesamenumber of elements to corresponding array dimensions F or numeric arrays all rows must have the same number of elements all pages must have the same number of rows and columns and so on 12 5 12 Multidimensional Arrays You can take advantage of MATLAB s scalar expansion capabilities together with the colon operator to fill an entire dimension with a single value Als 2 3 se 5 A 3 ans 5 5 5 5 5 5 5 5 5 To turn A into a 3 by 3 by 3 by 2 four dimensional array enter Alea se 2 gt Se PD DSi ADS 6 ade GNE Nits 2h SEG Be 6 bo Be B21 Aes tp 3y 2 PL oO Ts a ae OR Oa Ty Note that after the first two assignments MATLAB pads A with zeros as needed to maintain the corresponding sizes of dimensions Generating Arrays Using MATLAB Functions You can use MATLAB functions such asrandn ones andzeros to generate multidimensional arrays in the same way you use them for two dimensional arrays Each argument you supply represents the size of the corresponding dimension in the resulting array F or example to create a 4 by 3 by 2 array of normally distributed random numbers B randn 4 3 2 To generate an array filled with a single constant value use ther ep mat function repmat replicates an array in this case a 1 by 1 array through a vector of array dimensions B repmat 5 3 4 2 B I 1 Fs 5 5 5 5 5 5 5 5 5 5 5 5
92. the change occurs break the simulation interval into two pieces and call the solvers twice If you do not know the time at which the change occurs try reducing the error tolerances Rel Tol andAbsTol UseMaxStep as a last resort InitialStep Initial Step has a positive scalar value This property sets an upper bound on the magnitude of the first step size the solver tries Generally the automatic procedure works very well However the initial step size is based on the slope of the solution at the initial timetspan 1 and if the slope of all solution components is zero the procedure might try a step size that is much too large If you know this is happening or you want to be sure that the solver resolves important behavior at the start of the integration help the code start by providing a suitable ni tial Step Mass Matrix Properties The solvers of the ODE suite can solve problems of the form M t y y F t y with a mass matrix M that is nonsingular and usually sparse Useodeset to set Mass to M Mt or M t y ifthe ODE fileF mis coded so that F t y mass returns a constant time dependent or time and state dependent mass matrix respectively The default value of Mass iS none The ode23s solver can only solve problems with a constant mass matrix M For examples of mass matrix problems seef emlode fem2ode or bat onode If M is singular then M t y F t y is a differential algebraic equation DAE DAEs hav
93. the field name str patient 2 name str Ann Lane To access elements within fields append the appropriate indexing mechanism tothe field name That is if the field contains an array use array subscripting if the field contains a cell array use cell array subscripting and so on test2b patient 3 test 2 2 test2b 153 Use the same notations to assign values to structure fields for example patient 3 test 2 2 7 13 7 13 Structures and Cell Arrays 13 8 You can extract field values for multiple structures at a time For example the line below creates a 1 by 3 vector containing all of thebi I ing fields bills patient billing bills 127 0000 28 5000 504 7000 Similarly you can create a cell array containing thet est data for the first two structures tests patient 1 2 test tests 3x3 double 3x3 double Accessing Field Values Using setfield and getfield Direct indexing is usually the most efficient way to assign or retrieve field values If however you only know the field name as a string for example if you have used thef i el dnames function to obtain the field name within an M file you can usethes et field andgetfi eld functionstodothesamething getfield obtains a value or values from a field or field element f getfield array array_index field field_index where thefield_index is optional andarray_index is optional for a 1 by 1 structure array The fu
94. the function value The default valueis1 e 4 This parameter is used by f mi nsearch but not fminbnd e options MaxFunEvals iS the maximum number of function evaluations allowed The default valueis 500 forf mi nbnd and200 engt h x0 for fminsearch 7 9 7 Function Functions 7 10 The number of function evaluations the number of iterations and the algorithm are returned in the structure out put when you providef mi nbnd or fmi nsearch with a fourth output argument as in x fval exitflag output fminbnd humps 0 3 1 or x fval exitflag output fminsearch three var v Finding Zeros of Functions Thef zero function attempts to find a zero of one equation with one variable This function can be called with either a one element starting point or a two element vector that designates a starting interval If you givef zero a starting point Xo f zero first searches for an interval around this point where the function changes sign If the interval is found then f zero returns a value near where the function changes sign If no such interval is found f zero returns NaN Alternatively if you Know two points where the function value differs in sign you can specify this starting interval using a two element vector f zero is guaranteed to narrow down the interval and return a value near a sign change Usef zero tofind a zero of thehumps function near 0 2 a fzero humps 0 2 a 0 1316 For thi
95. the license server to a different machine The MathWorks can give you new information by email or telephone Your system administrator should edit the License F ileto reflect your licensing information If you edit the file yourself follow the instructions in the Installation Guide or run mdown followed by mst art Torun mdown you must be a member of the UNIX group madmi n or a member of group 0 if madmi n does not exist Themat ab script in mat ab bin sets the environment variable LM_LICENSE_FILE to contain the pathname where i cense dat is stored This pathnameis normally mat ab etc license dat wherever MATLAB is stored If necessary you can change this environment variable to point to some other location The file usr tmp m_TMW5 og where the license daemon s output usually is redirected contains a log of all license check outs check ins and denials A new entry is recorded in the log each time a transaction occurs To save file space you can delete it occasionally Creating a Local Options File You can instruct the license manager to e Reserve one or more keys for a particular user group of users or host e Specify the users groups of users or hosts that have permission to access one or more products To use these options you can create a local options file and list its pathname as thefourth field on theDAEMON linein thel icense dat file Depending on the length of your path this line may become f
96. the subscripts 2 3 2 Aiea A 2 3 2 gt nN 1 As you add dimensions toan array you also add subscripts A four dimensional array for example has four subscripts The first two reference a row column pair the second two access the third and fourth dimensions of data Note The general multidimensional array functions residein thedatatypes directory Creating Multidimensional Arrays You can use the same techniques to create multidimensional arrays that you use for two dimensional matrices In addition MATLAB provides a special concatenation function that is useful for building multidimensional arrays This section discusses e Generating arrays using indexing e Generating arrays using MATLAB functions e Usingthecat function to build multidimensional arrays Multidimensional Arrays Generating Arrays Using Indexing One way to create a multidimensional array is to create a two dimensional array and extend it For example begin with a simple two dimensional array A A 5 7 8 019 4 3 6 A is a 3 by 3 array that is its row dimension is 3 and its column dimension is 3 To add a third dimension to A 2 10 4 3 5 6 9 8 7 MATLAB responds with econ ee 5 7 8 0 1 9 4 3 6 APE 2 es 1 0 4 3 5 6 9 8 7 You can continue to add rows columns or pages to the array using similar assignment statements Extending Multidimensional Arrays To extend A in any dimension e Increment or add
97. tspan y0 options init mu tspan 0 max 20 3 mu several periods yO 2 0 options odeset Vectorized on OG dap shies are ete lens tega lot ae ep a let ene ETE N Aa pea Such Epara AAD ia Grogs igttetiean ete oaa function dfdy jacobian t y mu dfdy 0 1 2 mu y 1 y 2 1 mu 1 y 1 2 1 Example 3 Large Stiff Sparse Problem Thisis an example of a potentially large stiff sparse problem Likevdpode the file is coded to use both the Vect ori zed andj acobi an properties but only one is used during the course of a simulation Like both previous examples brussode responds to the i ni t flag Thebrussode exampleis theclassic Brusselator system H airer and Wanner modeling diffusion in a chemical reaction 8 37 8 O rdinary Differential Equations 8 38 u 1 uv 4u a N Dras 2u U 1 and is solved on the time interval 0 10 with 1 50 and u 0 1 sin 2nx with x i N Dfor i 1 N There are 2N equations in the system but the J acobian is banded with a constant width 5 if the equations are ordered as u4 Vj U V2 brussode t y Orbrussode t y n returns the derivatives vector for the Brusselator problem The parameter n gt 2 is used to specify the number of grid points the resulting system consists of 2n equations By default n is2 The problem becomes increasingly stiff and the acobian increasingly sparse as n is increased brusso
98. variable y such that y y gt Y ou can then express this system as Y Y2 na 2 y2 W 1 yy Y2 Y1 Writing the O DE File The code below shows how to represent the van der Pol system in a MATLAB ODE file an M file that describes the system to be solved An ODE file always accepts at least two arguments t andy This simple two line file assumes a value of 1 for u yy and y become y 1 and y 2 elements in a two element vector function dy vdpl t y dy y 2 1 y 1 2 y 2 y 1 Note This ODE file does not actually use thet argument in its computations It is not necessary for it touse they argument either in some cases for example it may just return a constant Thet andy variables however must always appear in the input argument list Calling the Solver Oncethe ODE system is coded in an ODE file you can usethe MATLAB ODE solvers to solve the system on a given time interval with a particular initial condition vector For example to useode45 to solve the van der Pol equation on time interval 0 20 with an initial value of 2 for y 1 and an initial value of 0 for y 2 T Y ode45 vdpl 0 20 2 0 Representing Problems The resulting output T Y is a column vector of time points T and a solution array Y Each row in solution array Y corresponds toa time returned in column vector T Viewing the Results Use thep ot command to view solver output plot t y 1 t y i 2 title
99. well These sections step you through an example debugging session Although the example M files might be simpler than your own MATLAB code the debugging concepts demonstrated here remain the same Debugging An Overview Debugging is the process by which you isolate and fix problems with your code Debugging helps to correct two kinds of errors e Syntax errors such as misspelling a function name or omitting a parenthesis MATLAB detects most syntax errors and displays a message describing the error and showing its line number in the M file e Runtime errors These errors are usually algorithmicin nature for example you might modify the wrong variable or perform a calculation incorrectly Runtime errors are apparent when an M file produces unexpected results You can usually correct syntax errors easily based on MATLAB s error messages Runtime errors are more difficult to track down because the function s local workspace is lost when the error forces a return to the MATLAB base workspace Use any of the following techniques to isolate the cause of runtime errors e Remove selected semicolons from the statements in your M file Semicolons suppress the display of intermediate calculations in the M file By removing thesemicolons you instruct MATLAB todisplay these results on your screen as the M file executes e Addkeyboard statements to the M file Keyboard statements stop M file execution at the point where they appear and a
100. when height passes through zero in a decreasing direction and stop integration Also locate both decreasing and increasing zero crossings of velocity and don t stop integration value y height velocity isterminal 1 0 direction 1 0 8 45 8 O rdinary Differential Equations 8 46 Example 6 Advanced Event Location orbitode isa standard test problem for nonstiff solvers presented in Shampine and Gordon see reference that follows Theor bitode problem is a system of four equations yi Y3 Yo Y4 L Yz H u y1 u Va 2a ae ge as ry r gt u y gt uYy2 Y4 2Y3 Y2 gS 5 where u 1 82 45 ws l yu ri Eyt y 2 ra Ay y The first two solution components are coordinates of the body of infinitesimal mass so plotting one against the other gives the orbit of the body around the other two bodies The initial conditions have been chosen so as to make the orbit periodic This corresponds to a spaceship traveling around the moon and returning to the earth Moderately stringent tolerances are necessary to reproduce the qualitative behavior of the orbit Suitable values are 1e 5 for Rel Tol andile 4 for AbsTol The event functions implemented in this example locate the point of maximum distance from the earth and the time the spaceship returns to earth Examples Applying the O DE Solvers Reference Shampine L F and M K Gordon Computer Solution of Ordinary Differential Equ
101. where r is a column vector of residues p is a column vector of pole locations and k is a row vector of direct terms Consider the transfer function 5 7 5 Polynomials and Interpolation 24495 1 6s 14857 b 4 8 a 1 6 8 r p k residue b a r 12 p 4 2 k Given threeinput arguments r p andk r esi due converts back to polynomial form b2 a2 residue r p k 5 8 Interpolation Interpolation Interpolation is a process for estimating values that lie between known data points It has important applications in areas such as signal and image processing MATLAB provides a number of interpolation techniques that let you balance the smoothness of the data fit with speed of execution and memory usage The interpolation functions live in a directory called pol yf un inthe MATLAB Toolbox Function Description griddata Data gridding and surface fitting interpl One dimensional interpolation table lookup interp2 Two dimensional interpolation table lookup interp3 Three dimensional interpolation table lookup interpft One dimensional interpolation using FFT method interpn N D interpolation table lookup spline Cubic spline data interpolation One Dimensional Interpolation There are two kinds of one dimensional interpolation in MATLAB e Polynomial interpolation e FFT based interpolation Polynomial Interpolation The function i nt er p1 performs one dimensi
102. y x exp x The code below writes x and y into a newly created file namedexptable txt fid fopen exptable txt w fprintf fid Exponential Function n n forintf fid 6 2f 12 8f n y status fclose fid The first call tof printf outputs a title followed by two carriage returns The second call tof printf outputs the table of numbers The format control string specifies the format for each line of the table e A fixed point value of six characters with two decimal places e Two spaces e A fixed point value of twelve characters with eight decimal places fprintf converts the elements of array y in column order The function uses the format string repeatedly until it converts all the array elements Now usef scanf to read the exponential data file fid fopen exptable txt r title fgetl fid table count fscanf fid f 2 11 table table status fclose fid Formatted Files The second line reads the file title The third line reads the table of values two floating point values on each line until it reaches end of file count returns the number of values matched A function related tof printf sprintf outputs its results to a string instead of a file or the screen For example root2 sprintf The square root of f is 10 8e n 2 sqrt 2 15 17 13 File O 15 18 Symbols 1 2 25 10 7 10 9 amp 10 24 10 24 10 24 10 25 10 2
103. ymax quad8 Alternatively any user defined quadrature function name can be passed to dbl quad as long as the quadrature function has the same calling and return arguments as quad Ordinary Differential Equations Quick Start 4 zope we we Be A ae eh es 8 3 RepresentingProblems 8 5 ODE Solvers 20 0544 8 10 Creating ODE Files 2 2 2 8 14 Improving Solver Performance 8 17 Examples Applying the ODE Solvers 8 34 8 O rdinary Differential Equations 8 2 This chapter describes how to use MATLAB to solve initial value problems of ordinary differential equations ODEs and differential algebraic equations DAEs It discusses how to represent initial value problems IVPs in MATLAB and how to apply MATLAB s ODE solvers to such problems It explains how to select a solver and how to specify solver options for efficient customized execution This chapter alsoincludes a troubleshooting guidein the Questions and Answers section and extensive examples in the Examples Applying the ODE Solver section Category Function Description Ordinary differential ode45 equation solvers ode23 odell3 odel5s ode23s ode23t ode23tb ODE option handling odeset odeget ODE output functions odepl ot odephas2 odephas3 odeprint Nonstiff differential equations medium order method Nonstiff differential equations low order method Nonstiff
104. your M file How MATLAB Passes Function Arguments From the programmer s perspective MATLAB appears to pass all function arguments by value Actually however MATLAB passes by value only those arguments that a function modifies If a function does not alter an argument but simply uses it in a computation MATLAB passes the argument by reference to optimize memory use Function Workspaces Each M file function has an area of memory separate from MATLAB s base workspace in which it operates This area is called the function workspace with each function having its own workspace context 10 11 10 M File Programming 10 12 While using MATLAB theonly variables you can access arethosein thecalling context beit the base workspace or that of another function The variables that you pass to a function must be in the calling context and the function returns its output arguments to the calling workspace context You can however define variables as global variables explicitly allowing more than one workspace context to access them Checking the Number of Function Arguments Thenargin andnargout functions let you determine how many input and output arguments a function is called with You can then use conditional statements to perform different tasks depending on the number of arguments For example function c testargl a b if nargin 1 c a 2 elseif nargin 2 c a b end Given a single input argument this f
105. 0 01 L x 0 9 2 0 04 6 A second way to represent a mathematical function at the command lineis by creating an inline object from a string expression F or example you can create an inline object of the humps function f inline 1l x 0 3 2 0 01 L x 0 9 2 0 04 6 You can then evaluatef at 2 0 f 2 0 ans 4 8552 You can also create functions of more than one argument with i nl i ne by specifying the names of the input arguments along with the string expression For example the following function has two input arguments x andy f inline y sin x x cos y x y f pi 2 pi ans 3 1416 All of the functions described in this chapter are called function functions because they accept as one of their arguments either the name of an M file like humps or an inline object that defines a mathematical function 7 Function Functions Plotting Mathematical Functions Thef pl ot function plots a mathematical function between a given set of axes limits You can control the x axis limits only or both the x and y axis limits For example to plot thehumps function over the x axis range 5 5 use fplot humps 5 5 grid on 100 60 4 40F 4 20 J7 20 fi fi L L L fi L L L 5 4 3 2 1 0 1 A 3 4 5 7 4 Plotting Mathematical Functions You can zoom in on the function by selecting y axis limits of 10 and25 using fplot humps 5 5 10 25
106. 1 7 11 9 11 9 1l 11 11 11 11 12 11 13 11 14 11 Character Arrays Strings 11 2 This chapter explains MATLAB s support for string data It describes how to create character arrays and cell arrays of strings the two ways to represent strings It also discusses how to perform common string operations such as searching and replacing and how to convert between string and numeric formats Thestring functions arelocated in the directory namedst rf un inthe MATLAB Toolbox Category Function Description General char Create character array string double Convert string to numeric codes cellstr Create cell array of strings from character array blanks String of blanks debl ank Remove trailing blanks eval Execute string with MATLAB expression String Tests ischar True for character array iscellstr True for cell array of strings isletter True for letters of alphabet isspace True for whitespace characters String Operations strcat Concatenate strings strvcat Concatenate strings vertically strcmp Compare strings strncmp Compare first N characters of strings findstr Find one string within another strjust J ustify string strmatch Find matches for string Category Function Description strrep Replace string with another strtok Find token in string upper Convert string to uppercase ower Convert string to lowercase String to Number num2str Convert number to string Coerel int2str Convert in
107. 127 00 L billing 28 50 Vues Tt 79 75 73 RS t 68 70 68 180 178 177 5 118 118 119 220 210 205 172 170 169 M ultidimensional Structure Arrays Applying Functions to Multidimensional Structure Arrays Toapply functions to multidimensional structure arrays operate on fields and field elements using indexing F or example find the sum of the columns of the test arrayinpatient 1 1 2 sum patient 1 1 2 test Similarly add all thebi ling fields in thepati ent array total sum patient billing 12 21 12 Multidimensional Arrays 12 22 Structures and Cell Arrays Structures pO Ww Agi Aiea ag ok ce SBS Building Structure Arrays i a a Be a ate Bots Bice Gen Ge LBS Accessing Data in Structure Arrays 136 Using the size Function with Structure Arrays 13 9 Adding Fields to Structures 139 Deleting Fields from Structures 13 9 Applying Functions and Operators hoe ad e eee 13 9 Writing Functions to Operate on Structures i ERR Ale Sha 13 10 Organizing Data in Structure Arrays 13 11 Nesting Structures a oaoa a a a a a aaa a a 13 16 Cell Arrays Meh ee be GE si ay a es ae a BLO Creating Cell Arrays os Se waa a 2 ee 1319 Obtaining Data from Cell arrays Se 2 EAN es cathe amp he ABE2S Deleting Cals at pruo Gove Af Godlee 40 L329 Reshaping Cell Arrays D hayir ao a LB25 Replacing Lists of Variables with Cell arrays wee ee 13 25 A
108. 14 7 14 7 15 7 Function Functions This chapter describes function functions MATLAB functions that work with mathematical functions instead of numeric arrays These function functions include e Plotting e Optimization and zero finding e Numerical integration quadrature The function functions are located in the MATLAB f unf un directory This table provides a brief description of the functions discussed in this chapter Related functions are grouped by category Category Function Description Plotting fplot Plot function Optimization f mi nbnd Minimize function of one variable with and zero finding bound constraints fminsearch Minimize function of several variables fzero Find zero of function of one variable Numerical quad Numerically evaluate integral low integration order method quad8 Numerically evaluate integral higher order method dbl quad Numerically evaluate double integral Note F or details on another set of function functions ordinary differential equation solvers see Chapter 8 Representing Functions in MATLAB Representing Functions in MATLAB MATLAB represents mathematical functions by expressing them in M files For example consider the function 1 1 NO ae E ee eta ee x 0 3 0 01 x 0 9 0 04 This function can be used as input to any of the function functions Y ou can find it in the M file named humps m function y humps x y L x 0 3 2
109. 3 A abs 6 37 absolute accuracy ODE 8 22 accuracy of calculations 10 18 Acrobat Reader 2 23 ActiveX entries for MATLAB 2 4 adding cells to cell arrays 13 21 fields to structure arrays 13 9 addition of matrices 4 6 addpath 2 14 adjacency matrix 9 15 and graphing 9 15 Bucky ball 9 16 defined 9 15 distance between nodes 9 21 node 9 15 numbering nodes 9 17 administration of license 2 59 Adobe Acrobat Reader 2 23 advanced indexing 10 42 aggregation 14 36 airflow modeling 9 21 algorithms ODE solvers Adams Bashworth Moulton PECE 8 10 Bogacki Shampine 2 3 8 10 Dormand Prince 4 5 8 10 modified Rosenbrock formula 8 11 numerical differentiation formulas 8 11 alphanumeric characters See character AND operator rules for evaluating 10 24 angl e 6 37 annotating plots 2 18 ans 10 18 answer assigned toans 10 18 any 10 26 Application Program Interface API 1 5 arguments checking number of 10 12 defined for function 10 7 for ODE file 8 6 order 10 13 10 15 passed by reference 10 11 passed by value 10 11 passing variable number 10 14 arithmetic expressions 10 21 arithmetic operators 10 21 arithmetic operators overloading 14 29 arrays cell array of strings 11 7 character 10 19 11 4 dimensions inverse permutation 12 14 editing 2 52 elements 4 4 indexing advanced 10 42 multidimensional 12 3 numeric converting to cell array 13 32 of strings 11 5 storage 10 42 Index arrow icon in Debugger 3 7 arrow
110. 4 and the The Asset subsref Method on page 14 40 for a description of these methods Numericindices greater than the number returned byf i el dcount arehandled by the inner swi tch statement which maps the index value to the appropriate field in the stock structure 14 49 14 MATLAB Classes and O bjects Field name indexing assumes field names other than num shares and share_price areasset fields which eliminates the need for knowledge of asset fields by child methods The asset subsref method performs field name error checking See thesubsref help entry for general information on implementing this method The Stock set Method Theset method provides a property name interfaceliketheget method It is designed to update the number of shares the share value and the descriptor The current value and the date are automatically updated function s set s varargin SET Set stock properties to the specified values and return the updated object property_argin varargin while length property _argin gt 2 prop property argin 1 val property_argin 2 property_argin property_argin 3 end switch prop case NumberShares S num_ shares val case SharePrice s share price val case Descriptor S asset set s asset Descriptor val otherwise error Invalid property end end S asset set s asset CurrentValue s num_ shares s share_price Date date Note that
111. 553 0 4691 0 49011 0 4691 0 4901i 0 4248 0 6249 0 29971 0 62494 0 29971 D 3 0710 0 0 0 2 4645 17 6008i 0 0 0 2 4645 17 6008i The first eigenvector is real and the other two vectors are complex conjugates of each other All three vectors are normalized to have Euclidean length norm v 2 equal to one The matrix V D i nv V which can be written more succinctly as V D V is within roundoff error of A And i nv V A V or V A V iS within roundoff error of D Some matrices do not have an eigenvector decomposition These matrices are defective or not diagonalizable For example A 6 12 19 9 2 0 33 4 9 15 For this matrix V D eig A 4 35 4 Matrices and Linear Algebra produces V 0 4741 0 4082 0 4082 0 8127 0 8165 0 8165 0 3386 0 4082 0 4082 D 1 0000 0 0 0 1 0000 0 0 0 1 0000 There is a double eigenvalue at A 1 The second and third columns of V are negatives of each other they are merely different normalizations of the single eigenvector corresponding to 1 For this matrix a full set of linearly independent eigenvectors does not exist The optional Symbolic Math Toolbox extends MATLAB s capabilities by connecting to Maple a powerful computer algebra system One of the functions provided by the toolbox computes the J ordan Canonical Form This is appropriate for matrices like our example which is 3 by 3 and has exactly known integer elements
112. 6 46 47 33 64 50 51 61 32 18 19 29 53 59 58 56 21 27 26 24 57 55 54 60 25 23 22 28 52 62 63 49 20 30 31 17 This matrix is half way to being another magic square Its elements are a rearrangement of the integers 1 64 Its column sums are the correct value for an 8 by 8 magic square sum B ans 260 260 260 260 260 260 260 260 10 41 10 M File Programming 10 42 But its row sums sum B are not all the same Further manipulation is necessary to make this a valid 8 by 8 magic square Deleting Rows and Columns You can delete rows and columns from a matrix using just a pair of square brackets Start with X A Then to delete the second column of X use X 2 This changes X to L 162 13 511 8 ae i 414 1 If you delete a single element from a matrix the result isn t a matrix anymore So expressions like X 1 2 result in an error However using a single subscript deletes a single element or sequence of elements and reshapes the remaining elements into a row vector So X 2 2 10 results in X 169 27 13121 Advanced Indexing MATLAB stores each array as a column of values regardless of the actual dimensions This column consists of the array columns appended end to end For example MATLAB stores A 26 9 4 2 8 3 0 1 Indexing and Subscripting as e oOW OND WwW amp PY Accessing A with a single subscript indexes directly into the storage column A 3 acc
113. 74i Notice that although the sequence x is real y is complex The first component of the transformed data is the constant contribution and the fifth element corresponds to the Nyquist frequency The last three values of y correspond to negative frequencies and for the real sequencex they are complex conjugates of three components in the first half of y Suppose we want to analyzethe variations in sunspot activity over the last 300 years You are probably aware that sunspot activity is cyclical reachinga maximum about every 11 years Let s confirm that Astronomers have tabulated a quantity called the Wolfer number for almost 300 years This quantity measures both number and size of sunspots 6 33 6 Data Analysis and Statistics Load and plot the sunspot data load sunspot dat year Sunspot 1 wolfer sunspot 2 plot year wolfer title Sunspot Data Sunspot Data 200 T 180F al 1607 4 140 J 1207 A 100F 4 80 60 q 40 J 20 7 0 n 1 1 n 1700 1750 1800 1850 1900 1950 2000 Now take the FFT of the sunspot data Y fft wolfer The result of this transform is the complex vector Y The magnitude of Y squared is called the power and a plot of power versus frequency is a periodogram Remove the first component of Y whichis simply the sum of the data and plot the results 6 34 Fou
114. 97 te Now evaluate the model at regularly spaced points and overlay the original data in a plot Te Se Oar Oct deed bjt Y ones size T exp T T exp T a plot T Y t y o grid on 1 5 0 97 0 87 0 7 F 0 67 0 5 L L L L 0 0 5 1 1 5 2 2 5 This is a much better fit than the second order polynomial function 6 18 Regression and Curve Fitting Multiple Regression If yis a function of more than one independent variable the matrix equations that express the relationships among the variables can be expanded to accommodate the additional data Suppose we measure a quantity y for several values of parameters x and x The observations are entered as Or EE egmadi 69 7 26 28 XT cs Pe Y2 SS y 1 A multivariate model of the data is Multiple regression solves for unknown coefficients ag aq and a gt by performing a least squares fit Construct and solve the set of simultaneous equations by forming the regression matrix X and solving for the coefficients using the backslash operator X ones size xl x1 x2 a X y a 0 1018 0 4844 0 2847 The least squares fit model of the data is therefore y 0 1018 0 4844 x 0 2847 x Tovalidate the model find the maximum of the absolute value of the deviation of the data from the model Y X a MaxErr max abs Y y MaxErr 0 0038 This is sufficiently small to be confident the model reasonably fits the data
115. AB 10 18 inheritance example class 14 37 multiple 14 35 simple 14 34 initial condition 8 14 defined 8 5 example 8 6 initial condition vector 8 6 ODE 8 3 initial value problem 8 2 defined 8 5 initial values defining in ODE file 8 14 returned by 8 15 inner product 4 7 input from keyboard 10 61 obtaining from M file 10 61 input arguments defined by function 10 7 input output binary 15 2 error status 15 2 formatted 15 2 functions 15 2 inserting in Editor 2 48 integer data type 15 15 integers changing to strings 11 13 integration double 7 15 numerical 7 14 Seealso ordinary differential equation solvers interactive user input 10 61 interpl 5 9 interp2 5 12 interp3 5 15 interpft 5 11 interpn 5 16 interpolation comparing methods 5 13 cubic 5 10 cubic spline 5 10 defined 5 9 FFT based 5 11 1 13 Index memory 5 11 multidimensional 5 15 5 16 cubic 5 16 linear 5 16 nearest neighbor 5 16 one dimensional 5 9 cubic 5 10 cubic spline 5 10 linear 5 10 nearest neighbor 5 10 polynomial 5 9 smoothness 5 11 speed 5 11 three dimensional nearest neighbor 5 15 tricubic 5 15 trilinear 5 15 two dimensional 5 12 bicubic 5 12 5 13 bilinear 5 12 5 13 nearest neighbor 5 12 5 13 interpreter MATLAB 2 5 interrupting a running program 2 7 inv 4 20 inverse 4 20 inverse permutation of array dimensions 12 14 ipermute 12 2 12 14 isa 14 11 isempty 10 24 isinf 10 26 i snan 6 13 10 26 iterative methods for sparse matric
116. B 1 2 N 1 1 c 1 2 2 N 1 2 3 2 y i y i 1 i 3 2 y i y itl 4 2 c i 1 3 y i 2 2 c i 1 4 y i 2 3 2 N 5 B 3 2 N 5 C fdy spdiags B 2 2 2 N 2 N Note this is a SPARSE Jacobian unction S jpattern t y N ones 2 N 5 2 2 2 N 2 zeros N 1 1 2 2 N 1 4 zeros N 1 spdiags B 2 2 2 N 2 N f nargin lt 4 isempty N N 2 HW re ou vooo re ewswdWwWs WTO WoT Fr EHP 090 TS SE Example 4 Finite Element Discretization femlode t y orfemlode t y n returns the derivatives vector for a finite element discretization of a partial differential equation The parameter n controls the discretization and the resulting system consists ofn equations By default n is9 Examples Applying the O DE Solvers This example involves a mass matrix The system of ODE s comes froma method of lines solution of the partial differential equation tou _ au at x2 with initial condition u 0 x sin x and boundary conditions u t 0 u t z 0 An integer N is chosen h is defined as 1 N 1 and the solution of the partial differential equation is approximated at x kzh for k 0 1 N 1 by N U Lx KOH k 1 Here ox is a piecewise linear function that is 1 at x and 0 at all the other x A Galerkin discretization leads to the system of ODEs c t A t c Re where c t Cy t and the tridiagonal matrices A t and R are giv
117. BE EAE om aea E od ewan Seeded E EEE Na Meade ies ah ae function dydt f t y pl p2 dydt lt Insert a function of t and or y pl and p2 here gt function tspan y0 options init pl p2 tspan lt Insert tspan here gt yO lt Insert y0 here gt options lt Insert options odeset or here gt function dfdy jacobian t y pl p2 dfdy lt Insert Jacobian matrix here gt function S jpattern t y pl p2 S lt Insert Jacobian matrix sparsity pattern here gt 8 19 8 O rdinary Differential Equations 8 20 function M mass t y pl p2 M lt Insert mass matrix here gt Oy Rtn tar mie Merde em Selita iy ow aha Aerie ie Tata ns es tay ete Me E Schr whet este ea arte a niet oie Senate function value isterminal direction events t y pl p2 value lt Insert event function vector here gt isterminal lt Insert logical ISTERMI NAL vector here gt direction lt Insert DIRECTION vector here gt Creating an Options Structure The odeset Function Theodeset function creates an options structure that you can supply to any of the ODE solvers odeset accepts property name property value pairs using the syntax options odeset namel valuel name2 value2 This creates a structureopti ons in which the named properties have the specified values Any unspecified properties contain default values in the solvers For all properties it is suffici
118. C lass Overloading Arithmetic Operators Several arithmetic operations are meaningful on polynomials and should be implemented for the polynom class When overloading arithmetic operators keep in mind what data types you want to operate on In this section thep us minus and mti mes methods are defined for the polynom class to handle addition subtraction and multiplication on polynom polynom and polynom double combinations of operands Overloading the Operator If either p or q is a polynom the expression p q generates a call toa function pol ynom plus m if it exists unless p org isan object of a higher precedence as described in Object Precedence on page 14 66 The following M file redefines the operator for the polynom class function r plus p q POLYNOM PLUS Implement p q for polynoms p polynom p q polynom q k length q c length p c r polynom zeros 1 k p c zeros 1 k q c The function first makes sure that both input arguments are polynomials This ensures that expressions such as p 1 that involve both a polynom and a double work correctly The function then accesses the two coefficient vectors and if necessary pads one of them with zeros to make them the same length The actual addition is simply the vector sum of the two coefficient vectors Finally the function calls the pol ynom constructor a third time to create the properly typed result 14 29 14
119. Fourier transform Two dimensional discrete F ourier transform N dimensional discrete F ourier transform Inverse discrete F ourier transform Two dimensional inverse discrete F ourier transform N dimensional inverse discrete F ourier transform Magnitude Phase angle Unwrap phase angle in radians Move zeroth lag to center of spectrum Sort numbers into complex conjugate pairs Next higher power of two 6 31 6 Data Analysis and Statistics For length N input sequence x the DFT is a length N vector X fft andifft implement the relationships Jzn D 1 lt k lt N X k y x n e n 1 j2zk D 1 lt n lt N N 1 x N oy X k e k 1 Note As the first element of a MATLAB vector has an index 1 the summations in the equations above are from 1 to N These produce identical results as traditional Fourier equations with summations from 0 to N 1 If x n is real we can rewrite the above equation in terms of a summation of sine and cosine functions with real coefficients N x n L ack cos k 1 2n k 1 n 1 2m K 1 n 1 TaD epi where a k real X k b k imag X k 1 lt n lt N The FFT of a column vector x x 43 7 9 100 0 is found with y fft x 6 32 Fourier Analysis and the Fast Fourier Transform FFT which results in y 6 0000 11 4853 2 7574i 2 0000 12 0000i 5 4853 11 2426 18 0000 5 4853 11 2426 2 0000 12 0000i 11 4853 2 75
120. Hooe ee eee 4 o 48 15 F en ee ee 4 ee eee ego eee jooee o oo eee o i e eeiiy 48 2 eoo eee e 4 ee oe oo eseo oio ee AB 25 Re oo 0 0 oe zl Latitude 48 3 F oe 4 48 35 F i 48 4 F 4 48 45 L L L L L L L L 210 8 210 9 211 211 1 211 2 211 3 211 4 211 5 211 6 211 7 211 8 Longitude Note For information on seamount see Parker R L L Shure amp J Hildebrand The Application of Inverse Theory to Seamount Magnetism Reviews of Geophysics Vol 25 1987 pp 17 40 5 19 5 Polynomials and Interpolation Apply Delaunay triangulation and overplot the resulting triangles on the scatter plot ay CBD me X 48 3 A iN O tri delaunay x y hold on trimesh tri x y z hold off hidden off grid on xlabel Longitude ylabel Latitude RIIT KEES Q aay BSA VK YI Anv 1 fi J 21 211 6 211 7 211 8 aK V ERY AAA Longitude Interpolation Here s a contour plot xi yi meshgrid 210 8 01 211 8 48 5 01 47 9 zi griddata x y z xi yi cubic c h contour xi yi zi c clabel c h 47 9 48l 48 1 00007 48 2 F 48 3 F 3501 4000 AOO 48 4 48 5 1 1 f L L f L L 210 8 210 9 211 211 1 2112 211 3 211 4 211 5 211 6 2117 211 8 The arguments for mes hgr i d encompass the largest and smallest x and y values in the origina
121. Krogh for nonstiff solvers The analytical solutions areJ acobian elliptic functions accessible in MATLAB The interval here is about 1 5 periods Therigidode system consists of the Euler equations of a rigid body without external forces as proposed by Krogh ri gi dode is a system of three equations Y 1 Y2 3 y2 Y1 Y3 y3 0 51 y1 Y2 rigidode init returns the defaulttspan y0 andoptions values for this problem These values are retrieved by an ODE solver if the solver is invoked with empty t span or y0 arguments This example uses the default solver options so the third output argument is set to empty instead of an options structure created withodeset By means of the i ni t flag the entire initial value problem is defined in one file Reference Shampine L F and M K Gordon Computer Solution of Ordinary Differential Equations W H Freeman amp Co 1975 Examples Applying the O DE Solvers function varargout rigidode t y flag RI GI DODE Euler equations of a rigid body without external forces switch flag case Return dy dt f t y varargout 1 f t y case init Return default tspan y0 options varargout 1l 3 init otherwise error Unknown flag flag end function dydt f t y dydt y 2 y 3 y 1 y 3 0 51 y 1 y 2 function tspan y0 options init tspan 0 12 yO 0 1 1 options Example
122. LAB 7 3 Plotting Mathematical Functions 7 4 Minimizing Functions and Finding Zeros 7 7 Numerical Integration Quadrature 7 14 Ordinary Differential Equations 8 Quick Startu cocel ee he teed San a a a 8 3 Representing Problems uuaa 8 5 ODE Solvers see bbe cies Oona Pee ee ees 8 10 Creating ODE Files 00 0 cece eee 8 14 Improving Solver Performance 8 17 Examples Applying the ODE Solvers 8 34 iv Contents Questions and Answers 0 0 00 cc eee ee eee 8 50 9 10 Introduction u 9 5 Viewing Sparse Matrices uuaa aa 9 11 Example Adjacency Matrices and Graphs 9 15 Sparse Matrix Operations 20 0000 9 23 M File Programming MATLAB Programming A Quick Start 10 2 SCM Pts ee seen ee iets Dea ee id ieee ree deere 10 5 FUNCTIONS 2 x eee Sa EB ld Ee RS PES 10 6 Local and Global Variables 5 10 16 Data Types 056 esi en te Hed Ed Ge 10 19 Operators 29 0 a vei eee eee Deedee Pe ete eae Ps 10 21 Flow Control seara ey R ee ae Aaa 10 30 Subfunctions var precian EER ei RE EA AR 10 38 Indexing and Subscripting 0 05 10 40 String Evaluation 0 0 0 c cece eee 10 46 Command unction Duality 10 48 Empty Matric
123. LAB code 11 14 string containing MATLAB expression 10 46 values in Debugger 3 7 event location ODE 8 17 8 30 examples adjacency matrix Sparse 9 15 airflow modeling 9 21 brussode 8 37 Bucky ball 9 16 checking number of function arguments 10 12 container class 14 54 Delaunay triangulation 5 19 femlode 8 40 for 10 35 function 10 6 if 10 31 inheritance 14 37 interpolation 5 13 M file for structure array 13 11 ODE solvers 8 34 orbitode 8 46 polynomial class 14 23 rigidode 8 34 script 10 5 second difference operator 9 7 Sparse matrix 9 7 9 15 switch 10 33 theoretical graph sparse 9 15 van der Pol 8 5 extra parameters 8 16 stiff 8 8 vdpode 8 35 vectorization 10 63 while 10 34 exclusive 0R operator 10 26 execution Index pausing 10 61 script 10 5 exiting MATLAB 2 4 expanding cell arrays 13 21 structure arrays 13 4 exponential fit to data 6 25 exponentials matrix 4 31 exporting data 2 26 expressions and scalar expansion 10 22 arithmetic 10 21 involving empty arrays 10 24 logical 10 24 most recent answer 10 18 overloading 14 20 relational 10 23 external program running from MATLAB 10 62 eye 4 10 9 23 F factorization 9 27 Cholesky 4 24 for sparse matrices 9 27 Cholesky 9 30 LU 9 28 triangular 9 28 incomplete 9 32 LU 4 25 positive definite 4 24 QR 4 28 fast Fourier transform See Fourier transform fast fclose 15 2 15 3 15 5 femlode 8 40 f emlode example 8 40 feof 15 2 15 10 ferro
124. MATLAB Simulink Simulink Extensions e Simulink Accelerator e Real Time Workshop e Real Time Windows Target e Stateflow Blocksets e DSP e Fixed Point Power Systems e Nonlinear Control Design e FRE Extensions e MATLAB Compiler e MATLAB C C Math Libraries e MATLAB Web Server e MATLAB Report Generator Toolboxes Control System e Communications Database Financial Frequency Domain System Identification e Fuzzy Logic e Higher Order Spectral Analysis I mage Processing LMI Control Model Predictive Control u Analysis and Synthesis NAG Foundation Neural Network Optimization Partial Differential Equation QFT Control Design Robust Control Signal Processing Spline Statistics Symbolic Math System Identification Wavelet Contact The MathWorks or visit www mat hworks com for an up to date product list 1 Introduction 1 10 MATLAB Working Environment UsingtheEnvironment 2 2 The Command Window 2 5 TheFigureWindow 2 2 2 2 18 Help and Online Documentation 2 20 Disk File Manipulation and Shell Escape 2 25 Datalmport Export 4 2 26 Memory Utilization 2 2 2 2 33 Microsoft WindowsHandbook 2 35 UNIX Handbook 084 2 54 2 MATLAB W orking Environment Using the Environment MATLAB is both a language and a wo
125. MATLAB The Language of Technical Computing Computation 4 Visualization cal Programming I The WORKS Using MATLAB SS Version 5 koa How to Contact The MathW orks 508 647 7000 Phone 508 647 7001 Fax The MathWorks Inc Mail 24 Prime Park Way Natick MA 01760 1500 http www mathworks com Web ftp mathworks com Anonymous FTP server comp soft sys matlab Newsgroup support mathworks com Technical support suggest mathworks com Product enhancement suggestions bugs mat hworks com Bug reports doc mathworks com Documentation error reports subscribe mathworks com Subscribing user registration service mathworks com Order status license renewals passcodes inf o mathworks com Sales pricing and general information Using MATLAB COPYRIGHT 1984 1999 by The MathWorks Inc The software described in this document is furnished under a license agreement The software may be used or copied only under the terms of the license agreement No part of this manual may be photocopied or repro duced in any form without prior written consent from The MathWorks Inc U S GOVERNMENT If Licensee is acquiring the Programs on behalf of any unit or agency of the U S Government the following shall apply a For units of the Department of Defense the Government shall have only the rights specified in the license under which the commercial computer software or commercial software documentation was obtained as set
126. N 8 1 6 3 NaN 7 4 2 Compute a sum for each column in the matrix sum a ans 15 NaN 15 Any mathematical calculation involving NaNs propagates NaNs through tothe final result as appropriate Data Pre Processing You should remove Na Ns from the data before performing statistical computations H ere are some ways to remove NaNs from data i find 7snan x Find indices of elements in vector that are x x i not NaNs then keep only the non NaN elements x x find isnan x Remove Na Ns from vector x x 1snan x Remove Na Ns from vector faster x isnan x Remove Na Ns from vector X any isman X Remove any rows of matrix X containing NaNs Note You must use the special function i snan to find Na Ns because by IEEE arithmetic convention the logical comparison NaN NaN always produces 0 You cannot usex x NaN toremoveNaNs from your data If you frequently need to remove NaNs write a short M file function function X excise X X any isnan X Now typing X excise X accomplishes the same thing Removing Outliers You can remove outliers or misplaced data points from a data set in much the same manner as NaNs For the vehicle traffic count data the mean and standard deviations of each column of the data are mu mean count sigma std count 6 13 6 Data Analysis and Statistics The number of rows with outliers greater than t
127. Ordinary Differential Equations coding as M file 8 6 coding to return initial values 8 15 creating 8 14 defined 8 5 defining initial values 8 14 guidelines for creating 8 16 output 8 15 overview 8 14 passing additional parameters 8 16 template 8 18 rewriting for ODE solvers 8 5 organizing data cell arrays 13 28 multidimensional arrays 12 17 structure arrays 13 11 orthogonal matrix 4 27 orthogonalization 4 24 orthonormal columns 4 27 Out of Memory message 10 70 outer product 4 7 outliers 6 13 output controlling display format 2 7 displayed 14 27 suppressing 2 9 output arguments defined by function 10 7 output properties ODE solvers 8 22 overdetermined systems of simultaneous linear equations 4 15 overloading 14 14 arithmetic operators 14 29 functions 14 20 14 22 14 31 loadobj 14 61 operators 14 2 pie3 14 57 saveobj 14 61 overriding operator precedence 10 29 overtype mode 2 48 P pack 2 33 10 66 page subscripts 12 3 paging in the Command Window 2 7 parentheses for input arguments 10 7 to override operator precedence 10 29 partial fraction expansion 5 7 partial pivoting 4 26 parts of a function 10 6 passing arguments by reference 10 11 by value 10 11 path adding directories to 14 7 cache 2 15 changing 2 14 MATLAB 2 14 search 2 14 2 50 path 2 14 Path Browser 2 50 pathdef m 2 15 2 50 pathtool 2 50 pausing during M file execution 10 61 1 21 Index 1 22 pausing in Debugger 3 7 pcode 10 11 PCs an
128. TLAB Informally the terms matrix and array are often used interchangeably More precisely a matrix is a two dimensional rectangular array of real or complex numbers that represents a linear transformation The linear algebraic operations defined on matrices have found applications in a wide variety of technical fields The Symbolic M ath Toolboxes extend MATLAB s capabilities to operations on various types of nonnumeric matrices MATLAB has dozens of functions that create different kinds of matrices Two of them can be used to create a pair of 3 by 3 example matrices for use throughout this chapter The first example is symmetric A pascal 3 A The second example is not symmetric B magi c 3 B 8 1 6 3 5 7 4 9 2 Another example is a 3 by 2 rectangular matrix of random integers C fix 10 rand 3 2 C Matrices in MATLAB A column vector is an m by 1 matrix a row vector is a 1 by n matrix anda scalar is a 1 by 1 matrix The statements u S produce a column vector a row vector and a scalar u 3 1 4 2 0 1 7 1 4 5 4 Matrices and Linear Algebra Addition and Subtraction Addition and subtraction of matrices is defined just as it is for arrays element by element Adding A toB and then subtracting A from the result recovers B X A B X 9 2 4 10 5 12 Y X A Y 8 1 6 3 5 7 4 9 2 Addition and subtraction require both matrices to have the same dimension
129. The function must be capable of returning a vector output when given a vector input You must also consider which variable is in the inner integral and which goes in the outer integral In this example the inner variable is x and the outer variable is y the order in the integral is dxdy In this case the integrand function is function out integrnd x y out y sin x x cos y Toperform the integration two functions are availablein thef unf un directory The first db quad is called directly from the command line This M file evaluates the outer loop using quad At each iteration quad calls the second helper function that evaluates the inner loop To evaluate the double integral use result dbl quad integrnd xmin xmax ymin ymax 7 15 7 Function Functions 7 16 The first argument is a string with the name of the integrand function the second to fifth arguments are xmi n lower limit of inner integral xma x upper limit of the inner integral ymi n lower limit of outer integral ymax upper limit of the outer integral Hereis a numeric example that illustrates the use of dbl quad xmin pi xmax 2 pi ymin 0 ymax pi result dbl quad integrnd xmin xmax ymin ymax The result is 9 8698 By default db quad callsquad Tointegrate the previous example using quad8 with the default values for the tolerance and trace arguments use result dbl quad integrnd xmin xmax ymin
130. UNIX workstations you may not have file system permission to edit pathdef m In this case put path andaddpath commands in your startup m file to change your path defaults MATLAB also provides a Path Browser with a convenient interface for viewing and changing the search path Usepathtool to start the Path Browser Files on the Search Path To display the search path usepat h Usewhat to see all of the MATLAB files in a directory With no arguments what displays the files in the current directory With a full or partial path what lists thefiles in any directory on the path for example what matlab elfun To see the code in a specific M file use thet ype command for example type rank To edit the M file useedit for example edit rank Note Save any M files you create or any MATLAB supplied M files that you edit in a directory that is not in the MATLAB directory tree If you keep your files in the MATLAB directory tree they might be overwritten when you install a new version of MATLAB Another consideration is that files in the MATL AB tool box directory tree are loaded and cached into memory at the beginning of 2 15 2 MATLAB W orking Environment 2 16 each MATLAB session to improve performance This cache is not updated until MATLAB is restarted If you add any files or make changes to any files in thet ool box directory you will not be able to see the changes until you restart MATLAB Current Directory
131. a disk or CD in your local system Many of the underlying Help and Online Documentation documents use HyperText Markup Language HTML and are accessed with an Internet Web browser such as Netscape Navigator or Microsoft I nternet Explorer To access the Help Desk type hel pdesk i A On Windows platforms you can also access the Help Desk by selecting k the Help Desk option under the Help menu All of MATLAB s operators and functions haveonline reference pages in HTML format which you can reach from the Help Desk These pages provide more details and examples than the basichel p entries HTML versions of other documents are also available A search engine can query all the online reference material PDF formatted Documentation Versions of all MATLAB documentation are available in Portable Document F ormat PDF through the Help Desk These pages are processed by Adobe s Acrobat Reader They reproduce the look and feel of the printed page complete with fonts graphics formatting and images You can use links from one table of contents or index of a manual as well as internal links to go directly to the page of interest Acrobat Reader also allows you to print selected pages within a document This is the best way to get printed copies of the online MATLAB F unction Reference which is not otherwise available in hard copy MathWorks Web Site f your computer is connected to the Internet the Help Desk provides connections
132. a functions are located in the mat f un directory in the MATLAB Toolbox Category Function Description Matrix analysis norm Matrix or vector norm nor mest Estimate the matrix 2 norm rank Matrix rank det Determinant trace Sum of diagonal elements nul Null space orth Orthogonalization rref Reduced row echelon form subspace Angle between two subspaces Linear equations and Linear equation solution inv Matrix inverse cond Condition number for inversion condest 1 norm condition number estimate chol Cholesky factorization cholinc Incomplete Cholesky factorization lu LU factorization luinc Incomplete LU factorization 4 2 Matrices and Linear Algebra Category Function Description Eigenvalues and singular values Matrix functions qr nnis pinv Iscov eig svd ei gs svds poly pol yeig condeig hess qz schur expm logm sqrtm funm Orthogonal triangular decomposition Nonnegative least squares Pseudoinverse Least squares with known covariance Eigenvalues and eigenvectors Singular value decomposition A few eigenvalues A few singular values Characteristic polynomial Polynomial eigenvalue problem Condition number for eigenvalues Hessenberg form QZ factorization Schur decomposition Matrix exponential Matrix logarithm Matrix square root Evaluate general matrix function 4 3 4 Matrices and Linear Algebra 4 4 Matrices in MA
133. a in Multidimensional Arrays Organizing Data in Multidimensional Arrays You can use multidimensional arrays to represent data in two ways e As planes or pages of two dimensional data You can then treat these pages as matrices e As multivariate or multidimensional data For example you might havea four dimensional array where each element corresponds to either a temperature or air pressure measurement taken at one of a set of equally spaced points in a room For example consider an RGB image For a single image a multidimensional array is probably the easiest way to store and access data Array RGB 0 689 0 706 0 118 0 884 Page3 1 535 0 532 0 653 0 925 blue Jo 314 0 265 0 159 0 101 intensity 0 553 0 633 0 528 0 493 a 0 441 0 465 0 312 0 512 70 342 0 647 0 515 0 816 10 421 0 398 PageZ 111 0 300 0 205 0 526 0 912 0 713 fF ME 0 523 0 428 0 712 0 979 0 219 0 328 intensity 214 0 604 0 918 0 344 0 128 0 133 values 0 100 0 121 0 113 0 126 0 112 0 986 0 234 0 432 0 204 0 175 Pagel 0 765 0 128 0 863 0 521 0 760 0 531 red 1 000 0 985 0 761 0 698 10 997 0 910 intensity 9 455 0 783 0 224 0 395 0 995 0 726 0 021 0 500 0 311 0 123 1 000 1 000 0 867 0 051 1 000 0 945 0 998 0 893 0 990 0 941 1 000 0 876 0 902 0 867 0 834 0 798 To access an entire plane of the image use red_ plane RGB
134. a parent to the stock object The stock constructor must therefore call the asset constructor Thec ass function which is called to create the stock object defines the asset object as the parent Keep in mind that the asset object is created in the temporary workspace of the stock constructor function and is stored as a field asset in the stock structure The stock object inherits the asset fields but the asset object is not returned to the base workspace function s stock varargin STOCK Stock class constructor s stock descriptor num shares share price switch nargin case 0 if no input arguments create a default object s num shares 0 s share_ price 0 a asset none 0 s class s stock a case 1 if single argument of class stock return it if isa varargin l stock s varargin l else error Input argument is not a stock object end case 3 create object using specified values S num shares varargin 2 s share price varargin 3 a asset varargin l varargin 2 varargin 3 s class s stock a otherwise error Wrong number of input arguments end Example Assets and Asset Subclasses Constructor Calling Syntax The stock constructor method can be called in one of three ways e NoInput Argument If called with no arguments the constructor returns a default object with empty fields e Input Argument is a Stock Object If called with a sing
135. a structure array Plane organization Element by element organization A B 0 112 0 986 0 234 0 432 0 765 0 128 0 863 0 521 sf 1 000 0 985 0 761 0 698 Bi 1 1 B 1 2 Bi 1 3 T 0 112 af 0 986 of 0 234 7 342 0 647 0 515 0 816 9 peaks g ete 9 wee 0 111 0 300 0 205 0 526 T 0 523 0 428 0 712 0 929 b 0 689 b 0 706 b 0 118 B 2 1 B 2 2 B 2 3 895 855 Bagg Dooi Lp 78 o rie op 0863 a 0 314 0 265 0 159 0 101 g 0 111 g 0 300 g 0 205 b 0 535 b 0 532 b 0 653 1 by 1 structure array where each field is a 128 by 128 array 128 by 128 structure array where each field is a single data element Plane Organization In case 1 above each field of the structureis an entire plane of the image You can create this structure using A r RED A g GREEN A b BLUE This approach allows you to easily extract entire image planes for display filtering or other tasks that work on the entire image at once To access the entire red plane for example use red_ plane A red Plane organization has the additional advantage of being extensible to multiple images in this case If you have a number of images you can store them as A 2 A 3 and soon each containing an entire image 13 13 13 Stru
136. aFile i Temporary Files and Directories Binary Files Reading Binary Files Writing Binary Files Controlling Position in a File Understanding File Position Formatted Files Reading Strings LineBy Line from Text F iles Reading Formatted Text Writing Text Files 15 3 15 4 15 5 15 6 15 7 15 7 15 8 15 10 15 11 15 13 15 13 15 14 15 15 13 File O 15 2 MATLAB s file input and output I O functions read and write arbitrary binary and formatted text files They allow you to read data collected in other formats and to write out data for other programs or devices The low level file I O functions livein a directory called i of un in the MATLAB Toolbox Category Function Description FileOpeningand fopen Open file Closing fclose Close file Binary I O fread Read binary data from file fwrite Write binary data to file Formatted I O fscanf Read formatted data from file fprintf Write formatted data to file fget Read line from file discard newline character fgets Read line from file keep newline character String Conversion sprintf Write formatted data to string sscanf Read string under format control File Positioning ferror Inquire filel O error status feof Test for end of file fseek Set file position indicator ftell Get file position indicator frewind Rewind file Temporary Files tempdir Get temporary directory name tempname Get temporary file
137. aN 6 12 10 18 nargin 10 12 nargout 10 12 ndgrid 5 16 5 17 12 2 ndims 12 2 12 9 nearest neighbor interpolation 5 10 5 12 5 13 5 15 multidimensional 5 16 nesting cell arrays 13 29 for loops 10 35 functions 10 67 i f statements 10 31 structures 13 16 newlines in string arrays 11 11 nnz 9 11 9 13 nodes 9 15 distance between 9 21 numbering 9 17 nonzero elements number of 9 11 nonzero elements of sparse matrix 9 11 maximum number in sparse matrix 9 7 storage 9 5 9 11 values 9 11 visualizing with spy plot 9 19 nonzeros 9 11 norm4 12 norm minimal 4 22 normalizing data 6 21 nosplash startup option 2 3 not equal to operator 10 23 NOT operator rules for evaluating 10 25 Not a Number 10 18 now 10 60 null 4 19 number of arguments 10 12 numbers changing to strings 11 13 date 10 54 time 10 54 numeric data type 10 20 numeric format 2 7 2 38 numerical integration 7 14 nzmax 9 11 9 13 0 object oriented programming 14 2 converter functions 14 25 Index features of 14 2 inheritance multiple 14 35 inheritance simple 14 34 overloading 14 20 14 22 subscripting 14 14 overview 14 2 See also classes and objects objects accessing data in 14 13 as indices into objects 14 18 creating 14 4 invoking methods on 14 4 loading 14 61 overview 14 2 precedence 14 66 saving 14 61 ODE solver properties error tolerance 8 17 8 21 absolute accuracy 8 22 AbsTol 8 22 relative accuracy 8 21 Rel Tol 8 22 ev
138. adient stabilized cgs Conjugate gradient squared gmres Generalized minimum residual pcg Preconditioned conjugate gradient qmr Quasiminimal residual All six methods are designed to solve Ax b The preconditioned conjugate gradient method pcg is restricted to symmetric positive definite matrix A The other five can handle nonsymmetric square matrices Sparse M atrix O perations All six methods can make use of left and right preconditioners The linear system Ax b is replaced by the equivalent system M AM7 Mx Mj b The preconditioners M and M gt are chosen to accelerate convergence of the iterative method In many cases the preconditioners occur naturally in the mathematical model A partial differential equation with variable coefficients may be approximated by one with constant coefficients for example Incomplete matrix factorizations may be used in the absence of natural preconditioners The five point finite difference approximation to Laplace s equation on a square two dimensional domain provides an example The following statements use the preconditioned conjugate gradient method with an incomplete Cholesky factorization as a preconditioner A delsq numgrid S 50 b ones size A 1 1 tol 1 e 3 maxit 10 R cholinc A tol x flag err iter res pcg A b tol maxit R R Only four iterations are required to achieve the prescribed accuracy Background information on these ite
139. age for the object and creating a class directory containing M files that operate on the object These M files contain the methods for the class The class directory can also include functions that define the way various MATLAB operators including arithmetic operations subscript referencing and concatenation apply to the objects Redefining how a built in operator works for your class is Known as overloading the operator Features of Object Oriented Programming When using well designed classes object oriented programming can significantly increase code reuse and make your programs easier to maintain and extend Programming with classes and objects differs from ordinary structured programming in these important ways e Function and operator overloading You can create methods that override existing MATLAB functions When you call a function with a user defined object as an argument MATLAB first checks toseeif thereisa Classes and O bjects An O verview method defined for the object s class If there is MATLAB calls it rather than the normal MATLAB function Encapsulation of data and methods Object properties are not visible from the command line you can access them only with class methods This protects the object properties from operations that are not intended for the object s class Inheritance Y ou can create class hierarchies of parent and child classes in which the child class inherits data fields and methods from the
140. age requires memory proportional to the area of the image on the screen A color image of 500 by 500 pixels uses one megabyte of memory To limit the amount of memory required for these operations limit the size of the images you display Resolving Memory Errors If you do not havea chunk of memory large enough to allocate a matrix an out of memory error may occur even though you seem to have enough available memory To consolidate the fragmented memory you can use the MATLAB pack command or you can allocate larger matrices earlier in the MATLAB session If you run out of memory often use these tips i A For Windows increase virtual memory by using System Properties for ai Performance which you can access from the Control Panel X For UNIX ask your system manager to increase your swap space F or VAXNMS ask your system manager to increase your working set and or pagefile quota MATLAB s Memory Management MATLAB uses the standard C functions mal oc andfree to allocate dynamic memory These routines maintain a pool of memory that is allocated from the operating system relatively slowly mal oc andf ree allocate memory from this pool for MATLAB much more quickly If the pool runs low mal oc asks the operating system for another large chunk of memory to replenish the pool As MATLAB releases memory the pool can grow very large To maintain speed mal oc andfree donot return the additional memory to the operating
141. airly long In the following example this line is shown on two lines however you should keep it all on one line DAEMON MLM usr local matlab etc m matlab fusr local matlab etc local options A local options file is not required If it does exist it can have one line or many lines reflecting your special needs The license manager allocates keys according to these options until all keys are in use If you try to reserve more than the authorized number of keys in the options file a warning message appears in thel icense log file UN IX Handbook A local options file might look similar to this one RESERVE 1 MATLAB USER patricia RESERVE 3 MATLAB HOST pegasus RESERVE 1 CONTROL GROUP devels INCLUDE SIGNAL HOST abrea INCLUDE SIGNAL USER tom EXCLUDE SI MULINK GROUP devels GROUP devels andrea tom fred The lines starting with RESERVE contain the number of keys for a particular product set aside for a specific user group or host This does not limit the number of keys for that group or host it simply ensures that a key will be available when you want it unless the specified number of reserved keys has already been reached The lines starting with NCLUDE contain the products to be restricted toa particular user group or host only that user group or host is allowed to use this product You can have multiple NCLUDE lines for the same feature including different users groups or hosts The lines starting with EXCLUDE c
142. al analysis George Forsythe and J H Wilkinson worked on the problem in the 1950s See G E Forsythe and W R Wasow Finite Difference Methods for Partial Differential Equations Wiley 1960 One of the authors of this guide Moler used finite differences by combinations of distinguished fundamental solutions to the underlying differential equation formed from Bessel and trigonometric functions The idea is a generalization of the fact that the real and imaginary parts of complex analytic functions are solutions to Laplace s equation In the early 1970s new matrix algorithms particularly Gene Golub s orthogonalization techinques for least squares problems provided further algorithmic improvements Today MATLAB allows us to express the entire algorithm in a few dozen lines to compute the solution with great accuracy in a few minutes on a computer at home and to readily manipulate color three dimensional displays of the results We have included our MATLAB program membrane m with the M files supplied along with MATLAB What Is MATLAB MATLAB is a high performance language for technical computing It integrates computation visualization and programming in an easy to use environment where problems and solutions are expressed in familiar mathematical notation Typical uses include e Math and computation e Algorithm development e Modeling simulation and prototyping e Data analysis exploration and visualization e Scient
143. alculates the magnitude and phase of the transformed sequence It uses the abs function to obtain the magnitude of the data theang e function to obtain the phase information and unwrap to remove phase jumps greater than pi to their 2 pi complement y fft x m abs y p unwrap angle y Now create a frequency vector for the x axis and plot the magnitude and phase f O length y 1 99 length y subplot 2 1 1 plot f m ylabel Abs Magnitude grid on subplot 2 1 2 plot f p 180 pi ylabel Phase Degrees grid on xlabel Frequency Hertz 6 37 6 Data Analysis and Statistics 6 38 a 40 ne 30 D 20 38 Saot 0 10 20 30 40 50 60 70 80 90 100 400 600 Phase Degrees 8007 1000F 1200 fi fi fi fi fi i fi fi 0 L 10 20 30 40 50 60 70 80 90 100 Frequency Hertz The magnitude plot is perfectly symmetrical about the Nyquist frequency of 50 Hertz The useful information in the signal is found in the range 0 to 50 H ertz FFT Length Versus Speed You can add a second argument tof f t to specify a number of points n for the transform y fft x n With this syntax f f t pads x with zeros if it is shorter than n or truncates it if it is longer than n If you donot specifyn f ft defaults tothe length of theinput sequence The execution time for f f t depends on the length of the transform e For anyn that is a power of two f f t us
144. and Loading Objects Object Precedence How MATLAB Determines Which Method to Call 14 2 14 9 14 20 14 23 14 34 14 37 14 54 14 61 14 66 14 68 14 MATLAB Classes and O bjects Classes and Objects An Overview This chapter describes how to define classes in MATLAB Classes and objects enable you to add new data types and new operations to MATLAB The class of a variable describes the structure of the variable and indicates the kinds of operations and functions that can apply tothe variable An object is an instance of a particular class The phrase object oriented programming describes an approach to writing programs that emphasizes the use of classes and objects You can view classes as new data types having specific behaviors defined for the dass For example a polynomial class might redefine the addition operator so that it correctly performs the operation of addition on polynomials Operations defined to work with objects of a particular class are know as methods of that class You can also view classes as new items that you can treat as single entities An example is an arrow object that MATLAB can display on graphs perhaps composed of MATLAB line and patch objects and that has properties like a Handle Graphics object Y ou can create an arrow simply by instantiating the arrow class You can add classes to your MATLAB environment by specifying a MATLAB structure that provides data stor
145. and controls the numeric format of the values displayed on the screen The command affects only how numbers are displayed not how MATLAB computes or saves them et On Windows platforms you can change the default format by selecting a Preferences from the File menu and selecting the desired format from the General tab 2 MATLAB W orking Environment Here are various formats and the output produced from a two element vector with components of different magnitudes x 4 3 1 2345e 6 format short 1 3333 0 0000 format short e 1 3333e 000 1 2345e 006 format short g 1 3333 1 2345e 006 format long 1 33333333333333 0 00000123450000 format long e 1 333333333333333e 000 1 234500000000000e 006 format long g 1 33333333333333 1 2345e 006 format bank B33 0 00 2 8 The Command W indow format format rat 4 3 1 810045 format hex 3f 5555555555555 Zeb4b623labfd271 If the largest element of a matrix is larger than 10 or smaller than 10 MATLAB applies a common scale factor for the short and long formats In addition to thef or mat commands shown above format compact suppresses many of the blank lines that appear in the output This lets you view more information on a screen or window To show the blank lines use format loose If you want more control over the output format usethes printf andf printf functions Suppressing Output If you simply type a statement and press Return or Enter MATLAB automat
146. ange points receive a value of NaN Cubic splineinterpolation met hod spline This method uses a series of functions to obtain interpolated data points determining separate functions between each pair of existing data points At its endpoint an existing data point each function has at least the same first and second derivatives as the function following it Cubic interpolation met hod cubic This method fits a cubic function through y and returns the value of this function at the points specified by xi Out of range points receive a value of NaN All of these methods require that x be monotonic that is either always increasing or always decreasing from point to point Each method works with non uniformed spaced data If x is already equally spaced you can speed execution time by prepending an asterisk to the met hod string for example cubic Interpolation Speed Memory and Smoothness Considerations When choosing an interpolation method keep in mind that some require more memory or longer computation time than others However you may need to trade off these resources to achieve the desired smoothness in the result e Nearest neighbor interpolation is the fastest method However it provides the worst results in terms of smoothness Linear interpolation uses more memory than the nearest neighbor method and requires slightly more execution time Unlike nearest neighbor interpolation its results are
147. anization Ann Lones Dan Smith L name Kim Lee B 1 B 2 B 3 name Ann Jones name Dan Smith name Kim Lee 90 Pack st L address il6 Elm st j address 80 Park St L address_j 5 Lake Ave address 116 Elm St L amount _ 12 50 amount 81 29 amount 30 00 12 50 L amount i 50 B 1 name Ann Jones B 1 address 80 Park St B 1 amount 12 5 A name strvcat Ann Jones Dan Smith A Dan io ole e A address strvcat 80 Park St 5 Lake Ave B 2 a Sees 5 La e Ave A amount 12 5 81 29 30 B 2 amount 81 29 Each of the possible organizations has advantages depending on how you want to access the data e Plane organization makes it easier to operate on all field values at once F or example to find the average of all the values in theamount field Using plane organization avg mean A amount Using denent by el ement organization avg mean B amount 13 15 13 Structures and Cell Arrays 13 16 e Element by element organization makes it easier to access all the information related to a single client Consider an M file cl i ent m which displays the name and address of a given client on screen Using plane organization pass individual fidds function client name address di sp name disp address unction client B isp B Using dement by dement organization pass an entire structure f d To call thec i ent functi
148. ar The toolbar in the Command Window provides easy access to popular operations Hold the cursor over a button a tooltip appears describing the button and additional information appears in the status bar Open Workspace New File Co Undo Browser Path Simulink cut i Paste Browser Model Hel a Xl o S Tooltip Describes Button orkepace Browser Show or Hide the Toolbar To remove the toolbar from the Command Window select Toolbar from the View menu the menu item becomes unchecked To display the toolbar select Toolbar from the View menu the menu item becomes checked and the toolbar appears This does not affect the setting for Show Toolbar in the Preferences dialog box The preferences setting pertains to the toolbar status when you first start MATLAB Move the Toolbar You can move the toolbar to another position Click and hold on any separator bar the line between groups of buttons and then drag the toolbar to a new location If you move the toolbar to an inside edge of the Command Window it will become docked meaning it will not overlap the contents of the Command Window The new toolbar position is maintained when you restart MATLAB Microsoft W indows Handbook Menus The Command Window menus provide access to some operations not available from the toolbar Hold the cursor on a menu item and a description of the item appears in the status bar Edit View Window Help New Open apen Selection Bun Scr
149. arameters in the ODE solvers are selected to handle common problems In some cases however tuning the parameters for a specific problem can improve performance significantly You do this by supplying the solvers with one or more property values contained within an options argument T Y solver F tspan y0 options The property values within theopt i ons argument are created with theodes et function in which named properties are given specified values Category Property Name Page Error tolerance Re Tol AbsTol 8 21 Solver output Output Fen OutputSel Refine Stats 8 22 J acobian matrix J acobian Constant Pattern Vectorized 8 25 Step size InitialStep MaxStep 8 28 Mass matrix Mass MassSingul ar 8 29 Event location Events 8 30 odel5s MaxOrder BDF 8 32 Special Purpose ODE Files and the flag Argument The MATLAB ODE solvers are capable of using additional information provided in the ODE file In this more general use an ODE file is expected to respond totheargumentsodefile t y flag pl p2 wheret andy arethe integration variables flag is a string indicating the type of information that 8 17 8 O rdinary Differential Equations the ODE fileshould return and p1 p2 areany additional parameters that the problem requires This table shows the currently supported flags Flags Return Values mo empty init jacobian jpattern mass events F t y tspan y0 andoptions fo
150. arentheses In the code above the indexing expression i accesses thei th cell of varargin The expression 2 represents the second element of the cell contents Functions Packing varargout Contents When allowing any number of output arguments you must pack all of the output intothevarargout cell array Usenar gout todetermine how many output arguments the function is called with For example this code accepts a two column input array where the first column represents a set of x coordinates and the second represents y coordinates It breaks the array into separate xi yi vectors that you can pass intothet est var function on the previous page function varargout testvar2 arrayin for i 1 nargout Cell array assignment varargout i arrayin i end The assignment statement inside thef or loop uses cell array assignment syntax The left side of the statement the cell array is indexed using curly braces to indicate that the data goes inside a cell For complete information on cell array assignment see Structures and Cell Arrays in Chapter 13 Here s how tocalltestvar2 a 1 2 3 4 5 6 7 8 9 0 p1 p2 p3 p4 p5 testvar2 a varargin and varargout in Argument Lists varargin Orvarargout must appear last in the argument list following any required input or output variables That is the function call must specify the required arguments first For example these function declaration
151. array toa single cell of a cell array G cell 1 16 for m 1 16 G m NUM m end Cell Arrays Cell Arrays of Structures Use cell arrays to store groups of structures with different field architectures c_str cell 1 2 c_str l label 12 2 94 12 5 94 c_str l obs 47 52 55 48 17 22 35 11 c_str 2 xdata 0 03 0 41 1 98 2 12 17 11 c_str 2 ydata 3 5 18 0 9 c_str 2 zdata 0 6 0 8 1 2 2 3 4 cell 1 cell 2 c_str 1 c_str 2 L label 12 2 94 12 5 94 esime 0 03 0 41 1 98 2 12 17 11 L test 47 52 55 48 L billing 3 5 18 0 9 TERENAS L test 0 6 0 812 2 3 4 Cell lof thec_str array contains a structure with two fields one a string and the other a vector Cell 2 contains a structure with three vector fields When building cell arrays of structures you must use content indexing Similarly you must use content indexing to obtain the contents of structures within cells The syntax for content indexing is cell _array index field For example to access the abe field of the structure in cell 1 use c_str l label 13 33 13 Structures and Cell Arrays 13 34 MATLAB Classes and Objects Classes and Objects An Overview Designing User Classes in MATLAB Overloading Operators and Functions Example A Polynomial Class Building on Other Classes Example Assets and Asset Subclasses Example The Portfolio Container Saving
152. ary 11 13 to hexadecimal 11 13 decomposition eigenvalue 4 34 Schur 4 37 singular value 4 38 deconv 5 4 deconvolution 5 4 defaults setting 2 3 Delaunay triangulation 5 19 closest point searches 5 22 delete 2 25 deleting cells from cell array 13 25 fields from structure arrays 13 9 deletion operator 10 42 delimiter in string 11 12 demos MATLAB 1 6 density of sparse matrix 9 6 derivative of polynomial 5 5 descriptive statistics 6 7 det 4 20 determinant 4 20 diag 9 24 diagonal creating sparse matrix from 9 8 of a matrix 4 10 diary 2 28 2 32 diff 611 difference between successive vector elements 6 11 difference equations 6 29 differential equations See ODE solvers di m argument for cat 12 7 Index dimension compatibility 4 13 dimensions deleting 13 25 permuting 12 13 removing singleton 12 12 dir 2 25 direct methods for systems of equations 9 33 directories adding to path 14 7 class 14 6 Contents mfile 10 9 help for 10 9 MATLAB 1 5 private 10 39 14 5 temporary 15 6 directory current 2 16 discrete F ourier transform 6 31 disp 13 16 dispatch type 14 69 dispatching priority 10 10 displ ay method 14 12 examples 14 13 displayed output 14 27 displaying cell arrays 13 21 error and warning messages 10 52 field names for structure array 13 4 sparse matrices 9 13 distance between nodes 9 21 division matrix 4 13 of polynomials 5 4 dl mr ead 2 27 dl mwr i t e 2 28 doc 2 23 documentation how to use 1 6 document
153. as BDFs also known as Gear s method that are usually less efficient Like ode113 o0de15s isa multistep solver If you suspect that a problem is stiff or ifode45 failed or was very inefficient try odel5s ode23s iS based on a modified Rosenbrock formula of order 2 Becauseit isa one step solver it may be more efficient than ode15s at crude tolerances It can solve some kinds of stiff problems for which ode15s is not effective ode23t is an implementation of the trapezoidal rule using a free interpolant Use this solver if the problem is only moderately stiff and you need a solution without numerical damping ode23tb isan implementation of TR BDF 2 an implicit Runge Kutta formula with a first stage that is a trapezoidal rule step and a second stage that is a backward differentiation formula of order two By construction the same iteration matrix is used in evaluating both stages Likeode23s this solver may be more efficient than ode15s at crude tolerances ODE Solver Basic Syntax All of the ODE solver functions share a syntax that makes it easy to try any of the different numerical methods if it is not apparent which is the most appropriate To apply a different method to the same problem simply change the ODE solver function name The simplest syntax common to all the solver functions is T Y solver F tspan y0 wheresol ver is one of the ODE solver functions listed previously 38 11 8 O rdinary Differential Eq
154. ate a 100 byte binary file containing the 25 elements of the 5 by 5 magic square each stored as 4 byte integers fwriteid fopen magic5 bin w count fwrite fwriteid magic 5 int32 status fclose fwriteid Binary Files In this case f write setsthecount variable to25 unless an error occurs in which case the value is less 15 9 13 File 1 O Controlling Position in a File 15 10 Once you open a file with f open MATLAB maintains a file position indicator that specifies a particular location within a file MATLAB uses thefile position indicator to determine where in the file the next read or write operation will begin The table below summarizes MATLAB functions for controlling the file position indicator Function Purpose feof Determine if file position indicator is at end of file fseek Set file position indicator ftell Get file position indicator frewind Reset file position indicator to beginning of file Thefseek andftell functions let you set and query the position in the file at which the next input or output operation takes place e Thefseek function repositions the file position indicator letting you skip over data or back up to an earlier part of the file e Theftel function gives the offset in bytes of the file position indicator for a specified file The syntax for f seek is status fseek fid offset origin fid isthe fileidentifier for thefile of f set iS a posi
155. ate and time as serial date number date Current date as date string clock Current date and time as date vector Conversion datenum Convert to serial date number datestr Convert to string representation of date datevec Date components Utility calendar Calendar weekday Day of the week eomday End of month datetick Date formatted tick labes Timing cputi me CPU timein seconds Hicy toc Stopwatch timer et i me Elapsed time Date Formats MATLAB works with three different date formats date strings serial date numbers and date vectors When dealing with dates you typically work with date strings 16 Sep 1996 MATLAB works internally with serial datenumbers 729284 A serial date represents a calendar date as the number of days that has passed since a fixed base date In MATLAB serial date number 1 isJ anuary 1 0000 MATLAB also uses serial time to represent fractions of days beginning at midnight for 10 54 Times and Dates example 6 p m equals 0 75 serial days Sothestring 16 Sep 1996 6 00 pm in MATLAB is date number 729284 75 All functions that require dates accept either date strings or serial date numbers If you are dealing with a few dates at the MATLAB command line level date strings are more convenient If you are using functions that handle large numbers of dates or doing extensive calculations with dates you will get better performance if you use date numbers Date vectors are an internal format for som
156. ate the menu text for this expression by typing amp Censor in the Menu Text box In the example we have provided when you run Censor from the Tools menu text you have highlighted in the Editor is replaced with X s Microsoft W indows Handbook General Options Choose Options from the Tools menu and select the General tab im ks ka m jm a e Show worksheet style tabs controls if the Editor window displays tabs for navigating among open files 2 47 2 MATLAB W orking Environment e Show DataTips When the cursor remains positioned over a variable the item is expanded and the results are displayed in the Editor window MATLAB Editor Debugger File Edit View Debug Tools Window Help osle Hae e Sa S22 w Fi Untitled2 e Automatically reload externally modified files F iles changed outside of the Editor are automatically reloaded Otherwise the Editor asks if you want to reload the file e Disable overtype mode Insert key Disables the Insert key which switches overtype mode on and off Append m to filename on Save As The m extension is appended to the filename provided in the Save As dialog box if that extension is not already present If this option is not checked the file is saved with the filename exactly as typed e Printin color Prints the M filein color when a color printer is used e Recently used file list Controls number of files listed on the recen
157. ated in class directories with the same name and enables MATLAB to determine which method to use based on the type or class of the variables passed to the function For example if p is a portfolio object then pi e3 p calls port folio pie3 m because the argument is a portfolio object The Process of Selecting a Method When you call a method for which there are multiple versions with the same name MATLAB determines the method to call by e Looking at the classes of the objects in the argument list to determine which argument has the highest object precedence the class of this object controls the method selection and is called the dispatch type e Applying the function precedence order to determine which of possibly several implementations of a method to call This order is determined by the location and type of function How MATLAB Determines W hich M ethod to Call Determining the Dispatch Type MATLAB first determines which argument controls the method selection The class type of this argument then determines the class in which MATLAB searches for the method The controlling argument is either e The argument with the highest precedence or e The leftmost of arguments having equal precedence User defined objects take precedence over MATLAB s built in classes such as doubl e orchar You can set therelative precedence of user defined objects with thei nferi orto andsuperi orto functions as described in Object Precedenc
158. ation online 2 23 dos 2 25 DOS window starting MATLAB from 2 4 dot product 4 8 double data type 10 20 double precision 15 8 double precision matrix 10 19 dsearch 5 22 E echoing 2 38 edit 2 15 2 39 editing arrays 2 52 commands 2 5 M files 2 39 editor 2 55 accessing 10 4 command line 2 5 default 2 55 for creating M files 10 2 10 4 preference 2 38 See also Editor MATLAB s Editor MATLAB s customizing menus 2 41 font used in 2 50 indenting 2 39 matching quotes automatically 2 49 options 2 55 printing in color 2 41 saving files 2 40 syntax formatting 2 49 See also editor Editor Debugger 2 39 3 2 3 4 caution 2 39 highlighting 2 39 indenting 2 39 Index See also editor ei g 4 34 5 3 12 16 eigenvalue decomposition 4 34 eigenvalues 4 34 of sparse matrix 9 36 eigenvector 4 34 element by element organization for structures 13 14 else 10 30 10 31 elseif 10 30 10 31 empty arrays andi f statement 10 32 and relational operators 10 24 and whi e loops 10 34 empty matrices 10 49 end method 14 18 end of file 15 10 ending MATLAB 2 4 environment 2 2 eps 10 18 epsilon 10 18 equal to operator 10 23 error bound for data fit 6 28 handling 10 51 tolerance ODE 8 17 8 21 error 10 6 errors 10 51 displaying 10 52 finding 3 2 input output 15 2 opening files 15 4 eval for error handling 10 51 evaluating polynomials in matrix sense 5 4 string containing function name 10 46 string containing MAT
159. ations W H Freeman amp Co 1975 p 246 8 47 8 O rdinary Differential Equations function varargout orbitode t y flag ORBI TODE Restricted three body problem yO 1 2 0 0 1 04935750983031990726 switch flag case Return dy dt f t y varargout 1 f t y case init Return default tspan y0 options varargout 1 3 init y0 case events Return value isterminal direction varargout 1 3 events t y y0 otherwise error Unknown flag flag end Oo sin tetetias ies deta tn cote tah etal let edema E AAE AE a facie a A us deg A R A igh ete a 4s Jere Pete function dydt f t y mu 1 82 45 mustar 1 mu r13 y 1 mu 2 y 2 2 1 5 r23 y 1 mustar 2 y 2 2 1 5 dydt y 3 y 4 2 y 4 y 1 mustar y 1 mu 1r13 mu y 1 mustar r23 2 y 3 y 2 mustar y 2 r13 mu y 2 r23 J Gf Aina a Ee ate EN S BA Wye nde E bad yet aine a en eastty MO A og EE eee amar function tspan y0 options init y tspan 0 6 19216933131963970674 y0 y options odeset RelTol 1e 5 AbsTol 1e 4 Of aaa ne a ea tl ae e Gist lh Did steed a aa i a A a ered ties Ee a el function value isterminal direction events t y y0 Locate the time when the object returns closest to the initial point yO and starts to move away and stop integration Also locate the time when the object is far
160. atrix permute it and perform a sparse triangular solve for each column of B e If A is symmetric or Hermitian and has positive real diagonal elements find a symmetric minimum degree order p and attempt to compute the Cholesky factorization of A p p If successful finish with two sparse triangular solves for each column of B e Otherwise if A is not triangular or is not Hermitian with positive diagonal or if Cholesky factorization fails find a column minimum degree order p 9 33 9 Sparse Matrices 9 34 Compute the LU factorization with partial pivoting of A p and perform two triangular solves for each column of 8 For a square matrix MATLAB tries these possibilities in order of increasing cost The tests for triangularity and symmetry are relatively fast and if successful allow for faster computation and more efficient memory usage than the general purpose method For example consider the sequence below L U lu A yo SAADE x U y In this case MATLAB uses triangular solves for both matrix divisions since L is a permutation of a triangular matrix and U is triangular Use the function spparms to turn off the minimum degree preordering if a better preorder is known for a particular matrix Iterative Methods Six functions are available that implement iterative methods for sparse systems of simultaneous linear systems Function Description bicg Biconjugate gradient bicgstab Biconjugate gr
161. attern to generate a sparse J acobian matrix numerically If the J acobian matrix is large size greater than approximately 100 by 100 and sparse this can accelerate execution greatly For an example using the J Pattern property seethebrussode example on 8 37 8 27 8 O rdinary Differential Equations 8 28 Vectorized Set Vectorized on toinform the stiff solver that the ODE file is coded so that F t yl y2 returns F t y1 F t y2 When computing J acobians numerically the solver passes this information to the nu mj ac routine This allows numj ac to reduce the number of function evaluations required to compute all the columns of the J acobian matrix and may reduce solution time significantly With MATLAB s array notation it is typically an easy matter to vectorize an ODE file For example the stiff van der Pol example shown previously can be vectorized by introducing colon notation into the subscripts and by using the array power and array multiplication operators function dy vdpl000 t y dy y 2 1000 1 y 1 2 y 2 y 1 Step Size Properties The step size properties let you specify the first step size tried by the solver potentially helping it to recognize better the scale of the problem In addition you can specify bounds on the sizes of subsequent time steps Property Value Description MaxStep Positive scalar U pper bound on solver step size InitialStep Positive s
162. ay switch var case 1 disp 1 case 2 3 4 disp 2 or 3 or 4 case 5 disp 5 otherwise disp something else end while Thewhi e loop executes a statement or group of statements repeatedly as long as the controlling expression is true 1 Its syntax is while expression statements end If the expression evaluates toa matrix all its elements must be1 for execution to continue To reduce a matrix toa scalar value usetheal andany functions For example this whi e loop finds the first integer n for which n n factorial is a 100 digit number n 1 while prod 1 n lt 1e100 n n l end Exit awhile loop at any time using the br eak statement while Statements and Empty Arrays A whi e condition that reduces to an empty array represents a false condition That is while A S1 end never executes statement S1 when A is an empty array 10 34 Flow Control for Thef or loop executes a statement or group of statements a predetermined number of times Its syntax is for index start increment end statements end Thedefault increment is1 You can specify any increment including a negative one For positive indices execution terminates when the value of the index exceeds theend value for negative increments it terminates when the index is less than the end value For example this loop executes five times for i 2 6 x i 2 x i 1 end You can nest multiple f or loop
163. ay contents Apply a cell function toa cell array Display graphical depiction of cell array Deal inputs to outputs True for cell array Convert numeric array into cell array Structures Structures Structures are MATLAB arrays with named data containers called fid ds The fields of a structure can contain any kind of data For example one field might contain a text string representing a name another might contain a scalar representing a billing amount a third might hold a matrix of medical test results and so on patient name John Doe C billing _ 127 00 test 79 5 73 180 178 177 5 220 210 205 Like standard arrays structures are inherently array oriented A single structure is a 1 by 1 structure array just as the value 5 is a 1 by 1 numeric array You can build structure arrays with any valid size or shape including multidimensional structure arrays Note The examples in this section focus on two dimensional structure arrays For examples of higher dimension structure arrays see Chapter 12 Building Structure Arrays You can build structures in two ways e Using assignment statements e Usingthestruct function Building Structure Arrays Using Assignment Statements You can build a simple 1 by 1 structure array by assigning data to individual fields MATLAB automatically builds the structure as you go along F or 13 3 13 Structures and Cell Arrays exampl
164. ays of the same size or vectors If they are vectors of different sizes i nter pn passes them tondgri d and then uses the resulting arrays There are three different interpolation methods for multidimensional data e Nearest neighbor interpolation method nearest This method chooses the value of the nearest point e Linear interpolation met hod linear This method uses piecewise linear interpolation based on the values of the nearest two points in each dimension e Cubicinterpolation method cubi c This method uses piecewise cubic interpolation based on the values of the nearest four points in each dimension All of these methods require that X1 X2 X3 be monotonic In addition you should prepare these matrices using the ndgri d function or else be sure that the pattern of the points emulates the output of ndgrid Each method automatically maps the input toan equally spaced domain before interpolating If X is already equally spaced you can speed execution time by prepending an asterisk tothe met hod string for example cubic Interpolation Multidimensional Data Gridding Thendgri d function generates arrays of data for multidimensional function evaluation and interpolation ndgr i d transforms the domain specified by a series of input vectors into a series of output arrays Thei th dimension of these output arrays are copies of the elements of input vector x The syntax for nd
165. be aware that e f you delete a variable from the workspace the amount of free memory indicated by whos usually does not get larger unless the deleted variable occupied the highest memory addresses The number actually indicates the amount of contiguous unused memory Clearing the highest variable makes the number larger but clearing a variable beneath the highest variable has no effect This means that you might have more free memory than is indicated by whos Computers with virtual memory do not display the amount of free memory remaining because neither MATLAB nor the hardware imposes limitations Removing a Function From Memory MATLAB creates a list of M and ME X filenames at startup for all files that reside below themat ab tool box directories This list is stored in memory and O ptimizing the Performance of MATLAB Code is freed only when a new list is created during a call tothe path function Function M file code and ME X file relocatable code are loaded into memory when the corresponding function is called The M file code or relocatable code is removed from memory when e The function is called again and a new version now exists e The function is explicitly cleared with thec ear command e All functions are explicitly cleared with thec ear functions command e MATLAB runs out of memory Nested Function Calls The amount of memory used by nested functions is the same as the amount used by calling them on consec
166. bject e g when an expression is entered without terminating with a semicolon Accesses class properties Enables indexed reference and assignment for user objects Supports end syntax in indexing expressions using an object e g A 1 end Supports using an object in indexing expressions Methods that convert an object toa MATLAB data type The following sections discuss the implementation of each type of method as well as providing references to examples used in this chapter 14 MATLAB Classes and O bjects 14 10 The Class Constructor Method The directory for a particular class must contain an M file known as the constructor for that class The name of the constructor is the same as the name of the directory excluding the prefix and m extension that defines the name of the class The constructor creates the object by initializing the data structure and instantiating an object of the class Guidelines for Writing a Constructor Class constructors must perform certain functions so that objects behave correctly in the MATLAB environment In general a class constructor must handle three possible combinations of input arguments e Noinput arguments e An object of the same class as an input argument e The input arguments used to create an object of the class typically data of some kind No Input Arguments f there are no input arguments the constructor should create a default object Since there are no input
167. ble when a solution component is unexpectedly large The solvers require nonzero tolerances and use a mixed test to avoid these problems At each step the error e in thei th component of the solution is required to satisfy this condition Je i lt max RelTol abs y i AbsTol i The use of Rel To is clear to obtain p correct digits let Rel Tol 10 p or slightly smaller The use of Abs Tol depends on the problem scale Abs Tol is a threshold the solver does not guarantee correct digits for solution components smaller than AbsTol i Ifthe problem has a natural threshold use it as AbsTo A small value of Abs To doesnot adversely affect the computation but be aware that the problem s scaling might mean that an important component is smaller than the specified Abs Tol You might think that you computed the component with the relative accuracy of Re Tol when in fact it is below theAbsTo threshold and you have few if any correct digits Even if you are not interested in correct digits in this component failing to compute it accurately may harm the accuracy of components you do care about Generally the solvers handle this situation automatically but not always You can get close to machine precision but not that close The solvers do not allow Rel Tol near eps because they try to approximate a continuous function At tolerances comparable to eps the machine arithmetic causes all functions to look discontinuous
168. bles it needs in the MATLAB workspace When execution completes the variables i theta and rho remain in the workspace To see a listing of them enter whos at the command prompt 10 5 10 M File Programming 10 6 Functions Functions are M files that accept input arguments and return output arguments They operate on variables within their own workspace This is separate from the workspace you access at the MATLAB command prompt Simple Function Example Theaverage function is a simple M file that calculates the average of the elements in a vector function y average x AVERAGE Mean of vector elements AVERAGE X where X is a vector is the mean of vector elements Non vector input results in an error m n size x if m 1 n 1 m 1 amp n 1 error Input must be a vector end y sum x length x Actual computation If you would like try entering these commands in an M file called average m Theaverage function accepts a single input argument and returns a single output argument To call theaverage function enter z 1 99 average z ans 50 Basic Parts of a Function M File A function M file consists of e A function definition line e AH1line e Help text e The function body e Comments Functions Function Definition Line The function definition line informs MATLAB that the M file contains a function and specifies the argument calling se
169. calar Suggested initial step size Generally it is not necessary for you to adjust MaxStep andi nitialStep because the ODE solvers implement state of the art variable time step control algorithms Adjusting these properties without good reason may result in degraded solver performance MaxStep MaxStep has a positive scalar value This property sets an upper bound on the magnitude of the step size the solver uses If the differential equation has periodic coefficients or solution it may be a good idea to set MaxSt ep to some fraction such as 1 4 of the period This guarantees that the solver does not enlarge the time step too much and step over a period of interest Improving Solver Performance Do not reduce Max St ep to produce more output points This can slow down solution time significantly Instead useRef i ne 8 24 to compute additional outputs by continuous extension at very low cost Do not reduce Max St ep when the solution does not appear to be accurate enough Instead reduce the relative error tolerance Re Tol and use the solution you just computed to determine appropriate values for the absolute error tolerance vector Abs Tol See Error Tolerance Properties on page 8 21 for a description of the error tolerance properties Generally you should not reduce MaxSt ep to make sure that the solver doesn t step over some behavior that occurs only once during the simulation interval If you know the time at which
170. cess the portfolio name field function b subsref p index SUBSREF Define field name indexing for portfolio objects switch index 1 type case switch index 1 subs case name if length index b p name else switch index 2 type case b p name index 2 subs end end end end Saving and Loading O bjects Note that the portfolio implementation ofs ubsr ef is designed to provide access to specific elements of thename field it is not a general implementation that provides access to all structure data such as the stock class implementation of subref Seethesubsref help entry for more information about indexing and objects New Portfolio Class Behavior With the additions and changes made in this example the portfolio class now e Includes a field for an account number e Adds the account number when a portfolio object is saved for the first time e Automatically updates the older version of portfolio objects when you load them into the MATLAB workspace 14 65 14 MATLAB Classes and O bjects Object Precedence Object precedence is a means to resolve the question of which of possibly many versions of an operator or function to call in a given situation Object precedence enables you to control the behavior of expressions containing different classes of objects For example consider the expression objectA obj ectB Ordinarily MATLAB assumes that the objects have equal precedence and cal
171. ch diagonals of to populate That is the elements in column j of 8B fill the diagonal specified by element of d AS an example consider the matrix B and the vector d B 41 11 0 52 22 0 63 33 13 14 44 24 d 2 Use these matrices to create a 7 by 4 sparse matrix A A spdiags B d 7 4 A 1 1 r1 4 1 41 2 2 22 5 2 52 K3 13 3 3 33 6 3 63 2 4 24 4 4 44 7 4 74 9 9 9 Sparse Matrices 9 10 In its full form A looks like this full A ans 13 0 24 33 0 44 0 63 0 74 spdi ags can also extract diagonal elements from a sparse matrix or replace matrix diagonal elements with new values Typehe p spdi ags for details Importing Sparse Matrices from Outside MATLAB You can import sparse matrices from computations outside MATLAB Usethe spconvert function in conjunction with the oad command to import text files containing lists of indices and nonzero elements F or example consider a three column text fileT dat whose first column is a list of row indices second column is a list of column indices and third column is a list of nonzero values These statements load T dat into MATLAB and convert it into a sparse matrix load T dat S spconvert T Thesave and oad commands can also process sparse matrices stored as binary data in MAT files Finally a Fortran utility routinehbo2mat is available to convert a file containing a sparse matrix in the Harwell Boeing format into a MAT file tha
172. changing context in Debugger 3 8 context 10 11 loading 2 11 2 12 of individual functions 10 11 saving 2 11 viewing contents 2 52 viewing during execution 3 2 workspace 2 52 Workspace Browser 2 52 writing data 2 26 Index sound files 2 29 www mat hwor ks com 2 23 X Xdefaults file 2 58 xor 10 26 Y yellow arrow in Debugger 3 7 Z zeros 9 23 12 6 1 31
173. ches the maximum number of 10 recursive calls the method returns a value of nf indicating possible singularity You can includea fourth argument for quad or quad8 that specifies a relative error tolerance for the integration If this fourth argument is a two element vector its first element specifies a relative tolerance and its second an absolute tolerance If a nonzero fifth argument is passed to quad or quad8 the function evaluations are traced with a point plot of the integrand Example Computing the Length of a Curve You can usequad or quad8 tocompute the length of a curve Consider the curve parameterized by the equations x t sin 2t y t cos t z t t where te 0 32 A three dimensional plot of this curve is t 0 0 1 3 pi plot3 sin 2 t cos t t Thearc length formula says the length of the curveis the integral of the norm of the derivatives of the parameterized equations N umerical Integration Q uadrature 3n 4cos 2t sin t 1 dt 0 The function hcurve computes the integrand function f hcurve t f sqrt 4 cos 2 t 2 sin t 2 1 Integrate this function with a call toquad len guad hcurve 0 3 pi len 1 7222e 01 The length of this curveis about 17 2 Example Double Integration Consider the numerical solution of ymax xmax f x y dxdy ymin xmin For this example f x y ysin x xcos y The first step is to build the function to be evaluated
174. column vector formed from the columns of the original matrix So for our magic square A 8 is another way of referring to the value 14 stored inA 4 2 If you try to use the value of an element outside of the matrix it is an error t A 4 5 Index exceeds matrix dimensions However if you store a value in an element outside of the matrix the size of the matrix increases to accommodate the new element xX A x 4 5 17 X 16 2 3 13 0 5 11 10 8 0 9 7 6 12 0 4 14 15 1 17 Subscript expressions involving colons refer to portions of a matrix A 1 k j is the first k elements of the j th column of A So sum A 1 4 4 Indexing and Subscripting computes the sum of the fourth column But there is a better way The colon by itself refers to all the elements in a row or column of a matrix and the keyword end refers to the last row or column So sum A end computes the sum of the elements in the last column of A ans 34 Concatenation Concatenation is the process of joining small matrices together to make bigger ones In fact you made your first matrix by concatenating its individual elements The pair of square brackets is the concatenation operator For an example start with the 4 by 4 magic square A and form B A A 32 A 48 At 16 The result is an 8 by 8 matrix obtained by joining the four submatrices B 16 2 3 13 48 34 35 45 5 11 10 8 37 43 42 40 9 7 6 12 41 39 38 44 4 14 E5 1 3
175. con in the Workspace Browser and click Open or double click the icon The variable is displayed in the Editor Debugger window where you can edit it You can only use this feature with numeric arrays Microsoft W indows Handbook Current Values Change Any Value By Editing It in the Cell MATLAB Editor D ebugger BI Ei sA ER EEEE Current Dimensions Add or Remove Rows and Columns By Editing these Dimensions Current Cell 2 MATLAB W orking Environment UNIX Handbook This section describes several MATLAB environment tools and their use with the UNIX operating system Editor Debugger MATLAB provides a built in combined Editor Debugger that offers basic text editing operations as well as access to M file debugging tools This is essentially the same E ditor Debugger as the one provided for Windows platforms See the section Microsoft Windows Handbook for a discussion of file editing with this tool The Editor Debugger offers a graphical user interface To use the E ditor Debugger typeedi t at the MATLAB prompt Y ou can also use debugging commands in the Command Window Refer to Chapter 3 for a discussion of MATLAB s debugging capabilities File Edit View Debug Tools Window Help polynom m home ron polynom m function p polynom a if nargin 00 pec Pp elseif isa a polynom Untitled1 home class p polynom polynom class p pol
176. count Number is a MATLAB function that returns an account number composed of the first three letters of the client name prepended to a series of digits To illustrate implementation techniques get Account Number is not a portfolio method so it cannot access the portfolio object data directly Therefore it is necessary to define a portfolios ubsr ef method that enables access to the name field in a portfolio object s structure 14 63 14 MATLAB Classes and O bjects For this example get Account Number simply generates a random number which is formatted and concatenated with elements 1 to3 from the portfolio name field function n getAccountNumber p provides a account number for object p n upper p name 1 3 strcat numdstr round rand 1 7 10 Note that the portfolio object is indexed by field name and then by numerical subscript to extract thefirst threeletters Thesubsref method must be written to support this form of subscripted reference The Portfolio subsref Method When MATLAB encounters a subscripted reference such as that made in the get Account Number function p name 1 3 MATLAB calls the portfolios ubsref method to interpret the reference If you do not defineasubsref method the above statement is undefined for portfolio objects recall that here p is an object not just a structure The portfolios ubsref method must support field name and numeric indexing for the get Account Number function to ac
177. ctly into MATLAB using the oad function This creates a variable whose name is the same as the filename See Loading and Saving the Workspace for details on l oad You can also use d mr ead if you need to specify alternate value delimiters dl mr ead is discussed in Delimiter Separated Text Files This method is useful for loading data files from other applications that have their own established file formats These functions are discussed in detail in Chapter 15 dl mread Read ASCII data file textread Read string and numeric data froma fileinto MATLAB variables using conversion specifiers wklread Read spreadsheet WK 1 file imread Read image from graphics file auread Read Sun au sound file Wwavread Read Microsoft WAVE wav sound file 2 27 2 MATLAB W orking Environment Develop a MEX file to read the data Develop a Fortran or C translation program This is the best method if C or Fortran routines are already available for reading data files from other applications See the MATLAB Application Program Interface Guide for more information Develop a program to translate your data into MAT file format and then read the MAT file into MATLAB with the load command See the MATLAB Application Program Interface Guidefor more information Ex porting Data from MATLAB There are several methods for getting MATLAB data to other applications Method Usethedi ary command Save
178. ctory called pol yf un inthe MATLAB Toolbox Function Description conv Multiply polynomials deconv Divide polynomials poly Polynomial with specified roots pol yder Polynomial derivative polyfit Polynomial curve fitting pol yval Polynomial evaluation polyvalm Matrix polynomial evaluation residue Partial fraction expansion residues roots Find polynomial roots The Symbolic Math Toolbox contains additional specialized support for polynomial operations Representing Polynomials MATLAB represents polynomials as row vectors containing coefficients ordered by descending powers For example consider the equation p x x 2x 5 This is the celebrated example Wallis used when he first represented Newton s method to the French Academy To enter this polynomial into MATLAB use p 10 2 5 Polynomials Polynomial Roots Theroots function calculates the roots of a polynomial r roots p 2 0946 1 0473 1 1359i 1 0473 1 1359i By convention MATLAB stores roots in column vectors The function pol y returns to the polynomial coefficients p2 poly r p2 1 8 8818e 16 2 5 poly androots are inverse functions up to ordering scaling and roundoff error Characteristic Polynomials Thepol y function also computes the coefficients of the characteristic polynomial of a matrix A 1 2 3 0 9 51 756 90 1 pol y A ans 1 0000 3 9500 1 8500 163 2750 The roots of this poly
179. ctures and Cell Arrays 13 14 The disadvantage of plane organization is evident when you need to access subsets of the planes To access a subimage for example you need to access each field separately red sub A r 2 12 13 30 grn_sub A g 2 12 13 30 blue sub A b 2 12 13 30 Element by Element Organization Case 2 has the advantage of allowing easy access to subsets of data To set up the data in this organization use for i 1 size RED 1 for j 1 size RED 2 Bii j r RED i j Bli j g GREEN i j B i j b BLUE i j end end With element by element organization you can access a subset of data with a single statement Bsub B 1 10 1 10 To access an entire plane of the image using the element by element method however requires a loop red plane zeros 128 128 for i 1 128128 red plane i Bli r end Element by element organization is not the best structure array choice for most image processing applications however it can be the best for other applications wherein you will routinely need to access corresponding subsets of structure fields The example in the following section demonstrates this type of application Structures Example A Simple Database Consider organizing a simple database A 1 Plane organization B 2 Element by element org
180. d This function accepts an object and a property name and uses as witch statement to determine which field to access This method is called by the subclass get methods when accessing the data in the inherited properties The Asset subsref Method Thesubsref method provides access to the data contained in an asset object using one based numeric indexing and structure field name indexing The outer s wi t ch statement determines if the index is a numeric or field name syntax The inner swi tch statements map the index to the appropriate value MATLAB callssubsref whenever you make a subscripted reference to an object e g A i A i or A fiel dname Seethesubsref help entry for more information on object subscripted reference Example Assets and Asset Subclasses function b subsref a index SUBSREF Define field name indexing for asset objects switch index type case switch index subs case 1 b a descriptor case 2 b a date case 3 b a current_ value otherwise error I ndex out of range end case switch index subs case descriptor b a descriptor case date b a date case current value b a current_value otherwise error Invalid field name end case error Cell array indexing not supported by asset objects end The Asset set Method The asset class set method is called by subclass s et methods This method accepts an asset object and variable lengt
181. d MATLAB use of system resources 10 68 PDF files 2 23 percent sign comments 10 9 performance improving for M files 3 17 improving for solvers 8 17 obtaining statistics for ODE solvers 8 13 permission string 15 3 permutations 9 24 permute 12 2 12 13 permuting array dimensions 12 13 inverse 12 14 persistent variables 10 17 phase 6 37 pi 10 18 pi e3 function overloaded 14 57 pinv 4 21 pivoting partial 4 26 plane organization for structures 13 13 platforms exchanging files between 2 31 plot 2 18 Plot Editor 2 18 plots annotations in 2 18 plotting mathematical functions 7 4 polar 10 5 poly 5 3 polyder 5 5 polyfit 5 6 6 20 6 23 6 25 6 28 pol ynomial fits to data 6 20 interpolation 5 9 regression 6 15 polynomials 5 1 and curve fitting 5 6 basic operations 5 2 characteristics 5 3 derivative of 5 5 dividing 5 4 evaluating in matrix sense 5 4 example class 14 23 multiplying 5 4 representing 5 2 roots 5 3 polyval 5 4 6 23 6 25 6 28 positive definite factorization 4 24 power operator 10 21 powers matrix 4 31 preallocation 10 65 cell array 10 65 13 23 structure array 10 65 precedence object 14 66 within expression 10 29 precision char 15 8 double 15 8 float 15 8 for data types 15 8 long 15 8 short 15 8 single 15 8 uchar 15 8 preconditioner for sparse matrix 9 32 preferences 2 37 2 58 pre processing data 6 12 6 21 primary function 10 38 printing documentation 2 23 Index
182. d reveal that they are integers divided by 360 4 20 Inverses and Determinants If A is square and nonsingular then without roundoff error X inv A B would theoretically be the same as X A B andY B inv A would theoretically be thesameasY B A But the computations involving the backslash and slash operators are preferable because they require less computer time less memory and have better error detection properties Pseudoinverses Rectangular matrices do not haveinverses or determinants At least one of the equations AX I and XA I does not havea solution A partial replacement for theinverseis provided by the MooreP enrose pseudoinverse which is computed by thepi nv function X pinv C X 0 1159 0 0729 0 0171 0 0534 0 1152 0 0418 The matrix Q X C Q 1 0000 0 0000 0 0000 1 0000 is the 2 by 2 identity but the matrix P C X P 0 8293 0 1958 0 3213 0 1958 0 7754 0 3685 0 3213 0 3685 0 3952 is not the 3 by 3 identity However P acts like an identity on a portion of the space in the sense that P is symmetric P C is equal toC and X P is equal tox 4 21 4 Matrices and Linear Algebra 4 22 If A is m by n with m gt n and full rank n then each of the three statements x A b x pinv A b X inv A A A b theoretically computes the same least squares solution x although the backslash operator does it faster However if A does not have full rank the solution t
183. de jpattern orbrussode jpattern n returns a Sparse matrix of 1s and 0s showing the locations of nonzeros in the acobian oF oy By default the stiff ODE solvers generate J acobians numerically as full matrices However if Pattern is seton withodeset a solver calls the ODE file with the flag j pattern This provides the solver with a sparsity pattern that it uses to generate the J acobian numerically as a sparse matrix Providing a sparsity pattern can significantly reduce the number of function evaluations required to generate the acobian and can accelerate integration For the Brusselator problem if the sparsity pattern is not supplied 2n evaluations of the function are needed to compute the 2n by 2n J acobian matrix If the sparsity pattern is supplied only four evaluations are needed regardless of the value of n Reference H airer E and G Wanner Solving Ordinary Differential Equations II Stiff and Differential Algebraic Problems Springer Verlag Berlin 1991 pp 5 8 Examples Applying the O DE Solvers function varargout brussode t y flag N BRUSSODE Stiff problem modeling a chemical reaction if nargin lt 4 isempty N N 2 end switch flag case Return dy dt f t y varargout l f t y N case init Return default tspan y0 options varargout 1l 3 init N case jpattern Return sparsity pattern of df dy varargout 1 jpattern t y N
184. del For more information about this model type ot kademo to run a demonstration t y ode23 lotka 0 2 20 20 3 19 3 Debugger and Profiler 3 20 3 Generate the profile report and save the results to thefilel ot kaprof profile report lotkaprof This suspends the profiler displays the profile report in a Web browser window and saves the results See Viewing Profiler Results for more information 4 Restart the profiler without clearing the existing statistics profile resume The profiler is now ready to continue gathering statistics for any more M files you run It will add these new statistics to those generated in the previous steps 5 Stop the profiler when you are finished gathering statistics profile off Viewing Profiler Results Profile Reports To display profiler results type profile report profile report Suspends the profiler The results appear in a report ina Web browser window Thereport opens with asummary report and from that page you can access the detail and history reports Summary Profile Report The summary report presents statistics about the overall execution and provides summary statistics for each function called Values reported include e Clock precision the precision of the profiler s time measurement When Time for a function is 0 it is actually a positive value but smaller than the profiler can detect given the clock precision e Time columns the total time sp
185. determined linear systems involve a rectangular matrix with more rows than columns that is m by n with m gt n The full size QR factorization produces a square m by m orthogonal Q and a rectangular m by n upper triangular R Q R gr C Q 0 8182 0 3999 0 4131 0 1818 0 8616 0 4739 0 5455 0 3126 0 7777 11 0000 8 5455 0 7 4817 0 0 In many cases the last m n columns of Q are not needed because they are multiplied by the zeros in the bottom portion of R So the economy size QR factorization produces a rectangular m by n Q with orthonormal columns and a square n by n upper triangular R For our 3 by 2 example this is not much of asaving but for larger highly rectangular matrices the savings in both time and memory can be quite important Q R qr C 0 Q 0 8182 0 3999 0 1818 0 8616 0 5455 0 3126 11 0000 8 5455 0 7 4817 LU QR and Cholesky Factorizations In contrast to the LU factorization the QR factorization does not require any pivoting or permutations But an optional column permutation triggered by the presence of a third output argument is useful for detecting singularity or rank deficiency At each step of the factorization the column of the remaining unfactored matrix with largest norm is used as the basis for that step This ensures that the diagonal elements of R occur in decreasing order and that any linear dependence among the columns will almos
186. differential equations variable order method Stiff differential equations and DAEs variable order method Stiff differential equations low order method Moderately stiff differential equations and DAEs trapezoidal rule Stiff differential equations low order method Create alter ODE OPTI ONS structure Get ODE 0PTI ONS parameters Time series plots Two dimensional phase plane plots Three dimensional phase plane plots Print to command window Quick Start Quick Start 1 Write the ordinary differential equation y f ty Ys Y N D asa system of first order equations by making the substitutions 7 n 1 Y1 Y Y25Y 5 Yay Then y1 Yo y2 Y3 Yn f t Yr Y Yn is a system of n first order ODEs For example consider the initial value problem y 3y yy 0 y 0 0 y 0 1 y 0 1 Irr Solve the differential equation for its highest derivative writing y in terms of t and its lower derivatives y 3y y y Ifyoulet y Y Y2 Y and y3 y then y Y2 Y3 Y3 3 3 Y2Y is a system of three first order ODEs with initial conditions 8 3 8 O rdinary Differential Equations 8 4 y 0 0 y 0 1 y3 0 1 Note that the IVP now has theform Y F t Y Y 0 Yo where Y Y Y2 Y3 Code the first order system in an M file that accepts two arguments t and y and returns a column vector function dy F t y dy y 2 y 3 3
187. ditional 1s to indexing expressions for the array To access the value8 in the four dimensional case use C 1 2 1 2 Singleton dimension index Multidimensional Arrays Getting Information About Multidimensional Arrays You can use MATLAB functions and commands to get information about multidimensional arrays you have created Information Function Example Array size size si ze C ans 2 2 1 2 rows columns dim3 dim4 Array ndi ms ndi ms C dimensions ans 4 Array whos whos storage and Name Size Bytes Class format A 2x2x2 64 double array B 2x2x2 64 double array C 4 D 64 double array D 4 D 192 double array Grand total is 48 elements using 384 bytes Working with Multidimensional Arrays Many of the concepts that apply to two dimensional matrices extend to multidimensional arrays as well This section describes how to apply basic indexing and reshaping techniques to multidimensional arrays Consider a 10 by 5 by 3 array nddata of random integers nddata fix 8 randn 10 5 3 12 9 12 Multidimensional Arrays 12 10 Indexing Toaccess a single element of a multidimensional array use integer subscripts Each subscript indexes a dimension the first indexes the row dimension the second indexes the column dimension the third indexes the first page dimension and so on To access element 3 2 on page 2 of nddat a for example usenddat a 3 2 2 You can use vectors as array subscripts In this cas
188. docode 10 11 canonical class 14 9 case 10 32 case conversion 11 3 cat 9 24 12 2 12 7 12 8 12 19 catch 10 36 for error handling 10 51 cd 2 16 2 25 cell indexing 13 20 13 24 cell arrays accessing a subset of cells 13 24 accessing data 13 23 Index applying functions to 13 27 cell indexing 13 20 concatenating 13 22 content indexing 13 20 converting to numeric array 13 32 creating 13 19 using assignments 13 20 with cel s function 13 23 with curly braces 13 22 defined 13 2 deleting cells 13 25 deleting dimensions 13 25 displaying 13 21 expanding 13 21 flat 13 29 indexing 13 20 multidimensional 12 19 nested 13 29 building with thece s function 13 30 indexing 13 30 of strings 11 7 comparing strings 11 10 of structures 13 33 organizing data 13 28 overview 13 19 preallocation 10 65 13 23 reshaping 13 25 to replace comma separated list 13 25 visualizing 13 21 cel datatype 10 20 cell disp 13 21 cell plot 13 21 cells 13 19 13 23 13 29 char data type 10 20 15 8 character arrays 10 19 11 4 two dimensional 11 5 characteristic polynomial 5 3 characteristic roots of matrix 5 3 characters corresponding ASCII values 11 5 finding in string 11 11 chol 9 24 9 30 Cholesky factorization 4 24 for sparse matrices 9 30 class directories 14 6 class function 14 11 classes 10 19 clearing definition 14 6 constructor method 14 10 designing 14 9 methods object oriented 14 2 methods debugging 14 6 methods requi
189. e on page 14 66 MATLAB searches for functions by name When you call a function MATLAB knows the name number of arguments and the type of each argument MATLAB uses the dispatch type to choose among multiple functions of the same name but does not consider the number of arguments Function Precedence Order The function precedence order determines the precedence of one function over another based on the type of function and its location on the MATLAB path From the perspective of method selection MATLAB contains two types of functions those built into MATLAB and those written as M files MATLAB treats these types differently when determining the function precedence order MATLAB selects the correct function for a given context by applying the following function precedence rules in the order given 1 Overloaded Built in Function If thereis a method in the class directory of the dispatching argument that has the same name as a MATLAB function then this method is called instead of the MATLAB function 2 Nonoverloaded MATLAB Functions If there is no overloaded method then the MATLAB function is called Note that MATLAB built in functions take precedence over both subfunctions and private functions but MATLAB M file functions do not 3 Subfunctions Subfunctions take precedence over other functions on the path with the same name Note that subfunctions with the same name as MATLAB built in functions or overloaded built in functi
190. e attempt was unsuccessful MATLAB automatically closes all open files when you exit from MATLAB Itis still good practice however to close a file explicitly with f cl ose when you are finished using it Not doing so can unnecessarily drain system resources Note Closing a file does not clear the file identifier variable f i d However subsequent attempts to access a file through this file identifier variable will not work 15 5 13 File 1 O Temporary Files and Directories Thetempdir andtempname commands assist in locating temporary data on your system Function Purpose tempdir Get temporary directory name tempname Get temporary filename You can create temporary files Some systems deletetemporary files every time you reboot the system On other systems designating a file as temporary may mean only that the fileis not backed up A function namedt empdi r returns the name of the directory or folder that has been designated to hold temporary files on your system For example issuing tempdir on a UNIX system returns the t mp directory MATLAB also provides at empname function that returns a filenamein the temporary directory The returned filename is a suitable destination for temporary data F or example if you need to store some data in a temporary file then you might issue the following command first fid fopen tempname w Note The filename that t empname generates is not guaranteed to be
191. e create the 1 by 1 pati ent structure array shown at the beginning of this section patient name John Doe patient billing 127 00 patient test 79 75 73 180 178 177 5 220 210 205 Now entering patient at the command line results in name John Doe billing 127 test 3x3 double patient is an array containing a structure with three fields To expand the structure array add subscripts after the structure name patient 2 name Ann Lane patient 2 billing 28 50 patient 2 test 68 70 68 118 118 119 172 170 169 Thepatient structure array now has size 1 2 Notethat once a structure array contains more than a single element MATLAB does not display individual field contents when you type the array name Instead it shows a summary of the kind of information the structure contains patient patient 1x2 struct array with fields name billing test You can also usethef i el dnames function to obtain this information fieldnames returns a cell array of strings containing field names 13 4 Structures As you expand the structure MATLAB fills in unspecified fields with empty matrices so that e All structures in the array have the same number of fields e All fields have the same field names For example entering pati ent 3 name Alan Johnson expands the patient array tosize 1 3 Nowbothpatient 3 billing and patient 3 test contain empty matrices Note Field si
192. e each vector element must bea valid subscript that is within the bounds defined by the dimensions of the array To access elements 2 1 2 3 and 2 4 on page3 of nddata use nddata 2 1 3 4 3 The Colon and Multidimensional Array Indexing MATLAB s colon indexing extends to multidimensional arrays F or example to access the entire third column on page 2 of nddata usenddata 3 2 The colon operator is also useful for accessing other subsets of data For example nddata 2 3 2 3 1 results in a 2 by 2 array a subset of the data on page 1 of nddata This matrix consists of the data in rows2 and 3 columns 2 and 3 on the first page of the array The colon operator can appear as an array subscript on both sides of an assignment statement For example to create a 4 by 4 array of zeros C zeros 4 4 Now assign a 2 by 2 subset of array nddata tothe four elements in the center of C C 2 3 2 3 nddata 2 3 1 2 2 Avoiding Ambiguity in Multidimensional Indexing Some assignment statements such as Alsacty 2c cS E are ambiguous because they do not provide enough information about the shape of the dimension to receive the data In the case above the statement tries to assign a one dimensional vector to a two dimensional destination MATLAB produces an error for such cases To resolve the ambiguity be sure Multidimensional Arrays you provide enough information about the destination for the assigned data and tha
193. e nonstiff 8 5 simple stiff 8 8 varargi n 10 14 13 27 in argument list 10 15 unpacking contents 10 14 varargout 10 15 in argument list 10 15 packing contents 10 15 variables containing filenames 2 13 currently in the workspace 2 10 deleting and memory use 10 66 1 29 Index global 10 16 rules for use 10 17 in dispatching priority 10 10 local 10 16 memory usage 10 67 naming 10 16 persistent 10 17 replacing list with a cell array 13 25 storage in memory 10 66 viewing values during execution 3 2 vdpode 8 35 vector column 4 5 initial condition ODE 8 6 of dates 10 57 preallocation 10 65 row 4 5 time span vector ODE 8 12 vector products 4 7 vectorization 10 63 example 10 63 for J acobian matrix computation ODE 8 28 replacingf or loops 10 30 version 10 18 version obtaining 10 18 visualization 2 19 visualizing cell array 13 21 ODE solver results 8 7 sparse matrix spy plot 9 19 voronoi 5 22 Voronoi diagrams 5 22 Ww warnings 10 51 displaying 10 52 wavread 2 27 2 29 Web site The MathWorks 1 8 2 23 what 2 15 whi ch used with methods 14 71 while 10 34 and empty arrays 10 34 example 10 34 syntax 10 34 white space finding in string 11 11 who 2 10 whos 2 10 2 52 9 11 12 9 interpreting memory use 10 66 wildcards for oad 2 13 Windows and MATLAB use of system resources 10 68 environment for MATLAB 2 35 registry 2 4 wklread 2 27 wklwrite 2 29 working environment MATLAB 1 4 2 2 workspace 2 10
194. e the result is an m by n numeric array Filenames Stored in String Variables If the filenames and variable names you are working with are stored in string variables you can use command function duality to call oad and save as functions In this case the input arguments appear in the same order as they would at the command line For example the statements save myfile VARI VAR2 A myfile oad A are the same as save myfile VAR1 VAR2 load myfile Toload or save multiple files with the same prefix and successive integer suffixes use a loop For example this code saves the squares of the numbers 1 through 10 in files dat a1 through data10 file data for i 1 10 j i 2 save file int2str i j end Wildcards Thel oad andsave commands let you specify a wildcard character to search for patterns of variable names F or example Save rundate x 2 13 2 MATLAB W orking Environment saves all variables in the workspace that start with x in thefilerundata mat Similarly load testdata ex1 95 loads fromt est data mat all the variables whose first three characters are ex1 and last two characters are 95 regardless of the characters between them Search Path MATLAB uses a Search path to find M files MATLAB s M files are organized in directories or folders on your file system Many of these directories of M files are provided along with MATLAB while others ar
195. e Command W indow Loading and Saving the Workspace MATLAB Ss ave and oad commands let you save the contents of the MATLAB workspace at any time during a session and then reload the data back into MATLAB duringthat session or a later one oad ands ave can alsoimport and export text data files Saving the Workspace Thesave command saves the contents of the workspace into a binary MAT file that you can read back later with the oad command For example Save j unel0 saves the entire workspace contents in the filej une10 mat If desired you can save only certain variables by specifying the variablenames after the filename For example Save junel0 x y z saves only variables x y andz On Windows platforms thes ave operation is also available by selecting Save Workspace As from the File menu Note The MATLAB Application Program Interface Guide provides details on reading and writing MAT files from external C or Fortran programs 2 11 2 MATLAB W orking Environment 2 12 Specifying File Format You can control the format in whichs ave stores data by appending flags tothe filename variable name list mat Use binary MAT file form default ascii Use 8 digit ASCII form ascii double Use 16 digit ASCII form ascii double tabs Delimit array elements with tabs v4 Save in format that MATLAB version 4 can open append Append data to existing MAT file If you use the v4 flag you can only save
196. e Documentation Set MATLAB comes with an extensive set of documentation consisting of an online Help facility and online MATLAB Function Reference as well as printed manuals The full set of printed documentation includes the following titles e The MATLAB Installation Guide describes how to install MATLAB on your platform Getting Started with MATLAB explains how to get started with the fundamentals of MATLAB Using MATLAB provides in depth material on the MATLAB language working environment and mathematical topics Using MATLAB Graphics describes how to use MATLAB s graphics and visualization tools The MATLAB Application Program Interface Guide explains how to write C or Fortran programs that interact with MATLAB MATLAB New Features provides information useful in making the transition from the latest release of MATLAB What Want What I Should Do need toinstall MATLAB See the Installation Guide for your platform I m new to MATLAB and Start by reading Getting Started with MATLAB The most want to learn it quickly important things to learn are how to enter matrices how to use the colon operator and how to invoke functions After you master the basics you can access the rest of the documentation as needed or you can use online help and the demonstrations to learn other commands I m upgrading from an Read the New Features document to find out about the new earlier release features in the latest release
197. e Help button to access help for those dialog boxes Disk File Manipulation and Shell Escape Disk File Manipulation and Shell Escape Thecommandsdir type del ete andcd implement a set of generic operating system commands for manipulating files This table indicates how these commands map to other operating systems MATLAB Commands Windows UNIX dir dir ls type type cat delete del orerase rm cd chdir cd For most of these commands you can use pathnames wildcards and drive designators in the usual way Running External Programs The exclamation point character is a shal escapeand indicates that the rest of the input line is a command to the operating system This is quite useful for invoking utilities or running other programs without quitting from MATLAB On UNIX for example Ivi darwin m invokes the vi editor for a file named dar wi n m After you quit the program the operating system returns control to MATLAB See the commands uni x and dos in online help to run external programs that return results and status 2 25 2 MATLAB W orking Environment Data Import Ex port There are many ways to move data between MATLAB and other applications In most cases you can simply use M ATLAB s native data exchange capabilities to read in or write out files For more complicated data sets you may want to create your own C or Fortran program to read or write a file Importing Data into MATLAB You can introd
198. e Lotka Volterra predator prey model y Y1 4y1y2 Y2 Y2 BY1y2 Create an M file ot ka m function yp lotka t y LOTKA Lotka Volterra predator prey model global ALPHA BETA yp y 1 ALPHA y 1 y 2 y 2 BETA y 1 y 2 1 Local and Global Variables Then interactively enter the statements global ALPHA BETA ALPHA 0 01 BETA 0 02 t y ode23 lotka 0 10 1 1 plot t y The two global statements make the values assigned toALPHA and BETA at the command prompt available inside the function defined by ot ka m They can be modified interactively and new solutions obtained without editing any files For your MATLAB application to work with global variables e Declarethevariableasg obal in every function that requires access toit To enable the workspace to access the global variable also declare it as gl obal from the command line e in each function issuetheg obal command beforethe first occurrence of the variable name The top of the M file is recommended MATLAB global variable names aretypically longer and more descriptive than local variable names and sometimes consist of all uppercase characters These are not requirements but guidelines to increase the readability of MATLAB code and reduce the chance of accidentally redefining a global variable Persistent Variables A variable may be defined as persistent sothat it does not change value from one call to another Persistent var
199. e MATLAB functions you do not typically use them in calculations A date vector contains the elements y ear month day hour minute second MATLAB provides functions that convert date strings to serial date numbers and vice versa Dates can also be converted to date vectors Here are examples of the three date formats used by MATLAB Date string 02 Oct 1996 Serial date number 729300 Date vector 1996 10200 0 Conversions Between Date Formats Functions that convert between date formats are datenum Convert date string to serial date number datestr Convert serial date number to date string datevec Split date number or date string into individual date elements 10 55 10 M File Programming 10 56 Here are some examples of conversions from one date format to another dl datenum 02 Oct 1996 dl 729300 d2 datestr d1 10 d2 12 Oct 1996 dvl datevec d1l dvl 199610 2 0 0 0 dv2 datevec d2 dv2 199610 12 0 0 0 Date String Formats Thedat enum function is important for doing date calculations efficiently dat enum takes an input string in any of several formats with dd mmm yyyy mm dd yyyy Or dd mmm yyyy hh mm ss ss most common You can form up to six fields from letters and digits separated by any other characters e The day field is an integer from 1 to 31 e The month field is either an integer from 1 to 12 or an alphabetic string with at least three characters e The year f
200. e a parent object which facilitates the development of related classes that behave similarly but are implemented differently There are two kinds of inheritance e Simpleinheritance in which a child object inherits characteristics from one parent class e Multiple inheritance in which a child object inherits characteristics from more than one parent class This section also discusses a related topic aggregation Aggregation allows one object to contain another object as one of its fields Simple Inheritance A class that inherits attributes from a single parent class and adds new attributes of its own uses simple inheritance Inheritance implies that objects belonging to the child class have the same fields as the parent class as well as additional fields Therefore methods associated with the parent class can operate on objects belonging tothe child class The methods associated with the child class however cannot operate on objects belonging to the parent class You cannot access the parent s fields directly from the child class you must use access methods defined for the parent The constructor function for a class that inherits the behavior of another has two special characteristics Building on O ther Classes e t calls the constructor function for the parent class to create the inherited fields e Thecalling syntax for thecl ass function is slightly different reflecting both the child class and the parent cla
201. e available separately as toolboxes If you enter the namef oo at the MATLAB prompt the MATLAB interpreter 1 Looks for foo asa variable 2 Checks for f oo as a built in function 3 Looks in the current directory for a file named f oo m 4 Searches the directories on the search path for f 00 m Although the actual search rules are more complicated because of the restricted scope of private functions subfunctions and object oriented functions this simplified perspective is accurate for the ordinary M files that you usually work with If you have more than one function with the same name only the first onein the search path order is found other functions with the same name are considered to be shadowed and cannot be executed Changing the Search Path You can display and change the search path for the duration of your current session using thepath addpath andr mpath functions e path by itself returns the current search path e path s wheres isa string sets the path tos The Command W indow e addpath home lib andpath path home lib both append a new directory to the path e rmpath home ib removes the path home lib The default search path remembered between sessions is defined in the file pat hdef min the directory named ocal on your system pat hdef executes automatically each time you start MATLAB A On Windows platforms you can directly edit pat hdef m with your text en editor X On
202. e implemented in a way that is very similar to thes t ock class as would other types of asset subclasses Designing the Asset Class The asset class provides storage and access for information common to all asset children It is not intended to be instantiated directly soit does not require an extensive set of methods To serve its purpose the class needs to contain the following methods e Constructor e get andset e subsref andsubsasgn e display The Asset Constructor Method The asset dass is based on a structure array with four fields e descriptor Identifier of the particular asset e g stock name savings account number etc e date The date the object was created calculated by the date command e current_value The current value of the asset calculated from subclass data This information is common to asset child objects stock bond and savings so it is handled from the parent object to avoid having to define the same fields in each child class This is particularly helpful as the number of child classes increases Example Assets and Asset Subclasses function a asset varargin ASSET Constructor function for asset object a asset descirptor current_value switch nargin case 0 if no input arguments create a default object a descriptor none a date date a current_value 0 a class a asset case 1 if single argument of class asset return it if isa varargin l
203. e matrix A 9 4 6 8 2 7 the full singular value decomposition is U 5 V svd A U 0 6105 0 7174 0 3355 0 6646 0 2336 0 7098 0 4308 0 6563 0 6194 5 14 9359 0 0 5 1883 0 0 V 0 6925 0 7214 0 7214 0 6925 4 39 4 Matrices and Linear Algebra You can verify that U S V is equal toA to within roundoff error F or this small problem the economy size decomposition is only slightly smaller U S V svd A 0 ice 0 6105 0 7174 0 6646 0 2336 0 4308 0 6563 ae 14 9359 0 0 5 1883 V 0 6925 0 7214 0 7214 0 6925 Again U S V is equal to A to within roundoff error 4 40 Polynomials and nterpolation Polynomials i Representing Polynomials i Polynomial Roots Characteristic Polynomials Polynomial Evaluation Convolution and Deconvolution Polynomial Derivatives Polynomial Curve Fitting Partial Fraction Expansion Interpolation One Dimensional Interpolation Two Dimensional Interpolation Comparing Interpolation Methods Interpolation and Multidimensional Arrays Triangulation and Interpolation of Scattered Data 5 Polynomials and Interpolation 5 2 Polynomials MATLAB provides functions for standard polynomial operations such as polynomial roots evaluation and differentiation In addition there are functions for more advanced applications such as curve fitting and partial fraction expansion The polynomial functions live in a dire
204. e matrix 9 5 sparse and full comparison 9 11 viewing for sparse matrix 9 11 Index strcmp 11 9 string 10 19 and relational operators 11 11 array 11 5 comparing values on cell arrays 11 10 converting to cell arrays 11 7 padding for equal row length 11 5 arrays cell array 11 7 categorizing characters 11 11 comparing 11 9 concatenation 10 46 conversion 11 3 11 13 15 2 creating 11 4 delimiting character 11 12 evaluating 11 14 evaluation 10 46 finding starting position of substring 11 12 functions that test 11 2 operations 11 2 removing trailing blanks 11 6 representation 11 4 scalar 11 11 searching and replacing 11 12 token 11 12 struct data type 10 20 structs 13 3 13 5 13 16 structure arrays accessing data 13 6 usinggetfield 13 8 adding fields 13 9 applying functions to 13 9 building 13 3 usingstructs 13 5 data organization 13 11 defined 13 2 deleting fields 13 9 element by element organization 13 14 expanding 13 4 13 5 fields assigning data to 13 3 assigning usingset field 13 8 defined 13 3 indexing nested structures 13 17 within fields 13 7 multidimensional 12 20 applying functions 12 21 nesting 13 16 obtaining field names 13 4 organizing data 13 11 example 13 15 overview 13 3 plane organization 13 13 preallocation 10 65 size 13 9 subscripting 13 4 within cell arrays 13 33 writing M files for 13 10 example 13 11 structures See also structure arrays structures used with classes 14 7 subarrays acce
205. e possible to do better than a simple polynomial fit with this data set Case Study Curve Fitting Ex ponential Fit By looking at the population data plots on the previous pages the population data curve is somewhat exponential in appearance To take advantage of this let s try to fit the logarithm of the population values again working with normalized year values logpl polyfit sdate og1l0 pop 1 logpredl 10 polyval logpl sdate semilogy cdate ogpred1 cdate pop grid on 3 0 fi fi fi fi 1750 1800 1850 1900 1950 2000 6 25 6 Data Analysis and Statistics Now try the logarithm analysis with a second order model logp2 polyfit sdate log1l0 pop 2 logpred2 10 polyval logp2 sdate semi logy cdate ogpred2 cdate pop grid on 10 z 10 10 10 L i L j 1750 1800 1850 1900 1950 2000 This is a more accurate model The upper end of the plot appears to taper off while the polynomial fits in the previous section continue concave up to infinity 6 26 Case Study Curve Fitting Compare the residuals for the second order logarithmic model Residuals in Log Population Scale Residuals in Population Scale logres2 10g10 pop polyval logp2 sdate plot cdate logres2 0 02 0 03F 4 4 0 04 i i i 1750 1800 1850 1900 1950 2000 r pop 10 polyval logp2 sdate plot cdate r
206. e solutions only when y0 is consistent that is if there is a vector yp0O such that M t0 yO f tO yO Theodel5s andode23t solvers can solve DAEs of index 1 provided that M is not state dependent and y0 is 8 29 8 O rdinary Differential Equations 8 30 sufficiently close to being consistent If there is a mass matrix you can use odeset toset theMassSingular property to yes no or maybe The default value of maybe causes the solver to test whether the problem is a DAE If it is the solver treats yO as a guess attempts to compute consistent initial conditions that are close toy 0 and continues to solve the problem When solving DAEs it is very advantageous to formulate the problem so that M isa diagonal matrix a semi explicit DAE For examples of DAE problems see hbldae Orampldae Property Value Description Mass none M Indicate whether the ODE file returns M t a mass matrix M t y MassSingular yes no Indicate whether the mass matrix is maybe singular Mass Change this property from none if the ODE file is coded so that F t y mass returns amass matrix M indicates a constant mass matrix M t indicates a time dependent mass matrix and M t y indicates a time and state dependent mass matrix MassSingular Set this property to no if the mass matrix is not singular For an example of an ODE file with a mass matrix see Example 4 Finite Element
207. e this approach your ODE file must check the value of the third argument and return the appropriate output For example you can modify the van der Pol ODE filevdp1 m to check the third argument f ag and return either the default vector F t y or tspan y0 options depending on the value of fl ag function outl out2 out3 vdpl t y flag if stremp flag Return dy dt F t y outl y 2 1 y 1 2 y 2 y 1 elseif strcmp flag init Return tspan y0 options outl 0 20 tspan out2 2 0 initial conditions out3 odeset RelTol le 4 options end Note The third argument referred to as thef ag argument is a special argument that notifies the ODE file that the solver is expecting a specific kind of information The i nit string for initial values is just one possible value for this flag For complete details on thef ag argument see Special Purpose ODE Files and the flag Argument on page 8 17 8 15 8 O rdinary Differential Equations 8 16 Passing Additional Parameters to the ODE File In some cases your ODE system may require additional parameters beyond the requiredt andy arguments For example you can generalize the van der Pol ODE file by passing it amu parameter instead of specifying a value for mu explicitly in the code function outl out2 out3 vdpode t y flag mu if margin lt 4 isempty mu mu 1 end if strcemp flag Return
208. ears first in the list Additional optional arguments are appended to the list 10 13 10 M File Programming Cell array indexing 10 14 Passing Variable Numbers of Arguments Thevarargin andvarargout functions let you pass any number of inputs or return any number of outputs toa function MATLAB packs all of the specified input or output into a cell array a special kind of MATLAB array that consists of cells instead of array elements Each cell can hold any size or kind of data one might hold a vector of numeric data another in the same array might hold an array of string data and soon Here s an example function that accepts any number of two element vectors and draws a line to connect them function testvar varargin for i l length varargin x i varargin i 1 y i varargin i 2 end xmin min 0 min x ymin min 0 min y axis xmin fix max x 3 ymin fix max y 3 plot x y Coded this way thet est var function works with various input lists for example testvar 2 3 1 5 4 8 6 5 4 2 2 3 testvar 1 0 3 5 4 2 1 1 Unpacking varargin Contents Because var ar gin contains all the input arguments in a cell array it s necessary to use cell array indexing to extract the data For example y i varargin i 2 Cell array indexing has two subscript components e The cell indexing expression in curly braces e The contents indexing expression s in p
209. easing the number of output points by a factor of n This feature is especially useful when using a medium or high order solver such as 0de45 for which solution components can change substantially in the course of a single step To obtain smoother plots increase the Ref i ne property Note In all the solvers the default value of Refi ne is1 Withinode45 however Refine is4 tocompensate for the solver s large step sizes To override this and see only the time steps chosen by 0de45 set Refine tol Improving Solver Performance The extra values produced for Ref i ne are computed by means of continuous extension formulas These are specialized formulas used by the ODE solvers to obtain accurate solutions between computed time steps without significant increase in computation time Stats TheSt ats property specifies whether statistics about the computational cost of the integration should be displayed By default Stats isoff Ifitison after solving the problem the integrator displays e The number of successful steps e The number of failed attempts e The number of times the ODE file was called to evaluate F t y e The number of times that the partial derivatives matrix dF dy was formed e The number of LU decompositions e The number of solutions of linear systems You can obtain the same values by including a third output argument in the call tothe ODE solver T Y S ode45 myfun This statement produces a vector
210. ecl ass function to instantiate the object and return the object as the output argument If necessary place the object in an object hierarchy using thesuperiorto andi nferiorto functions Using the class Function in Constructors Within a constructor method you use thec ass function to associate an object structure with a particular class This is done using an internal class tag that is only accessible using thec ass andisa functions For example this call to thecl ass function identifies the object p to be of type pol ynom p class p polynom Examples of Constructor M ethods See the following sections for examples of constructor methods e The Polynom Constructor Method on page 14 24 e The Asset Constructor Method on page 14 38 e The Portfolio Constructor Method on page 14 55 Identifying O bjects Outside the Class Directory Thecl ass andi sa functions used in constructor methods can also be used outside of the class directory The expression isa a class_ name checks whether a is an object of the specified class For example if p isa polynom object each of the following expressions is true isa pi double isa hello char isa p polynom 14 11 14 MATLAB Classes and O bjects 14 12 Outside of the class directory thec ass function takes only one argument it is only within the constructor that cl ass can have more than one argument The expression class a ret
211. ectory for object a If the input object is already of typecl ass_ name then MATLAB callstheconstructor which just returns the input argument Examples of Converter Methods See the following sections for examples of converter methods e The Polynom to Double Converter on page 14 25 e The Polynom to Char Converter on page 14 25 14 19 14 MATLAB Classes and O bjects Overloading Operators and Functions 14 20 In many cases you may want to change the behavior of MATLAB s operators and functions for cases when the arguments are objects You can accomplish this by overloading the relevant functions Overloading enables a function to handle different types and numbers of input arguments and perform whatever operation is appropriate for the highest precedence object See Object Precedence on page 14 66 for more information on object precedence Overloading Operators Each built in MATLAB operator has an associated function name e g the operator has an associated p us m function You can overload any operator by creating an M file with the appropriate name in the class directory For example if either p or q is an object of typecl ass_ name the expression p q generates a call toa function cl ass_name plus m if it exists Ifp and q are both objects of different classes then MATLAB applies the rules of precedence to determine which method to use Examples of Overloaded Operators See the following sec
212. ectory newmat h is on the MATLAB search path A subdirectory of newmath called private can contain functions that only the functions innewmat h can call Because private functions areinvisible outside of the parent directory they can use the same names as functions in other directories This is useful if you want to create your own version of a particular function while retaining the original in another directory Because MATLAB looks for private functions before standard M file functions it will find a private function named t est m before a nonprivate M filenamedt est m You can create your own private directories simply by creating subdirectories calledpri vate using the standard procedures for creating directories or folders on your computer Do not place these private directories on your path 10 39 10 M File Programming Indexing and Subscripting 10 40 Subscripts The element in rowi and column j of A is denoted by A i j For example supposeA magic 4 ThenA 4 2 isthe number in the fourth row and second column For our magicsquare A 4 2 iS15 Soitis possi bleto compute the sum of the elements in the fourth column of A by typing A 1 4 A 2 4 A 3 4 Al4 4 It is also possible to refer to the elements of a matrix with a single subscript A k This is the usual way of referencing row and columns vectors But it can also apply to a fully two dimensional matrix in which case the array is regarded as one long
213. ed with plot3 X 1 X 2 X 3 0 M atrix Powers and Exponentials shows the solution spiraling in towards the origin This behavior is related to the eigenvalues of the coefficient matrix which are discussed in the next section 4 33 4 Matrices and Linear Algebra 4 34 Eigenvalues An e amp genvalue and eigenvector of a square matrix A area scalar A and a vector v that satisfy Av Av With the eigenvalues on the diagonal of a diagonal matrix A and the corresponding eigenvectors forming the columns of a matrix V we have AV VA If V is nonsingular this becomes the a genvalue decomposition A VAV A good exampleis provided by the coefficient matrix of the ordinary differential equation in the previous section A 0 6 l 6 2 16 5 20 10 The statement lambda ei g A produces a column vector containing the eigenvalues For this matrix the eigenvalues are complex lambda 3 0710 2 4645 17 6008i 2 4645 17 6008i The real part of each of the eigenvalues is negative so et approaches zero as tincreases The nonzero imaginary part of two of the eigenvalues contributes the oscillatory component sin at to the solution of the differential equation Eigenvalues With two output arguments ei g computes the eigenvectors and stores the eigenvalues in a diagonal matrix V D ei g A V 0 8326 0 1203 0 2123i 0 1203 0 2123i 0 3
214. el5s ode23s ode23t orode23tb Yes Thesolversode15s andode23t can solve some DAEs of the form M t y f t y where M t is singular the DAEs must be index 1 For examples seeampldae andhbidae 8 53 8 O rdinary Differential Equations Solving Different Kinds of Systems Question Answer Can integrate a set of Not directly You have to represent the data as a function by sampled data interpolation or some other scheme for fitting data The smoothness of this function is critical A piecewise polynomial fit like a spline can look smooth to the eye but rough toa solver the solver will take small steps where the derivatives of the fit have jumps Either use a smooth function to represent the data or use one of the lower order solvers 0de23 0de23s o0de23t ode23tb that is less sensitive to this Can I solve Not directly In some cases it is possible to use the initial value delay differential problem solvers to solve delay differential equations by breaking equations the simulation interval into smaller intervals the length of a single delay For more information about this approach see Shampine L F Numerical Solution of Ordinary Differential Equations Chapman amp Hall Mathematics 1994 What do do when have ode45 and the other solvers that are available in this version of the final and not theinitial the MATLAB ODE suite allow you to solve backwards or value forwards in time The syntax for the solver
215. en by exp t 2h 3 ifi J 2 h_ ifi Aij ep t h 6 ifi j 1 and Ry 1 h ifi j 1 0 otherwise 0 otherwise The initial values c 0 are taken from the initial condition for the partial differential equation The problem is solved on the time interval 0 z femlode t mass orfemlode t mass n returns the time dependent mass matrix M evaluated at timet By default ode15s solves systems of theform y F t y However if theMass property is changed from none to M M t or Mt y withodeset thesolver calls the ODE file with the flag mass The ODE file returns a mass matrix which the solver uses to solve M t y y F t y If the mass matrix isa constant M the problem can be also be solved with ode23s 8 41 8 O rdinary Differential Equations For example to solve a system of 20 equations use T Y odel5s femlode odeset Mass M t 20 8 42 Examples Applying the O DE Solvers femlode also responds tothe flag i nit seetheri gi dode example for details function varargout femlode t y flag N MFEMIODE Stiff problem with a time dependent mass matrix if margin flag demo end if nargin lt 4 isempty N N 9 end switch flag Case Return dy dt f t y varargout l f t y N case init Return default tspan y0 options varargout 1l 3 init N case mass Return mass matrix
216. ength x tot sqsum x mu Dy tot length x 1 Line 4 3 24 AM 74 Check the values of mu andt ot from the Debugger Highlight the text of each variable and right click to bring up the context menu and choose Evaluate Selection Or alternatively choose Evaluate Selection from the View menu Both the selection and the result are displayed in the Command Window K gt gt mu mu 3 K gt gt tot tot 4 The problem is in thes qs um function 3 7 3 Debugger and Profiler 3 8 Changing Workspace Context Use the Stack pull down menu in the upper right corner of the Debugger window to change workspace contexts To step out from thevar i ance function and see the base workspace contents select Base Workspace from the menu MATLAB Editor Debugger C WINDOWS Desktop variance m Iof x File Edit View Debug Window Help 5 x l Base Workspace variance Base Workspace pem He e Sle aE stack function y variance x mu sum x length x tot sqsum x mu Oy tot length x 1 10 54 AM 4 Check the workspace context using whos or the graphical Workspace Browser The variables v and myvar1 as well as any other variables you may have created show up in the listing To return to thevari ance workspace context select Variance from the menu Stepping Through Code and Continuing Execution Clear the breakpoint at line 4in vari ance m by placing the c
217. ension of cells using a single statement Like standard array deletion use vector subscripting when deleting a row or column of cells and assign the empty matrix to the dimension A cell_ subscripts When deleting cells curly braces donot appear in the assignment statement at all Reshaping Cell Arrays Like other arrays you can reshape cell arrays using ther eshape function The number of cells must remain the same after reshaping you cannot user eshape to add or remove cells A cell 3 4 size A ans 3 4 B reshape A 6 2 size B ans Replacing Lists of Variables with Cell Arrays Cell arrays can replace comma separated lists of MATLAB variables in e Function input lists e Function output lists e Display operations e Array constructions Square brackets and curly braces 13 25 13 Structures and Cell Arrays 13 26 If you use the colon to index multiple cells in conjunction with the curly brace notation MATLAB treats the contents of each cell as a separate variable For example assume you have a cell array T where each cell contains a separate vector The expression T 1 5 is equivalent to a comma separated list of the vectors in the first five cells of T Consider the cell array C C 1 1 2 3 C 2 1 0 1 C 3 1 10 C 4 9 8 7 Cae 3 To convolve the vectors inC 1 andc 2 usingconv d conv C 1 2 d 1 2 4 2 3 Display vectors two three and fo
218. ent in a function including all child functions called Becausethetime for a function includes time spend on child M File Profiler functions the times do not add up to the Total recorded time and the percentages add up to more than 100 e Self time columns total time spent in a function not including time for any child functions called Adding the Self time values for all functions listed equals the Total recorded time The Self time percentages for all functions add up to approximately 100 Below is the summary report for the Lotka Volterra model described in An Example Using the Profiler 3 21 3 Debugger and Profiler View Summary View Details for View Sequence of shown here All Functions Function Calls B Profiler Report Netscape JUX Fle Eal vew Go Comnicaor Hep Summary Function Details Function Call Histo MATLAB Profile Report Summary Report generated 07 Aug 2998 27 38 33 Total Time that the Profiler was Recording Total recorded time 0 20 s Number of M functions 4 Number of M subfunctions 1 Number of builtins 23 Clock precision Function List View Details for Each Name Time Calls Time call Self time Location Function ode23 0 20 1000 1 0 2000 0 13 650 Devtools mat lab5S night o ode23 eva lODEF i le 0 04 20 0 34 0 0012 0 01 5 0 Devtools mat ab5 night Indudes Built in 003 15 0 34 0 0009 0 02 10
219. ent location 8 17 8 30 Events 8 31 J acobian matrix 8 17 8 25 Jacobian 826 J Constant 8 26 8 27 J Pattern 826 Vectorized 8 26 8 28 mass matrix 8 17 8 29 Mass 8 30 MassSingular 8 30 modifying property structure 8 21 odel5s 8 17 BDF 8 32 MaxOrder 8 32 odeset function 8 20 querying property structure 8 21 smoothing output 8 24 solution components for output function 8 24 solver output 8 17 8 23 Out put Fen 8 23 OutputSel 8 23 8 24 Refine 8 23 8 24 Stats 8 23 8 25 specifying overview 8 13 step size 8 17 8 28 Initial Step 8 28 8 29 MaxStep 8 28 Seealso ODE solvers ODE solvers basic example nonstiff problem 8 6 stiff problem 8 8 boundary conditions 8 41 calling 8 6 different kinds of systems 7 13 examples 8 34 flag argument 8 18 multistep solver 8 10 nonstiff solvers 8 10 obtaining performance statistics 8 13 obtaining solutions at specific times 8 12 one step solver 8 10 overview 8 10 quick start 8 3 representing problems 8 5 rewriting problem as first order system 8 6 solution array 8 12 stability 8 32 stiff problems 8 8 stiff solvers 8 10 syntax basic 8 11 time interval 8 6 1 19 Index 1 20 time span vector 8 12 van der Pol example extra parameters 8 16 nonstiff 8 5 stiff 8 8 See also ODE solver properties ODE See Ordinary Differential Equations ode113 8 10 description 8 10 odel5s 8 8 8 11 8 32 8 41 8 51 8 53 description 8 11 properties 8 17 ode23 8 10 8 53 description 8
220. ent to type only the leading characters that uniquely identify the property name odeset ignores case for property names With no input arguments odeset displays all property names and their possible values indicating defaults with AbsTol positive scalar or vector le 6 BDF on off Events on off InitialStep positive scalar Jacobian on off JConstant on off J Pattern on off Mass none M M t M t y MassSingular yes no maybe MaxOrder 1 21 3 44 5 MaxStep positive scalar OutputFcen string OutputSel vector of integers Refine positive integer RelTol positive scalar le 3 Stats on off Vectorized on off Improving Solver Performance Modifying an Existing O ptions Structure To modify an existing opti ons argument use options odeset oldopts namel valuel This sets opti ons equal tothe existing structureol dopt s overwriting any values in ol dopts that are respecified using name value pairs and adding to the structure any new pairs The modified structure is returned as an output argument In the same way the command options odeset oldopts newopts combines the structures ol dopts andnewopts In the output argument any values in the second argument other than the empty matrix overwrite those in the first argument Querying Options The odeget Function Thesolvers usetheodeget fu
221. erence might well be less than measurement errors in the original data A rectangular matrix A is rank deficient if it does not have linearly independent columns If A is rank deficient the least squares solution to AX B is not unique The backslash operator A B issues a warningif A is rank deficient and produces a basic solution that has as few nonzero elements as possible 0 9 T Undetermined Systems Underdetermined linear systems involve more unknowns than equations When they are accompanied by additional constraints they are the purview of linear programming By itself the backslash operator deals only with the unconstrained system The solution is never unique MATLAB finds a basic solution which has at most m nonzero components but even this may not be unique The particular solution actually computed is determined by the QR factorization with column pivoting see a later section on the QR factorization 4 17 4 Matrices and Linear Algebra Hereis a small random example R fix 10 rand 2 4 R 6 8 7 3 3 5 4 1 b fix 10 rand 2 1 1 2 The linear system Rx b involves two equations in four unknowns Since the coefficient matrix contains small integers it is appropriate to display the solution in rational format The particular solution is obtained with format rat p R b p 0 5 7 0 11 7 One of the nonzero components is p 2 becauseR 2 isthecolumn of R with largest nor
222. ers 1 Introduction 1 4 The MATLAB System The MATLAB system consists of five main parts The MATLAB Language This is a high level matrix array language with control flow statements functions data structures input output and object oriented programming features It allows both programming in the small to rapidly create quick and dirty throw away programs and programming in the large to create complete large and complex application programs The language features are organized into six directories in the MATLAB Toolbox ops Operators and special characters lang Programming language constructs strfun Character strings iofun File input output timefun Time and dates datatypes Data types and structures The MATLAB Working Environment This is the set of tools and facilities that you work with as the MATLAB user or programmer It includes facilities for managing the variables in your workspace and importing and exporting data It also includes tools for developing managing debugging and profiling M files MATLAB s applications The working environment features are located in a single directory general General purpose commands Handle Graphics This is the MATLAB graphics system It includes high level commands for two dimensional and three dimensional data visualization image processing animation and presentation graphics It also includes low level commands that allow you to fully custo
223. erty isseton with odeset a solver calls the ODE file with the flag events This provides the solver with information that it uses to locate zero crossings of the elements in the val ue vector Theval ue vector may be any length It is evaluated at the beginning and end of a step and if any elements change sign with the directionality specified in direction the zero crossing point is located Thei ster mi nal vector consists of logical 1s and 0s enabling you to specify whether or not a zero crossing of the corresponding value element halts the integration The di rection vector enables you to specify a desired Examples Applying the O DE Solvers directionality positive 1 negative 1 or don t care 0 for each val ue element ball ode also responds to the flag i nit Seetheri gi dode example for details function varargout ballode t y flag BALLODE Equations of motion for a bouncing ball switch flag case Return dy dt f t y varargout 1 f t y case init Return default tspan y0 options varargout 1 3 init case events Return value isterminal direction varargout 1l 3 events t y otherwise error Unknown flag flag end function dydt f t y dydt y 2 9 8 function tspan y0 options init tspan 0 10 y0 0 20 s odeset Events on function value isterminal direction events t y Locate the time
224. erval of 0 3000 For scaling purposes plot just the first component of y t t y odel5s vdp1000 0 3000 2 0 plot t y 1 0 title Solution of van der Pol Equation mu 1000 xl abel time t ylabel solution y 1 Representing Problems solution y 1 Solution of van der Pol Equation mu 1000 2 5 T T T 0 O ie D eo O 0 5F O00000 O00 0 O O COCR gmm000 0 0 0 0000000 88 O O OO 00000 ES i L L L 0 500 1000 1500 2000 time t L 2500 3000 8 9 8 O rdinary Differential Equations 8 10 ODE Solvers The MATLAB ODE solver functions implement numerical integration methods Beginning at the initial time and with initial conditions they step through the time interval computing a solution at each time step If the solution for a time step satisfies the solver s error tolerance criteria itisa successful step Otherwise it is a failed attempt the solver shrinks the step size and tries again This section describes how to represent problems for use with the MATLAB solvers and how to optimize solver performance You can also use the online help facility to get information on the syntax for any function as well as information on demo files for these solvers Nonstiff Solvers There are three solvers designed for nonstiff problems e ode45 is based on an explicit Runge Kutta 4 5 formula the Dormand Prince pair It is a one step solver in compu
225. es 0 000 ee 10 49 Errors and Warnings 0000 cee eee eee 10 51 Times and Dates 0000 ee 10 54 Obtaining User Input 0 0 000000 e eee 10 61 Shell Escape Functions 0000000 eeu 10 62 Optimizing the Performance of MATLAB Code 10 63 Character Arrays Strings 1 Character Arrays 0006 ee 11 4 Cell Arrays of Strings 0 0 0 c eee eas 11 7 String Comparisons 0 00000 cece eee 11 9 Searching and Replacing 0 005 11 12 String Numeric Conversion 2 0 005 11 13 2 Multidimensional Arrays auauua 12 3 vi Contents Computation with Multidimensional Arrays 12 15 Organizing Data in Multidimensional Arrays 12 17 Multidimensional Cell Arrays 12 19 Multidimensional Structure Arrays 12 20 Structures and Cell Arrays B Structures x2 set pied A T aA A ET ai Ana acts ats 13 3 Cell Arrays i becouse bo etait dd Meee eee eset 13 19 MATLAB Classes and Objects 4 Classes and Objects An Overview 5 14 2 Designing User Classesin MATLAB 14 9 Overloading Operators and Functions 14 20 Example A Polynomial Class 14 23 Building on Other Classes 000 0c eae 14 34 Example Assets and
226. es 9 34 for systems of equations 9 33 J J acobian matrix ODE 8 17 8 25 constant J acobian 8 27 evaluated analytically 8 27 sparsity pattern 8 27 vectorized computation 8 28 J avaand MATLAB OOP 14 7 joining matrices 10 41 J ordan Canonical Form 4 36 K keyboard statements 3 2 keys 2 59 arrow 2 6 control 2 6 keyword search 2 22 kinds of M files 10 3 kron 411 Kronecker tensor product 4 11 L lasterr 10 51 and error handling 10 51 least squares 9 31 less than operator 10 23 less than or equal to operator 10 23 library mathematical function 1 5 license manager 2 35 2 59 license dat 2 59 line breaks 2 31 line number 2 39 linear algebra and matrices 4 2 linear interpolation 5 10 multidimensional 5 16 linear systems of equations Index direct methods 9 33 iterative methods 9 33 sparse 9 33 linear transformation 4 4 linear in the parameters regression 6 17 LM LICENSE FILE 2 60 load 2 11 2 27 9 10 10 66 loading ASCII files 2 13 2 27 data 2 26 MAT files 2 12 objects 14 61 using wildcards 2 13 oadobj example 14 62 local variables 10 16 log file 2 3 0g10 6 25 logarithm analysis with a second order model 6 26 ogfile startup option 2 3 logical expressions 10 24 and subscripting 10 27 logical functions 10 26 logical operators 10 24 rules for evaluation 10 24 long 15 8 long integer 15 8 lookfor 2 22 10 3 10 7 and H1 line 10 3 loops for 10 35 while 10 34 u 9 28 9 29 LU factorization 4 25 for
227. es in the array weather 1 andweather 2 are initialized to the empty matrix All structures in theweather array are initialized using one set of field values The structures in theweat her array are initialized with distinct field values specified with cell arrays Accessing Data in Structure Arrays Using structure array indexing you can access the value of any field or field element in a structure array Likewise you can assign a value to any field or field element For the examples in this section consider this structure array patient array patient 1 patient 2 patient 3 John Doe name Ann Lane name AL Smith Name billing 127 00 atest J 79 15 180 178 220 210 13 177 5 205 atest L billing 28 50 billing 504 70 118 172 70 68 matest 80 80 80 118 119 153 153 154 170 169 181 190 182 Structures You can access subarrays by appending standard subscripts to a structure array name For example the line below results in a 1 by 2 structure array mypatients patient 1 2 1x2 struct array with fields name billing test The first structure in themypatients array is the same as the first structure inthepati ent array mypatients 1 ans name John Doe billing 127 test 3x3 double Toaccess afield of a particular structure includea period after the structure name followed by
228. es the high speed radix 2 algorithm This results in the fastest execution time Additionally the algorithm for power of twon is highly optimized for real x providing a 40 increase in speed over the complex case e For any composite number n that is not a power of two f f t uses a prime factor algorithm The speed of this algorithm depends on both the size of n and the number of prime factors it has Although 1013 and 1000 are close in Fourier Analysis and the Fast Fourier Transform FFT magnitude f f t transforms a sequence of length 1000 much more quickly than a sequence of length 1013 e For aprimenumber n f ft cannot usean FFT algorithm It instead performs the slower computation intensive DFT directly The inverse FFT funcion i f f t also accepts a transform length argument For practical application of the FFT the Signal Processing Toolbox includes numerous functions for spectral analysis 6 39 6 Data Analysis and Statistics 6 40 Function Functions Representing Functionsin MATLAB Plotting Mathematical Functions Minimizing Functions and Finding Zeros Minimizing Functions of One Variable F Minimizing Functions of Several Variables Setting Minimization Options Finding Zeros of Functions TIPS a el aS ee Troubleshooting Numerical Integration Quadrature Example Computing the Length of a Curve Example Double Integration ee 7 4 7 7 7 7 7 8 7 9 7 10 7 12 7 13 7
229. esl pop popl popl polyval pl sdate figure plot cdate resl plot cdate popl cdate pop 250 f 3 r 50 soo Linear fit appears unsatisfactory s wp Residuals of linear fit show note negative population strongly patterned 3 vol Values at lower end of scale 20 Es 100 10 4 50 D ii 10 E F 0 7 7 e ee Be S50 T800 1850 7300 1250 2000 50 7800 7850 7900 7950 2000 p polyfit sdate pop 2 res2 pop pop2 pop2 polyval p sdate figure plot cdate res2 plot cdate pop2 cdate pop t 250 soo Quadratic polynomial pro vides better fit to data 150 100 50 Residuals still appear strongly el patterned i i i L 1800 1850 1900 1950 2000 o f i i 8 1750 1800 1850 1900 1950 2000 1750 6 23 6 Data Analysis and Statistics 6 24 Fit Residuals p polyfit sdate pop 4 res4 pop pop4 pop4 polyval p sdate figure plot cdate res4 plot cdate pop4 cdate pop 300 i T T r 6 Fourth order model provides little 2507 improvement note that curve ao still begins to turn upward at al 5 20 F lower end of plot ot Ree i al 150 7 ne 2b 2 igol Residuals still appear strongly pat ai terned SOP 6 ey Q t i i i 3 i i i 7 1750 1800 1850 1900 1950 2000 1750 1800 1850 1900 1950 2000 It s evident from studying the fit plots and residuals that it should b
230. esses the third value in the column the number 3 A 7 accesses the seventh value 9 and soon If you supply more subscripts MATLAB calculates an index into the storage column based on the dimensions you assigned to the array For example assume a two dimensional array likeA has size d1 d2 whered1 isthe number of rows in the array andd2 is the number of columns If you supply two subscripts i j representing row column indices the offset is j 1 dl4i Given the expression A 3 2 MATLAB calculates the offset into A s storage column as 2 1 3 3 or 6 Counting down six elements in the column accesses the value 0 This storage and indexing scheme also extends to multidimensional arrays In this case MATLAB operates on a page by page basis to create the storage column again appending elements columnwise For example consider a 5 by 4 by 3 by 2 array C 10 43 10 M File Programming MATLAB stores C as MATLAB displays C as Caw cd eraenownwonswencorarove ere NNW OD page 1 1 ll a D D a page 2 1 6 7 0 9 1 page 3 1 aowavanwvoo vr od NARKO page 2 2 page 3 2 10 44 Indexing and Subscripting Again a single subscript indexes directly into this column For example C 4 produces the result ans 0 If you specify two subscripts i indicating row column indices MATLAB calculates the offset as described above Two subscripts always access the
231. essions you can create using the Customize command and several substitution variables Observe the differing results when you invoke the expression in the Editor Path Browser or Array Editor Ex pression Result DOS copy Y Pathname Backup file Pathname UNIX bin cp Pathname Pathname mex Pathname which File doc Sel edit Sel disp Selection BeginSel through EndSel File Sel File Sel 2 Runs Mex script on a C file Display pathname of function Call Help Desk for selected function Edit selected function Indicates in Command Window beginning and end of selected E ditor text When invoked from Array Editor doubles the values of selected array elements 2 45 2 MATLAB W orking Environment 2 46 Here is a longer example demonstrating how to create a custom tool using MATLAB to script the Editor First create an M file calledcensor m function censor pathname selstart sel end Read the specified file into an array fp fopen pathname rt txt fread fp fseek fp 0 1 txt char txt Insert your code here For example the next line replaces the selection with X s txt selstart selend X Write the modified array back to the file fwrite fp txt fclose fp Now use Customize from the Tools menu to create the MATLAB expression censor Pathname BeginSel EndSel Also cre
232. ex 1 26 values 9 11 nonzero elements of sparse matrix number of 9 11 operations 9 23 permutation 9 24 preconditioner 9 32 propagation through computations 9 23 QR factorization 9 30 reordering 9 3 9 24 storage 9 5 for various permutations 9 26 viewing 9 11 theoretical graph 9 15 triangular factorization 9 28 viewing contents graphically 9 13 viewing storage 9 11 visualizing 9 19 working with 9 2 sparse ODE example 8 37 spconvert 9 10 spdiags 9 8 special values 10 18 speye 9 23 9 26 9 29 splash screen at startup 2 3 spones 9 26 spparms 9 27 9 34 sprand 9 23 spreadsheet files writing 2 29 sprintf 15 2 15 17 spy 9 13 spy plot 9 19 sqrtm4 31 square brackets for output arguments 10 7 squeeze 12 2 12 12 12 16 sscanf 15 2 15 15 stability ODE solvers 8 32 stack Debugger 3 2 3 8 starting MATLAB 2 2 from DOS window 2 4 startup files 2 2 startup options 2 3 for UNIX 2 4 startup m2 2 statements conditional 10 12 on multiple lines 2 10 statistics analyzing residuals 6 22 correlation coefficients 6 9 covariance 6 9 descriptive 6 7 pre processing data 6 21 status bar in command window 2 36 step in Debugger 3 5 through code using commands 3 12 step size ODE 8 17 8 28 first step 8 29 upper bound 8 28 stiff ODE example 8 37 stiffness ODE defined 8 8 stopping a running program 2 7 stopping in Debugger 3 6 storage array 10 42 data type 10 20 for various permutations of sparse matrix 9 26 of spars
233. except the one directly opposite it on the sphere in eight steps An Airflow Model A calculation performed at NASA s Research Institute for Applications of Computer Science involves modeling the flow over an airplane wing with two trailing flaps o s Ava TA a leer wae J 07 ae enn eee gt Y aa o 5 2 cee 06S a Renee EES ROO 0 5 0 4 q 4 0 3 0 2 0 1 In a two dimensional model a triangular grid surrounds a cross section of the wing and flaps The partial differential equations are nonlinear and involve several unknowns including hydrodynamic pressure and two components of velocity Each step of the nonlinear iteration requires the solution of a sparse linear system of equations Since both the connectivity and the geometric location of the grid points are known thegp ot function can producethegraph shown above 9 21 9 Sparse Matrices In this example there are 4253 grid points each of which is connected to between 3 and 9 others for a total of 28831 nonzeros in the matrix anda density equal to 0 0016 This spy plot shows that the node numbering yields a definite band structure The Laplacian of the mesh 0 RP i T T 500 1000 1500 2000 2500 3000 3500 4000 0 1000 2000 3000 4000 nz 28831 9 22 Sparse M atrix O perations Sparse Matrix Operations Most of MATLAB s standard mathematical functions work on sparse matrices
234. expression on theright side of the assignment in curly braces This indicates that you are assigning cell contents not the cells themselves Consider the 2 by 2 cell array N N 1 1 1 2 4 5 N 1 2 Name N 2 1 2 4i N 2 2 7 13 23 13 Structures and Cell Arrays 13 24 You can obtain the string in N 1 2 using c N 1 2 Na me Note In assignments you can use content indexing to access only a single cell not a subset of cells For example the statements A 1 value and B A 1 are both invalid However you can use a subset of cells any place you would normally use a comma separated list of variables for example as function inputs or when building an array See Replacing Lists of Variables with Cell Arrays on page 13 25 for details To obtain subsets of a cell s contents concatenate indexing expressions F or example to obtain element 2 2 of the array in cell N 1 1 use d N 1 1 2 2 d Accessing a Subset of Cells Using Cell Indexing Use cell indexing to assign any set of cells to another variable creating a new cell array Usethe colon operator to access subsets of cells within a cell array cell 1 1 cell 1 2 cell 1 3 3 5 9 cell 1 1 cell 1 2 6 0 cell 2 1 cell 2 2 cell 2 3 cell 2 1 cell 2 2 7 2 cell 3 1 cell 3 2 cell 3 3 4 Fi 2 Cell Arrays Deleting Cells You can delete an entire dim
235. f Each Value An optional third argument tof read controls the data type of the input The data type argument controls both the number of bits read for each value and the interpretation of those bits as character integer or floating point values MATLAB supports a wide range of precisions which you can specify with MATLAB specific strings or their C or Fortran equivalents Some common precisions include e char and uchar for signed and unsigned characters usually 8 bits e short and ong for short and long integers usually 16 and 32 bits respectively e float and double for single and double precision floating point values usually 32 and 64 bits respectively Note The meaning of a given precision can vary across different hardware platforms For example a uchar is not always 8 bits fread also provides a number of more specific precisions such as i nt 8 and fl oat 32 1fin doubt use these precisions which are not platform dependent Look upfread in online help for a complete list of precisions For example if f i d refers to an open file containing single precision floating point values then the following command reads the next 10 floating point values into a column vector A A fread fid 10 float Writing Binary Files Thef write function writes the elements of a matrix toa file in a specified numeric precision returning the number of values written For instance these lines cre
236. first page of a multidimensional array provided they are within the range of the original array dimensions If more than one subscript is present all subscripts must conform to the original array dimensions For example C 6 2 is invalid because all pages of C have only five rows If you specify more than two subscripts MATLAB extends its indexing scheme accordingly For example consider four subscripts i j k intoa four dimensional array with size d1 d2 d3 d4 MATLAB calculates the offset into the storage column by 1 1 d3 d2 d1 k 1 d2 d1 j 1 01 1 For example if you index the array C using subscripts 3 4 2 1 MATLAB returns the value 5 index 38 in the storage column In general the offset formula for an array with dimensions d d2 d3 dpl using any subscripts s4 5S2 53 Spn is Sn 1 dn 1 dn 2 d1 HSn 1 1 dn 2 d1 HS2 1 d1 51 Because of this scheme you can index an array using any number of subscripts You can append any number of 1s to the subscript list because these terms become zero F or example C 3 25 14 a S is equivalent to C 3 2 10 45 10 M File Programming String Evaluation 10 46 String evaluation adds power and flexibility tothe MATLAB language letting you perform operations like executing user supplied strings and constructing executable strings through concatenation of strings stored in variables eval Theeval function evaluates a string
237. form the stiff solver that the ODE file is coded so that F t yl y2 returns CRC yh E EYZ esx 8 26 Improving Solver Performance J Constant Set Constant on iftheJ acobian matrix dF dy is constant does not depend on t ory Whether computing the acobians numerically or evaluating them analytically the solver takes advantage of this information to reduce solution time For the stiff van der Pol example the J acobian matrix is J f 0 1 2000 y 1 y 2 1 1000 1 y 1 2 not constant sothe Const ant property does not apply Jacobian Set acobi an on toinform the solver that the ODE file is coded such that F t y J acobian returns oF oy By default acobi an isoff and J acobians are generated numerically Coding the ODE file to evaluate the J acobian analytically often increases the speed and reliability of the solution for the stiff problem The acobian shown above for the stiff van der Pol problem can be coded into the ODE file as function outl vdp1l000 t y flag if stremp flag return dy outl y 2 1000 1 y 1 2 y 2 y 1 elseif strcmp flag j acobian return J outl 0 1 2000 y 1 y 2 1 1000 1 y 1 2 end J Pattern Set Pattern on if dF dy is a sparse matrix and the ODE file is coded so that F Pattern returns a sparsity pattern matrix This is a sparse matrix with 1s where there are nonzero entries in the J acobian numj ac uses the sparsity p
238. forth in subparagraph a of the Rights in Commercial Computer Software or Commercial Software Documentation Clause at DFARS 227 7202 3 therefore the rights set forth herein shall apply and b For any other unit or agency NOTICE Notwithstanding any other lease or license agreement that may pertain to or accompany the delivery of the computer software and accompanying documentation the rights of the Government regarding its use reproduction and disclo sure are as set forth in Clause 52 227 19 c 2 of the FAR MATLAB Simulink Stateflow Handle Graphics and Real Time Workshop are registered trademarks and Target Language Compiler is a trademark of The MathWorks Inc COPYRIGHT 1995 Bristol Technology Inc All rights reserved COPYRIGHT 1995 Microsoft Corporation All rights reserved Other product or brand names are trademarks or registered trademarks of their respective holders Printing History December 1996 First printing for MATLAB 5 0 J une 1997 Revised for MATLAB 5 1 J anuary 1998 Revised for MATLAB 5 2 January 1999 Revised for MATLAB 5 3 Release 11 Introduction 1 MATLAB Working Environment 2 Using the Environment 000000 ea ee 2 2 The Command Window 000 0 eee aee 2 5 The Figure Window 0 00 cee eee eee 2 18 Help and Online Documentation 2 20 Disk File Manipulation and Shell Escape 2 25 Data Import Export
239. fread 2 27 15 2 15 7 free 2 33 frewind 15 2 15 10 fscanf 15 2 15 13 fseek 15 2 15 10 ftell 15 2 15 10 full 9 24 9 27 function 10 6 appl ying to multidimensional structure arrays 12 21 to structure contents 13 9 body 10 3 10 8 calling context 10 11 characteristics 10 3 clearing from memory 10 11 contents 10 6 defined 10 3 dispatching priority 10 10 example 10 6 executing function name string 10 46 logical 10 26 minimizing 7 7 multiple output values 10 7 name 10 9 name resolution 10 10 optimization 10 63 order of arguments 10 13 10 15 passing variable number of arguments 10 14 primary 10 38 private 10 39 storing as pseudocode 10 11 to create arrays 12 6 what happens when called 10 10 workspace 10 11 function call history 3 24 function definition line 10 7 defined 10 3 for subfunction 10 38 function details report 3 23 function functions 7 1 function reference pages 2 23 1 11 Index 1 12 functions applying to cell arrays 13 27 class 14 11 computational applying to structure fields 13 9 converters 14 25 inferiorto 14 66 isa 14 11 nested 10 67 optimization 7 7 overloading 14 20 14 22 14 31 subassign 14 17 subsref 14 14 superiorto 14 66 which 14 71 fwrite 2 28 15 2 15 8 fzero 7 10 G Gaussian elimination 4 24 4 25 general preferences 2 38 geodesic dome 9 16 get method 14 13 getfield 13 8 global variables 10 16 rules for use 10 17 gplot 9 15 graph characteristics 9 20 defined 9 15
240. ft division Matrix power and a matrix A magi c 3 A 8 1 3 5 4 9 3 A ans 24 3 9 15 12 27 18 21 O perators Relational Operators MATLAB provides these relational operators lt Less than lt Less than or equal to gt Greater than gt Greater than or equal to E qual to Not equal to Relational O perators and Arrays MATLAB s relational operators compare corresponding elements of arrays with equal dimensions Relational operators always operate element by element In this example the resulting matrix shows where an element of A is equal to the corresponding element of B AvtS 22 SPO 9 085 3 05 5 6 B 8 7 0 3 2 5 4 1 7 A B ans 0 1 0 0 0 1 0 0 0 For vectors and rectangular arrays both operands must be the same size unless oneis a scalar F or the case where one operand is a scalar and the other is not MATLAB tests the scalar against every element of the other operand Locations where the specified relation is true receive the value1 Locations where the relation is false receive the value 0 10 23 10 M File Programming 10 24 Relational Operators and Empty Arrays The relational operators work with arrays for which any dimension has size zero as long as both arrays are the same size or one is a scalar However expressions such as A return an error if A is not 0 by 0 or 1 by 1 To test for empty arrays use the function isempt y A Lo
241. g the Environment The filest artup mis for you to use You can set default paths define Handle Graphics defaults or predefine variables in your workspace For example creating astartup m with the line addpath home me mytools adds a tools directory to your default search path ss On Windows platforms placethest artup m filein the folder named Be local inthetool box folder xX On UNIX workstations placethest art up mfilein the directory named matlab off of your home directory eg mat ab Startup O ptions You can specify startup options for MATLAB aq Add these options to the target path for your Windows shortcut for i a MATLAB If you run MATLAB from a DOS window include these options with the startup command Startup Option Description automation Start MATLAB as an automation server minimized and without the MATLAB splash screen For more information see Chapter 7 of the Application Program Interface Guide logfile logfilename Automatically write output from MATLAB tothe specified log file mi ni mi ze Start MATLAB minimized and without the MATLAB splash screen nosplash Start MATLAB without displaying the MATLAB splash screen r M file Automatically run the specified M file immediately after MATLAB starts 2 MATLAB W orking Environment 2 4 regserver Modify the Windows registry with the appropriate Activex entries for MATLAB For more information see Chapter 7 of the Applicati
242. gger while you are running an M filein the Command Window or you will get an error When you run the Editor Debugger without MATLAB open it becomes a pure text editor you cannot use it as a debugger To use the debugger launch the Editor Debugger from within MATLAB or with MATLAB open 2 39 2 MATLAB W orking Environment Menus Provide Access to Operations Toolbar Some Menu Items are Also Available from the Toolbar MATLAB Editor Debugger matlabrc m D nightiy toolbox local matlabrc Ea al File Edit View Debug Tools Window Help la x Cah 6 AA G3 oe 2 Stacks led Automatic Color ies Set up path A al he if exist pathdef file mat labpath pathdef Automatic Indenting ii Toolbar al matlabre m Ready Lne16 358 PM Multiple Files Open i Status Bar Hold the Cursor on a Menu Item or Toolbar Button a Current Line Number the Editor Description of it then Appears in the Status Bar New Save Copy Print Set Clear Step In Continue Function List of Functions on the Call File to Disk Breakpoint Stack Editor Debugger H A S amp 8 BG LH B Stack finishcont zl PT E E Open Cut Paste Help Clear All Single Quit File Breakpoints Step Debugging Tite Bar The name of the file you are editing appears in the title bar If there is an asterisk appearing after the name in thetitle bar it indicates that yo
243. gical Operators MATLAB provides these logical operators 6 AND OR NOT Note In addition to these logical operators theops directory contains a number of functions that perform bitwise logical operations See online help for more information Each logical operator has a specific set of rules that determines the result of a logical expression e An expression usingtheAND operator amp is trueif both operands are logically true In numeric terms the expression is true if both operands are nonzero O perators This example shows the logical AND of the elements in the vector u with the corresponding elements in the vector v 1 02 3 0 5 5 6100 7 V D ngri a i IS 1 0 1 0 0 1 An expression using the0R operator is trueif oneoperand is logically true or if both operands are logically true An OR expression is false only if both operands are false In numeric terms the expression is false only if both operands are zero This example shows the logical OR of the elements in the vector u and with the corresponding elements in the vector v u v 1 1 1 1 0 1 An expression using theNO0T operator negates the operand This produces a false result if the operand is true and true if it is false In numeric terms any nonzero operand becomes zero and any zero operand becomes one This example shows the negation of the elements in the vector u Logical Operators and Arrays MATLAB s logical operators com
244. grid is X1 X2 X3 ndgrid x1 x2 x3 For example assume that you want to evaluate a function of three variables over a given range Consider the function X7 X3 X3 Z X for 2r lt x lt 0 27 lt Xp lt 4r and 0 lt X3 lt 2r To evaluate and plot this function XI o 220 2923 x2 2 0 25 2 x3 2 0 16 2 X1 X2 X3 ndgrid x1 x2 x3 z X2 exp X1 2 X2 2 X3 2 slice X2 X1 X3 z 1 2 8 2 2 2 0 2 5 17 5 Polynomials and Interpolation 5 18 Triangulation and Interpolation of Scattered Data MATLAB provides routines that aid in the analysis of closest point problems and geometric analysis Function Description convhul Convex hull del aunay Delaunay triangulation dsearch Search Delaunay triangulation for nearest point inpolygon True for points inside polygonal region polyarea Area of polygon rectint Area of intersection for two or more rectangles tsearch Closest triangle search voronol Voronoi diagram Interpolation Delaunay Triangulation Thedel aunay function returns a set of triangles such that no data points are contained in any triangle s circumcircle Totry del aunay load theseamount data set supplied with MATLAB and view the data as a simple scatter plot load seamount plot x y markersize 12 xlabel Longitude ylabel Latitude grid on 47 95 48 A A 4 48 05 j See ar 3 KABA
245. h the same name MATLAB stops searching when it finds the first implementation of the method on the path regardless of the implementation type ME X file P code M file Selection from Multiple Implementation Types There are four file precedence types MATLAB uses file precedence to select between identically named functions in the same directory The order of precedence for file types is 1 MEX files 2 MDL file Simulink model How MATLAB Determines W hich M ethod to Call 3 P code 4 M file For example if MATLAB finds a P code and an M file version of a method ina class directory then the P code version is used It is therefore important to regenerate the P code version whenever you edit the M file Querying Which Method MATLAB Will Call Y ou can determine which method MATLAB will call using thewhi ch command For example which pie3 your_matlab_path tool box matl ab specgraph pie3 m However ifp is a portfolio object which pie3 p dir_on_your_path portfolio pie3 m portfolio method Thewhi ch command determines which version of pi e3 MATLAB will call if you passed a portfolio object as the input argument To see a list of all versions of a particular function that are on your MATLAB path use the al option See thewhi ch reference page for more information on this command 14 71 14 MATLAB Classes and O bjects 14 72 Filel O Opening and Closing Files Using the File Identifier fid Closing
246. h argument list of property name property value pairs and returns the modified object Subclasss et methods call the asset set method and require the capability to return the modified object 14 41 14 MATLAB Classes and O bjects since MATLAB does not support passing arguments by reference See The Stock set Method on page 14 50 for an example function a set a varargin SET Set asset properties and return the updated object property_argin varargin while length property _argin gt 2 prop property argin 1 val property _argin 2 property_argin property_argin 3 end switch prop case Descriptor a descriptor val case Date a date val case Current Val ue a current_ value val otherwise error Asset properties Descriptor Date CurrentVal ue end end The Asset subsasgn Method Thesubsasgn method is the assignment equivalent of thes ubsref method This version enables you to change the data contained in an object using one based numeric indexing and structure field name indexing The outer switch statement determines if the index is a numeric or field name syntax Theinner switch statements map the index value to the appropriate value in the stock structure MATLAB callssubsas gn whenever you execute an assignment statement e g A i val A i val orA fieldname val Seethesubsasgn help entry for more information on assignment statements in MATLAB Example Assets
247. h specialized and general mathematical operations The sparse matrix functions are located in thes par fun directory in the MATLAB toolbox directory Category Function Description Elementary sparse speye Sparse identity matrix matrices sprand Sparse uniformly distributed random matrix sprandn Sparse normally distributed random matrix sprandsym Sparse random symmetric matrix spdiags Sparse matrix formed from diagonals Full to sparse sparse Create sparse matrix conversion full Convert sparse matrix to full matrix find Find indices of nonzero elements spconvert Import from sparse matrix external format Working with nnz Number of nonzero matrix elements Sparse matrices nonzeros Nonzero matrix elements nzmax Amount of storage allocated for nonzero matrix elements Spones Replace nonzero sparse matrix elements with ones spall oc issparse Allocate space for sparse matrix True for sparse matrix Category Function Description spfun Apply function to nonzero matrix elements spy Visualize sparsity pattern gplot Plot graph as in graph theory Reordering col mmd Column minimum degree permutation algorithms s y mmmd Symmetric minimum degree permutation symrcm Symmetric reverse Cuthill M cK ee permutation col perm Column permutation randperm Random permutation dmperm Dulmage M endelsohn permutation Linear algebra ei gs A few eigenvalues svds A few singular values luinc Incomplete LU factorizati
248. haracteristic roots 5 3 double precision 10 19 elements 4 4 exponentials 4 31 iterative methods 9 34 multiplication 4 8 power operator 10 22 powers 4 31 See also matrices max 9 24 mean 12 15 measuring performance of M files 3 17 memory and function workspace 10 11 cache 2 15 management 10 66 Out of Memory message 10 70 reducing fragmentation 10 65 use of variables 10 67 utilization 2 33 menus 2 37 meshgrid 5 12 5 15 5 21 methods 14 2 converters 14 19 determining which is called 14 71 display 14 12 end 14 18 get 14 13 invoking on objects 14 4 listing 14 32 precedence 14 68 required by MATLAB 14 9 set 14 13 subsasgn 14 14 subsref 14 14 M files 2 31 comments 10 9 Index contents 10 3 corresponding to functions 14 21 creating quick start 10 2 creating with text editor 10 4 debugging 3 2 dispatching priority 10 10 editing 2 39 for entering data 2 26 for ODE solvers ODE file 8 6 impact of clearing on breakpoints 3 3 kinds 10 3 naming 10 2 obtaining input interactively 10 61 obtaining keyboard input 10 61 optimization 10 63 overview 10 3 pausing during execution 10 61 performance of 3 17 primary function 10 38 profiling 3 17 running at startup 2 3 search path 2 14 subfunction 10 38 superseding existing names 10 39 to operate on structures 13 10 to represent mathematical functions 7 3 Microsoft Windows and MATLAB use of system resources 10 68 environment for MATLAB 2 35 min 6 9
249. he class directories are subdirectories of directories on the MATLAB search path but are not themselves on the path For instance t he new pol ynom directory could be a subdirectory of MATLAB s working directory or your own personal directory that has been added to the search path Classes and O bjects An O verview Adding the Class Directory to the MATLAB Path After creating the dass directory you need toupdatethe MATLAB path sothat MATLAB can locate the class source files The class directory should not be directly on the MATLAB path Instead you should add the parent directory to the MATLAB path For example if the po ynom class directory is located at c my_classes pol ynom you add the class directory to the MATLAB path with theaddpat h command addpath c my_classes If you create a class directory with the same name as another class MATLAB treats the two class directories as a single directory when locating class methods F or more information see How MATLAB Determines Which Method to Call on page 14 68 Data Structure One of the first steps in the design of a new class is the choice of the data structure to be used by the class Objects are stored in MATLAB structures The fields of the structure and the details of operations on the fields are visible only within the methods for the class The design of the appropriate data structure can affect the performance of the code Tips for C and Java Programmer
250. here are two edit fields in the Customize Tools Menu Menu Text Supply the name for the menu item that you want to add tothe Tools menu To create a mnemonic for the menu command precede a character with amp Use this mnemonic by typing Alt followed by t thet indicates the Tools menu followed by the mnemonic character Note The Alt key need not be held down while pressing the keys that follow F or instance in the example above you can execute the Run command by choosing Run on the Tools menu or by typing Alt t and r in sequence MATLAB Expression In this box type the expression you want MATLAB to evaluate when you select the custom tool from the Tools menu Any valid MATLAB expression is allowed You can also include any of six special substitution variables The variables may be typed directly or inserted via the submenu as shown above When your custom tool is chosen information about the current file is substituted into the expression replacing the substitution variables and the expression is evaluated in MATLAB Different information is Microsoft W indows Handbook substituted for the substitution variables depending upon whether the expression is executed by the Editor Path Browser or Array Editor The table summarizes the variables and the substitutions made for them within the various environment tools Submenu Variable Invoked Invoked Invoked Item Inserted from Editor from Path from Array Brow ser Edito
251. hogonal matrices U and V we have AV Ur ATU Vr Since U and V are orthogonal this becomes the singular value decomposition Aas The full singular value decomposition of an m by n matrix involves an m by m U an m by n x and an n by n V In other words U and V are both square and Lis thesamesizeas A If A has many more rows than columns the resulting U can be quite large but most of its columns are multiplied by zeros in In this situation the economy sized decomposition saves both time and storage by producing an m by n U an n by n and the same V The eigenvalue decomposition is the appropriate tool for analyzing a matrix when it represents a mapping from a vector space into itself as it does for an ordinary differential equation On the other hand the singular value decomposition is the appropriate tool for analyzing a mapping from one vector Space into another vector space possibly with a different dimension Most systems of simultaneous linear equations fall into this second category If A is square symmetric and positive definite then its eigenvalue and singular value decompositions arethesame But as A departs from symmetry and positive definiteness the difference between the two decompositions increases In particular the singular value decomposition of a real matrix is always real but the eigenvalue decomposition of a real nonsymmetric matrix might be complex Singular Value Decomposition For the exampl
252. hree standard deviations is obtained with n p size count outliers abs count mu ones n 1 gt 3 sigma ones n 1 nout sum outliers nout 1 0 0 There is one outlier in the first column Remove this entire observation with count any outliers 6 14 Regression and Curve Fitting Regression and Curve Fitting It is often useful to find functions that describe the relationship between some variables you have observed Identification of the coefficients of the function often leads to the formulation of an overdetermined system of simultaneous linear equations These coefficients can be efficiently found using the MATLAB backslash operator Suppose you measure a quantity y at several values of timet t 0 3 8 1 1 1 6 2 3 y 0 5 0 82 1 14 1 25 1 35 1 40 plot t y 0o grid on 0 97 0 87 0 77 0 67 0 5 Polynomial Regression Based on the plot it is possible that the data may be modeled by a polynomial function y a at at The unknown coefficients ag a4 and a can be computed by doing a least squares fit which minimizes the sum of the squares of the deviations of the data from the model There are six equations in three unknowns 6 15 6 Data Analysis and Statistics es Sl og 1t ti y 2 gt LEE y 1t t2 BL agaa 3x ay Y4 1tyty la 5 tt Y6 2 1 te tg represented by the 6 by 3 matrix X ones size t t
253. i nd contains theintegers from 1 through the row sizeof A Theni nd variable contains the integers from 1 through the column size of A 3 above uses a MATLAB vectorization trick to replicate a single column of data through any number of columns The code is B A ones 1 n_cols wheren_cols isthe desired number of columns in the resulting matrix 10 64 O ptimizing the Performance of MATLAB Code The final line of repmat uses array indexing to create the output array Each element of the row index array mi nd is paired with each element of the column index array ni nd 1 Thefirst element of mi nd the row index is paired with each element of ni nd MATLAB moves through theni nd matrix in a columnwise fashion so mi nd 1 1 goes with ni nd 1 1 then ni nd 2 1 and soon The result fills the first row of the output array 2 Moving columnwisethrough mi nd each element is paired with the elements of ni nd as above Each complete pass through theni nd matrix fills one row of the output array Array Preallocation You can often improve code execution time by preallocating the arrays that store output results Preallocation prevents MATLAB from having to resize an array each time you enlarge it Use the appropriate preallocation function for the kind of array you are working with Array Type Function Examples Numeric Zeros y zeros 1 100 array Cell array cel B cell 2 3 B 1 3 1 3 B 2 2 string
254. iables may be used within a function only Persistent variables remain in memory until the M file is cleared or changed persistent is exactly likeg obal except that the variable name is not in the global workspace and the value is reset if the M file is changed or cleared Three MATLAB functions support the use of persistent variables ml ock Prevents an M file from being cleared mun ock Unlocks an M file that had previously been locked by ml ock mi sl ocked Indicates whether an M file can be cleared or not 10 17 10 M File Programming 10 18 Special Values Several functions return important special values that you can usein your M files ans eps real max real min computer flops version Most recent answer variable If you do not assign an output variable to an expression MATLAB automatically stores the result in ans Floating point relative accuracy This is the tolerance MATLAB uses in its calculations Largest floating point number your computer can represent Smallest floating point number your computer can represent 3 1415926535897 Imaginary unit Infinity Calculations liken 0 wheren is any nonzero real value result ini nf Not a Number an invalid numeric value Expressions like 0 0 andi nf inf result in aNaN as do arithmetic operations involving aNaN n 0 wheren is complex also returns NaN Computer type Count of floating point operations MATLAB version string Al
255. ic vector That is di ff X is X 2 X 1 X 3 X 2 X n X n 1 So for a vector A A 9 2 3015 4 dif f A ans 11 5 3 1 4 zal Besides computing the first difference di f f is useful for determining certain characteristics of vectors For example you can use di f f to determine if a vector is monotonic elements are always either increasing or decreasing or if a vector has equally spaced elements This table describes a few different ways to usediff with a vector x diff x Tests for repeated elements all diff x gt 0 Tests for monotonicity all diff diff x 0 Tests for equally spaced vector elements 6 11 6 Data Analysis and Statistics 6 12 Data Pre Processing Missing Values The special value NaN stands for Not a Number in MATLAB IEEE floating point arithmetic convention specifies NaN as the result of undefined expressions such as 0 0 The correct handling of missing data is a difficult problem and often varies in different situations For data analysis purposes it is often convenient to use Na Ns to represent missing values or data that are not available MATLAB treats Na Ns in a uniform and rigorous way they propagate naturally through to the final result in any calculation Any mathematical calculation involving NaNs will produce NaNs in the results For example consider a matrix containing the 3 by 3 magic square with its center element set to NaN a magic 3 a 2 2 Na
256. ically displays the results on screen However if you end the line with a semicolon MATLAB performs the computation but does not display any output This is particularly useful when you generate large matrices F or example A magic 100 2 9 2 MATLAB W orking Environment Long Command Lines If a statement does not fit on one line use an ellipsis three periods followed by Return or Enter to indicate that the statement continues on the next line For example Sls Tide Dp soe La te Tbs TP OT ae 1 8 1 9 1 10 1 11 1 12 Blank spaces around the and signs are optional but they improve readability The maximum number of characters allowed on a single line is 4096 MATLAB Workspace The MATLAB workspace contains a set of variables named arrays that you can manipulate from the MATLAB command line You can usethewho and whos commands to see what is currently in the workspace The who command gives a short list while the whos command also gives size and data type information Hereis the output produced by whos on a workspace containing eight variables of different data types whos Name Size Bytes Class A 4x4 128 double array D 3x5 120 double array M 10x1 40 cell array S 1x3 628 struct array h 1x11 22 char array n 1x1 8 double array S 1x5 10 char array V 1x14 28 char array Grand total is 93 elements using 984 bytes To delete all existing variables from the workspace enter clear 2 10 Th
257. id Notice that this code does not use any loops Instead thef scanf function continues to read in text as long as the input format is compatible with the format specifier An optional size argument controls the number of matrix elements read F or example iff i d refers to an open file containing strings of integers then this line reads 100 integer values into the column vector A A fscanf fid 5d 100 This line reads 100 integer values into the 10 by 10 matrix A A fscanf fid 5d 10 10 A related function s scanf takes its input from a string instead of a file For example this line returns a column vector containing 2 and its square root root2 numdstr 2 sqrt 2 rootvalues sscanf root2 f Writing Text Files Thef printf function converts data to character strings and outputs them to thescreen or a file A format control string containing conversion specifiers and 15 15 13 File 1 O 15 16 any optional text specify the output format The conversion specifiers control the output of array elements f pr i ntf copies text directly Common conversion specifiers include e e for exponential notation e f for fixed point notation e g to automatically select the shorter of e and f Optional fields in the format specifier control the minimum field width and precision For example this code creates a text file containing a short table of the exponential function xX 0 0 1 1
258. identity involving sums of products of binomial coefficients The Cholesky factorization also applies to complex matrices Any complex matrix which has a Cholesky factorization satisfies A A and is said to be Hermitian positive definite The Cholesky factorization allows the linear system A x b to be replaced by R R x b Because the backslash operator recognizes triangular systems this can be solved quickly with x R R b If A is n by n the computational complexity of chol A is O n3 but the complexity of the subsequent backslash solutions is only O n2 LU Factorization Gaussian elimination or LU factorization expresses any square matrix as the product of a permutation of a lower triangular matrix and an upper triangular matrix A LU 4 25 4 Matrices and Linear Algebra 4 26 where L is a permutation of a lower triangular matrix with ones on its diagonal and U is an upper triangular matrix The permutations are necessary for both theoretical and computational reasons The matrix 0 1 1 0 cannot be expressed as the product of triangular matrices without interchanging its two rows Although the matrix 1 1 0 can be expressed as the product of triangular matrices when e is small the elements in the factors are large and magnify errors so even though the permutations are not strictly necessary they are desirable Partial pivoting ensures that the elements of L are bounded by one in magnitude
259. ield architectures Such functions can access structure fields and elements for processing Note When writing M file functions to operate on structures you must perform your own error checking That is you must ensure that the code checks for the expected fields Asan example consider a collection of data that describes measurements at different times of the levels of various toxins in a water source The data consists of fifteen separate observations where each observation contains three separate measurements You can organize this data into an array of 15 structures where each structure has three fields one for each of the three measurements taken Structures The function concen shown below operates on an array of structures with specific characteristics Its arguments must contain the fields ead mercury and chromium function rl r2 concen toxtest Create two vectors rl contains the ratio of mercury to lead at each observation r2 contains the ratio of lead to chromium rl toxtest mercury toxtest lead r2 toxtest lead toxtest chromium Plot the concentrations of lead mercury and chromium on the same plot using different colors for each lead toxtest lead mercury toxtest mercury chromium toxtest chromi um plot lead r hold on plot mercury b plot chromium y hold off Try this function with a sample structure array liket est test 1
260. ield is a non negative integer if only two digits are specified then a year 19yy is assumed if the year is omitted then the current year is used as a default e The hours minutes and seconds fields are optional They are integers separated by colons or followed by AM or PM Times and Dates For example if the current year is 1996 then these are all equivalent 17 May 1996 17 May 96 17 May May 17 1996 5 17 96 5 17 and both of these represent the same time 17 May 1996 18 30 5 17 96 6 30 pm Note that the default format for numbers only input follows the American convention Thus 3 6 is March 6 not J une 3 If you create a vector of input date strings use a column vector and be sure all strings are the same length Fill in with spaces or zeros Output Formats The function dat est r D dateform converts a serial date D to one of 19 different date string output formats showing date time or both The default output for dates is a day month year string 01 Mar 1996 You select an alternative output format by using the optional integer argument dat ef orm 10 57 10 M File Programming 10 58 dateform Format Description 0 01 Mar 1996 15 45 17 day month year hour minute second 1 01 Mar 1996 day month year 2 03 01 96 month day year 3 Mar month three letters 4 M month single letter 5 3 month 6 03 01 month day 7 1 day of month 8 Wed day of week three letters
261. ific and engineering graphics e Application development including graphical user interface building MATLAB is an interactive system whose basic data element is an array that does not require dimensioning This allows you to solve many technical computing problems especially those with matrix and vector formulations in a fraction of thetime it would taketowritea programin a scalar noninteractive language such as C or Fortran The name MATLAB stands for matrix laboratory MATLAB was originally written to provide easy access to matrix software developed by the LINPACK and EISPACK projects which together represent the state of the art in software for matrix computation MATLAB has evolved over a period of years with input from many users In university environments it is the standard instructional tool for introductory and advanced courses in mathematics engineering and science In industry MATLAB is thetool of choice for high productivity research development and analysis MATLAB features a family of application specific solutions called tool boxes Very important to most users of MATLAB toolboxes allow you to learn and apply specialized technology Toolboxes are comprehensive collections of MATLAB functions M files that extend the MATLAB environment to solve particular classes of problems Areas in which toolboxes are available include signal processing control systems neural networks fuzzy logic wavelets simulation and many oth
262. imization routine appears to enter an infinite loop or returns a solution that is not a minimum or not a zeroin the case of f zero Thereis no guarantee that you havea global minimum unless your problem is continuous and has only one minimum Starting the optimization from a number of different starting points or intervals in the case of f mi n bnd may help to locate the global minimum or verify that thereis only one minimum Use different methods where possible to verify results Modify your function to include a penalty function to give a large positive value tof when infeasibility is encountered Your objective function may be returning nf NaN or complex values The optimization routines expect only real numbers to be returned Any other values may cause unexpected results Insert code into your objective function M file to verify that only real numbers are returned use the functionsi sr eal and isfinite 7 13 7 Function Functions Numerical Integration Quadrature The area beneath a section of a function F x can be determined by numerically integrating F x a process referred to as quadrature The two MATLAB functions for one dimensional quadrature are e quad Use Adaptive Simpson s rule e quad8 Use Adaptive Newton Cotes 8 pane rule To integrate the function defined by humps m from 0 tol use q guad humps 0 1 q 29 8583 Both quad and quad8 operate recursively If either method rea
263. in a way that reflects composition of the underlying linear transformations and allows compact representation of systems of simultaneous linear equations The matrix product C AB is defined when the column dimension of A is equal to the row dimension of B or when one of them is a scalar If A is m by p and B is p by n their product C is m by n The product can actually be defined using MATLAB s for loops colon notation and vector dot products Matrices in MATLAB for i 1 m for j 1 n C i j ACi B s 4 end end MATLAB uses a single asterisk to denote matrix multiplication The next two examples illustrate the fact that matrix multiplication is not commutative AB is uSually not equal to BA X A B X 15 15 15 26 38 26 41 70 39 15 28 47 15 34 60 15 28 43 A matrix can be multiplied on the right by a column vector and on the left by a row vector x A u X 8 17 30 y v B y 12 10 4 9 4 Matrices and Linear Algebra 4 10 Rectangular matrix multiplications must satisfy the dimension compatibility conditions X AFC X 17 19 31 41 pi 70 Y C A Error using gt Inner matrix dimensions must agree Anything can be multiplied by a scalar w s y The Identity Matrix Generally accepted mathematical notation uses the capital letter to denote identity matrices matrices of various sizes with ones on the main diagonal and zeros elsewhere These matrices have the p
264. in color from Editor 2 41 2 48 private directory 10 39 in dispatching priority 10 10 private function 10 39 private functions precedence of 14 70 private methods 14 5 product dot 4 8 inner 4 7 outer 4 7 profile 3 17 example 3 19 3 21 syntax 3 18 profiling 3 17 details report 3 23 function call history 3 24 reports 3 20 programming debugging 3 2 programming object oriented 14 2 programs running external 10 62 running from MATLAB 2 25 stopping while running 2 7 property structure ODE creating 8 20 modifying 8 21 querying 8 21 ps 2 34 pseudocode 10 11 pseudoinverses 4 21 qr 428 QR factorization 4 28 9 30 quad 7 14 8 50 quad8 7 14 8 50 quadrature 7 14 question mark button 2 22 questions and answers ODE solvers different kinds of systems 7 13 quick start MATLAB programming 10 2 ODE solvers 8 3 quit 10 66 quitting MATLAB 2 4 Seealsofinish m quotes for creating strings 11 4 R r M file startup option 2 3 rand 9 23 randn 12 6 rank 9 3 deficiency 4 29 9 31 rational format 4 18 reading data 2 26 sound files 2 27 spreadsheet files 2 27 values from files 15 7 real max 10 18 real min 10 18 red stop sign in Debugger 3 6 reducing memory fragmentation 10 65 reference passing arguments by 10 11 reference documentation 2 23 reference pages 2 23 reference subscripted 14 14 references for OO design 14 8 1 23 Index registry file 2 55 for Windows 2 4 resetting 2 4 regression 6 15 linear in
265. in the MATLAB environment and provides useful functionality for a polynomial data type the polynom class implements the following methods e A constructor method pol ynom m e A polynom to double converter e A polynom to char converter e Adisplay method e Asubsref method e Overloaded and operators e Overloaded roots pol yval plot and diff functions The following sections describe the implementation of these methods 14 23 14 MATLAB Classes and O bjects 14 24 The Polynom Constructor Method Here is the polynom class constructor pol ynom pol ynom m function p polynom a POLYNOM Polynomial class constructor p POLYNOM v creates a polynomial object fromthe vector v containing the coefficients of descending powers of x if margin p c p class p polynom elseif isa a polynom p a else p c a p class p polynom end iT iT a Constructor Calling Syntax You can call the polynom constructor method with one of three different arguments e Nol nput Argument If you call the constructor function with noarguments it returns a polynom object with empty fields e Input Argument is an Object If you call the constructor function with an input argument that is already a polynom object MATLAB returns theinput argument Thei sa function pronounced is a checks for this situation e Input Argument is a coefficient vector If the input argument is a va
266. in the first index element and there are two index elements The class method for end must then return the index value for the last element of the first dimension When you implement the end method for your class you must ensure it returns a value appropriate for the object Indexing an Object with Another Object When MATLAB encounters an object as an index it calls thes ubsi ndex method defined for the object For example suppose you have an object a and you want to use this object to index into another object b c bla Designing User Classes in MATLAB A subsi ndex method might do something as simple as convert the object to double format to be used as an index as shown in this sample code function d subsindex a SUBSI NDEX convert the object a to double format to be used as an index in an indexing expression d double a subsi ndex values are 0 based not 1 based Converter Methods A converter method is a class method that has thesamenameas another class such as char or doubl e Converter methods accept an object of one class as input and return an object of another class Converters enable you to e Use methods defined for another class e Ensure that expressions involving objects of mixed class types execute properly A converter function call is of the form b class_name a where a is an object of a class other than class_name In this case MATLAB looks for a method called class_name in the class dir
267. ing and frequency analysis Spline Curve fitting and regression Statistics Advanced statistical analysis nonlinear curve fitting and regression System Identification Parametric ARMA modeling Wavelet Wavelet analysis Column riented Data Sets Column Oriented Data Sets Univariate statistical data is typically stored in individual vectors The vectors can be either 1 by n or n by 1 For multivariate data a matrix is the natural representation but there are in principle two possibilities for orientation By MATLAB convention however the different variables are put into columns allowing observations to vary down through the rows Therefore a data set consisting of twenty four samples of three variables is stored in a matrix of size 24 by 3 Consider a sample data set comprising vehicle traffic count observations at three locations over a twenty four hour period Time Location 1 Location 2 Location 3 01h00 11 11 9 02h00 7 13 11 03h00 14 17 20 04h00 11 13 9 05h00 43 51 69 06h00 38 46 76 07h00 61 132 186 08h00 75 135 180 09h00 38 88 115 10h00 28 36 55 11h00 12 12 14 12h00 18 27 30 13h00 18 19 29 14h00 17 15 18 15h00 19 36 48 6 3 6 Data Analysis and Statistics Time Location 1 Location 2 Location 3 16h00 32 47 10 17h00 42 65 92 18h00 57 66 151 19h00 44 55 90 20h00 114 145 257 21h00 35 58 68 22h00 11 12 15 23h00 13 9 15 24h00 10 9 7 6 4 Column riented Data Sets
268. ing each coefficient by its original degree function q diff p POLYNOM DIFF ODIFF p is the derivative of the polynom p C Spee d length c 1 degree q polynom p c 1l d d 1 1 Listing Class Methods The function call methods class_name or its command form methods class_name shows all the methods available for a particular class For the pol ynom example the output is methods polynom Methods for class polynom char display mi nus plot polynom roots diff double mt i mes plus polyval subsref Example A Polynomial C lass Plotting the two polynom objects x and p calls most of these methods x polynom 1 0 p polynom 1 0 2 5 plot diff p p 10 p 20 x 20 6 x 16 x 8 x 8 T T 44 a 14 33 14 MATLAB Classes and O bjects Building on Other Classes A MATLAB object can inherit properties and behavior from another MATLAB object When one object the child inherits from another the parent the child object includes all the fields of the parent object and can call the parent s methods The parent methods can access those fields that a child object inherited from the parent class but not fields new to the child class Inheritance is a key feature of object oriented programming It makes it easy to reuse code by allowing child objects to take advantage of code that exists for parent objects Inheritance enables a child object to behave exactly lik
269. ion precedence causes a b amp to be equivalent to a b amp However in most programming languages a b amp c is equivalent to a b amp c that is amp takes precedence over To ensure compatibility with future versions of MATLAB you should use parentheses to explicity specify the intended precedence of statements containing combinations of amp and O perators Overriding Default Presedence The default precedence can be overridden using parentheses as shown in this example A 3 9 5 Be i 2 41 Se C A B 2 C 0 7500 9 0000 0 2000 C A B 2 C 2 2500 81 0000 1 0000 Expressions can also include values that you access through subscripts b sqrt A 2 2 B 1 b 10 29 10 M File Programming Flow Control 10 30 There are six flow control statements in MATLAB e f together withelse andel sei f executes a group of statements based on some logical condition swi tch together with case and ot her wi se executes different groups of statements depending on the value of some logical condition whi e executes a group of statements an indefinite number of times based on some logical condition f or executes a group of statements a fixed number of times break terminates execution of af or or whi e loop etry catch changes flow control if an error is detected during execution ret urn causes execution to return to the invoking function All flow constructs use e
270. ipt Load Workspace Save Workspace As Show Workspace Show Graphics Property Editor Show GUI Layout Tool Set Path Preferences Print Setup Print Print Selection 1 open m 2 demoinsert m 3 demoinsert2 m 4 insertproblem mat Exit MATLAB Ctrl O Preferences Set preferences to control the appearance and operation of the Command Window Select Preferences from the File menu The Preferences dialog box 2 37 2 MATLAB W orking Environment appears from which you set General Command Window Font and Copying Options preferences Preferences General Command Window Font Copying Options Numeric Format Short E Long E Short G Long G Rational Ol Iie lite lite lite lie lie lle lie Loose default C Compact Editor Preference Built in Editor C Browse m Help Directory D matlab help Browse I Echo On M Show Toolbar IV Enable Graphical Debugging Cancel Apply G eneral Preferences Numeric Format Specify the default numeric format F or more information see The format Command or the reference page for f or mat Editor Preference Use MATLAB s Editor or specify another F or more information about MATLAB s Editor see E ditor Debugger Help Directory Specify the directory in which MATLAB help files reside Echo On Turn M file echoing on see theecho reference page for
271. ith them You must convert such arrays to double via thedoubl e function before doing any math operations You can define user classes and objects in MATLAB that are based on the struct datatype For moreinformation about creating classes and objects see Classes and Objects An Overview in Chapter 14 10 19 10 M File Programming This table describes the data types in more detail Class Example Description array virtual data type cell 17 hello eye 2 Cell array Elements of cell arrays contain other arrays Cell arrays collect related data and information of a dissimilar size together char Hello Character array each character is 16 bits long Also referred to as a string double 1 2 3 4 Double precision numeric array this is the most 5 61 common MATLAB variable type numeric virtual data type Sparse speye 5 Sparse double precision matrix 2 D only The sparse matrix stores matrices with only a few nonzero elements in a fraction of the space required for an equivalent full matrix Sparse matrices invoke special methods especially tailored to solve sparse problems storage virtual data type struct a day 12 Structure array Structure arrays have field names a color Red The fields contain other arrays Like cell arrays a mat magic 3 structures collect related data and information together uint8 ui nt 8 magic 3 Unsigned 8 bit integer array Theunit 8 array user object inline
272. itions for ODEs boundary 8 41 initial 8 3 8 5 confidence intervals 6 28 constructor methods 14 10 guidelines 14 10 usingcl ass in 14 11 containment 14 36 content indexing 13 20 to access cell contents 13 23 contents of sparse matrix 9 11 Contents m file 10 9 continuous extension ODE solvers 8 12 contour 5 21 contour plots to compare interpolation methods 5 14 control keys for editing commands 2 6 conv 5 4 13 26 conversion specifiers 2 27 converter methods 14 19 14 25 converting and handling dates 10 54 base numbers 11 3 cases of strings 11 3 numbers 11 13 strings 11 3 11 13 15 2 convex hull 5 23 convhull 5 23 convolution 5 4 copying options 2 39 corrcoef 6 10 correlation coefficients 6 9 cos 10 5 cov 6 9 covariance 6 9 creating cell array 13 19 multidimensional array 12 4 ODE files 8 14 sparse matrix 9 7 Index string 11 4 string array 11 6 structure array 13 3 cross 12 15 cubic interpolation multidimensional 5 16 one dimensional 5 10 cubic spline interpolation 5 10 curly braces for cell array indexing 13 20 13 22 to build cell arrays 13 22 to nest cell arrays 13 29 current directory 2 16 curve fitting 5 6 confidence intervals 6 28 customizing menus in Editor 2 41 Cuthill McKee reverse ordering 9 27 D data binary 15 2 exchanging between platforms 2 31 exporting 2 26 filter 6 29 H arwell Boeing format 9 10 importing 2 26 monotonic 5 12 multivariate 6 3 pre processing 6 12 sorting 6 7
273. ization for rectangular matrices These three factorizations are availablethrough thechol u andqr functions All three of these factorizations make use of triangular matrices where all the elements either above or below the diagonal are zero Systems of linear equations involving triangular matrices are easily and quickly solved using either forward or back substitution Cholesky Factorization The Cholesky factorization expresses a symmetric matrix as the product of a triangular matrix and its transpose A R R where R is an upper triangular matrix Not all symmetric matrices can be factored in this way the matrices that have such a factorization are said to be positive definite This implies that all the diagonal elements of A are positive and that the offdiagonal elements are not too big The Pascal matrices provide an interesting example Throughout this chapter our example matrix A has been the 3 by 3 Pascal matrix Let s temporarily switch to the 6 by 6 A pascal 6 Me wr gt uw a 10 20 35 56 15 35 70 126 21 56 126 252 e ee ee ey ao amp e WwW PH LU QR and Cholesky Factorizations The elements of A are binomial coefficients Each element is the sum of its north and west neighbors The Cholesky factorization is R chol A Res ooo coc OF Sooo Ore SO OrFPN Be SOF Ww PFE Cor RD Fe 1 The elements are again binomial coefficients The fact that R R is equal to A demonstrates an
274. ize n by n For example the covariance matrix for count cov count is arranged as 2 2 2 O11 512 013 DP B63 921 522 923 2 2 2 921 932 933 corrcoef produces a matrix of correlation coefficients for an array of data where each row is an observation and each column is a variable The correlation coefficient is a normalized measure of the strength of the linear relationship between two variables Uncorrelated data results in a correlation coefficient of 0 equivalent data sets have a correlation coefficient of 1 For an m by n matrix the correlation coefficient matrix has size n by n The arrangement of the elements in the correlation coefficient matrix corresponds to the location of the elements in the covariance matrix described above For our traffic count example corrcoef count results in ans 1 0000 0 9331 0 9599 0 9331 1 0000 0 9553 0 9599 0 9553 1 0000 Clearly there is a strong linear correlation between the three traffic counts observed at the three locations as the results are close to 1 Basic Data Analysis Functions Finite Differences MATLAB provides three functions for finite difference calculations Function Description di ff Difference between successive elements of a vector Numerical partial derivatives of a vector gradient Numerical partial derivatives a matrix del 2 Discrete Laplacian of a matrix Thedi f f function computes the difference between successive elements in a numer
275. k four bytes status fseek fid 4 cof File Position bof 1 2 3 4 5 6 7 8 9 10 eof File Contents Qo O S 20 Be ne kde Oy 2 5 File Position Indicator uN Calling fread again reads in the next value 3 three fread fid 1 short 15 12 Formatted Files Formatted Files This section explains how to read from and write to formatted text files Function Purpose fget Read line from file discard newline character fgets Read line from file keep newline character fscanf Read formatted data from file fprintf Write formatted data to file Reading Strings Line By Line from Text Files MATLAB provides two functions f get and f gets that read lines from formatted text files and store them in string vectors The two functions are almost identical the only difference is that f get s copies the newline character to the string vector but f get does not The following M file function demonstrates a possible use of f get This function uses f get toread an entire file one line at a time For each line the function determines whether an input literal string i t eral appears in the line 15 13 13 File 1 O If it does the function prints the entire line preceded by the number of times the literal string appears on the line function y litcount filename literal Search for number of string matches per line fid fopen filename rt eee while feof fid line fgetl fid
276. keys for editing commands 2 6 ASCII files 15 13 loading data from 2 27 loading into workspace 2 13 reading formatted text 15 14 saving 2 12 writing 15 15 assignment statements 4 4 10 15 10 16 using to build structure arrays 13 3 A stable differentiation formulas 8 32 auread 2 27 automatic highlighting in E ditor Debugger 2 39 automatic indenting in Editor Debugger 2 39 automatic scalar expansion 10 22 automation server 2 3 automation startup option 2 3 auwrite 2 29 B ballode 8 44 bandwidth of sparse matrix reducing 9 27 base numeric converting 11 13 base date 10 54 base number conversion 11 3 bicubic interpolation 5 12 5 13 bilinear interpolation 5 12 5 13 binary files 2 31 15 7 controlling data type of values read 15 8 controlling number of values read 15 7 reading 15 7 writing to 15 8 binary from decimal conversion 11 13 bits for precision 15 8 blanks finding in string arrays 11 11 removing from strings 11 6 boundary condition ODE 8 41 breakpoints 3 3 clearing 3 8 3 14 setting 3 5 3 11 Brusselator system ODE example 8 37 brussode 8 37 Buckminster Fuller dome 9 16 Bucky ball 9 16 buttons in command window 2 36 C C functions used for memory management 2 33 C programs for reading files 2 28 for writing files 2 29 C and MATLAB OOP 14 7 cache path 2 15 calling context 10 11 calling MATLAB functions compilation for later use 10 10 how MATLAB searches for functions 10 10 ODE solvers 8 6 storing as pseu
277. l of these special functions and constants reside in MATLAB s el mat directory and provide online help Here are several examples that usethem in MATLAB expressions x 2 pi A 342i 7 81 tol 3 eps Data Types Data Types There are six fundamental data types classes in MATLAB each onea multidimensional array The six classes aredouble char sparse storage cell andstruct The two dimensional versions of these arrays are called matrices and are where MATLAB gets its name array char numeric cel struct user object double storage ints uint8 int16 APSE SS uint16 int32 wint32 You will probably spend most of your time working with only two of these data types the double precision matrix doubl e and the character array char or string This is because all computations are donein double precision and most of the functions in MATLAB work with arrays of double precision numbers or strings The other data types are for specialized situations like image processing uni t 8 sparse matrices s parse and large scale programming cel and struct You can t create variables with the types numeric array orstorage These virtual types serve only to group together types that share some common attributes Thest orage data types are for memory efficient storage only You can apply basic operations such as subscripting and reshaping to these types of arrays but you can t perform any math w
278. l seamount data To obtain these values use mi n mi n x max max x and mi n mi n y max max y 5 21 5 Polynomials and Interpolation 5 22 Closest Point Searches YOu can search through the Delaunay triangulation data with two functions e dsearch finds the point closest to a point you specify e tsearch given a point xi yi returns an index intothede aunay output that specifies the enclosing triangle for the point Voronoi Diagrams Vornoi diagrams are a closest point plotting technique related to Delaunay triangulation The Voronoi diagram for theseamount datais load seamount voronoi x y grid on 48 05 48 1 H 48 15 48 2 48 25 48 3 48 35 48 4 210 9 211 211 1 211 2 211 3 211 4 211 5 211 6 Interpolation Convex Hulls Theconvhul function returns the indices of the points in a data set that comprise the convex hull for the set For example to view the convex hull for theseamount data load seamount plot x y markersize 10 k convhull x y hold on plot x k y k hold off grid on 47 95 48 F 48 05 48 1 F 48 15 48 2 48 25 48 3 F 48 35 F 48 4 F 48 45 L 1 L 1 ii fi L 1 if 210 8 210 9 211 21711 211 2 211 3 211 4 211 5 211 6 211 7 211 8 5 23 5 Polynomials and Interpolation 5 24 Data Analysis and Statistics Column Oriented Data Sets Basic Data Analysis Functi
279. lass In general the rules for indexing objects are the same as the rules for indexing structure arrays For details see Chapter 13 Structures and Cell Arrays Handling Subscripted Reference The use of a subscript or field designator with an object on the right hand side of an assignment statement is known as a subscripted reference MATLAB calls a method namedsubsref inthesesituations Object subscri pted references can Designing User Classes in MATLAB be of three forms an array index a cell array index and a structure field name A 1 A I A field Each of these results in a call by MATLAB tothesubsref method in the dass directory MATLAB passes two arguments tosubsref B subsref A S The first argument is the object being referenced The second argument S isa structure array with two fields e S type isastringcontaining or specifying the subscript type The parentheses represent a numeric array the curly braces a cell array and the dot a structure array e S subs iSacell array or string containing the actual subscripts A colon used as a subscript is passed as the string For instance the expression A 1 2 causes MATLAB tocall subsref A 5 wheres isa 1 by 1 structure with S type S subs 1 2 Similarly the expression A 1 2 uses S type S subs 1 2 The expression A field 14 15 14 MATLAB Classes and O bjects 14 16 callssubs
280. lass2 Location in Hierarchy If obj ectA is aboveobj ect B in the precedence hierarchy then the expression obj ectA obj ectB calls cl assA plus m Conversely if obj ect B is aboveobj ectA inthe precedence hierarchy then MATLAB calls c ass B pl us m See How MATLAB Determines Which Method to Call on page 14 68 for related information 14 67 14 MATLAB Classes and O bjects How MATLAB Determines Which Method to Call In MATLAB functions exist in directories in the computer s file system A directory may contain many functions M files Function names are unique only within a single directory e g more than one directory many contain a function called pi e3 When you type a function name on the command line MATLAB must search all the directories it is aware of to determine which function to call This list of directories is called the MATLAB path When looking for a function MATLAB searches the directories in the order they are listed in the path and calls the first function whose name matches the name of the specified function If you write an M file called pi e3 m and put it in a directory that is searched before thes pecgraph directory that contains MATLAB s pi e3 function then MATLAB uses your pi e3 function instead note that this is not true for built in functions like p ot which are always found first Object oriented programming allows you to have many methods MATLAB functions loc
281. le input argument that is a stock object the constructor returns the input argument A single argument that is not a stock object generates an error e Threel nput Arguments Iftherearethreeinput arguments the constructor uses them to define the stock object e Otherwise If none of the above three conditions are met return an error For example this statement creates a stock object to record the ownership of 100 shares of XYZ corporation stocks with a price per share of 25 dollars XYZ_stock stock XYZ 100 25 The Stock get Method Theget method provides a way to access the data in the stock object using a property name style interface similar to Handle Graphics While in this example the property names are similar to the structure field name they can be quite different You could also choose to exclude certain fields from access via the get method or return the data from the same field for a variety of property names if such behavior suits your design 14 47 14 MATLAB Classes and O bjects function val get s prop_name GET Get stock property fromthe specified object and return the value Property names are NumberShares SharePrice Descriptor Date Type CurrentValue switch prop_name case NumberShares val num shares case SharePrice val s share price case Descriptor val get s asset Descriptor call asset get method case Date val get s asset Date
282. le of contents with functions grouped by subject area or use the Help Desk typehel pdesk or select from Help menu to see the F unction Reference grouped by subject For tips and troubleshooting problems use the Help Desk type hel pdesk or select from Help menu to visit the Technical Support section of The MathWorks Web site www mat hworks com and use the Solution Search Engine to search the Technical Support database of problem solutions Use the Help Desk typehe pdesk or select from Help menu or send e mail tobugs mat hworks comorsuggest mat hworks com Use the Help Desk typehe pdesk or select from Help menu to submit an e mail help request form describing your question or problem 1 Introduction 1 8 About Simulink Simulink a companion program to MATLAB is an interactive system for simulating nonlinear dynamicsystems It isa graphical mouse driven program that allows you to model a system by drawing a block diagram on the screen and manipulating it dynamically It can work with linear nonlinear continuous time discrete time multivariable and multirate systems Blocksets are add ins to Simulink that provide additional libraries of blocks for specialized applications like communications signal processing and power systems Real time Workshop is a program that allows you to generate C code from your block diagrams and to run it on a variety of real time systems About Toolboxes MATLAB featu
283. le on Starts the profiler clearing previously recorded statistics detail level Specifies the level of function to be profiled history Specifies that the exact sequence of function calls is to be recorded profile report Suspends the profiler generates a profile report in HTML format and displays the report in your Web browser basename Saves the report in the file basename in the current directory profile plot Suspends the profiler and displays in a figure window a bar graph of the functions using the most execution time 3 18 M File Profiler profile resume Restarts the profiler without clearing previously recorded statistics profile clear Clears the statistics recorded by the profiler profile off Suspends the profiler profile status Displays a structure containing the current profiler status stats profile info Suspends the profiler and displays a structure containing profiler results An Example Using the Profiler This example demonstrates how to run the profiler 1 Start the profiler profile on detail builtin history The detail builtin option instructs the profiler to gather statistics for built in functions in addition to the default M functions M subfunctions and ME X functions The history option instructs the profiler to track the exact sequence of entry and exit calls 2 Execute an M file This example runs the Lotka Volterra predator prey population mo
284. lead 007 test 2 lead 031 test 3 lead 019 test 1 mercury 0021 test 2 mercury 0009 test 3 mercury 0013 test 1 chromium 025 test 2 chromium 017 test 3 chromium 10 Organizing Data in Structure Arrays The key to organizing structure arrays is to decide how you want to access subsets of the information This in turn determines how you build the array that holds the structures and how you break up the structure fields 13 11 13 Structures and Cell Arrays For example consider a 128 by 128 RGB image stored in three separate arrays RED GREEN and BLUE 0 689 0 706 0 118 0 884 Blueintensity 535 0 532 0 653 0 925 values 0 314 0 265 0 159 0 101 0 553 0 633 0 528 0 493 0 441 0 465 0 512 0 512 0 398 0 401 0 421 0 398 0 342 0 647 0 515 0 816 912 0 713 Greenintensity 311 9 300 0 205 0 526 219 0 328 values 0 523 0 428 0 712 0 929 128 0 133 0 214 0 604 0 918 0 344 0 100 0 121 0 113 0 126 0 288 0 187 0 204 0 175 0 112 0 986 0 234 0 432 760 0 531 Red intensity f0 765 0 128 0 863 0 521 997 0 910 values 1 000 0 985 0 761 0 698 995 0 726 0 455 0 783 0 224 0 395 0 021 0 500 0 311 0 123 1 000 1 000 0 867 0 051 1 000 0 945 0 998 0 893 0 990 0 941 1 000 0 876 0 902 0 867 0 834 0 798 13 12 Structures There are at least two ways you can organize such data into
285. les Microsoft Windows system resources Windows uses system resources to track fonts windows and screen objects Resources can be depleted by using multiple figure windows multiple fonts or several Uicontrols The best way to free up system resources is to close all inactive windows conified windows still use resources O ptimizing the Performance of MATLAB Code e The performance of a permanent swap file is typically better than a temporary swap file e Typically a swap file twice the size of the installed RAM is sufficient UNIX Specific Topics e Memory that MATLAB requests from the operating system is not returned to the operating system until the MATLAB process in finished e MATLAB requests memory from the operating system when there is not enough memory available in the MATLAB heap to store the current variables It reuses memory in the heap as long as the size of the memory segment required is available in the MATLAB heap For example on one machine these statements use approximately 15 4 MB of RAM a rand le6 1 b rand le6 1 These statements use approximately 16 4 MB of RAM c rand 2 1e6 1 These statements use approximately 32 4 MB of RAM a rand le6 1 b rand le6 1 clear c rand 2 1e6 1 This is because MATLAB is not able to fit a 2 1 MB array in the space previously occupied by two 1 MB arrays The simplest way to prevent overallocation of memory is to preallocate the largest vecto
286. linear interpolation met hod linear This method uses piecewise linear interpolation based on the values of the nearest eight points e Tricubicinterpolation method cubic This method uses piecewise cubic interpolation based on the values of the nearest sixty four points All of these methods requirethat X Y andZ be monotonic that is either always increasing or always decreasing in a particular direction In addition you should preparethese matrices using themes hgri d function or else be sure that the pattern of the points emulates the output of meshgrid 5 15 5 Polynomials and Interpolation 5 16 Each method automatically maps the input toan equally spaced domain before interpolating If x is already equally spaced you can speed execution time by prepending an asterisk tothe met hod string for example cubic Interpolation of Higher Dimensional Data The function i nt er pn performs multidimensional interpolation finding interpolated values between points of a multidimensional set of samples V The most general form fori nter pn is VI interpn X1 X2 X3 V 1 2 Y3 met hod 1 2 3 are matrices that specify the points for which values of V are given V is a matrix that contains the values corresponding to these points 1 2 3 are the points for which i nt er pn returns interpolated values of V For out of range values i nter pn returns NaN Y1 Y2 Y3 must be either arr
287. lio objects in the mat file being loaded If oadobj exists MATLAB passes the portfolio object to oadobj which must then return the Saving and Loading O bjects modified object as an output argument The output argument is then loaded into the workspace If the input object does not match the current definition as specified by the constructor function then MATLAB converts it to a structure containing the same fields and the object s structure with all the values intact that is you now have a structure not an object The following implementation of oadobj first uses i sa to determine whether the input argument is a portfolio object or a structure If theinput is an object it is simply returned since no modifications are necessary If the input argument has been converted to a structure by MATLAB then the new account _number field is added tothe structure and is used to create an updated portfolio object function b oadobj a loadobj for portfolio class if isa a portfolio b a else ais an old version a account number getAccount Number a b class a portfolio end Changing the Portfolio Constructor The portfolio structure array needs an additional field to accommodate the account number To create this field add the line p account_number to portfolio portfolio min both thezero argument and variable argument sections The getAccountN umber Function In this example get Ac
288. ll on page 14 68 14 MATLAB Classes and O bjects Debugging Class Methods You can use the MATLAB debugging commands with object methods in the same way that you use them with other M files The only differenceis that you need to include the class directory name before the method name in the command call as shown in this example using dbst op dbstop pol ynom char While debugging a class method you have access to all methods defined for the class including inherited methods private methods and private functions For more information about debugging MATLAB functions see Chapter 3 Debugger and Profiler Changing Class Definition If you change the class definition such as the number or names of fields in a class you must issue a clear classes command to propagate the changes to your MATLAB session This command also clears all objects from the workspace See the cl ear command help entry for more information Setting Up Class Directories The M files defining the methods for a class are collected together in a directory referred to as the class directory The directory name is formed with the class name preceded by the character For example one of the examples used in this chapter is a dass involving polynomials in a single variable The name of the class and the name of the class constructor is pol ynom The M files defining a polynomial class would be located in directory with the name pol ynom T
289. llow you to examine and change the function s local workspace This mode is indicated by a special prompt K gt gt Resume function execution by typingr et urn and pressing the Return key MATLAB Debugger e Comment out the leading function declaration and run the M fileas a script This makes the intermediate results accessible in the base workspace e Usethe MATLAB Debugger The Debugger is useful for correcting runtime errors precisely because it enables you to access function workspaces and examine or changethe values they contain The Debugger allows you to set and clear breakpoints specific lines in an M file at which execution halts It also lets you change workspace contexts view the function call stack and execute the lines in an M file one by one This section describes these tasks in detail Note TheM file breakpoint information is closely associated with the copy of the M file that MATLAB holds in memory If you clear the M file by editing or by issuingcl ear Mfile all of Mfil e s breakpoints are also cleared M Files For An Example Session Totry the Debugger first create an M file called vari ance m that accepts an input vector and returns an unbiased variance estimate This file calls another M file sqs um that computes the mean removed squared sum for the input vector function y variance x mu sum x length x tot sqsum x mu y tot length x 1 Create thes qsum m file exactly
290. ls the method associated with the leftmost object H owever there are two exceptions e User defined classes have precedence over MATLAB built in classes e User defined classes can specify their relative precedence with respect to other user defined classes using thei nf eri orto andsuperiorto functions For example in the section Example A Polynomial Class on page 14 23 the polynom class defines ap us method that enables addition of polynom objects Given the polynom object p p polynom 1 0 2 5 p x 3 2 x 5 The expression 1 p ans xX 3 2 x4 calls the polynom pl us method which converts the double 1 to a polynom object and then adds it top The user defined polynom class has precedence over the MATLAB double class Specifying Precedence of User Defined Classes You can specify the relative precedence of user defined classes by calling the inferiorto orsuperi orto function in the class constructor Thei nf eri orto function places a class below other classes in the precedence hierarchy The calling syntax for thei nf eri orto function is inferiorto classl class2 O bject Precedence You can specify multiple classes in the argument list placing the class below many other classes in the hierarchy Similarly thes uperi orto function places a class above other classes in the precedence hierarchy The calling syntax for thes uperi orto function is Superiorto classl c
291. lue function display p POLYNOM DISPLAY Command window display of a polynom displ yy disp inputname 1 disp disp char p disp The statement p polynom 1 0 2 5 14 27 14 MATLAB Classes and O bjects 14 28 creates a polynom object Since the statement is not terminated with a semicolon the resulting output is p KOS A DE E The Polynom subsref Method Suppose the design of the polynom class specifies that a subscripted reference toapol ynom object causes the polynomial to be evaluated with the value of the independent variable equal to the subscript That is for a polynom object p p polynom 1 0 2 5 the following subscripted expression returns the value of the polynomial at x 3 andx 4 p 3 4 ans 16 51 subsref Implementation Details This implementation takes advantage of thechar method already defined in the polynom class to produce an expression that can then be evaluated function b subsref a s SUBSREF switch s type case ind s subs for i l length ind b i eval strrep char a x num2str ind i end otherwise error Specify value for x as p x end Once the polynomial expression has been generated by the char method the strrep function is used to swap the passed in value for the character x The eval function then evaluates the expression and returns the value in the output argument Example A Polynomial
292. ly apply the original function to the coefficient field Overloading roots for the Polynom Class The method po ynom roots m finds the roots of polynom objects function r roots p POLYNOM ROOTS ROOTS p is a vector containing the roots of p r roots p c The statement roots p results in ans 2 0946 1 0473 1 1359 1 0473 1 1359 Overloading polyval for the Polynom Class The function pol yval evaluates a polynomial at a given set of points pol ynom pol yval muses nested multiplication or Horner s method to reduce thenumber of multiplication operations used to compute the various powers of x function y polyval p x POLYNOM POLYVAL POLYVAL p x evaluates p at the points x y 0 for a op c y y x a end Overloading plot for the Polynom Class The overloaded p ot function uses bothr oot andpol yval The function selects the domain of the independent variable to be slightly larger than an interval 14 31 14 MATLAB Classes and O bjects 14 32 containing all real roots Then pol yval is used to evaluate the polynomial at a few hundred points in the domain function plot p POLYNOM PLOT PLOT p plots the polynom p r max abs roots p Xx 1 1 0 01 1 1 r y polyval p x plot x y title char p grid on 3s Overloading diff for the Polynom Class The method pol ynom diff m differentiates a polynomial by reducing the degree by 1 and multiply
293. m The other nonzero component is p 4 becauseR 4 dominates after R 2 is eliminated 4 18 Solving Linear Equations The complete solution to the overdetermined system can be characterized by adding an arbitrary vector from the null space which can be found using the nul function with an option requesting a rational basis Z null R r Z 1 2 7 6 1 2 1 2 1 0 0 1 It can be confirmed that A Z is zero and that any vector of the form xX p Z q for an arbitrary vector q satisfies R x b 4 19 4 Matrices and Linear Algebra Inverses and Determinants If A is square and nonsingular the equations AX I and XA I have the same solution X This solution is called the inverse of A is denoted by A and is computed by the function i nv The determinant of a matrix is useful in theoretical considerations and some types of symbolic computation but its scaling and roundoff error properties make it far less satisfactory for numeric computation Nevertheless the function det computes the determinant of a square matrix d det A X inv A d 1 Yrs 3 33 1 33 5 2 1 2 1 Again because A is symmetric has integer elements and has determinant equal to one so does its inverse On the other hand d det B X inv B d 360 X 0 1472 0 1444 0 0639 0 0611 0 0222 0 1056 0 0194 0 1889 0 1028 Closer examination of the elements of X or use off or mat rat woul
294. manager considers that one key in use When the total number of licensed keys are in use no more users can invoke MATLAB FLEXIm consists of a license daemon and a product daemon that run ona server node On UNIX computers the server node is usually the file server on which MATLAB is installed Throughout this section references tothe mat ab directory refer to the directory where the contents of the MATLAB CD is installed The license and product daemons run in the background on the server node They are responsible for checking in and out licenses as users invoke and quit MATLAB License Administration A number of license administration tools are available in matl ab etc including mdown Shut down all license daemons mhostid Display hostid of the machine on which you are running mst at Show the current status of all network licensing activities The command mst at a displays all information Use the switch f instead of a todisplay a list of whois using what features mst at h displays usage help Imstart Start license daemons If license daemons are already running you must first use mdown to shut them down Understanding the License File The License File icense dat contains the details of your license such as the number of keys you have for MATLAB the toolboxes that you purchased and the hostids of the licensed CPUs If you upgrade your license or need to move 2 59 2 MATLAB W orking Environment
295. matrix corresponds to a sample point with each input appearing as columns of the matrix The function y filter b a x processes the data in vector x with the filter described by vectors a andb creating filtered data y Thefi ter command can be thought of as an efficient implementation of the difference equation The filter structure is the general tapped delay line filter described by the difference equation below where n is the index of the current sample na is the order of the polynomial described by vector a and nbis the order of the polynomial described by vector b The output y n is a linear combination of current and previous inputs x n x n 1 and previous outputs y n 1 y n 2 a 1 y n b 1 x n b 2 x n 1 b nb x n nb 1 a 2 y n 1 a na y n na 1 Suppose for example we want to smooth our traffic count data with a moving average filter to see the average traffic flow over a 4 hour window This process is represented by the difference equation y n 5x n 5x n 1 5x n 2 3x n 3 The corresponding vectors are 1 1 4 1 4 1 4 1 4 a b 6 29 6 Data Analysis and Statistics 6 30 Executing the command load count dat creates the matrix count in the workspace For this example extract the first column of traffic counts and assign it to the vector x X count 1 The four hour moving average of the data is efficiently calculated with y filter b a x
296. ments and return a default object See Guidelines for Writing a Constructor on page 14 10 for more information Modifying Objects During Save or Load When you issue asave or oad command on objects MATLAB looks for class methods called saveobj and oadobj in the class directory You can overload these methods to modify the object before the save or load operation F or example you could defineas aveobj method that saves related data along with the object or you could write al oadobj method that updates objects to a newer version when this type of object is loaded into the MATLAB workspace Example Defining saveobj and loadobj In the section Example The Portfolio Container on page 14 54 portfolio objects are used to collect information about a client s investment portfolio Now suppose you decide to add an account number to each portfolio object that is saved You can define a portfolios aveobj method to carry out this task automatically during the save operation Suppose further that you have already saved a number of portfolio objects without the account number Y ou want to update these objects during the load operation so that they are still valid portfolio objects You can do this by defining al oadobj method for the portfolio class 14 61 14 MATLAB Classes and O bjects 14 62 Summary of Code Changes Toimplement the account number scenario you need to add or change the following functions e portfolio
297. mes known as the Q less QR factorization is available Sparse M atrix O perations With one sparse input argument and one output argument R qr S returns just the upper triangular portion of the QR factorization The matrix R provides a Cholesky factorization for the matrix associated with the normal equations ee However the loss of numerical information inherent in the computation of S S is avoided With two input arguments having the same number of rows and two output arguments the statement C R qr S B applies the orthogonal transformations toB producingC Q B without computing Q The Q less QR factorization allows the solution of sparse least squares problems minimizel A x b with two steps c R gr A b x R c IfA is sparse but not square MATLAB uses these steps for the linear equation solving backslash operator x A b Or you can do the factorization yourself and examine R for rank deficiency It is also possible to solve a sequence of least squares linear systems with different right hand sides b that are not necessarily known whenk qr A is computed The approach solves the semi normal equations R R x A b with x R R A b 9 31 9 Sparse Matrices 9 32 and then employs one step of iterative refinement to reduce roundoff error r b A x e R R A r Xx x e Incomplete Factorizations Thel ui nc andcholinc functions provide ap
298. minimal norm 4 22 mi ni mi ze startup option 2 3 minimizing functions of one variable 7 7 of several variables 7 8 setting minimization options 7 9 minimum degree ordering 9 27 mi sl ocked 10 17 missing values 6 12 ml ock 10 17 monotonic data for interpolation 5 12 Moore Penrose pseudoinverse 4 21 more 2 7 multidimensional arrays applying functions 12 15 element by element functions 12 15 matrix functions 12 16 vector functions 12 15 cell arrays 12 19 computations on 12 15 creating 12 4 at the command line 12 5 with functions 12 6 with thecat function 12 7 defined 12 2 extending 12 5 format 12 9 indexing 12 10 avoiding ambiguity 12 10 with the colon operator 12 10 interpolation 5 15 5 16 number of dimensions 12 9 organizing data 12 17 permuting dimensions 12 13 removing singleton dimensions 12 12 reshaping 12 11 size of 12 9 storage 12 9 structure arrays 12 20 applying functions 12 21 subscripts 12 3 multidimensional data gridding 5 17 1 17 Index 1 18 multidimensional interpolation 5 15 5 16 cubic 5 16 linear 5 16 nearest neighbor 5 16 multiple conditions for s wi t ch 10 34 multiple inheritance 14 35 multiple lines for a single statement 2 10 multiple output values 10 7 multiple regression 6 19 multiplication matrix 4 8 of polynomials 5 4 multistep solver ODE 8 10 multivariate data 6 3 munl ock 10 17 N names for functions 10 9 for variables 10 16 structure fields 13 4 superseding 10 39 N
299. mize the appearance of graphics as well as to build complete graphical user interfaces GUIs for your MATLAB applications The graphics functions are organized into five directories in the MATLAB Toolbox graph2d graph3d specgraph graphics ui tools Two dimensional graphs Three dimensional graphs Specialized graphs Handle Graphics Graphical user interface tools The MATLAB Mathematical Function Library This is a vast collection of computational algorithms ranging from elementary functions like sum sine cosine and complex arithmetic to more sophisticated functions like matrix inverse matrix eigenvalues Bessel functions and fast Fourier transforms The math and analytic functions are organized into eight directories in the MATLAB Toolbox el mat el fun specfun mat fun datafun polyfun funfun sparfun Elementary matrices and matrix manipulation Elementary math functions Specialized math functions Matrix functions numerical linear algebra Data analysis and Fourier transforms Interpolation and polynomials Function functions and ODE solvers Sparse matrices The MATLAB Application Program Interface API This is a library that allows you to write C and Fortran programs that interact with MATLAB It includes facilities for calling routines from MATLAB dynamic linking calling MATLAB as a computational engine and for reading and writing MAT files 1 Introduction How to Use th
300. more information Show Toolbar Show or hide the toolbar in the Command Window EnableGraphical Debugging At each breakpoint automatically show the Debugger Microsoft W indows Handbook Command Window Font Specify the font characteristics for the text displayed in the Command Window Copying Options Specify options used when copying items from MATLAB tothe Windows clipboard for pasting into other applications Editor Debugger The Editor Debugger provides basic text editing operations as well as access to M file debugging tools The Editor Debugger offers a graphical user interface It supports automatic indenting and syntax highlighting for details see the General Options section under View Menu You can also use debugging commands in the Command Window See Chapter 3 for more information about MATLAB s debugging capabilities To specify the default editor for MATLAB select Preferences from the File menu in the Command Window On the General page select MATLAB s Editor Debugger or specify another Starting the Editor Debugger Tostart the Editor Debugger select New from the File menu or click the new file page icon button on the toolbar or typeedi t at the command line To start the Editor Debugger opening it to a particular file select Open from the File menu or click the open file folder icon button on the toolbar or type edit filename at the command line Do not use the Editor Debu
301. n elseif n lt p B ntl p end C A B else C 0 end This example uses the two argument form of eval with thecatchf cn function shown above clear Ave i 2 32 6 T 20 kk SA B 9 5 6 0 4 9 eval A B catch A B A 1 7 B randn 9 9 eval A B catchfcn A B Displaying Error and Warning Messages Usetheerror andf printf functions to display error information on the screen Theerror function has the syntax error error string If you call theerror function from inside an M file error displays the text in the quoted string and causes the M file to stop executing F or example suppose the following appears inside the M file myfi le m if n lt l error n must be 1 or greater end 10 52 Errors and W arnings For n equal to0 the following text appears on the screen and the M file stops 22 Error using gt myfile n must be 1 or greater In MATLAB warnings are similar to error messages except program execution does not stop Use the war ni ng function to display warning messages warning warning string The function ast warn displays the last warning message issued by MATLAB 10 53 10 M File Programming Times and Dates MATLAB provides functions for time and date handling These functions are in a directory called ti mef un in the MATLAB Toolbox Category Function Description Current time and date now Current d
302. n myfun 8 23 8 O rdinary Differential Equations 8 24 the solver calls it with myfun tspan y0 init before beginning the integration so that the output function can initialize Subsequently the sol ver callsstatus myfun t y after each step In addition to your intended use of t y codemyf un sothat it returns ast atus output value of 0 or 1 If status 1 integration halts This might be used for instance to implement a STOP button When integration is complete the solver calls the output function with myfun done Some example output functions are included with the ODE solvers e odepl ot time series plotting e odephas2 two dimensional phase plane plotting e odephas3 threedimensional phase plane plotting e odeprint print solution as it is computed Use these as models for your own output functions odep ot is the default output function for all the solvers It is automatically invoked when the solvers are called with no output arguments O utputSel The Out put Sel property is a vector of indices specifying which components of the solution vector are to be passed to the output function For example if you want tousetheodep ot output function but you want to plot only thefirst and third components of the solution you can do this using options odeset OutputFcn odeplot OutputSel 1 3 Refine TheRefi ne property an integer n produces smoother output by incr
303. n The statements size A nnz A show that A is a matrix of order 2945 with 14 473 nonzero elements The statement v d eigs A 1 0 computes the smallest eigenvalue and eigenvector Finally L L gt 0 full v L L gt 0 Xx 1 1 32 1 contour x x L 15 axis square distributes the components of the eigenvector over the appropriate grid points and produces a contour plot of the result 1 0 87 0 67 0 4 0 2 zi 1 0 5 0 0 5 1 Thenumerical techniques usedinei gs andsvds are described in a paper by D C Sorensen Implicitly Restarted Arnoldi Lanczos Methods for Large Scale 9 37 9 Sparse Matrices EigenvalueCalculations A copy of the paper is availablethrough the MATLAB Help Desk 9 38 M File Programming MATLAB Programming A Quick Start Scripts Functions Local and Global Variables Data Types Operators Flow Control Subfunctions Indexing and Subscripting String E valuation Command F unction Duality Empty Matrices Errorsand Warnings Times and Dates Obtaining User Input Shell Escape Functions Optimizing the Performance of MATLAB Code 10 2 10 5 10 6 10 16 10 19 10 21 10 30 10 38 10 40 10 46 10 48 10 49 10 51 10 54 10 61 10 62 10 63 10 M File Programming MATLAB Programming A Quick Start Files that contain MATLAB language code are called M files M files can be functions that accept arguments and
304. n About M ultidi mensional Arrays Working with Multidimensional Arrays y Indexing af Ry anri e a Reshaping Permuting Array Dimensions Computation with Multidimensional Arrays Functions that Operate on Vectors i Functions that Operate Element by E lement Functions that Operate on Planes and Matrices Organizing Data in Multidimensional Arrays Multidimensional Cell Arrays Multidimensional Structure Arrays Applying Functions to Multidimensional Structure Arrays i 12 3 12 4 12 9 12 9 12 10 12 11 12 13 12 15 12 15 12 15 12 16 12 17 12 19 12 20 12 21 12 Multidimensional Arrays This chapter discusses multidimensional arrays MATLAB arrays with more than two dimensions Multidimensional arrays can be numeric character cell or structure arrays Multidimensional arrays are broadly useful for example in the representation of multivariate data or multiple pages of two dimensional data MATLAB provides a number of functions that directly support multidimensional arrays Y ou can extend this support by creating M files that work with your data architecture Function Description cat Concatenate arrays ndgrid Generate arrays for N D functions and interpolation ndi ms Number of array dimensions i permute Inverse permute array dimensions per mute Permute array dimensions shiftdim Shift dimensions squeeze Remove singleton dimensions 12 2 M ultidimensional Array
305. n with a single argument S sparse A For example A 0 0 0 5 0 2 0 0 1 3 0 0 0 0 4 O01 S sparse A produces S 3 1 1 2 2 2 3 2 3 4 3 4 1 4 5 The printed output lists the nonzero elements of S together with their row and column indices The elements are sorted by columns reflecting the internal data structure You can convert a sparse matrix to full storage using thef ul function provided the matrix order is not too large For exampleA full S reverses the example conversion 9 6 Introduction Converting a full matrix to sparse storage is not the most frequent way of generating sparse matrices If the order of a matrix is small enough that full storage is possible then conversion to sparse storage rarely offers significant savings Creating Sparse Matrices Directly Y ou can create a sparse matrix froma list of nonzero elements usingthes parse function with five arguments S sparse i S m n i andj are vectors of row and column indices respectively for the nonzero elements of the matrix s is a vector of nonzero values whose indices are specified by the corresponding i j pairs mis the row dimension for the resulting matrix and n is the column dimension The matrix S of the previous example can be generated directly with S sparse 3 2 3 4 1 1 2 2 3 4 1 2 3 4 5 4 4 S 3 1 2 2 3 2 4 3 1 4 on Fe wr Pe Thes parse command has a number of alternate
306. nal function gar eqn to find the solution to Garfield s equation function y garfield a b q r save gardata a b q r gareqn load gardata This M file 1 Saves theinput argumentsa b q andr toa MAT filein the workspace using thes ave command 2 Uses the shell escape operator to access a C or Fortran program called gareqn that uses the workspace variables to perform its computation gareqn writes its results to the gar data MAT file 3 Loads thegardat a MAT file to obtain the results Reading and Writing MAT Files The MAT file subroutine library described in the MATLAB Application Program Interface Guide provides routines for reading and writing MAT files Also see this document for information on ME X files and C or Fortran functions that you can call directly from MATLAB O ptimizing the Performance of MATLAB Code Optimizing the Performance of MATLAB Code This section describes techniques that often improve the execution speed and memory management of MATLAB code e Vectorization of loops e Vector preallocation MATLAB is a matrix language which means it is designed for vector and matrix operations For best performance you should take advantage of this where possible Vectorization of Loops You can speed up your M file code by vectorizing algorithms Vectorization means converting for and while loops to equivalent vector or matrix operations A Simple Example Here is one way to compute the sine of
307. name O pening and Closing Files Opening and Closing Files Files are opened with thef open command and closed with thef close command Function Purpose fopen Open file fclose Close file Before reading or writing a text or binary file you must open it with thef open command fid fopen filename permission The per mi ssi on string specifies the kind of access you require Possible permission strings include e r for reading only e w for writing only e a for appending only e for both reading and writing Note Systems such as Microsoft Windows that distinguish between text and binary files may require additional characters in the permission string such as rb toopena binary file for reading f open returns a fileidentifier f i d the value you use to access the open file Thisf open statement opens the data file named penny dat for reading fid fopen penny dat r MATLAB File I O Functions and ANSI Standard C Many of the MATLAB file 1 O functions are based on the I O functions of the ANSI Standard C Library If you know C therefore you are probably familiar with these routines However not all MATLAB file I O commands work the 15 3 13 File 1 O 15 4 same way as their C language counterparts Check MATLAB command syntax and functionality usingtheonline help facility or theonline MATLAB Function Reference Using the File Identifier fid The file iden
308. nce function and approximate derivative Largest component Average or mean value Median value Smallest component Product of elements Sort in ascending order Sort rows in ascending order Standard deviation Sum of elements Trapezoidal numerical integration For vector input arguments to these functions it does not matter whether the vectors are oriented in row or column direction For array arguments however the functions operate column by column on the data in the array This means 6 7 6 Data Analysis and Statistics 6 8 for example that if you apply max toan array the result is a row vector containing the maximum values over each column Note You can add more functions to this list using M files but when doing so you must exercise care to handle the row vector case If you are writing your own column oriented M files check other M files for example mean m and diff m Continuing with the vehicle traffic count example the statements mx max count mu mean count sigma std count result in mx 114 145 257 mu 32 0000 46 5417 65 5833 sigma 25 3703 41 4057 68 0281 Tolocatetheindex at which the minimum or maximum occurs a second output parameter can be specified For example mx indx min count mx 7 9 7 i ndx 2 23 24 shows that the lowest vehicle count is recorded at 02h00 for the first observation point column one and at 23h00 and 24h00 for the other
309. nction syntax corresponds to f array array_index field field_index For example to access thename field in the second structure of the pati ent array use str getfield patient 2 name Similarly set field lets you assign values to fields using the syntax f setfield array array_index field field_index value Structures Using the size Function with Structure Arrays Usethesi ze function to obtain the sizeof a structure array or of any structure field Given a structure array name as an argument si ze returns a vector of array dimensions Given an argument in the formarray n field thesize function returns a vector containing the size of the field contents For example for the 1 by 3 structure array patient size patient returns thevector 1 3 Thestatementsize patient 1 2 name returnsthelength of thename string for element 1 2 ofpatient Adding Fields to Structures You can add afield to every structurein an array by adding the field toa single structure For example to add a social security number field to the pati ent array use an assignment like patient 2 ssn 000 00 0000 Now pati ent 2 ssn has the assigned value Every other structure in the array also has thes s n field but these fields contain the empty matrix until you explicitly assign a value to them Deleting Fields from Structures You can remove a given field from every structure within a structure array using ther mf
310. nction toextract property values from ano pti ons structure created with odeset o odeget options name This returns the value of the specified property or an empty matrix if the property value is unspecified in theopti ons structure Aswithodeset itis sufficient to type only the leading characters that uniquely identify the property name case is ignored for property names Error Tolerance Properties The solvers use standard local error control techniques for monitoring and controlling the error of each integration step At each step the local error e in thei th component of the solution is estimated and is required to be less than or equal to the acceptable error which is a function of two user defined tolerances Rel Tol andAbsTol Je i lt max RelTol abs y i AbsTol i e Rel Tol isthe rdative accuracy tolerance a measure of the error relative to the size of each solution component Roughly it controls the number of correct digits in the answer The default 1e 3 corresponds to 0 1 accuracy 8 21 8 O rdinary Differential Equations 8 22 e AbsTol isa scalar or vector of the absolute error tolerances for each solution component Abs Tol i is a threshold below which the values of the corresponding solution components are unimportant The absolute error tolerances determine the accuracy when the solution approaches zero The default value isle 6 Set tolerances using odeset either at the comma
311. nd line or in the ODE file Property Value Description Rel Tol Positive scalar A relative error tolerance that applies fle 3 to all components of the solution vector y Default value is10 3 0 1 accuracy AbsTol Positive scalar Absolute error tolerances that apply to or vector fle 6 the corresponding components of the solution vector If a scalar value is specified it applies to all components of the solution vector y Default value is10 6 The ODE solvers are designed to deliver for routine problems accuracy roughly equivalent to the accuracy you request They deliver less accuracy for problems integrated over long intervals and problems that are moderately unstable Difficult problems may require tighter tolerances than the default values For relative accuracy adjust Re Tol For the absolute error tolerance the scaling of the solution components is important if y is somewhat smaller than AbsTol the solver is not constrained to obtain any correct digits in y You might have to solve a problem more than once to discover the scale of solution components Solver Output Properties The solver output properties available with odeset let you control the output that the solvers generate With these properties you can specify an output function a function that executes if you call the solver with no output arguments In addition the ODE solver output options let you obtain Improving Solver Performance
312. nd to indicate the end of the flow control block Note You can often speed up the execution of MATLAB code by replacing f or and whi e loops with vectorized code See Optimizing the Performance of MATLAB Code on page 10 63 if else and elseif if evaluates a logical expression and executes a group of statements based on the value of the expression In its simplest form its syntax is if logical expression statements end If the logical expression is true 1 MATLAB executes all the statements between thei f andend lines It resumes execution at the line following thee nd statement If the condition is false 0 MATLAB skips all the statements between thei f andend lines and resumes execution at the line following the end statement Flow Control For example if rem a 2 disp a is even b a 2 end You can nest any number ofi f statements If the logical expression evaluates to a nonscalar value all the elements of the argument must be nonzero For example assume X is a matrix Then the statement if X statements end is equivalent to if all X statements end Theelse andelseif statements further conditionalize thei f statement e Theel se statement has nological condition The statements associated with it execute if the precedingi f and possibly elseif condition is false 0 e Theel sei f statement has a logical condition that it evaluates if the precedingif and possibly e
313. ndermonde matrix in its calculations seethepol yf it M file for details The spread of thecdate values results in scaling problems One way to deal with this is to normalize thecdate data Preprocessing Normalizing the Data Normalization is a process of scaling the numbers in a data set to improve the accuracy of the subsequent numeric computations A way tonormalizecdat e is to scale it for zero mean and unit standard deviation sdate cdate mean cdate std cdate Now try the fourth order polynomial model using the normalized data p polyfit sdate pop 4 p 0 7047 0 9210 23 4706 73 8598 62 2285 Evaluate the fitted polynomial at the normalized year values and plot the fit against the observed data points pop4 polyval p sdate plot cdate pop4 cdate pop t grid on 300 250 200 150F ge A 100F 50F B50 1800 1 350 1900 1 950 2000 6 21 6 Data Analysis and Statistics 6 22 Another way to normalize data is to use some knowledge of the solution and units F or example with this data set choosing 1790 to be year zero would also have produced satisfactory results Analyzing Residuals A measure of the goodness of fit is the residual the difference between the observed and predicted data Compare the residuals for the various fits using normalized c date values Case Study Curve Fitting Fit Residuals pl polyfit sdate pop 1 r
314. nes if the index is anumericor field name syntax Thefieldcount asset method determines how many fields there are in the asset structure and thei f statement calls the asset subs as gn method for indices 1 tofiel dcount See The Asset fieldcount Method on page 14 44 and the The Asset subsasgn Method on page 14 42 for a description of these methods Numeric indices greater than thenumber returned byf i el dcount are handled by theinner s wi t ch statement which maps the index value to the appropriate field in the stock structure Field name indexing assumes field names other than num_shares and share_price areasset fields which eliminates the need for knowledge of asset fields by child methods The asset s ubs as gn method performs field name error checking Thesubsasgn method enables you to assign values to stock object data structure using two techniques For example suppose you have a stock object s stock XYZ 100 25 You could changethedescri ptor field with either of the following statements s 1 ABC or s descriptor ABC Seethes ubsasgn help entry for general information on assignment statements in MATLAB The Stock display Method When you issue the statement without terminating with a semicolon XYZStock stock XYZ 100 25 Example Assets and Asset Subclasses MATLAB looks for a method in the st ock directory called di spl ay The displ ay method for the stock class prod
315. nomial computed with r oot s are the characteristic roots or eigenvalues of the matrix A Useei g tocompute the eigenvalues of a matrix directly 5 3 5 Polynomials and Interpolation Polynomial Evaluation Thepol yval function evaluates a polynomial at a specified value To evaluate p ats 5 use polyval p 5 ans 110 It is also possible to evaluate a polynomial in a matrix sense In this case p s x 2x 5 becomes p X X 2X 51 where X is a square matrix and istheidentity matrix F or example create a square matrix X and evaluatethe polynomial p at x Xs 2 4 56 T 0 36 7 1 5 Y polyval mp X Y 377 179 439 111 81 136 490 253 639 Convolution and Deconvolution Polynomial multiplication and division correspond to the operations convolution and deconvolution The functions conv anddeconv implement these operations Consider the polynomials a s s2 2s 3 and b s 4s2 5s 6 To compute their product 1 2 3 b 45 6 a c conv a b 5 4 Polynomials Use deconvolution to divide a s back out of the product q r deconv c a q 4 5 6 r 0 0 0 0 0 Polynomial Derivatives Thepol yder function computes the derivative of any polynomial To obtain the derivative of the polynomial p 1 0 2 5 q polyder p q 3 0 2 pol yder also computes the derivative of the product or quotient of two polynomials For example create two polynomials a andb a 1 3 5
316. ns n length u avg mean u n med median u n Subfunction _ function a mean v n Calculate average a sum v n Subfunction e function m medi an v n Calculate median w sort v if rem n 2 m w n 1 2 else m w n 2 w n 241 2 end The subfunctions mean and medi an calculate the average and median of the input list The primary function news t ats determines the length of the list and calls the subfunctions passing to them the list length n Functions within the same M file cannot access the same variables unless you declare them as global within the pertinent functions or pass them as arguments In addition the help facility can only access the primary function in an M file When you call a function from within an M file MATLAB first checks the file to see if the function is a subfunction It then checks for a private function described in the following section with that name and then for a standard M file on your search path Because it checks for a subfunction first you can 10 38 Subfunctions supersede existing M files using subfunctions with the same name for example mean in the above code Function names must be unique within an M file however Private Functions Private functions are functions that reside in subdirectories with the special namepri vate They are visible only to functions in the parent directory F or example assume the dir
317. nt catch statement statement end In this sequence the statements between tr y andcat ch are executed until an error occurs The statements between catch andend are then executed Use lasterr toseethe cause of the error If an error occurs between catch andend MATLAB terminates execution unless anothert ry cat ch sequence has been established return return terminates the current sequence of commands and returns control to the invoking function or to the keyboard r et urn is also used to terminate keyboard mode A called function normally transfers control to the function that invoked it when it reaches the end of the function r et urn may be inserted Flow Control within the called function to force an early termination and to transfer control to the invoking function 10 37 10 M File Programming Subfunctions Function M files can contain code for morethan onefunction The first function in the fileis the primary function the function invoked with the M file name Additional functions within the file are subfunctions that areonly visibletothe primary function or other subfunctions in the same file Each subfunction begins with its own function definition line The functions immediately follow each other The various subfunctions can occur in any order as long as the primary function appears first Primary function t function avg med newstats u NEWSTATS Find mean and median with internal functio
318. nt or the ot her wi se statement Only the first matchingcase is executed e An optional ot her wise group This consists of the word ot her wi se followed by the statements to execute if the expression s value is not handled by any 10 32 Flow Control of the preceding case groups Execution of theot her wi se group ends at the end statement e Anend statement swi tch works by comparing the input expression to each case value F or numeric expressions a case statement is true if val ue expressi on For string expressions a case statement is trueifstrcmp value expression The code below shows a simple example of thes wi t ch statement It checks the variablei nput_num for certain values Ifi nput_numis 1 0 or1 thecase statements display the value on screen as text Ifi nput _num is none of these values execution drops to the ot her wi se statement and the code displays the text other value switch input _num case 1 disp negative one case 0 disp zero case l disp positive one otherwise disp other value end Note for C Programmers Unlike the C languages wi t ch construct MATLAB s switch does not fall through That is if the first case statement is true other case statements do not execute Therefore break statements are not used 10 33 10 M File Programming switch can handle multiple conditions in a singlecase statement by enclosing the case expression in a cell arr
319. nt sets the number of digits in the output string or specifies an actual format p numdstr pi 9 p 3 14159265 Bothi nt2str andnum2str are handy for labeling plots For example the following lines usenum2str to prepare automated labels for the x axis of a plot function plotlabel x y plot x y strl num2str min x str2 num2str max x out Value of f from stri to strl xl abel out Another class of numeric string conversion functions changes numeric values into strings representing a decimal value in another base such as binary or hexadecimal representation For example thedec2hex function converts a decimal value into the corresponding hexadecimal string 11 13 11 Character Arrays Strings 11 14 dec_num 4035 hex_num dec2hex dec_num hex_num FC3 Seethestrfun directory for a complete listing of string conversion functions Array String Conversion TheMATLAB function mat 2st r changes an array toa stringthat MATLAB can evaluate This string is useful input for a function such as eval which evaluates input strings just as if they were typed at the MATLAB command line Create a 2 by 3 array A A 12 3 45 6 1 2 3 4 5 6 mat 2str returns a string that contains the text you would enter to created at the command line B mat2str A B 1 2 3 4 5 6 Multidimensional Arrays Multidimensional Arrays Creating Multidimensional Arrays q Getting Informatio
320. o access cell contents to access cells within cells You can also build nested cell arrays with direct assignments using the statements shown in step 3 above Indexing Nested Cell Arrays Toindex nested cells concatenate indexing expressions The first set of subscripts accesses the top layer of cells and subsequent sets of parentheses access successively deeper layers For example array A has three levels of nesting cell 1 1 cell 1 2 17 24 1 8 15 a 25 8 23 5 7 14 16 7 3 0 Test 1 4 6 13 20 22 6 7 3 10 12 19 21 3 11 18 25 2 9 2 41 5 7i 17 Cell Arrays e Toaccess the 5 by 5 array in cell 1 1 useA 1 1 e Toaccess the 3 by 3 array in position 1 1 of cell 1 2 useA 1 2 1 1 e Toaccess the 2 by 2 cell array in cell 1 2 useA 1 2 e Toaccess the empty cell in position 2 2 of cell 1 2 use A 1 2 2 2 1 2 13 31 13 Structures and Cell Arrays 13 32 Converting Between Cell and Numeric Arrays Usef or loops to convert between cell and numericformats F or example create acell arrayF F 1 1 1 2 3 4 F 1 2 1 0 0 1 F 2 1 7 8 4 1 F 2 2 4i 3421 1 8i 5 Now usethreef or loops to copy the contents of F into a numeric array NUM for k 1 4 for es Ta 2 forj 1 2 NUM i j k F k i j end end end Similarly you must usef or loops to assign each value of a numeric
321. o show representative usage of cell and content addressing You can use the two forms interchangeably Note If you already havea numeric array of a given name don t try to create a cell array of the same name by assignment without first clearing the numeric array If you do not clear the numeric array MATLAB assumes that you are trying to mix cell and numeric syntaxes and generates an error Similarly MATLAB does not clear a cell array when you make a single assignment to it If any of the examples in this section give unexpected results clear the cell array from the workspace and try again MATLAB displays the cell array A in a condensed form A 3x3 double Anne Smith 3 0000 7 0000 1x21 double To display the full cell contents usethecel disp function For a high level graphical display of cell architecture usecel plot If you assign data to a cell that is outside the dimensions of the current array MATLAB automatically expands the array to include the subscripts you 13 21 13 Structures and Cell Arrays 13 22 specify It fills any intervening cells with empty matrices For example the assignment below turns the 2 by 2 cell array A into a 3 by 3 cell array A 3 3 5 cell 1 1 cell 1 2 cell 1 3 1 4 3 05 8 Anne Smith 129 cell 2 1 cell 2 2 cell 2 3 347i 3 14 3 14 ire cell 3 1 cell 3 2 cell 3 3 p3 5 Cell Array S
322. o the least squares problem is not unique There are many vectors x that minimize norm A x b The solution computed by x A b is abasic solution it has at most r nonzero components wherer is therank of A The solution computed byx pinv A b is the minimal norm solution it also minimizes nor mi x An attempt to compute a solution with x i nv A A A b fails becauseA A is singular Hereis an example to illustrates the various solutions does not have full rank Its second column is the average of the first and third columns If b A 2 is the second column then an obvious solution toA x bisx 0 1 0 But none of the approaches computes that x The backslash operator gives x A b Warning Rank deficient rank 2 X 0 5000 0 5000 Inverses and Determinants This solution has two nonzero components The pseudoinverse approach gives y pinv A b y 0 3333 0 3333 0 33 33 Thereis no warning about rank deficiency Butnorm y 0 5774 is less than norm x 0 7071 Finally z inv A A A b fails completely Warning Matrix is singular to working precision Z nf nf nf 4 23 4 Matrices and Linear Algebra 4 24 LU QR and Cholesky Factorizations MATLAB s linear equation capabilities are based on three basic matrix factorizations e Cholesky factorization for symmetric positive definite matrices e Gaussian elimination for general square matrices e Orthogonal
323. of a sohere in such a way that the distance from any point to its nearest neighbors is the same for all the points Each point has exactly three neighbors The Bucky ball models four different physical objects e The geodesic dome popularized by Buckminster Fuller e The Ceo molecule a form of pure carbon with 60 atoms in a nearly spherical configuration e n geometry the truncated icosahedron e In sports the seams in a soccer ball The Bucky ball adjacency matrix is a 60 by 60 symmetric matrix B B has three nonzero elements in each row and column for a total of 180 nonzero values This matrix has important applications related to the physical objects listed earlier For example the eigenvalues of 8 areinvolved in studying the chemical properties of Ceo To obtain the Bucky ball adjacency matrix enter B bucky At order 60 and with a density of 5 this matrix does not require sparse techniques but it does provide an interesting example You can also obtain the coordinates of the Bucky ball graph using B v bucky Example Adjacency Matrices and Graphs This statement generates v a list of xyz coordinates of the 60 points in 3 space equidistributed on the unit sphere The function gp ot uses these points to plot the Bucky ball graph gpl ot B v axis equal 0 67 0 4 It is not obvious how to number the nodes in the Bucky ball so that the resulting adjacency matrix reflects the spherical and combinat
324. ols for error handling in MATLAB are e Theeval function which lets you execute a function and specify a second function to execute if an error occurs in the first e Thel ast err function which returns a string containing the last error generated by MATLAB Theeval function provides error handling capabilities using the two argument form eval trystring catchstring If the operation specified byt r yst ring executes properly eval simply returns Iftrystring generates an error the function evaluates cat chstring Use catchstring tospecify a function that determines the error generated by trystring and takes appropriate action Thetrystringfatchstring formofeval is especially useful in conjunction with thel asterr function ast err returns a string containing the last error message generated by MATLAB Usel asterr insidethecat chstring function to catch the error generated bytrystring For example this function uses ast err to check for a specific error message that can occur during matrix multiplication The error message indicates that matrix multiplication is impossible because the operands have different inner dimensions If the message occurs the code truncates one of the matrices to perform the multiplication 10 51 10 M File Programming function C catchfcn A B lasterr j findstr l Inner matrix dimensions if isempty j m n size A p q size B if n gt p A ptl
325. ommand Thehel p command is the most basic way to determine the syntax and behavior of a particular function Information is displayed directly in the Command Window For example help magic displays MAGIC Magic square MAGI C N is an N by N matrix constructed from the integers 1 through N 2 with equal row column and diagonal sums Produces valid magic squares for N 1 3 4 5 Help and Online Documentation Note MATLAB Command Window he p entries use uppercase characters for the function and variable names to make them stand out from the rest of the text When typing function names however always use the corresponding lowercase characters since MATLAB is case sensitive and all function names are actually in lowercase All the MATLAB functions are organized into logical groups and MATLAB s directory structure is based on this grouping For instance all the linear algebra functions reside in the mat f un directory To list the names of all the functions in that directory with a brief description of each use help matfun Matrix functions numerical linear algebra Matrix analysis norm Matrix or vector norm normest Estimate the matrix 2 norm The command hel p by itself lists all the directories with a description of the function category each represents matl ab general matl ab ops The helpwin Command The MATLAB Help Window is available by typing hel pwin The Help Window gives you acce
326. on Using plane organization client A name 2 A address 2 Using dement by el ement organization client B 2 e Element by element organization makes it easier to expand the string array fields If you do not know the maximum string length ahead of time for plane organization you may need to frequently recreate thename or address field to accommodate longer strings Typically your data does not dictate the organization scheme you choose Rather you must consider how you want to access and operate on the data Nesting Structures A structure field can contain another structure or even an array of structures Once you have created a structure you can usethestruct function or direct assignment statements to nest structures within existing structure fields Building Nested Structures with the struct Function To build nested structures you can nest calls tothest ruct function For example create a 1 by 1 structure array A struct data 3 4 7 8 0 1 nest struct testnum Test 1 xdata 4 2 8 ydata 7 1 6 Structures You can build nested structure arrays using direct assignment statements These statements add a second element to the array A 2 data 9 3 2 7 6 5 A 2 nest testnum Test 2 A 2 nest xdata 3 4 2 A 2 nest ydata 5 0 9 A 1 A 2 3 47 932 data 801 pa 765 nest testnum Test 1 L nes
327. on cholinc Incomplete Cholesky factorization normest Estimate the matrix 2 norm condest 1 norm condition number estimate sprank Structural rank Linear equations pcg Preconditioned Conjugate Gradients M ethod iterative methods bicg BiConjugate Gradients Method bicgstab BiConjugate Gradients Stabilized Method cgs Conjugate Gradients Squared M ethod gmres Generalized Minimum Residual Method 9 3 9 Sparse Matrices 9 4 Category Function Description qmr Quasi Minimal Residual M ethod Miscellaneous symbfact Symbolic factorization analysis sppar ms Set parameters for sparse matrix routines spaugment Form least squares augmented system Introduction Introduction Sparse matrices are a special class of matrices that contain a significant number of zero valued elements This property allows MATLAB to e Store only the nonzero elements of the matrix together with their indices e Reduce computation time by eliminating operations on zero elements Sparse Matrix Storage For full matrices MATLAB stores internally every matrix element Zero valued elements require the same amount of storage space as any other matrix element For sparse matrices however MATLAB stores only the nonzero elements and their indices F or large matrices with a high percentage of zero valued elements this scheme significantly reduces the amount of memory required for data storage MATLAB uses three arrays internally to store spa
328. on Program Interface Guide unregserver Modify the Windows registry to remove the Activex entries for MATLAB Use this to reset the registry For more information see Chapter 7 of the Application Program Interface Guide For example to start MATLAB and automatically run the filer esul ts m use this target path for your Windows shortcut D bin nt matlab exe r results For the same example if you start MATLAB from a DOS window automatically run the filer es ul ts m upon startup by typing matlab r results For alist of MATLAB startup options available for UNIX at the UNIX prompt type matlab h Quitting MATLAB To quit MATLAB at any time typequit at the MATLAB prompt aq On Windows platforms you can also quit by selecting Exit from the File menu or by using the close box quit runsthescriptfi nish m iffinish m exists anywhere on the MATLAB path fi nish misa file you create that contains commands you want to run when MATLAB terminates For example include as ave command in your finish mfiletosavethe workspace when MATLAB quits Orin yourfi nish m file include code that will display a confirmation dialog box when you quit MATLAB Twosamplefi nish mfilesarein tool box local e finishsav m Saves the workspace to a MAT file when MATLAB quits e finishdi g m displays a dialog allowing you to cancel quitting The Command W indow The Command Window The Command Window is the main window in which you communicate with
329. on uses more than 200 time steps it s likely that your ts pan is too long or your problem is stiff Dividet span into pieces or tryodel5s The biggest startup cost occurs as the solver attempts to finda step size appropriate to the scale of the problem If you happen to know an appropriate step size usethe ni tial Step property For example if you repeatedly call the integrator in an event location loop the last step that was taken before the event is probably on scale for the next integration Seebal ode for an example Time Steps for Integration Question Answer The first step size that the integrator takes is too large and it misses important behavior Can I integrate with fixed step sizes You can specify the first step size with the nitial Step property The integrator tries this value then reduces it if necessary No 8 51 8 O rdinary Differential Equations 8 52 Error Tolerance and Other Options Question Answer How dol chooseRel Tol and AbsTol want answers that are correct to the precision of the computer Why can t simply set Rel Tol toeps Rel Tol the relative accuracy tolerance controls the number of correct digits in the answer AbsTo the absolute error tolerance controls the difference between the answer and the solution A relative error tolerance gets into trouble when a solution component vanishes An absolute error tolerance gets into trou
330. onal interpolation an important operation for data analysis and curve fitting This function uses polynomial techniques fitting the supplied data with polynomial functions between data points and evaluating the appropriate function at the desired interpolation points Its most general form is yi interpl x y xi met hod 5 9 5 Polynomials and Interpolation 5 10 y is a vector containing the values of a function and x is a vector of the same length containing the points for which the values in y are given xi is a vector containing the points at which to interpolate met hod is an optional string specifying an interpolation method There are four different interpolation methods for one dimensional data e Nearest naghbor interpolation met hod nearest This method sets the value of an interpolated point to the value of the nearest existing data point It uses the same algorithm as ther ound function to determine which value to choose values of xi with decimal portion less than 0 5 receive the preceding value values of xi with decimal portion greater than or equal to 0 5 receive the succeeding value Out of range points receive a value of NaN Not a Number Linear interpolation met hod linear This method fits separate functions between each pair of existing data points and returns the value of therelevant function at the points specified by xi This is the default method for thei nt erp1 function Out of r
331. ons Covariance and Correlation Coefficients Finite Differences Data Pre Processing Missing Values Removing Outliers Regression and Curve Fitting Polynomial Regression i Linear in the Parameters Regression Multiple Regression TEE Case Study Curve pie Polynomial Fit Analyzing Residuals Exponential Fit Error Bounds Difference Equations and Filtering 6 3 6 7 6 9 611 6 12 6 12 6 13 6 15 6 15 6 17 6 19 6 20 6 20 6 22 625 6 28 6 29 Fourier Analysis and the Fast Fourier Transform FFT 6 31 Magnitude and Phase of Transformed Data FFT Length Versus Speed 6 37 6 38 6 Data Analysis and Statistics 6 2 This chapter introduces MATLAB s data analysis capabilities It discusses how to organize arrays for data analysis how to use simple descriptive statistics functions and how to perform data pre processing tasks in MATLAB It also discusses other data analysis topics including regression curve fitting data filtering and fast F ourier transforms FFTs The data analysis and statistics functions are in the directory dat afun in the MATLAB Toolbox Use online help to get a complete list of functions A number of related toolboxes provide advanced functionality for specialized data analysis applications Toolbox Data Analysis Application Optimization Nonlinear curve fitting and regression Signal Processing Signal processing filter
332. ons can never be 14 69 14 MATLAB Classes and O bjects 14 70 called because of rules 1 and 2 However subfunctions always take precedence over M file and overloaded M file functions 4 Private Functions A function in a private directory i e a directory named private that is below the directory containing the calling function takes precedence over other functions on the path having the same name Note that private functions with the same name as MATLAB built in functions or overloaded built in functions can never be called because of rules 1 and 2 However private functions always take precedence over M file and overloaded M file functions 5 Class Constructor Functions Constructor functions functions having names that are the same as the directory for example po ynom pol ynom m take precedence over other MATLAB functions Therefore if you create an M file called pol ynom mand put it on your path before the constructor pol ynom pol ynom m version MATLAB will always call the constructor version 6 Overloaded Methods MATLAB calls an overloaded method if it is not masked by a subfunction or private function 7 Current Directory A function in the current working directory is selected before one elsewhere on the path 8 Elsewhere On Path Finally a function anywhere else on the path is selected Selection from Multiple Directories There may be a number of directories on the path that contain methods wit
333. ons and Subscripting with the find Function Thefi nd function determines the indices of array elements that meet a given logical condition It s useful for creating masks and index matrices In its most general form f i nd returns a single vector of indices This vector can be used to index into arrays of any size or shape For example A magic 4 A 16 2 3 13 5 ll 10 8 9 7 6 12 4 14 15 1 i find A gt 8 A i 100 A 100 2 3 100 5 100 100 8 100 7 6 100 4 100 100 1 You can also usef i nd to obtain both the row and column indices for a rectangular matrix as well as the array values that meet the logical condition Use thehel p facility for more information on fi nd Operator Precedence You can build expressions that use any combination of arithmetic relational and logical operators Precedence levels determine the order in which MATLAB evaluates an expression Within each precedence level operators have equal precedence and are evaluated from left to right The precedence 10 27 10 M File Programming 10 28 rules for MATLAB operators are shown in this table ordered from highest precedence level to lowest precedence level Operator Precedence Level Highest precedence negation unary plus unary minus dae otil Salle p CRS NE An addition Subtraction lt lt gt gt Lowest precedence Precedence of amp and MATLAB s left to right execut
334. ons where the unknown matrix appears on the left or right of the coefficient matrix X A B denotes the solution to the matrix equation AX B X B A denotes the solution to the matrix equation XA B You can think of dividing both sides of the equation AX B or XA B byA The coefficient matrix A is always in the denominator The dimension compatibility conditions for X A B require the two matrices A andB to have the same number of rows The solution X then has the same number of columns as8 and its row dimension is equal tothe column dimension of A For X B A theroles of rows and columns are interchanged 4 13 4 Matrices and Linear Algebra 4 14 In practice linear equations of the form AX B occur more frequently than those of the form XA B Consequently backslash is used far more frequently than slash The remainder of this section concentrates on the backslash operator the corresponding properties of the slash operator can be inferred from the identity B A A B The coefficient matrix A need not be square If A is m by n there are three cases m n Square system Seek an exact solution m gt n Overdetermined system Find a least squares solution m lt n Underdetermined system Find a basic solution with at most m nonzero components The backslash operator employs different algorithms to handle different kinds of coefficient matrices The various cases which are diagnosed automa
335. ontain the features to be restricted from a particular user host or group that user group or host is not allowed to use that product You may have multiple XCLUDE lines for the same feature as well Any line starting with GROUP defines the members of a group name used in the previous lines of this file License manager groups are distinct from UNIX protection groups and any other groups defined outside of MATLAB If a group name is used in aRESERVE NCLUDE or EXCLUDE line the group membership must be defined in a GROUP line 2 61 2 MATLAB W orking Environment 2 62 Debugger and Profiler MATLAB Debugger Debugging An Overview M Files For An Example Session Trial Run Se Debugging Using the Graphical User nterface Debugging from the Command Line M File Profiler Profiling An Overview How the Profiler Works The profile Command i An Example Using the Profiler Viewing Profiler Results Saving Profile Reports 3 2 3 2 3 4 3 4 3 10 3 17 3 17 3 17 3 18 3 19 3 20 3 25 3 Debugger and Profiler MATLAB Debugger 3 2 The MATLAB Debugger helps you identify programming errors in your MATLAB code Using the Debugger you can view the contents of the workspace at any time during function execution view the function call stack and execute M file code line by line MATLAB s Debugger has a graphical user interface and provides a command line interface as
336. ore each character as a 16 bit numeric value Use the double function to convert strings to their numeric values and char to revert to character representation name doubl e name name Columns 1 through 12 84 104 111 109 97 115 32 82 46 32 76 101 Column 13 101 name char name name Thomas R Lee Creating Two Dimensional Character Arrays When creating a two dimensional character array be sure that each row has the same length For example this line is legal because both input rows have exactly 13 characters name Thomas R Lee Sr Developer name Thomas R Lee Sr Developer When creating character arrays from strings of different lengths you can pad the shorter strings with blanks to force rows of equal length name Thomas R Lee Senior Developer 11 5 11 Character Arrays Strings 11 6 A simpler way to create string arrays is to usethechar function char automatically pads all strings to the length of the longest input string In this example char pads the 13 character input string Thomas R Lee with three trailing blanks so that it will be as long as the second string name char Thomas R Lee Senior Developer name Thomas R Lee Senior Devel oper When extracting strings from an array usethedebl ank function toremoveany trailing blanks trimname debl ank name 1 trimname Thomas R Lee size trimname ans Cell Arrays
337. ores your options in a registry file located under the directory HOME wi ndu Communication with this file is handled through a daemon called windu_registryd41 orwindu_registryd40 which is started automatically when you start MATLAB and is shut down shortly after you quit MATLAB 2 55 2 MATLAB W orking Environment 2 56 Toview theregistry type the commandr egedit Ther egedi t command starts the registry editor Your settings are located under HKEY_CURRENT_USER Software MathWorks This registry is not portable across different UNIX platforms The registry can only be read and written to on the platform on which it was created If you start the MATLAB Editor on a different platform it will start a registry daemon with the default options These options can be saved and retrieved from the registry daemon but once the daemon is shut down the options will be lost When you start the Editor if you get messages about missing options it is possible that your registry has somehow been corrupted To fix this problem kill your registry daemon and delete the registry database directory Then restart MATLAB and it will create a new registry with the default options Path Browser The Path Browser lets you view and modify MATLAB s search path and see all of its files To open the Path Browser typepat ht ool at thecommand line This is essentially the same Path Browser as the one provided for Windows platforms See the section
338. orial symmetries of the graph The numbering used by bucky mis based on the pentagons inherent in the ball s structure The vertices of one pentagon are numbered 1 through 5 the vertices of an adjacent pentagon are numbered 6 through 10 and so on The picture on the following page shows the numbering of half of the nodes one hemisphere the numbering of the other hemisphere is obtained by a reflection about the 9 17 9 Sparse Matrices equator Usegp ot toproducea graph showing half the nodes Y ou can add the node numbers using af or loop k 1 30 gplot B k k v axis Square for j 1 30 text v j 1 v j 2 int2str j end 0 87 0 67 0 4 02t 0 2F 0 4 F 9 18 Example Adjacency Matrices and Graphs To view a template of the nonzero locations in the Bucky ball s adjacency matrix use thes py function spy B Oe ee ee e oo e e ee ee e ere e 10a eS ee o ee e ee e e ne ke eee of re 207 e ee ee co o ee e ee e eee e e o oo o o 3 30 e e oo e ee e ee eee eve eee ee e oreo 40 o o s e ee e ee eee ee e o o e ee s MTI 50f oS E e ee o ee e ee ove e e o oe ee 60t l i i oo eo 0 10 20 30 60 nz 180 The node numbering that this model uses generates a spy plot with twelve groups of five elements corresponding to the twelve pentagons in the structure Each node is connected to two othe
339. orm all debugging tasks from the command line To follow this example use the M files from the beginning of this chapter function y variance x function tot sqsum x mu mu sum x length x tot 0 tot sqsum x mu for i 1 1 ength mu y tot length x 1 tot tot end Setting Breakpoints x i mu 2 dbstop inserts a breakpoint at a specified line The M file halts before the line actually executes Set breakpoints in variance after the computation of the mean line 2 and after the computation of the mean removed squared sum line 3 using dbstop variance 3 dbstop variance 4 3 11 3 Debugger and Profiler 3 12 Stepping Through Code and Using Keyboard Mode Take the vectorv 1 2 3 4 5 as example input The expected values for the mean and the mean removed squared sum are3 and1 0 respectively See if you get the expected results at the breakpoints E xecute the function from the command line variance v MATLAB displays the next line to be executed and its line number 3 tot sqsum x mu K gt gt When execution stops at a breakpoint you re automatically in keyboard mode as indicated by thek gt gt prompt At this prompt you can type standard MATLAB commands When you re finished typedbcont to resume execution Note On some platforms a debugging window may automatically appear when a function stops at a breakpoint When execution halts at the first breakpoint
340. orm v 1 normv normv inf ans 3 0000 2 2361 2 0000 The p norm of a matrix A max Alp IAx IX can be computed for p 1 2 andeoby norm A p Again the default valueis p 2 norm C 1 norm C norm C inf ans 19 0000 14 8015 13 0000 4 12 Solving Linear Equations Solving Linear Equations One of the most important problems in technical computing is the solution of simultaneous linear equations In matrix notation this problem can be stated as follows Given two matrices A and B does there exist a unique matrix X so that AX B or XA B It is instructive to consider a 1 by 1 example Does the equation 7X 21 have a unique solution The answer of course is yes The equation has the unique solution x 3 The solution is easily obtained by division x 21 7 3 The solution is not ordinarily obtained by computing the inverse of 7 that is 71 0 142857 and then multiplying 7t by 21 This would be more work and if 71 is represented to a finite number of digits less accurate Similar considerations apply to sets of linear equations with more than one unknown MATLAB solves such equations without computing the inverse of the matrix Although it is not standard mathematical notation MATLAB uses the division terminology familiar in the scalar case to describe the solution of a general system of simultaneous equations The two division symbols slash and backslash are used for the two situati
341. os ones rand andeye always return full results This is necessary to avoid introducing sparsity unexpectedly The sparse analog of zeros m n is simply s parse m n Thesparse analogs of rand andeye aresprand and speye respectively There is no sparse analog for the function ones 9 23 9 Sparse Matrices 9 24 Unary functions that accept a matrix and return a matrix or vector preserve the storage class of the operand If S is a sparse matrix then chol S is also a sparse matrix and di ag S iS a Sparse vector Columnwise functions such as max ands um also return sparse vectors even though these vectors may be entirely nonzero Important exceptions to this rulearethes parse andf ul functions Binary operators yield sparse results if both operands are sparse and full results if both are full For mixed operands the result is full unless the operation preserves sparsity IfS is sparse andF is full then S F S F and F S arefull whiles F andS amp F are sparse In some cases the result might be sparse even though the matrix has few zero elements Matrix concatenation using either thecat function or square brackets produces sparse results for mixed operands Submatrix indexing on the right side of an assignment preserves the storage format of the operand T S i j produces a sparse result if S is sparse whether i andj are scalars or vectors Submatrix indexing on the left asin T i j does not change the storage format of
342. our xi yi zi2 contour xi yi zi 3 nearest bilinear bicubic Notice that the bicubic method in particular produces smoother contours This is not always the primary concern however For some applications such as medical image processing a method like nearest neighbor may be preferred because it doesn t generate any new data values 5 14 Interpolation Interpolation and Multidimensional Arrays Several interpolation functions operate specifically on multidimensional data Function Description interp3 Three dimensional data interpolation interpn Multidimensional data interpolation ndgrid Multidimensional data gridding ndf un directory Interpolation of Three Dimensional Data The function i nt er p3 performs three dimensional interpolation finding interpolated values between points of a three dimensional set of samples V You must specify a set of known data points e X Y andZ matrices specify the points for which values of V are given e A matrixV contains values corresponding to the points in X Y andZ The most general form for i nt er p3 is VI interp3 X Y Z V XI YI ZI met hod XI Y andZl arethe points at which i nt er p3 interpolates values of V For out of range values i nt erp3 returns NaN There are three different interpolation methods for three dimensional data e Nearest neighbor interpolation method nearest This method chooses the value of the nearest point e Tri
343. owser from the View menu Double click on a File to Open It Path Browser olx Fie Edit View Path Tools Help 2 lx 2 Current Directory Devtools matlabS nightly bin nt Browse B addpath m Path E binpatch m Devtools matlabS nightly toolbox matlab general fe cd m Devtools matlabS nightly toolbox matlab ops le clear m Directories Devtools matlabS nightly toolbox matlab lang B computer m on Search Devtools matlabS nightly toolbox matlab elmat Devtools mat labS night ly toolbox mat lab el fun E sancti Path Devtools matlabS nightly toolbox matlab specfun E copyfile m Devtools matlabS nightly toolbox matlab mat fun E dbclear m Devtools matlabS nightly toolbox matlab datafun B dbcont m Devtools matlabS nightly toolbox matlab polyfun B dbdown m Devtpols matlabS nightly toolbox matlab funfun E dbmex m Devtpols matlabS nightly toolbox matlab spar fun i E dbquit m gt JA Tart ha Lelmat Lah Ol wi aht Irritanlharimat Lahi aenn A B dbstack m To Move a Directory Drag It to the Desired Position Use the menus in the Path Browser to e Add a directory to the front of the path e Remove a selected directory from the path e Save settings tothe pat hdef m file e Restore default settings Click Browse to find a directory to add to the path The Browse for Folder dialog box opens Note that the current directory displayed in this dialog box is the directory you had selected in
344. p to break t span into pieces on which the ODE function is smooth If the function is smooth and the code is taking extremely small steps you are probably trying to solve a stiff problem with a solver not intended for this purpose Switch toode15s 0de23s or ode23tb 8 55 8 O rdinary Differential Equations Troubleshooting Question Answer My integration proceeds very slowly using too many time steps know that the solution undergoes a radical change at timet where t0 lt t lt tfinal but the integrator steps past without seeing it First check that your t span is not too long Remember that the solver will use as many time points as necessary to produce a smooth solution If the ODE function changes on a time scale that is very short compared to thet s pan then the solver will use a lot of time steps Long time integration is a hard problem Break tspan into smaller pieces If the ODE function does not change noticeably on thet span interval it could be that your problemis stiff Try usingode15s or ode23s Finally makesurethat the ODE function is written in an efficient way The solvers evaluate the derivatives in the ODE function many times The cost of numerical integration depends critically on the expense of evaluating the ODE function Rather than recompute complicated constant parameters every evaluation store them in globals or calculate them once outside the function and pass them
345. pare corresponding elements of arrays with equal dimensions For vectors and rectangular arrays both operands must be the same size unless one is a scalar For the case where one operand is a scalar and the other is not MATLAB tests the scalar against every element of the other operand Locations where the specified relation is true receive the value 1 Locations where the relation is false receive the value 0 10 25 10 M File Programming Logical Functions In addition tothe logical operators MATLAB provides a number of logical functions Function Description Examples xor Performs an exclusive 0R on its a l operands xor returns true if one b 1 operand is true and the other false xor a b In numeric terms the function returns 1 if one operand is nonzero ans and the other operand is zero 0 al Returns 1 if all of the elements ina u 01 2 0 vector aret rue or nonzero al all u operates columnwise on matrices ans 0 A 01 2 3 5 0 all A ans 0 1 0 any Returns 1 if any of the elements of v 5 0 8 its argument are true or nonzero any v otherwise it returns0 Likeal the any function operates ans columnwise on matrices A number of other MATLAB functions perform logical operations F or example thei snan function returns1 for NaNs thei si nf function returns for nf s See theops directory for a complete listing of logical functions 10 26 O perators Logical Expressi
346. pended For example average m If the filename and the function definition line nameare different the internal name is ignored 10 9 10 M File Programming 10 10 Thus whilethefunction name specified on the function definition line does not have to be the same as the filename we strongly recommend that you use the same name for both How Functions Work You can call function M files from either the MATLAB command line or from within other M files Be sure to include all necessary arguments enclosing input arguments in parentheses and output arguments in square brackets Function Name Resolution When MATLAB comes upon a new name it resolves it into a specific function by following these steps 1 Checks to see if the name is a variable 2 Checks to see if the nameis a subfunction a MATLAB function that resides in the same M file as the calling function Subfunctions are discussed on page 10 38 3 Checks to see if the name is a private function a MATLAB function that resides in a private directory a directory accessible only to M files in the directory immediately above it Private directories are discussed on page 10 39 4 Checks to see if the name is a function on the MATLAB search path MATLAB uses the first file it encounters with the specified name If you duplicate function names MATLAB executes the one found first using the above rules It is also possible to overload function names This uses additional
347. pose Matrix transpose Display method Horizontal concatenation Vertical concatenation Subscripted reference Subscripted assignment Subscript index Overloading Functions You can overload any function by creating a function of the same name in the class directory When a function is invoked on an object MATLAB always looks in the class directory before any other location on the search path To overload thep ot function for a class of objects for example simply place your version of pl ot min the appropriate class directory Examples of Overloaded Functions See the following sections for examples of overloaded functions e Overloading Functions for the Polynom Class on page 14 31 e The Portfolio pie3 Method on page 14 57 Example A Polynomial C lass Example A Polynomial Class This example implements a MATLAB data type for polynomials by defining a new class called polynom The class definition specifies a structure for data storage and defines a directory po ynom of methods that operate on polynom objects Polynom Data Structure The polynom class represents a polynomial with a row vector containing the coefficients of powers of the variable in decreasing order Therefore a polynom object p is a structure with a single field p c containing the coefficients This field is accessible only within the methods in the po ynom directory Polynom Methods Tocreate a class that is well behaved with
348. pplying Functions and Operators 2 oe ee 13 27 Organizing Data in Cell Arrays 13 28 Nesting Cell Arrays ee es 13 29 Converting Between Cell and Numeric Arrays a 13 32 Cell ArraysofStructures 13 33 13 Structures and Cell Arrays 13 2 Structures are collections of different kinds of data organized by named fields Cdl arrays area special class of MATLAB array whose elements consist of cells that themselves contain MATLAB arrays Both structures and cell arrays provide a hierarchical storage mechanism for dissimilar kinds of data They differ from each other primarily in the way they organize data Y ou access data in structures using named fields whilein cell arrays data is accessed through matrix indexing operations This table describes the MATLAB functions for working with structures and cell arrays Category Function Description Structure functions fieldnames getfield isfield isstruct rmfield setfield struct struct2cel Cell array functions ce cell 2struct cell disp cell fun cell plot dea iscel num2cel Get structure field names Get structure field contents True if field is in structure array True for structures Remove structure field Set structure field contents Create or convert to structure array Convert structure array into cell array Create cell array Convert cell array into structure array Display cell arr
349. produce output or they can be scri pts that execute a series of MATLAB statements F or MATLAB to recognize a file as an M file its name must end in m You create M files using a text editor then use them as you would any other MATLAB function or command The process looks like this 1 Create an M file using a text function c myfile a b editor c sqrt a 2 b 2 2 Call the M file from the ae command line or from within b 3 342 another M file c myfile a b C 8 2109 10 2 M ATLAB Programming A Quick Start Kinds of M Files There are two kinds of M files Script M Files Function M Files e Donot accept input arguments or Can accept input arguments and return output arguments return output arguments e Operate on data in the workspace Internal variables are local tothe function by default e Useful for automating a series of Useful for extending the steps you need to perform many MATLAB language for your times application Whats in an M File This section shows you the basic parts of a function M file so you can familiarize yourself with MATLAB programming and get started with some examples Function definition line _ function f fact n H1 help 1 line FACT Factorial FACT N returns the factorial of N usually denoted by N L Put simply FACT N is PROD 1 N Help text gt Function body prod 1 n Thi
350. proximate incomplete factorizations which are useful as preconditioners for sparse iterative methods Thel ui nc function produces two different kinds of incomplete LU factorizations one involving a drop tolerance and one involving fill in level If A is a Sparse matrix andt 0 is asmall tolerance then L U luinc A tol computes an approximate LU factorization where all elements less than t ol times the norm of the relevant column are set to zero Alternatively L U luinc A 0 computes an approximate LU factorization where the sparsity pattern of L U is a permutation of the sparsity pattern of A For example load west0479 A west 0479 nnz A nnz lu A nnz luinc A le 6 nnz luinc A 0 shows that A has 1887 nonzeros its complete LU factorization has 16777 nonzeros its incomplete LU factorization with a drop tolerance of 1e 6 has 10311 nonzeros and its u 0 factorization has 1886 nonzeros Thel ui nc function has a few other options See the online help for details Thecholi nc function provides drop tolerance and level 0 fill in Cholesky factorizations of symmetric positive definite sparse matrices See the online help for more information Sparse M atrix O perations Simultaneous Linear Equations Systems of simultaneous linear equations can be solved by two different classes of methods e Direct methods These are usually variants of Gaussian elimination and are often expressed a
351. quence of the function The function definition line for theaverage function is function y average x core function name output argument keyword All MATLAB functions havea function definition line that follows this pattern If the function has multiple output values enclose the output argument list in square brackets nput arguments if present are enclosed in parentheses Use commas to separate multiple input or output arguments Here s a more complicated example function x y z sphere theta phi rho If there is no output leave the output blank function printresults x or use empty square brackets function printresults x The variables that you pass to the function do not need to have the same name as those in the function definition line H1 Line The H1 line so named because it is the first help text line is a comment line immediately following the function definition line Because it consists of comment text the H1 line begins with a percent sign For theaverage function the H1 lineis AVERAGE Mean of vector elements This is thefirst line of text that appears when a user typeshel p function_name at the MATLAB prompt Further the ookf or command searches on and displays only the H1 line Because this line provides important summary 10 7 10 M File Programming information about the M file it is important to make it as descriptive as possible Help Text You can c
352. r Pathname Pat hname Inserts Inserts Inserts complete complete name of pathname of pathname variable this file of file selected in right hand panel File File Inserts Inserts Inserts name ofthis name of name of function file i variable without its selected in extension right hand panel without extension Selection Sel Inserts the Inserts Inserts text that is complete indexing highlighted pathname expression in this view of file correspond selected in ing to right hand selection panel numbers in bottom right panel 2 43 2 MATLAB W orking Environment 2 44 Submenu Variable Invoked Invoked Invoked Item Inserted from Editor from Path from Array Browser Editor Quoted QuotedSel Adds Adds Selection additional single single quotes to quotes to selected strings to pathname preserve quotes e g and transmits who s becomes to who s MATLAB Beginning BeginSel Inserts of Selection character offset of first character in the selection End of EndSel Inserts Selection character offset of last character in the selection When you invoke your custom tool 1 All open files are saved 2 Appropriate values are substituted for all substitution variables in the tool s MATLAB expression 3 The expression is evaluated in MATLAB 4 The current file is reloaded if it has changed Microsoft W indows Handbook Here are some examples of expr
353. r This series of statements uses approximately 32 4 MB of RAM a rand le6 1 b rand le6 1 clear c rand 2 1le6 1 10 69 10 M File Programming 10 70 while these statements use only about 16 4 MB of RAM c rand 2 1e6 1 clear a rand le6 1 b rand 1le6 1 Allocating the largest vectors first allows for optimal use of the available memory What Does Out of Memory Mean Typically the Out of Memory message appears because MATLAB asked the operating system for a segment of memory larger than what is currently available Use any of the techniques discussed in this section to help optimize the available memory If theOut of Memory message still appears e Increase the size of the swap file e Make sure that there are no external constraints on the memory accessible to MATLAB on UNIX systems use the limit command to check e Add more memory to the system e Reduce the size of your data Character Arrays Strings Character Arrays Converting Between Characters and Nu umeric cValues Creating Two Dimensional Character Arrays Cell Arrays of Strings Converting Between Character Arrays and Cell Arrays of Strings String Comparisons Comparing Strings F or E quality Comparing Characters for Equality with Operators Categorizing Characters Within a String Searching and Replacing String Numeric Conversion Array String Conversion 11 4 11 5 11 5 1
354. r 15 2 fft 6 37 6 38 FFT See Fourier transform fast fget 15 2 15 13 fgets 15 2 15 13 fid 15 3 fieldnames 13 4 fields 13 3 accessing data within 13 6 accessing with get field 13 8 adding to structure array 13 9 applying functions to 13 9 all likenamed fields 13 10 assigning data to 13 3 assigning withsetfield 13 8 defined 13 3 deleting from structures 13 9 indexing within 13 7 names 13 4 size 13 9 writing M files for 13 10 fields 13 4 figure 2 18 figure window 2 18 file identifier 15 3 15 4 15 5 filenames contained in variables 2 13 wildcards for loading 2 13 files ASCII 15 13 loading 2 13 2 27 reading formatted text 15 14 reading line by line 15 13 saving 2 28 writing 15 15 Index 1 10 beginning of 15 10 binary 15 7 controlling data type values read 15 8 controlling number of values read 15 7 reading 15 7 writing to 15 8 closing 15 2 15 3 15 5 Contents m 10 9 current position 15 10 editing M files 2 39 end of 15 2 15 10 exchanging between platforms 2 31 failing to open 15 4 formatted 15 13 identifiers 15 4 input output 15 2 license dat 2 59 local options 2 60 log 2 3 manipulating 2 25 MAT files 2 11 2 31 matl abrc m2 2 M files 2 31 debugging 3 2 opening 2 16 15 2 15 3 permissions 15 3 position 15 2 15 10 reading with C or Fortran programs 2 28 registry 2 55 saving ASCII 2 12 search path default 2 15 sound reading 2 27 sound writing 2 29 spreadsheet 2 27
355. r example each of the parent objects could have inherited fields from multiple 14 35 14 MATLAB Classes and O bjects 14 36 grandparent objects and so on Multiple inheritance is implemented in the constructors by callingc ass with more than three arguments obj class structure class_name parentl parent2 You can append as many parent arguments as desired to the class input list Multiple parent classes can have associated methods of the same name In this case MATLAB calls the method associated with the parent that appears first in thecl ass function call in the constructor function There is no way to access subsequent parent function of this name Aggregation In addition to standard inheritance MATLAB objects support containment or aggregation That is one object can contain embed another object as one of its fields For example a rational object might use two polynom objects one for the numerator and one for the denominator You can call a method for the contained object only from within a method for the outer object When determining which version of a function to call MATLAB considers only the outermost containing class of the objects passed as arguments the classes of any contained objects are ignored See Example The Portfolio Container on page 14 54 for an example of aggregation Example Assets and Asset Subc lasses Example Assets and Asset Subclasses Asan example of simple
356. r nodes within its pentagon and one node in some other pentagon Since the nodes within each pentagon have consecutive numbers most of the elements in the first super and sub diagonals of B are nonzero In addition the symmetry of the numbering about the equator is apparent in the symmetry of the spy plot about the antidiagonal 9 19 9 Sparse Matrices 9 20 Graphs and Characteristics of Sparse Matrices Spy plots of the matrix powers of B illustrate two important concepts related to sparse matrix operations fill in and distance s py plots help illustrate these concepts spy B 2 spy B 3 spy B 4 spy B 8 nz 420 nz 780 0 20 40 60 0 20 40 60 nz 1380 nz 3540 Fill in is generated by operations like matrix multiplication The product of two or more matrices usually has more nonzero entries than the individual terms and so requires more storage As p increases B p fills in ands py B p gets more dense Example Adjacency Matrices and Graphs The distance between two nodes in a graph is the number of steps on the graph necessary to get from one node to the other The spy plot of the p th power of B shows the nodes that area distancep apart Asp increases it is possible to get tomoreand morenodes inp steps For the Bucky ball B 8 is almost completely full Only the antidiagonal is zero indicating that it is possible to get from any node to any other node
357. r this problem J acobian matrix J t y oF dy Matrix showing the acobian sparsity pattern Mass matrix M for solving M t y y F t y Information to define an event location problem The template below illustrates how to code an extended ODE file that uses the switch construct and the ODE file s third input argument f ag to supply additional information For illustration the file also accepts two additional input parameters p1 and p2 8 18 Improving Solver Performance Note The example below is only a template In your own coding you should not include all of the cases shown For example j acobi an information is used for evaluating J acobians analytically and j pattern information is used for generating J acobians numerically function varargout odefile t y flag pl p2 switch flag case Return dy dt f t y varargout l f t y p1 p2 case init Return default tspan y0 options varargout 1 3 init pl p2 case jacobi an Return Jacobian matrix df dy varargout l j acobian t y pl p2 case j pattern Return sparsity pattern matrix S varargout l j pattern t y pl p2 case mass i Return mass matrix varargout 1 mass t y pl p2 case events Return value isterminal direction varargout 1 3 events t y pl p2 otherwise error Unknown flag flag end Qf anaa aer EAE Ea ae i Gant town eta Gh E E
358. raph lends itself to matrix representation The adjacency matrix of an undirected graph is a matrix whose i j th and j i th entries are 1 if nodei is connected to nodej and 0 otherwise For example the adjacency matrix for a diamond shaped graph looks like Since most graphs have relatively few connections per node most adjacency matrices are sparse The actual locations of the nonzero elements depend on how the nodes are numbered A changein the numbering leads to permutation of the rows and columns of the adjacency matrix which can havea significant effect on both the time and storage requirements for sparse matrix computations Graphing Using Adjacency Matrices MATLAB s gp ot function creates a graph based on an adjacency matrix anda related array of coordinates Totry gp ot create the adjacency matrix shown above by entering A 0101 1010 0101 101 0 9 15 9 Sparse Matrices 9 16 Thecolumns of gp ot s coordinate array contain the Cartesian coordinates for the corresponding node F or the diamond example create the array by entering Ky Ss ide Bie ee Te B73 dB This places the first node at location 1 3 the second at location 2 1 the third at location 3 3 and the fourth at location 2 5 To view the resulting graph enter gpl ot A xy The Bucky Ball One interesting construction for graph analysis is the Bucky ball This is composed of 60 points distributed on the surface
359. rative methods and incomplete factorizations is available in Saad Yousef Iterative Methods for Sparse Linear E quations PWS Publishing Company 1996 Barrett Richard et al Templates for the Solution of Linear Systems Building Blocks for Iterative Methods Society for Industrial and Applied Mathematics 1994 9 35 9 Sparse Matrices 9 36 Eigenvalues and Singular Values Two functions are available which compute a few specified eigenvalues or singular values Function Description ei gs F ew eigenvalues svds F ew singular values These functions are most frequently used with sparse matrices but they can be used with full matrices or even with linear operators defined by M files The statement V lambda ei gs A k sigma finds thek eigenvalues and corresponding eigenvectors of the matrix A which are nearest the shift si gma Ifsi gma is omitted the eigenvalues largest in magnitude are found If si gma is zero the eigenvalues smallest in magnitude are found A second matrix B may be included for the generalized eigenvalue problem Av ABv The statement U V svds A k finds thek largest singular values of A and U S V svds A k 0 finds thek smallest singular values For example the statements L numgrid L 65 A delsq L Sparse M atrix O perations set up the five point Laplacian difference operator on a 65 by 65 grid in an L shaped two dimensional domai
360. rch attempts to return a vector x that is a local minimizer of the mathematical function near this starting vector Totryfminsearch createan M filet hree_var m that defines a function of three variables x y andz function b three var v x v 1 y v 2 z v 3 b x 2 2 5 sin y z 2 x 2 y 2 7 8 Minimizing Functions and Finding Zeros Now find a minimum for this function usingx 0 6 y 1 2 and z 0 135 as the starting values v 0 6 1 2 0 135 a fminsearch three var v 0 0000 1 5708 0 1803 Setting Minimization Options You can specify a vector of control options that sets some minimization parameters by calling f mi nbnd with the syntax x fminbnd fun x1 x2 options or f minsearch with the syntax x fminsearch fun x0 options options is a structure used by specialized Optimization Toolbox functions To set values as needed use options optimset Display iter to generate output at each iteration f mi nbnd and f mi nsearch use only four of theopti ons parameters e options Display isa flag that determines if intermediate steps in the minimization appear on the screen If set to i ter intermediate steps are displayed if set to of f no intermediate solutions are displayed if set to final displays just the final output e options Tol X is the termination tolerance for x Its default value is 1 e 4 e options Tol Fun is the termination tolerance for
361. re are some things you can do e List the names of the files in your current directory what e List the contents of M filef act m type fact e Call thef act function fact 5 ans 120 Scripts Scripts Scripts are the simplest kind of M file they have no input or output arguments They re useful for automating series of MATLAB commands such as computations that you have to perform repeatedly from the command line Scripts operate on existing data in the workspace or they can create new data on which to operate Any variables that scripts create remain in the workspace after the script finishes so you can use them for further computations Simple Script Example These statements calculaterho for several trigonometric functions oft heta then create a series of polar plots Comment line ______ An M file script to produce flower petal plots rtheta pi 0 01 pi rho 1 2 sin 5 theta 2 Computations H w rho 2 cos 10 theta 3 rho 3 sin theta 2 Lrho 4 5 cos 3 5 theta 3 for i 1 4 Graphical output polar theta rho i commands pause end Try entering these commands in an M file called pet als m This file is now a MATLAB script Typing pet al s at the MATLAB command line executes the statements in the script After the script displays a plot press Return to move to the next plot There are noinput or output arguments pet al s creates the varia
362. reate online help for your M files by entering text on one or more comment lines beginning with the line immediately following the H1 line The help text for the average function is AVERAGE X where X is a vector is the mean of vector elements Nonvector input results in an error When you typehel p function_name MATLAB displays the comment lines that appear between the function definition line and the first non comment executable or blank line The help system ignores any comment lines that appear after this help block For example typinghelp sin results in SIN Sine SIN X is the sine of the elements of X Function Body The function body contains all the MATLAB code that performs computations and assigns values to output arguments The statements in the function body can consist of function calls programming constructs like flow control and interactive input output calculations assignments comments and blank lines For example the body of the aver age function contains a number of simple programming statements m n size x Flow control if m 1 n 1 m 1 amp n 1 Error message display error Input must be a vector end Computation and assignment y sym x length x 10 8 Functions Comments As mentioned earlier comment lines begin with a percent sign Comment lines can appear anywhere in an M file and you can append commen
363. red by MATLAB 14 9 overview 14 2 clc 2 6 cl ear 2 10 10 11 10 66 clear class definition 14 6 clearing M files 3 3 clearing the Command Window 2 6 closest point searches 5 22 closing files 15 2 15 3 15 5 closing MATLAB 2 4 col mmd 9 27 9 28 colon operator 4 8 10 21 12 10 for multidimensional array subscripting 12 10 in subscripts 8 28 to access subsets of cells 13 24 used to index a page 12 16 used with scalar expansion 12 6 color printing from Editor 2 41 2 48 col perm 9 26 column vector 4 5 for polynomial roots representation 5 3 indexing as 10 43 Index of event locations ODE 8 31 columns 4 27 deleting 10 42 comma to separate function arguments 10 7 command line editing 2 5 command to operating system 2 25 Command Window 2 5 2 35 commands on multiple lines 2 10 comma separated list using cell arrays 13 25 comments in code 10 9 comparing interpolation methods 5 13 sparse and full matrix storage 9 11 strings 11 9 complex conjugate transpose 4 8 complex conjugate transpose operator 10 22 complex values in sparse matrix 9 5 computational functions applying to cell arrays 13 27 applying to multidimensional arrays 12 15 applying to sparse matrices 9 23 applying to structure fields 13 9 in M file 10 3 computer 10 18 computer type 10 18 concatenating cell arrays 13 22 strings 10 46 concatenation 10 41 condest 9 3 condition dimension compatibility 4 13 conditional statements 10 12 cond
364. ref A S where S type S subs field These simple calls are combined for more complicated subscripting expressions In such cases ength S isthenumber of subscripting levels For example A 1 2 name 3 4 callssubsref A S wheres is a 3 by 1 structure array with the values S 1 type 2 type 3 type abe 1 subs 1 2 S 2 subs name S 3 subs 3 4 How to Write subsref Thesubsref method must interpret the subscripting expressions passed in by MATLAB A typical approach is to use thes wi t ch statement to determine the type of indexing used and to obtain the actual indices The following three code fragments illustrate how to interpret the input arguments In each case the function must return the value B For an array index switch S type case B A S subs end For a cell array switch S type case B A S subs Bis a cell array end Designing User Classes in MATLAB For a structure array switch S type case switch S subs case fieldl B A fieldl case field2 B A field2 end end Examples of the subsref Method See the following sections for examples of thesubsref method e The Polynom subsref Method on page 14 28 e The Asset subsref Method on page 14 40 e The Stock subsref Method on page 14 48 e The Portfolio subsref Method on page 14 64 Subscripted Assignment The use of a subscrip
365. res a family of application specific solutions called toolboxes Very important to most users of MATLAB toolboxes allow you to learn and apply specialized technology Toolboxes are comprehensive collections of MATLAB functions M files that extend the MATLAB environment in order to solve particular classes of problems M any toolboxes are available from The MathWorks Some of these are listed on the following page contact The MathWorks or visit www mat hwor ks com for a complete up to date list The MATLAB Product Family How The MathWorks products fit together MATLAB is the foundation for all The MathWorks products MATLAB combines numeric computation 2 D and 3 D graphics and language capabilities in a single easy to use environment MATLAB Extensions are optional tools that support the implementation of systems developed in MATLAB Toolboxes are libraries of MATLAB functions that customize MATLAB for solving particular classes of problems Toolboxes are open and extensible you can view algorithms and add your own Simulink is a system for nonlinear simulation that combines a block diagram interface and live simulation capabilities with the core numeric graphics and language functionality of MATLAB Simulink Extensions are optional tools that support the implementation of systems developed in Simulink Blocksets are collections of Simulink blocks designed for use in specific application areas
366. rgument Thei sa function checks for this case e More than two input arguments If there are more than two input arguments the constructor assumes the first is the client s name and the rest are asset subclass objects A more thorough implementation would perform more careful input argument checking for example using thei sa function to determine if the arguments are the correct class of objects The Portfolio display Method The portfolio di sp ay method lists the contents of each contained object by calling the object s di sp ay method It then lists the client name and total asset value function display p DISPLAY Display a portfolio object for i l length p ind assets display p ind_assets i end stg sprintf nAssets for Client s nTotal Value 9 2f n p name p total value disp stg The Portfolio pie3 Method The portfolio class overloads the MATLAB pi e3 function to accept a portfolio object and display a 3 D pie chart illustrating the relative asset mix of the client s portfolio MATLAB calls the port fol io pi e3 m version of pi e3 whenever the input argument is a single portfolio object 14 57 14 MATLAB Classes and O bjects function pie3 p PIE3 Create a 3 stock_amt 0 for i l length p if isa p ind_a stock_amt D pie chart of a portfolio bond amt 0 savings amt 0 ind assets ssets i stock stock amt ind_assets i C nd_assets i bo bond amt bond amt
367. riable that is not a polynom object reshape it to be a row vector and assign it tothe c field of the object s structure Thec ass function creates thepol ynom object which is then returned by the constructor An example use of the po ynom constructor is the statement p polynom 1 0 2 5 This creates a polynomial with the specified coefficients Example A Polynomial C lass Converter Methods for the Polynom Class A converter method converts an object of one class to an object of another class Two of the most important converter methods contained in MATLAB classes aredouble andchar Conversion todoubl e produces MATLAB s traditional matrix although this may not be appropriate for some classes Conversion to char is useful for producing printed output The Polynom to Double Converter The double converter method for the polynom class is a very simple M file pol ynom doubl e m which merely retrieves the coefficient vector function c double p POLYNOM DOUBLE Convert polynom object to coefficient vector c DOUBLE p converts a polynomial object to the vector c containing the coefficients of descending powers of x C p c On the object p p polynom 1 0 2 5 the statement doubl e p returns ans 1 0 2 The Polynom to Char Converter The converter to char is a key method because it produces a character string involving the powers of an independent variable x Therefore once you ha
368. rier Analysis and the Fast Fourier Transform FFT N length Y Y 1 power abs Y 1 N 2 2 nyquist 1 2 freq 1 N 2 N 2 nyquist plot freq power grid on xlabel cycles year title Periodogram x10 Periodogram 2 T T T T T ESF EES EN a E S a 0 0 05 0 1 0 15 0 2 0 25 0 3 035 04 0 45 0 5 cycles year 6 35 6 Data Analysis and Statistics Thescalein cycles year is somewhat inconvenient Let s plot in years cycle and estimate what one cycle is For convenience plot the power versus period whereperiod 1 freq from 0 to 40 years cycle period l freq plot period power axis 0 40 0 2e7 grid on yl abel Power xl abel Period Years Cycle x10 0 5 10 15 20 25 30 35 40 Period Years Cvcle In order to determine the cycle more precisely mp index max power peri od index ans 11 0769 6 36 Fourier Analysis and the Fast Fourier Transform FFT Magnitude and Phase of Transformed Data Important information about a transformed sequence includes its magnitude and phase The MATLAB functions abs andangle calculate this information Totry this create a time vector t and use this vector to create a sequence x consisting of two sinusoids at different frequencies t 0 1 99 1 x sin 2 pi 15 t sin 2 pi 40 t Now usethef ft function to compute the DFT of the sequence The code below c
369. ring r symrcm B m symmmd B The three spy plots produced by the lines below show the three adjacency matrices of the Bucky Ball graph with these three different numberings The local pentagon based structure of the original numbering is not evident in the other three spy B spy B r r spy B m m Sparse M atrix O perations Original Reverse Cuthill McKee Minimum Degree nz 180 nz 180 nz 180 The reverse Cuthill McGee ordering r reduces the bandwidth and concentrates all the nonzero elements near the diagonal The minimum degree ordering m produces a fractal like structure with large blocks of zeros Tosee the fill in generated in the LU factorization of the Bucky ball use speye n n the sparse identity matrix to insert 3s on the diagonal of B B B 3 speye n n Since each row sum is now zero this new B is actually singular but it is still instructive to compute its LU factorization When called with only one output argument u returns the two triangular factors L and U in a single sparse matrix The number of nonzeros in that matrix is a measure of the time and storage required to solve linear systems involving B Here are the nonzero counts for the three permutations being considered Original u B 1022 Reverse Cuthill M cK ee lu B r r 968 Minimum degree lu B m m 660 Even though this is a small example the results are typical The original numbering
370. risons Comparing Characters for Equality with Operators You can use MATLAB relational operators on character arrays as long as the arrays you are comparing have equal dimensions or one is a scalar F or example you can use the equality operator to determine which characters in two strings match A fate B cake A B ans 0 1 0 1 All of the relational operators gt gt lt lt compare the values of corresponding characters Categorizing Characters Within a String There are two functions for categorizing characters inside a string e isletter determines if a character is a letter e isspace determines if a character is whitespace blank tab or new line For example create a string named mystring mystring Room 401 isl etter examines each character in the string producing an output vector of the same length as mystring A isletter mystring A 1 1 1 1 0 0 0 0 The first four elements in A arel true because the first four characters of mystring areletters 11 11 11 Character Arrays Strings Searching and Replacing MATLAB provides several functions for searching and replacing characters in a string Consider a string named abel label Sample 1 10 28 95 The strrep function performs the standard search and repl ace operation Use strrep tochange the date from 10 28 to 10 30 newlabel strrep label 28 30 newlabel
371. rking environment This chapter focuses on the MATLAB working environment As a working environment MATLAB includes facilities for managing the variables in your workspace and for importing and exporting data MATLAB alsoincludes tools for developing and managing M files MATLAB s applications The first part of this chapter describes general aspects of using the MATLAB working environment In this and subsequent chapters when it is necessary in this general material to call out features specific to a particular platform we use icons in the text margin to highlight the information pertinent to your platform Look for E for Microsoft Windows information for UNIX information X Additional sections at the end of this chapter discuss further platform specific MATLAB environment features e Microsoft Windows Handbook e UNIX Handbook Starting MATLAB ze On Windows platforms the installer creates a shortcut to the program 4844 file in the installation directory You can move this shortcut to your desktop if you want Double click on this shortcut icon to start MATLAB To start MATLAB ona UNIX system type mat ab at the operating system prompt Startup Files At startup MATLAB automatically executes the master M file mat abrc m and if it exists st art up m The file mat abrc m which lives in the ocal directory is reserved for use by The MathWorks and on multiuser systems by your system manager Usin
372. rocessing and data visualization Its most general form is Z interp2 X Y Z XI Yl met hod Z is arectangular array containing the values of a two dimensional function and X and Y are arrays of the same size containing the points for which the values inZ aregiven XI andY are matrices containing the points at which to interpolate the data met hod is an optional string specifying an interpolation method There are three different interpolation methods for two dimensional data e Nearest naghbor interpolation method nearest This method fits a piecewise constant surface through the data values The value of an interpolated point is the value of the nearest point Bilinear interpolation method linear This method fits a bilinear surface through existing data points The value of an interpolated point is a combination of the values of the four closest points This method is piecewise bilinear and is faster and less memory intensive than bicubic interpolation Bicubicinterpolation met hod cubic This method fits a bicubic surface through existing data points The value of an interpolated point is a combination of the values of the sixteen closest points This method is piecewise bicubic and produces a much smoother surface than bilinear interpolation This can be a key advantage for applications like image processing Use bicubic interpolation when the interpolated data and its derivative must be continuou
373. roperty that Al A and IA A whenever the dimensions are compatible The original version of MATLAB could not use for this purpose because it did not distinguish between upper and lowercase letters andi already served double duty as a subscript and as the complex unit So an English language pun was introduced The function eye m n returns an m by n rectangular identity matrix and eye n returns an n by n square identity matrix Matrices in MATLAB The Kronecker Tensor Product The Kronecker product kr on X Y of two matrices is the larger matrix formed from all possible products of the elements of X with those of Y If X is m by n andy is p by q then kron X Y iS mp by ng The elements are arranged in the order X 1 1 Y X 1 2 X 1 n Y X m 1 Y X m 2 Y En X m n Y The Kronecker product is often used with matrices of zeros and ones to build up repeated copies of small matrices For example if X is the 2 by 2 matrix X 1 2 3 4 and eye 2 2 isthe 2 by 2 identity matrix then the two matrices kron X 1 and kron l X are 1 0 2 0 0 1 0 2 3 0 4 0 0 3 0 4 and 1 2 0 0 3 4 0 0 0 0 1 2 0 0 3 4 4 11 4 Matrices and Linear Algebra Vector and Matrix Norms The p norm of a vector x slp PP is computed by nor m x p This is defined by any value of p gt 1 but the most common values of p are 1 2 and The default value is p 2 which corresponds to Euclidean length n
374. rse matrices with real elements Consider an m by n sparse matrix with nnz nonzero entries e The first array contains all the nonzero elements of the array in floating point format The length of this array is equal tonnz e The second array contains the corresponding integer row indices for the nonzero elements This array also has length equal tonnz e Thethird array contains integer pointers to the start of each column This array has length equal ton This matrix requires storage for nnz floating point numbers and nnz n integers At 8 bytes per floating point number and 4 bytes per integer thetotal number of bytes required to store a sparse matrix is 8 nnz 4 nnztn Sparse matrices with complex elements are also possible In this case MATLAB uses a fourth array with nnz elements to store the imaginary parts of the nonzero elements An element is considered nonzero if either its real or imaginary part is nonzero 9 5 9 Sparse Matrices Creating Sparse Matrices MATLAB never creates sparse matrices automatically Instead you must determine if a matrix contains a large enough percentage of zeros to benefit from sparse techniques The density of a matrix is the number of non zero elements divided by the total number of matrix elements Matrices with very low density are often good candidates for use of the sparse format Converting Full to Sparse You can convert a full matrix to sparse storage using thes parse functio
375. s Multidimensional Arrays Multidimensional arrays in MATLAB are an extension of the normal two dimensional matrix Matrices have two dimensions the row dimension and the column dimension column 1 1 1 2 1 3 1 4 ae 2 1 2 2 2 3 2 4 3 1 3 2 3 3 3 4 4 1 1 4 2 1 4 3 1 4 4 Y ou can access a two dimensional matrix element with two subscripts the first representing the row index and the second representing the column index Multidimensional arrays use additional subscripts for indexing A three dimensional array for example uses three subscripts e The first references array dimension 1 the row e The second references dimension 2 the column e The third references dimension 3 This guide uses the concept of a page to represent dimensions 3 and higher 1 1 3 1 2 3 1 3 3 1 4 3 261 38 2 62939 2 3 3 2 4 3 3 1 3 3 2 3 3 3 3 3 4 3 bel re 2 1 3 1 Olun 2 1 2 2 1 3 2 1 4 2y 2 2 2 2 2 3 2 2 4 2 2 3 2 2 3 3 2 3 4 2 hele dy Qh 2e0 tse Ds le sg 25 Deeb 2 Py PY 23 2 4 Ce R Qed Ue DS Ag 4 1 1 4 2 1 4 3 1 4 4 row gt 1 42 2 4 3 2 4 4 2 K 2 3 4 3 3 4 4 3 1 1 1 12 3 12 Multidimensional Arrays 12 4 To access the element in the second row third column of page 2 for example you use
376. s All of these methods require that X and Y be monotonic that is either always increasing or always decreasing from point to point You should prepare these matrices using the meshgri d function or else be sure that the pattern of the points emulates the output of mes hgri d In addition each method automatically maps the input to an equally spaced domain before interpolating If X andY are already equally spaced you can speed execution time by prepending an asterisk to the met hod string for example cubic Interpolation Comparing Interpolation Methods This example compares two dimensional interpolation methods on a 7 by 7 matrix of data 1 Generate the peaks function at low resolution x y meshgrid 3 1 3 z peaks x y surf x y z 2 Generate a finer mesh for interpolation xi yi meshgrid 3 0 25 3 3 Interpolate using nearest neighbor interpolation zil interp2 x y z xi yi nearest 4 interpolate using bilinear interpolation zi2 interp2 x y z xi yi bilinear 5 I nterpolate using bicubic interpolation zi3 interp2 x y z xi yi bicubic 5 13 5 Polynomials and Interpolation 6 Compare the surface plots for the different interpolation methods surf xi yi zil nearest surf xi yi zi2 bilinear surf xi yi zi3 bicubic 7 Compare the contour plots for the different interpolation methods contour xi yi zil cont
377. s for i 1 m for j lin A i j Uti j 1 end end Note You can often speed up the execution of MATLAB code by replacingf or and whi e loops with vectorized code See page 10 63 for details Using Arrays as Indices The index of af or loop can bean array For example consider an m by n array A The statement for i A statements end setsi equal tothe vector A k For the first loop iteration k is equal to1 for the second k is equal to2 and soon until k equals n That is the loop iterates 10 35 10 M File Programming 10 36 for a number of times equal to the number of columns in A For each iteration i is a vector containing one of the columns of A break Thebreak statement terminates the execution of af or loop or whi e loop When abreak statement is encountered execution continues with the next statement outside of the loop In nested loops br eak exits from the innermost loop only The example below shows a whi e loop that reads the contents of thefilef f t m intoa MATLAB character array A break statement is used to exit the whi e loop when the first empty line is encountered The resulting character array contains the M file help for thef ft program fid fopen fft m r reer while feof fid line fgetl fid if isempty line break end s strvcat s line end disp s try catch The general form of atry catch statement sequence is try statement stateme
378. s This section provides more detail and describes how to create more advanced ODE files that can accept additional input parameters and return additional information ODE File Overview Look at the simple ODE filevdp1 m from earlier in this chapter function dy vdpl t y dy y 2 1 y 1 2 y 2 y 1 Although this is a simple example it demonstrates two important requirements for ODE files e The first two arguments must bet andy e By default the ODE file must return a column vector F t y Defining the Initial Values in the ODE File It is possible to specify default tspan y0 andoptions in the ODE file defining the entire initial value problem in the one file In this case the solver can be called as T Y solver F 1 1 The solver extracts the default values from the ODE file You can also omit empty arguments at the end of the argument list For example T Y solver F When you call a solver with an empty or missingt span or y0 the solver calls the specified ODE file to obtain any values not supplied in the solver argument list It uses the syntax tspan y0 options F init Creating O DE Files The ODE file is then expected to return three outputs e Output lis thet span vector e Output 2 is the initial value y0 e Output 3 is either an options structure created with theodeset function or an empty matrix Coding the ODE File to Return Initial Values If you us
379. s If you are accustomed to programming in other object oriented languages such as C or J ava you will find that the MATLAB programming language differs from these languages in some important ways e In MATLAB method dispatching is not syntax based as it is in C and J ava When the argument list contains objects of equal precedence MATLAB uses the left most object to select the method to call e In MATLAB there is no equivalent to a destructor method To remove an object from the workspace use the c ear function e Construction of MATLAB data types occurs at runtime rather than compile time You register an object as belonging to a class by calling thec ass function e When using inheritance in MATLAB the inheritance relationship is established in the child class by creating the parent object and then calling 14 MATLAB Classes and O bjects theclass function For more information on writing constructors for inheritance relationships see Building on Other Classes on page 14 34 When using inheritance in MATLAB the child object contains a parent object in a property with the name of the parent class In MATLAB there is no passing of variables by reference When writing methods that update an object you must pass back the updated object and use an assignment statement For instance this call totheset method updates thename field of the object A and returns the updated object A set A name John Smith
380. s These functions operate on specific dimensions of multidimensional arrays that is they operate on individual elements on vectors or on matrices Functions that Operate on Vectors Functions that operate on vectors likes um mean and soon by default typically work on the first nonsingleton dimension of a multidimensional array Most of these functions optionally let you specify a particular dimension on which to operate There are exceptions however For example thecross function which finds the cross product of two vectors works on the first nonsingleton dimension having length three Note In many cases these functions have other restrictions on the input arguments for example some functions that accept multiple arrays require that the arrays be the same size Refer to the online help for details on function arguments Functions that Operate Element by Element MATLAB functions that operate element by element on two dimensional arrays likethe trigonometric and exponential functions in thee f un directory work in exactly the same way for multidimensional cases F or example thes i n function returns an array the samesizeas the function s input argument Each element of the output array is the sine of the corresponding element of the input array Similarly the arithmetic logical and relational operators all work with corresponding elements of multidimensional arrays that are the same size in every dimension If one
381. s you have no data from which to create the object so you simply initialize the object s data structures with empty or default values call thec ass function to instantiate the object and return the object as the output argument Support for this syntax is required for two reasons e When loading objects into the workspace the oad function calls the class constructor with no arguments e When creating arrays of objects MATLAB calls the class constructor to add objects to the array Object Input Argument If the first input argument in the argument list is an object of the same class the constructor should simply return the object Use thei sa function to determine if an argument is a member of a class See Overloading the Operator on page 14 29 for an example of a method that uses this constructor syntax Data Input Arguments f the input arguments exist and are not objects of the same class then the constructor creates the object using the input data Of course as in any function you should perform proper argument checking in your constructor function A typical approach is to usea varar gin input Designing User Classes in MATLAB argument and as wi tch statement to control program flow This provides an easy way to accommodate the three cases no inputs object input or the data inputs used to create an object It isin this part of the constructor that you assign values to the object s data structure call th
382. s designed to represent one particular asset in a person s investment portfolio This object contains two properties of its own and inherits three properties from its parent asset object Stock properties e NumberShares The number of shares for the particular stock object e SharePrice The value of each share Asset properties e Descriptor Theidentifier of the particular asset e g stock name savings account number etc e Date The date the object was created calculated by thedat e command e Current Value The current value of the asset Note that the property names are not actually the same as the field names of the structure array used internally by stock and asset objects The property name interface is controlled by the stock and assets et andget methods and is designed to resemble the interface of other MATLAB object properties Theasset field in the stock object structure contains the parent asset object and is used to access the inherited fields in the parent structure Stock Class Methods The stock class implements the following methods e Constructor e get andset e subsref andsubsasgn e display The Stock Class Constructor The stock constructor creates a stock object from three input arguments 14 MATLAB Classes and O bjects e The stock name e The number of shares e The share price The constructor must create an asset object from within the stock constructor to be able to specify it as
383. s function has some elements that are common to all MATLAB functions e A function definition line This line defines the function name and the number and order of input and output arguments e AH1 line H1 stands for help 1 line MATLAB displays the H1 line for a function when you use ookf or or request help on an entire directory e Hdp tet MATLAB displays the help text entry together with the H1 line when you request help on a specific function e The function body This part of the function contains code that performs the actual computations and assigns values to any output arguments 10 3 10 M File Programming 10 4 The Functions section coming up provides more detail on each of these parts of a MATLAB function Creating M Files Accessing Text Editors M files are ordinary text files that you create using a text editor MATLAB provides a built in editor although you can use any text editor you like Note To open the editor on the PC from the File menu choose New and then M File Another way to edit an M file is from the MATLAB command line using the edit command For example edit poof opens the editor on thefilepoof m Omitting a filename opens the editor on an untitled file You can create thef act function shown on the previous page by opening your text editor entering the lines shown and saving the text in afilecalledf act m in your current directory Once you ve created this file he
384. s is T Y ode45 ydot t0 tfinal y0 and the syntax acceptst0 gt tfinal Troubleshooting The following table provides troubleshooting questions and answers 8 54 Q uestions and Answers Troubleshooting Question Answer The solution doesn t look like what expected My plots aren t smooth enough I m plotting the solution as it is computed and it looks fine but the code gets stuck at some point If you re right about its appearance you need to reduce the error tolerances from their default values A smaller relative error tolerance is needed to compute accurately the solution of problems integrated over long intervals as well as solutions of problems that are moderately unstable Y ou should check whether thereare solution components that stay smaller than their absolute error tolerance for some time If so you are not asking for any correct digits in these components This may be acceptable for these components but failing to compute them accurately may degrade the accuracy of other components that depend on them Increase the value of Ref i ne from its default of 4 inode45 and1 in theother solvers The bigger the value of Ref i ne the more output points Execution speed is not affected much by the value of Refine First verify that the ODE function is smooth near the point where the code gets stuck If it isn t the solver must take small steps to deal with this It may hel
385. s matrix factorizations such as LU or Cholesky factorization The algorithms involve access to the individual matrix elements e Iterative methods Only an approximate solution is produced after a finite number of steps The coefficient matrix is involved only indirectly through a matrix vector product or as the result of an abstract linear operator Direct Methods Direct methods are usually faster and more generally applicable if thereis enough storage available to carry them out Iterative methods are usually applicable to restricted cases of equations and depend upon properties like diagonal dominance or the existence of an underlying differential operator Direct methods are implemented in the core of MATLAB and are made as efficient as possible for general classes of matrices Iterative methods are usually implemented in MATLAB M files and may make use of the direct solution of subproblems or preconditioners The usual way to access direct methods in MATLAB is not through the u or chol functions but rather with the matrix division operators and IfA is square the result of X A B isthe solution tothe linear systemA X B IfA is not square then a least squares solution is computed If A is a square full or sparse matrix then A B has the same storage class as B Its computation involves a choice among several algorithms e IfA istriangular perform a triangular solve for each column of B e IfA isa permutation of a triangular m
386. s reasonable Because programs often have several layers your code may not explicitly call the most expensive functions Rather functions within your code may be calling other time consuming functions that can be several layers down in the code In this case it s important to determine which of your functions are responsible for such calls The profiler often helps to uncover problems that you can solve by e Avoiding unnecessary computation which can arise from oversight e Changing your algorithm to avoid costly functions e Avoiding recomputation by storing results for future use When you reach the point where most of the time is spent on calls toa small number of built in functions you have probably optimized the code as much as you can expect How the Profiler Works Use the profile command to generate and view statistics Start the profiler usingprofile on and specify any options you want to use Then execute your M file The profiler counts how many seconds each line in the M files use The 3 17 3 Debugger and Profiler profiler works cumulatively that is adding to the count for each M file you execute until you clear the statistics Useprofile report todisplay the statistics gathered in an HTML formatted report in a Web browser The profile Command Hereis a summary of the main forms of pr of i e For details about these and other options typehel p profile ordoc profile Syntax Options Description profi
387. s starting point f zer o searches in the neighborhood of 0 2 until it finds a change of sign between 0 10949 and 0 264 This interval is then narrowed down to 0 1316 You can verify that 0 1316 has a function value very close to zero using humps a ans 8 8818e 16 Minimizing Functions and Finding Zeros Suppose you know two places where the function value of humps differs in sign such asx 1 andx 1 You can use humps 1 ans 16 humps 1 ans 5 1378 Then you can givef zero this interval to start with and f zero then returns a point near where the function changes sign Y ou can display information as f zero progresses with options optimset Display iter a fzero humps 1 1 options Func count X f x Procedure 1 1 5 13779 initial 1 1 16 initial 2 0 513876 4 02235 interpolation 3 0 243062 71 6382 bisection 4 0 473635 3 83767 interpolation 5 0 115287 0 414441 bisection 6 0 150214 0 423446 interpolation 7 0 132562 0 0226907 interpolation 8 0 131666 0 0011492 interpolation 9 0 131618 1 88371e 07 interpolation 10 0 131618 2 7935e 11 interpolation 11 0 131618 8 88178e 16 interpolation 12 0 131618 9 76996e 15 interpolation a 0 1316 7 11 7 Function Functions 7 12 The steps of the algorithm include both bisection and interpolation under the Procedure column If the example had started with a scalar starting point instead of an interval the first
388. set a breakpoint on line 4 of vari ance m Run variance once again variance v Execution pauses after control returns fromsqsum but beforevari ance uses the returned value oft ot From the Command Window set t ot toits correct value of 10 K gt gt tot 10 tot 10 Select Continue Execution from the Debug menu and the result is correct End the Debug Session Select the Exit Editor Debugger from the File menu to end the debugging session Debugging from the Command Line The MATLAB debugging commands are a set of tools that allow you to debug your M files from the command line The most general form for each debugging command is shown below Description Syntax Set breakpoint Remove breakpoint Stop on warning error or NaN nf generation Resume execution dbstop at line num in file name dbclear at line num in file name dstop if warning error nani nf inf nan dbcont MATLAB Debugger Description Syntax List function call stack List all breakpoints Execute one or more lines List M file with line numbers Change local workspace context down Change local workspace context up Quit debug mode dbstack dbstatus file name dbstep nlines dbtype file name dbdown dbup dbquit For more information about any of these debugging commands typehe p or doc followed by the command name Example Command Line Debugging Session You can perf
389. sets The constructor calculates this value from the objects passed in as arguments e account number The account number This field is assigned a value only when you save a portfolio object see Saving and Loading Objects on page 14 61 14 55 14 MATLAB Classes and O bjects function p portfolio name varargin PORTFOLIO Create a portfolio object containing the client s name and a list of assets switch nargin case 0 if no input arguments create a default object p name none p total value 0 p ind assets p account number p class p portfolio case 1 if single argument of class portfolio return it if isa name portfolio p name else disp inputname 1 is not a portfolio object return end otherwise create object using specified arguments p name name p total value 0 for i l length varargin p ind_assets i varargin i asset value get p ind_assets i Current Value p total_ value p total value asset value end p account number p class p portfolio end 14 56 Example The Portfolio C ontainer Constructor Calling Syntax The portfolio constructor method can be called in one of three different ways e Noinput arguments If called with no arguments it returns an object with empty fields Input argument is an object If the input argument is already a portfolio object MATLAB returns the input a
390. several ways to compare strings and substrings e You can compare two strings or parts of two strings for equality e You can compare individual characters in two strings for equality e You can categorize every element within a string determining whether each element is a character or whitespace These functions work for both character arrays and cell arrays of strings Comparing Strings For Equality There are two functions that determine if two input strings are identical e strcmp determines if two strings are identical e strncmp determines if the first n characters of two strings are identical Consider the two strings strl hello str2 help Stringsstr1 andstr2 are not identical soinvokingstrcmp returns 0 false For example C strcemp stri str2 C Note for C programmers This is an important difference between MATLAB s strcmp andC sstrcmp which returns 0 if the two strings are the same 11 9 11 Character Arrays Strings 11 10 The first three characters of str1 andstr2 areidentical so invoking st rnc mp with any value up to 3 returns 1 C strncmp stri str2 2 C 1 These functions work cell by cell on a cell array of strings Consider thetwo cell arrays of strings A pizza chips candy B pizza chocolate pretzels Now apply the string comparison functions strcmp A B oor jl strncmp A B 1 ans orr iit String Compa
391. size of the result should be that same function evaluated at zero For example horizontal concatenation C A B requires that A and8 have the same number of rows So if A is m by n andB is m by p then is m by n p This is still true ifm orn or p is zero Many operations in MATLAB produce row vectors or column vectors It is possible for the result to be the empty row vector r zeros 1 0 or the empty column vector C zeros 0 1 MATLAB 5 retains MATLAB 4 behavior for i f and whi e statements F or example if A S1 else S0 end executes statement 50 when A is an empty array 10 49 10 M File Programming Some MATLAB functions likes um and max are reductions For matrix arguments these functions produce vector results for vector arguments they produce scalar results Empty inputs produce the following results with these functions e sum is 0 e prod is1 e max is e min is 10 50 Errors and W arnings Errors and Warnings In many cases it s desirable to take specific actions when different kinds of errors occur For example you may want to prompt the user for more input display extended error or warning information or repeat a calculation using default values MATLAB s error handling capabilities let your application check for particular error conditions and execute appropriate code depending on the situation Error Handling with eval and lasterr The basic to
392. sparse matrices and reordering 9 28 M magnitude 6 37 malloc 2 33 manuals online 2 23 Maple 4 36 mass matrix ODE 8 17 8 29 constant mass matrix 8 30 returned by ODE file 8 30 matching quotes automatically 2 49 MAT files 2 11 2 31 loading 2 12 saving 2 11 mathematical functions finding zeros 7 7 library 1 5 minimizing 7 7 7 9 numerical integration 7 14 of one variable 7 7 finding zeros 7 10 of several variables 7 8 plotting 7 4 quadrature 7 14 representing in MATLAB 7 3 mathematical operations on sparse matrices 9 23 MathWorks Web site 2 23 MathWorks The home page 1 8 MATLAB Application Program Interface 1 5 Command Window 2 5 data type classes 14 3 demos 1 6 Handle Graphics 1 4 help for 2 20 history 1 3 interpreter 2 5 language 1 4 1 15 Index mathematical function library 1 5 overview 1 3 path 2 14 product family 1 9 programming functions 10 6 M files 10 2 quick start 10 2 scripts 10 5 quitting 2 4 representing functions 7 3 starting 2 2 structures 14 7 version 10 18 working environment 1 4 2 2 matlab 2 2 matlabrc m2 2 matrices addition 4 6 and linear algebra 4 2 concatenation 10 41 deleting rows and columns 10 42 diagonal of 4 10 dimension compatibility 4 13 division 4 13 empty 10 49 full to sparse conversion 9 2 9 6 identity 4 10 joining 10 41 multiplication 4 8 orthagonal 4 27 sparse 10 19 subtraction 4 6 symmetric 4 7 triangular 4 24 matrix as index for f or loops 10 35 c
393. splays theprompt string waits for keyboard input and then returns the value from the keyboard If the user inputs an expression the function evaluates it and returns its value This function is useful for implementing menu driven applications input can alsoreturn user input as a string rather than a numeric value To obtain string input append s tothe function s argument list name input Enter address s Pausing During Execution Some M files benefit from pauses between execution steps F or example the petals m script shown on page 10 5 pauses between the plots it creates allowing the user to display a plot for as long as desired and then press a key to move to the next plot Thepause command with noarguments stops execution until the user presses a key To pause for n seconds use pause n 10 61 10 M File Programming Shell Escape Functions 10 62 It is sometimes useful to access your own C or Fortran programs using shell escape functions Shell escape functions use the shell escape command to make external stand alone programs act like new MATLAB functions A shell escape M function is an M file that 1 Saves the appropriate variables on disk 2 Runs an external program which reads the data file processes the data and writes the results back out to disk 3 Loads the processed file back into the workspace For example look at the code for garfield m below This function uses an exter
394. ss The general syntax for establishing a simpleinheritance relationship usingthe class function is child_obj class child_obj child_ class parent obj Simple inheritance can span more than one generation If a parent class is itself an inherited class the child object will automatically inherit from the grandparent class Visibility of Class Properties and Methods The parent class does not have knowledge of the child properties or methods The child class cannot access the parent properties directly but must use parent access methods e g get orsubsref method to access the parent properties From the child class methods this access is accomplished via the parent field in the child structure For example when a constructor creates a child object c c class c child_class_ name parent object MATLAB automatically creates a field c parent class_name in the object s structure that contains the parent object You could then have a statement in the child s display method that calls the parent s display method display c parent_class_name See Designing the Stock Class on page 14 45 for examples that use simple inheritance Multiple Inheritance Inthe multipleinheritance case a class of objects inherits attributes from more than one parent class The child object gets fields from all the parent classes as well as fields of its own Multiple inheritance can encompass more than one generation Fo
395. ss to the same information as the he p command but the window interface provides convenient links to other topics 2 21 2 MATLAB W orking Environment 2 22 To use the Help Window on a particular topic type hel pwin topic wg On Windows platforms you can also access the Help Window by HA selecting the Help Window option under the Help menu or by clicking the question mark button on the menu bar The lookfor Command The ookf or command allows you to search for functions based on a keyword It searches through the first line of he p text which is known as the H1 line for each MATLAB function and returns the H1 lines containing a specified keyword For example MATLAB does not havea function namedi nverse So the response from help inverse is inverse m not found But lookfor inverse finds over a dozen matches Depending on which tool boxes you have installed you will find entries like I NVHILB Inverse Hilbert matrix ACOSH Inverse hyperbolic cosine ERFINV Inverse of the error function INV Matrix inverse PI NV Pseudoinverse FFT Inverse discrete Fourier transform FFT2 Two dimensional inverse discrete Fourier transform I CCEPS Inverse complex cepstrum DCT Inverse discrete cosine transform Adding all tothe ookf or command searches the entire help entry not just the H1 line The helpdesk Command The MATLAB Help Desk provides access to a wide range of help and reference information stored on
396. ssing 13 7 subassign 14 17 subfunctions 10 38 accessing 10 38 creating 10 38 debugging 10 39 function definition line 10 38 in dispatching priority 10 10 on search path 10 38 1 27 Index 1 28 subref 14 14 subsasgn 14 14 subscripted assignment 14 17 subscripting how MATLAB calculates indices 10 45 multidimensional arrays 12 3 overloading 14 14 structure arrays 13 4 with logical expression 10 27 with thef i nd function 10 27 subscripts 10 40 page 12 3 subsref method 14 14 substring within a string 11 12 subtraction of matrices 4 6 subtraction operator 10 21 sum 9 24 9 26 12 15 superi orto function 14 66 superseding existing M files names 10 39 suppressing output 2 9 surface plots to compare interpolation methods 5 14 svd 4 39 switch 10 32 case groupings 10 32 example 10 33 multiple conditions 10 34 Symbolic Math Toolbox 4 36 symmd 9 27 symmetric matrix 4 7 symmmd 9 28 symrcm 9 27 9 28 syntax errors finding 3 2 syntax formatting in Editor 2 49 syntax highlighting in E ditor Debugger 2 39 system dependent 2 55 systems of equations Seelinear systems of equations systems of ODEs 7 13 T tabs in string arrays 11 11 tempdir 15 2 15 6 tempname 15 2 15 6 temporary data 15 6 files 15 2 15 6 terminal events ODE 8 31 text files reading 15 14 in Command Window 2 39 in Editor 2 50 text editor See Editor textread 2 27 The MathWorks home page 1 8 theoretical graph 9 15 example 9 16 node 9 15 three
397. sum mfila A logical placetoinsert a breakpoint is after the computation of the mean andthe mean removed squared sum Open vari ance m and set a breakpoint at line 4 y tot length x 1 MATLAB Editor Debugger C WINDOWS Desktop variance m olx J Fie Edit View Debug Window Help lal x Disa Hee al ee ale function y variance x mu sum x length x tot sqsum x mu y tot length x 1 Line 4 511 PM IZ The line number is indicated at the bottom right of the status bar Set the breakpoint by positioning the cursor in the line of text and click on the breakpoint icon in the toolbar Alternatively you can choose Set Breakpoint from the Debug menu or right click to bring up the context menu and choose Set Clear Breakpoint MATLAB Debugger Examining Variables To get to the breakpoint and check the values of interest first execute the function from the Command Window variance v When execution of an M file pauses at a breakpoint the yellow arrow tothe left of the text E shows the next line to execute A downward yellow arrow 4 appearing to the left of the text indicates a pause at the end of the script or function This allows you to examine variables before returning to the calling function MATLAB Editor Debugger C WINDOWS Desktop variance m Oy x 3 Fie Edit View Debug Window Help 5 x Disa Ae Sle Sel aae stack raian function y variance x mu sum x l
398. system These routines make the assumption that if you need a large amount of memory once you will need it again A side effect of this algorithm is that once MATLAB has used a certain amount of memory it is no longer available to other programs even if MATLAB is nolonger using it The memory in the pool only returns to the operating system when MATLAB terminates 2 33 2 MATLAB W orking Environment X If you use an operating system tool such as ps on UNIX the display indicates the total sum of the memory allocated by MATLAB plus the contents of the pool This number can be deceiving because it indicates the highest level of memory use which may or may not be the current usage Microsoft W indows Handbook Microsoft Windows Handbook This section describes several MATLAB environment tools and their usein the Windows environment Command Window The Command Window appears when you first start MATLAB You useit to run MATLAB commands to launch MATLAB tools such as the Editor Debugger and to start toolboxes Use Menus as an Alternative to Typing Commands MATLAB Command Window iof x File Edit View Window Help Use Toolbar Del EBB O Ge we F for Easy Access magic 4 z to Popular Operations anga 16 2 3 13 5 11 168 8 9 7 6 12 4 14 15 1 ey n gt Status Bar feat al 4 Displays Description of Displays Status of Cap Num and Scroll Locks 2 35 2 MATLAB W orking Environment 2 36 Toolb
399. t _ testnum Test 2 xdata 4 2 8 L xdata 3 4 2 L ydata _l7 1 6 L ydata _9 0 9 Indexing Nested Structures Toindex nested structures append nested field names using dot notation The first text string in the indexing expression identifies the structure array and subsequent expressions access field names that contain other structures 13 17 13 Structures and Cell Arrays For example the array A created earlier has three levels of nesting e Toaccess the nested structure insideA 1 useA 1 nest e Toaccess thexdata field in the nested structure in A 2 use A 2 nest xdata e Toaccess element 2 of theydata fieldinA 1 uSeA 1 nest ydata 2 13 18 Cell Arrays Cell Arrays A cdl array is a MATLAB array for which the elements are cals containers that can hold other MATLAB arrays For example onecell of a cell array might contain a real matrix another an array of text strings and another a vector of complex values cell 1 1 cell 1 2 cell 1 3 Anne Smith 3 4 2 9 12 94 25 3i 8 16 9 7 6 Class 8 5 1 Obs 1 7 34 5i 7 92i Obs 2 i cell 2 1 cell 2 2 cell 2 3 i 4 2 7 I J4 text 15 1 43 2 98 8 3 45 5 67 52 16 3 4 2 7 02 8i You can build cell arrays of any valid size or shape including multidimensional structure arrays Note
400. t oad can process The H arwel Boeing data is available through anonymous ftp or the World Wide Web fromftp mathworks comin the directory pub mat hworks tool box mat ab sparfun Viewing Sparse M atrices Viewing Sparse Matrices MATLAB provides a number of functions that let you get quantitative or graphical information about sparse matrices General Storage Information The whos command provides high level information about matrix storage including size and storage class For example this whos listing shows information about sparse and full versions of the same matrix whos Name Size Bytes Class M_full 1100x1100 9680000 double array M sparse 1100x1100 4404 sparse array Grand total is 1210000 elements using 9684404 bytes Notice that the number of bytes used is much less in the sparse case because zero valued elements are not stored In this case the density of the sparse matrix is 4404 9680000 or approximately 00045 Information About Nonzero Elements There are several commands that provide high level information about the nonzero elements of a sparse matrix e nnz returns the number of nonzero elements in a sparse matrix e nonzeros returns a column vector containing all the nonzero elements of a sparse matrix e nzmax returns the amount of storage space allocated for the nonzero entries of a sparse matrix 9 11 9 Sparse Matrices To try some of these load the supplied sparse matrix west 0479 one of
401. t 2 XY 1 0000 0 0 1 0000 0 3000 0 0900 1 0000 0 8000 0 6400 1 0000 1 1000 1 2100 1 0000 1 6000 2 5600 1 0000 2 3000 5 2900 The solution is found with the backslash operator a X y 0 5318 0 9191 0 2387 The second order polynomial model of the data is therefore y 0 5318 0 919 1t 0 2387 t2 6 16 Regression and Curve Fitting Now evaluate the model at regularly spaced points and overlay the original data in a plot E Oi Te Dee ba ones size T T TT 2 a plot T Y t y 0 grid on lt 0 9 0 8 0 7 0 6 0 5 fi fi i fi 0 0 5 1 1 5 2 2 5 Clearly this fit does not perfectly approximate the data We could either increase the order of the polynomial fit or explore some other functional form to get a better approximation Linear in the Parameters Regression Instead of a polynomial function we could try using a function that is linear in the parameters In this case consider the exponential function t t y a aje ate The unknown coefficients ag a4 and a gt are computed by performing a least squares fit Construct and solve the set of simultaneous equations by forming 6 17 6 Data Analysis and Statistics the regression matrix X and solving for the coefficients using the backslash operator X ones size t exp t t exp t a X y a 1 3974 0 8988 0 4097 The fitted model of the data is therefore y 1 3974 0 8988 0 40
402. t Value 9 2f a descriptor a date a current value disp stg The stock class display method can now call this method to display the data stored in the parent class This approach isolates the stock di sp ay method from changes to the asset class The Asset fieldcount Method The asset f ie dcount method returns the number of fields in the asset object data structure fi el dcount enables asset child methods to determine the number of fields in the asset object during execution rather than requiring the child methods to have knowledge of the asset class This allows you to make changes tothe number of fields in the asset class data structure without having to change child class methods function num fields fieldcount asset_obj Determines the number of fields in an asset object Used by asset child class methods num fields length fieldnames struct asset_obj Thestruct function converts an object to its equivalent data structure enabling access to the structure s contents Other Asset Methods The asset class provides inherited data storage for its child classes but is not instanced directly Theset get and display methods provide access to the stored data It is not necessary to implement the full complement of methods Example Assets and Asset Subclasses for asset objects such as converters end and subsi ndex since only the child classes access the data Designing the Stock Class A stock object i
403. t both data and destination have the same shape F or example A 1 2 1 10 Reshaping Unless you change its shape or size a MATLAB array retains the dimensions specified at its creation You change array size by adding or deleting elements You change array shape by respecifying the array s row column or page dimensions while retaining the same elements Ther eshape function performs the latter operation For multidimensional arrays its form is B reshape A sl s2 s3 s1 s2 and soon represent the desired size for each dimension of the reshaped matrix Note that a reshaped array must have the same number of elements as the original array that is the product of the dimension sizes is constant M reshape M 6 5 aD wo Ww Ww e OM CO WO Fr ow HF UID Ww ODO UW ONNA A ooo N U e NUMA MA 12 11 12 Multidimensional Arrays Ther eshape function operates in a columnwise manner It creates the reshaped matrix by taking consecutive elements down each column of the original data construct C reshape C 6 2 19 E 1 6 11712 3 8 5 6 2 9 7 8 4 11 1 2 5 10 3 4 7 12 Here are several new arrays from reshaping nddata B reshape nddata 6 25 C reshape nddata 5 3 10 D reshape nddata 5 3 2 5 Removing Singleton Dimensions MATLAB creates singleton dimensions if you explicitly specify them when you create or
404. t certainly be revealed by examining these elements For our small example the second column of c has a larger norm than the first so the two columns are exchanged Q R P gr C Q 0 3522 0 8398 0 4131 0 7044 0 5285 0 4739 0 6163 0 1241 0 7777 11 3578 8 2762 0 7 2460 When the economy size and column permutations are combined the third output argument is a permutation vector rather than a permutation matrix 4 29 4 Matrices and Linear Algebra 4 30 Q R p qr C 0 Qs 0 3522 0 8398 0 7044 0 5285 0 6163 0 1241 R 11 3578 8 2762 0 7 2460 p 2 1 The QR factorization transforms an overdetermined linear system into an equivalent triangular system The expression norm A x b is equal to norm Q R x b Multiplication by orthogonal matrices preserves the Euclidean norm so this expression is also equal to norm R x y wherey Q b Sincethelast m n rows of R are zero this expression breaks into two pieces norm R lin 1lin x y lin and norm y n l m When A has full rank it is possible to solve for x so that the first of these expressions is zero Then the second expression gives the norm of the residual When A does not have full rank the triangular structure of R makes it possible to find a basic solution to the least squares problem M atrix Powers and Exponentials Matrix Powers and Exponentals IfA is a square ma
405. t in the class hierarchy where you can insert you own classes Creating Objects You create an object by calling the class constructor and passing it the appropriate input arguments In MATLAB constructors have the same name as the class name For example the statement p polynom 1 0 2 5 creates an object named p belonging to the class po ynom Once you have created a polynom object you can operate on the object using methods that are defined for the pol ynom class See Example A Polynomial Class on page 14 23 for a description of the po ynom class Invoking Methods on Objects Class methods are M file functions that take an object as one of the input arguments The methods for a specific class must be placed in the class directory for that class the cl ass_name directory This is the first place that MATLAB looks to find a dass method The syntax for invoking a method on an object is similar to a function call Generally it looks like outl out2 method _name object argl arg2 For example suppose a user defined class called pol ynom has achar method defined for the class This method converts a polynom object to a character string and returns the string This statement calls thechar method on the polynom object p s char p Classes and O bjects An O verview Using thec ass function you can confirm that the returned values isa character string class s ans char x 3 2 x5
406. t or field designator with an object on the left hand side of an assignment statement is known as a subscripted assignment MATLAB calls a method named subs asgn in these situations Object subscripted assignment can be of three forms an array index a cell array index anda structure field name A I B A I B A field B Each of these results in a call tosubsasgn of the form A subsasgn A S B The first argument A is the object being referenced The second argument S has the same fields as those used with subsref Thethird argument B is the new value 14 17 14 MATLAB Classes and O bjects 14 18 Examples of the subsasgn Method See the following sections for examples of thes ubsasgn method e The Asset subsasgn Method on page 14 42 e The Stock subsasgn Method on page 14 51 Defining end Indexing for an Object When you useend in an object indexing expression MATLAB calls the object s end class method If you want to be able to useend in indexing expressions involving objects of your class you must define anend method for your class Theend method has the calling sequence end a k n wherea isthe user object k istheindex in the expression wheretheend syntax is used andn is the total number of indices in the expression For example consider the expression A end l MATLAB calls theend method defined for the object A using the arguments end A 1 2 That is theend statement occurs
407. teger to string mat 2str Convert matrix toeval able string str2num Convert string to number sprintf Write formatted data to string sscanf Read string under format control Base Number hex2num Convert IEEE hexadecimal to double precision Conversion number hex2dec Convert hexadecimal string to decimal integer dec2hex Convert decimal integer to hexadecimal string bin2dec Convert binary string to decimal integer dec2bin Convert decimal integer to binary string base2dec Convert base B string to decimal integer dec2base Convert decimal integer to base B string 11 3 11 Character Arrays Strings Character Arrays 11 4 In MATLAB the term string refers to an array of characters MATLAB represents each character internally as its corresponding numeric value Unless you want to access these values however you can simply work with the characters as they display on screen Specify character data by placing characters inside a pair of single quotes F or example this line creates a 1 by 13 character array called name name Thomas R Lee In the workspace the output of whos shows Name Size Bytes Class name 1x13 26 char array You can see that a character uses two bytes of storage internally Theclass andischar functions show name s identity as a character array cl ass name ans char ischar name ans 1 Character Arrays Converting Between Characters and Numeric Values Character arrays st
408. ternate method of supplying the command modifiers makes them string arguments of functions load August17 dat hel p magic type rank This is MATLAB s command function duality Any command of the form command argument can also be written in the functional form command argument The advantage of the functional approach comes when the string argument is constructed from other pieces The following example processes multiple data files August1 dat August2 dat andsoon It uses the function i nt 2str which converts an integer to a character string to help build the filename for d 1 31 s August int2str d dat load s Process the contents of the d th file end Empty Matrices Empty Matrices Earlier versions of MATLAB allowed for only one empty matrix the 0 by 0 array denoted by MATLAB 5 provides for matrices and arrays where one but not all of the dimensions is zero For example 1 by 0 10 by 0 by 20 and 3 4 0 5 2 areall possible array sizes The two character sequence continues to denote the 0 by 0 matrix Empty arrays of other sizes can be created with the functions zeros ones rand or eye To create a 0 by 5 matrix for example use E zeros 0 5 The basic model for empty matrices is that any operation that is defined for m by n matrices and that produces a result whose dimension is some function of m and n should still be allowed when m or n is zero The
409. the Path Browser window If you want the Browse for Folder dialog box to open to the current directory click in the Current Directory field in the Path Browser window and then click Browse 2 51 2 MATLAB W orking Environment 2 52 Workspace Browser The Workspace Browser lets you view the contents of the current MATLAB workspace It provides a graphical representation of thewhos display To open the Workspace Browser select Show Workspace from the File menu click the Workspace Browser toolbar button or typeworkspace at the command line Workspace Browser Ioj x Bytes Class 8 double array 24 double array 48 double array 24 char array 84 sparse array 192 double array 132 struct array 600 cell array 830 inline object Erand total is 108 elements using 1942 bytes Open Delete Close You can resize the columns of information by dragging the column header borders The workspace is sorted by variable name Sorting by other fields is not supported To clear a variable select the variable and click Delete Shift click to select multiple variables To rename a variable first select it then click its name After a short delay type a new name and press Enter to complete the name change Editing Arrays The MATLAB Editor Debugger provides a visual representation of two dimensional numeric arrays which allows for easy editing To see and edit a graphical representation of a variable select a variable s i
410. the data in ASCII form Write the data ina special format When to Use For small arrays usethedi ar y command to create a diary fileand display the variables echoing them into this file The output of di ar y includes the MATLAB commands used during the session which is useful for inclusion in documents and reports You can use your text editor to edit the diary file removing unwanted text Use thesave command with the ascii option See Loading and Saving the Workspace for details ons ave You can also use d mwr i te if you need to specify alternate value delimiters dI mwr i te is discussed in Delimiter Separated Text Files Usef write and the other low level I O functions This method is useful for writing data files in the file formats required by other applications These functions are discussed in detail in Chapter 15 2 28 Data Import Export Use a specialized file write function for application specific formats Develop a MEX file to write the data Write out the data as a MAT file dl mwrite wklwrite imwrite auwrite wavwrite Write ASCII data file Write spreadsheet WK1 file Write image to graphics file Write Sun au sound file Write Microsoft WAVE wav sound file This is the best method if C or Fortran routines are already available for writing data files in the form needed by other applications See the MATLAB Application Program Interface Guide for more
411. the matrix on the left Permutation and Reordering A permutation of the rows and columns of a sparse matrixsS can be represented in two ways A permutation matrix P acts on the rows of S as P S or on the columns as SERR A permutation vector p which is a full vector containing a permutation of 1 n acts on the rows ofS as p or onthecolumnsas5 p For example the statements 13 425 eye 5 5 I ee ones 4 1 diag 11 11 55 diag e 1 diag e 1 p l P e S Sparse M atrix O perations produce p 1 3 4 2 5 P 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 S 11 1 0 0 0 1 22 1 0 0 0 1 33 1 0 0 0 44 1 0 0 0 1 55 You can now try some permutations using the permutation vector p and the permutation matrix P For example the statements p andP S produce ans 11 1 0 0 0 0 1 33 1 0 0 0 1 44 1 1 22 1 0 0 0 0 0 1 55 Similarly p andS P produce ans 11 0 0 1 0 1 1 0 22 0 0 33 1 1 0 0 1 44 0 1 0 0 1 0 55 9 25 9 Sparse Matrices 9 26 If P isa Sparse matrix then both representations use storage proportional ton and you can apply either tos in time proportional tonnz 5S The vector representation is slightly more compact and efficient so the various sparse matrix permutation routines all return full row vectors with the exception of the pivoting permutation in LU triangular factorization which returns a matrix compatible with earlier versions of MATLAB
412. the results Save profstats ode23 1x56 char M function 1 0 65100000000166 0 65100000000166 21x1 struct 0x1 struct 159x3 double 4 Toview the contents of an elementin thefunctionTabl e structure type for In a later session generate the profiler report using the saved results Type load profstats profreport stats MATLAB displays the profile report 3 27 3 Debugger and Profiler 3 28 Matrices and Linear Algebra Matrices and Linear Algebra Matrices in MATLAB Addition and Subtraction Vector Products and Transpose Matrix Multiplication 3 The Identity Matrix Vector and Matrix Norms Solving Linear Equations Square Systems Overdetermined Systems Undetermined Systems Inverses and Determinants Pseudoinverses LU QR and Cholesky F actorizations Cholesky F actorization ae LU Factorization QR Factorization Matrix Powers and E xponentials Eigenvalues Singular Value Decomposition 4 4 4 7 4 8 4 10 4 12 4 13 4 14 4 15 4 17 4 20 4 21 4 24 4 24 4 25 4 27 4 31 4 34 4 38 4 Matrices and Linear Algebra Matrices and Linear Algebra A matrix is a two dimensional array of real or complex numbers Linear algebra defines many matrix operations that are directly supported by MATLAB Matrix arithmetic linear equations eigenvalues singular values and matrix factorizations are included The linear algebr
413. the vector of event functions evaluated at t y Theval ue vector may be any length It is evaluated at the beginning and end of each integration step and if any elements maketransitions to from or through zero with the directionality specified in constant vector direction the solver uses the continuous extension formulas to determine the time when the transition occurred Terminal events halt the integration The argument i st er mi nal is a logical vector of 1s and 0s that specifies whether a zero crossing of the corresponding 8 31 8 O rdinary Differential Equations 8 32 value element is terminal 1 corresponds toa terminal event halting the integration 0 corresponds to a nonterminal event Thedirection vector specifies a desired directionality positive 1 negative 1 or don t care 0 for each val ue element The time an event occurs is located to machine precision within an interval of t t Nonterminal events are reported att For terminal events both t and t are reported For an example of an ODE file with an event location see E xample 5 Simple Event Location on page 8 44 odel15s Properties Theodel5s solver is a variableorder stiff solver based on the numerical differentiation formulas NDF s The NDFs are generally more efficient than the closely related family of backward differentiation formulas BDF s also known as Gear s methods Theode15s properties let you choose between
414. these formulas as well as specifying the maximum order for the solver Property Value Description Max Order 1 2 3 4 6 The maximum order formula used BDF on off Specifies whether the backward differentiation formulas are to be used instead of the default numerical differentiation formulas MaxO rder MaxOrder isan integer 1 through 5 used to set an upper bound on the order of the formula that computes the solution By default the maximum order is5 BDF Set BDF on tohaveodel5s usetheBDFs By default BDF isoff and the solver uses the NDFs For both the NDF s and BDFs the formulas of orders 1 and 2 are A stable the stability region includes the entire left half complex plane The higher order formulas are not as stable and the higher the order the worse the stability Improving Solver Performance Thereis a class of stiff problems stiff oscillatory that is solved more efficiently if Max Order is reduced for example to2 so that only the most stable formulas are used 8 33 8 O rdinary Differential Equations 8 34 Examples Applying the ODE Solvers This section contains several examples of ODE files These examples illustrate the kinds of problems you can solvein MATLAB For more examples see MATLAB s demos directory Example 1 Simple Nonstiff Problem ri gi dode iS a nonstiff example that can be solved with all five solvers of the ODE suite It is a standard test problem proposed by
415. these keys should be familiar to users of the E macs editor Arrow Key Control Key Operation M Ctrl p Recall previous line Y Ctrl n Recall next line lt Ctrl b Move back one character gt Ctrl f Move forward one character ctrl gt Ctrl r Move right one word ctrl lt Ctrl l Move left one word home Ctrl a Move to beginning of line end Ctrl e Move to end of line esc Ctrl u Clear line del Ctrl d Delete character at cursor backspace CtrlI h Delete character before cursor Ctrl k Delete kill to end of line Clearing the Command Window Usecl c toclear the Command Window This does not clear the workspace but only clears the view After usingc c you still can use the up arrow key to see the history of the commands one at a time The Command W indow Paging of Output in the Command Window To control paging of output in the Command Window use mor e By default more iSoff When you set mor e on a page a screen full of output displays at one time You then use Return To advance to the next line Space Bar To advance to the next page q To stop displaying the output Interrupting a Running Program You can interrupt a running program by pressing Ctrl c at any time n On Windows platforms you may have to wait until an executing built in function or ME X file has finished its operation X On UNIX systems program execution will terminate immediately The format Command Thef or mat comm
416. thest fromthe initial point yO and starts to move closer The current distance of the body is DSQ y 1 yO 1 2 y 2 y0 2 2 lt y 1 2 y0 y 1 2 y0 gt 8 48 Examples Applying the O DE Solvers A local minimumof DSQ occurs when d dt DSQ crosses zero heading in the positive direction We can compute d dt DSQ as lt d dt DSQ 2 y 1 2 y0 dy 1 2 dt 2 y 1 2 y0 y 3 4 dDSQdt 2 y 1 2 y0 1 2 y 3 4 value dDSQdt dDSQdt isterminal 1 0 stop at local minimum direction 1 1 local minimum local maxi mum 8 49 8 O rdinary Differential Equations 8 50 Questions and Answers This section contains a number of tables that answer questions about the use and operation of the MATLAB ODE solvers This section also contains a troubleshooting table The question and answer tables cover the following categories e General ODE Solver Questions e Problem Size Memory Use and Computation Speed e Time Steps for Integration e Error Tolerance and Other Options e Solving Different Kinds of Systems General ODE Solver Questions Question Answer How dothe ODE solvers differ from quad or quad8 Can I solve ODE systems in which there are more equations than unknowns or vice versa quad andquad8 solveproblems of the form y F t The ODE suite solves more general problems of the form y F t y No Problem Size
417. this function creates and returns a new stock object with the new values which you then copy over the old value For example given the stock object s stock XYZ 100 25 Example Assets and Asset Subclasses the followings et command updates the share price s set s SharePrice 36 It is necessary to copy over the original stock object i e assign the output to s because MATLAB does not support passing arguments by reference H ence thes et method actually operates on a copy of the object The Stock subsasgn Method Thesubsasgn method enables you to change the data contained in a stock object using numeric indexing and structure field name indexing MATLAB callssubsasgn whenever you execute an assignment statement e g A i val A i val orA fieldname val function s subsasgn s index val SUBSASGN Define index assignment for stock objects fc fieldcount s asset switch index type case if index subs lt fc S asset subsasgn s asset index val else switch index subs fc case 1 s num shares val case 2 s share price val otherwise error Index must be in the range 1 to mumdstr fc 2 end end case switch index subs case num shares S num shares val case share price s share price val otherwise S asset subsasgn s asset index val end end 14 51 14 MATLAB Classes and O bjects 14 52 The outer s wi tch statement determi
418. tically by examining the coefficient matrix include e Permutations of triangular matrices e Symmetric positive definite matrices e Square nonsingular matrices e Rectangular overdetermined systems e Rectangular underdetermined systems Square Systems The most common situation involves a square coefficient matrix A and a single right hand side column vector b The solution x A b is then the same size as b For example x Alu 10 12 Solving Linear Equations It can be confirmed that A x is exactly equal tou IfA and8 are square and the same size then X A B is alsothat size X A B X 19 3 1 17 4 13 6 0 6 It can be confirmed that A X is exactly equal toB Both of these examples have exact integer solutions This is because the coefficient matrix was chosen to bepasca 3 which has a determinant equal to one A later section considers the effects of roundoff error inherent in more realistic computation A square matrix A is singular if it does not have linearly independent columns If A is singular the solution to AX B either does not exist or is not unique The backslash operator A B issues a warning if A is nearly singular and raises an error condition if exact singularity is detected Overdetermined Systems Overdetermined systems of simultaneous linear equations are often encountered in various kinds of curve fitting to experimental data Hereis a hypothetical example A quantity
419. tifier that f open returns if successful is a nonnegative integer This integer acts as a handle to the file and is an argument to MATLAB file I O functions There are times when f open might fail For example f open fails if you try to open a file that does not exist If f open fails it does the following e It assigns 1 to the file identifier e It assigns an error message to an optional second output argument Note that the error messages are system dependent and are not provided for all errors on all systems The function f error may also provide information about errors It s good practice to test the file identifier each time you open a file For example this code loops until the user enters the name of a readable file fi d 0 while fid lt 1 filename i nput Open file s fid message fopen filename r if fid di sp message end end Now assumethat nof ile mat does not exist but thatgoodfi le mat does exist On one system the results are Open file nofile mat Cannot open file Existence Permissions Memory Open file goodfile mat O pening and Closing Files Closing a File When you finish reading or writing usef cl ose toclose the file For example this line closes the file associated with file identifier f i d status fclose fid This line closes all open files status fclose all Both forms return 0 if the file or files were successfully closed or 1 if th
420. ting y tp it needs only the solution at the immediately preceding time point y tp 1 1N general ode45 is the best function to apply as a first try for most problems ode23 is also based on an explicit Runge Kutta 2 3 pair of Bogacki and Shampine It may be more efficient than ode45 at crude tolerances and in the presence of mild stiffness Likeode45 0de23 is a one step solver ode113 isavariableorder Adams Bashforth M oulton PECE solver It may be more efficient than ode45 at stringent tolerances and when the ODE function is particularly expensive to evaluate 0de113 is a multistep solver it normally needs the solutions at several preceding time points to compute the current solution Stiff Solvers Not all difficult problems are stiff but all stiff problems are difficult for solvers not specifically designed for them Stiff solvers can be used exactly like the other solvers However you can often significantly improve the efficiency of the stiff solvers by providing them with additional information about the problem See I mproving Solver Performance on page 8 17 for details on how to provide this information and for details on how to change solver parameters such as error tolerances O DE Solvers There are four solvers designed for stiff or moderately stiff problems ode15s is a variable order solver based on the numerical differentiation formulas NDF s Optionally it uses the backward differentiation formul
421. tions for examples of overloaded operators e Overloading the Operator on page 14 29 e Overloading the Operator on page 14 30 e Overloading the Operator on page 14 30 Overloading O perators and Functions The following table lists the function names for most of MATLAB s operators Operation M File Description a plus a b Binary addition a mi nus a b Binary subtraction a umi nus a Unary minus ta uplus a Unary plus a b times a b Element wise multiplication a b mt i mes a b Matrix multiplication a b rdivide a b Right element wise division a b I divide a b Left element wise division a b mrdivide a b Matrix right division a b ml di vi de a b Matrix left division a b power a b Element wise power a b mpower a b Matrix power a lt It a b Less than a gt gt a b Greater than a lt e a b Less than or equal to a gt gela b Greater than or equal to a ne a b Not equal to a eq a b Equality a amp and a b Logical AND a or a b Logical OR a not a Logical NOT 14 21 14 MATLAB Classes and O bjects 14 22 Operation M File Description a d b colon a d b Colon operator a b colon a b a a command window output a b a b a sl s2 5n ctranspose a transpose a display a horzcat a b vertcat a b subsref a s subsasgn a s b subsindex a Complex conjugate trans
422. tive or negative offset value specified in bytes ori gin is an origin from which to calculate the move specified as a string cof Current position in file bof Beginning of file eof End of file Controlling Position in a File Understanding File Position Toseehowfseek andftel work consider this short M file A 1 5 fid fopen five bin w fwrite fid A short status fclose fid This code writes out the numbers 1 through 5 toa binary filenamedfi ve bin The call tof write specifies that each numerical element be stored as ashort Consequently each number uses two storage bytes Now reopen fi ve bin for reading fid fopen five bin r This call tof seek moves the file position indicator forward six bytes from the beginning of the file status fseek fid 6 bof File Position bof 1 2 3 4 5 6 7 8 9 10 eof File Contents 0 1 0 2 0 3 0 4 0 5 File Position Indicator uN This call tof read reads whatever is at file positions 7 and 8 and stores it in variablef our four fread fid 1 short The act of reading advances the file position indicator To determine the current file position indicator call f tell position ftell fid position 8 File Position bof 1 2 3 4 5 6 7 8 9 10 eof File Contents 0 1 0 2 0 3 0 4 0 5 File Position Indicator 4 15 11 13 File O This call tof seek moves the file position indicator bac
423. tly used file list under the File menu The Editor must be restarted for the new number to take effect 2 48 Microsoft W indows Handbook Editor Options Choose Options from the Tools menu and select the Editor tab Options Ea General Editor Mile syntax based formatting E i IV Auto indent on retum Auto indent size 3 I Emacs style tab key auto indenting m Bracket amp quote matching MV Bracket Matching M Quote Matching Show match for 25 sec Tab key settings atab character Tab key inserts Tab size 3 C spaces Search distance 50 lines Cancel e M file syntax based formatting Syntax highlighting As you type into the Editor the text is colored according to the kind of text entered Auto indent on return Pressing the Return or Enter key indents the current and next lines Auto indent size Type the number of columns to indent for each level of nested code The default accelerator for indenting in the MATLAB Editor is Ctrl l Emacs style tab key auto indenting The E macs editor uses the Tab key to indent the current line If you want the Tab key to indent the current line check this option Bracket and quote matching The Editor can indicate which bracket parenthesis brace or quote balances another in your file Four settings control this matching Bracket Matching Enables matching for brackets braces and parentheses Quote
424. to Start Plot Editor the Plot the Plot Plot Editor Features Mode Editor Figure No 1 File Edit Tools Wi dow Hep JOS US KA AS BOD 0 8 0 6 0 4 0 2 To save a figure select Save from the File menu To save it using a different format such as TIFF for use with other applications select Export from the File menu You can also save from the command line use thes aveas command including any options to save the figure in a different format MATLAB Graphics For more information about visualization with MATLAB see Using MATLAB Graphics 2 19 2 MATLAB W orking Environment 2 20 Help and Online Documentation Command Line Help There are several different ways to access online information about MATLAB functions Command Description help hel pwin lookfor helpdesk doc Display in the Command Window a description of the specified command Display a help window that describes the specified command and allows viewing help for other topics Display in the Command Window a brief description for all commands whose description includes the specified keyword Display the Help Desk page in a Web browser providing direct access to a comprehensive library of online help PDF formatted documentation troubleshooting information and The MathWorks Web site Display in a Web browser the reference page for the specified command providing a description additional remarks and examples The help C
425. to The MathWorks the home of MATLAB You can use electronic mail to ask questions make suggestions and report possible bugs You can also use the Solution Search Engine at The MathWorks Website to query an up to date data base of technical support information Alternatively you can point your Web browser directly at www mat hwor ks com to access The MathWorks Web site The doc Command Thedoc command accesses the HTML reference documentation for MATLAB functions and all installed toolboxes 2 23 2 MATLAB W orking Environment 2 24 For example if you have the Control System Toolbox and Symbolic Math Toolbox installed as well as MATLAB when you enter at the MATLAB command line doc eig you will seethe HTML reference documentation page reflecting the MATLAB version of ei g In addition in the Command Window you will see Overloaded functions doc control eig doc symbolic eig To see the documentation for either of those versions of theei g function issue the appropriatedoc command with the proper path as shown above Thedoc command starts your Web browser if it is not already running Help Menus and Help Buttons In the figure window use the Help menu to access help for the Plot Editor for MATLAB Graphics and to access the Help Window and Help Desk In the PrintFrame Editor window use the Help menu to access help for editing printframes In the Page Setup dialog box and in the Plot Editor dialog boxes click th
426. trix andp isa positive integer then A p multipliesA by itself p times X A 2 X 3 6 6 14 10 25 10 25 46 If A is square and nonsingular then A p multipliesi nv A by itself p times Y B 3 Y 0 0053 0 0034 0 0016 0 0068 0 0018 0 0001 0 0036 0 0070 0 0051 Fractional powers like A 2 3 are also permitted the results depend upon the distribution of the eigenvalues of the matrix Element by element powers are obtained with For example X A 2 A 1 1 1 4 1 9 The function sqrt mA computes A 1 2 byamoreaccuratealgorithm Them in sqrt m distinguishes this function fromsqrt A which likeA 1 2 does its job element by element 4 31 4 Matrices and Linear Algebra 4 32 A system of linear constant coefficient ordinary differential equations can be written dx dt Ax where x x t is a vector of functions of t and A is a matrix independent of t The solution can be expressed in terms of the matrix exponential x t 4x0 The function expm A computes the matrix exponential An example is provided by the 3 by 3 coefficient matrix A 0 6 l 6 2 16 5 20 10 and the initial condition x 0 x0 1 1 1 The matrix exponential is used to compute the solution x t to the differential equation at 101 points on the interval O lt t lt 1 with X for t 0 01 1 X X expm t A x0 end A three dimensional phase plane plot obtain
427. ts Step In Single Step Continue Quit Debugging Set or clear a breakpoint at the line containing the cursor Clear all breakpoints that are currently set Execute the current line of the M file and if the line is a call to another function step into that function Execute the current line of the M file Continue execution of M file until completion or until another breakpoint is encountered Exit the debugging state dbstop dbclear dbclear all dbstep in dbstep dbcont dbquit A right button mouse click in the Editor window produces a pop up menu of some of the options Setting Breakpoints Most debugging sessions start by setting a breakpoint Breakpoints stop M file execution at specified lines and allow you to view or change values in the 3 5 3 Debugger and Profiler 3 6 function s workspace before resuming execution A breakpoint is set or cleared at the line containing the cursor A red stop sign i next to a line indicates that a breakpoint is set at that line If the line selected for a breakpoint is not a valid executable line then the breakpoint is set at the next executable line Note The Debugger s Breakpoints menu also lets you halt M file execution if your code generates a warning error or NaN or nf value At the beginning of the debugging session you re not sure where the error in thevari ance function is or even if it s in thevari ance morsq
428. ts collected by this portfolio class Designing the Portfolio Class The portfolio class is designed to contain the various assets owned by a given individual and provide information about the status of his or her investment portfolio This example implements a somewhat over simplified portfolio class that e Contains an individual s assets e Displays information about the portfolio contents e Displays a 3 D pie chart showing the relative mix of asset types in the portfolio Required Portfolio Methods The portfolio class implements only three methods e portfolio The portfolio constructor e display Displays information about the portfolio contents e pi e3 Overloaded version of pi e3 function designed totakea single portfolio object as an argument Since a portfolio object contains other objects the portfolio class methods can use the methods of the contained objects For example the portfoliodis pl ay method calls the stock class disp ay method and so on Example The Portfolio C ontainer The Portfolio Constructor Method The portfolio constructor method takes as input arguments a client s nameand a variable length list of asset subclass objects stock bond and savings objects in this example The portfolio object uses a structure array with the following fields e name Theclient s name e ind assets The array of asset subclass objects stock bond savings e total value Thetotal value of all as
429. ts to the end of a line of code For example Add up all the vector elements y sum x Use the sum function The first comment line immediately following the function definition line is considered the H 1 line for the function The H1 line and any comment lines immediately following it constitute the online help entry for the file In addition to comment lines you can insert blank lines anywherein an M file Blank lines are ignored H owever a blank line can indicate the end of the help text entry for an M file Help for Directories You can make help entries for an entire directory by creating a file with the special name Cont ents m that resides in the directory This file must contain only comment lines that is every line must begin with a percent sign MATLAB displays the lines in a Contents m file whenever you type help directory name If a directory does not contain a Contents m file typinghel p directory_name displays the first help line the H1 line for each M filein the directory Function Names MATLAB function names have the same constraints as variable names MATLAB uses the first 31 characters of names Function names must begin with a letter the remaining characters can be any combination of letters numbers and underscores Some operating systems may restrict function names to shorter lengths The name of the text file that contains a MATLAB function consists of the function name with the extension m ap
430. u have made changes to the file but have not saved those changes 2 40 Microsoft W indows Handbook File Menu Use the File menu items to open save and print files and to exit from the Editor Debugger If you havea color printer and want to print in color before printing select Options from the Tools menu and check the Print in color checkbox Edit Menu Use the Edit menu items to find and replace text and to go to the line you specify View Menu Use Evaluate Selection to evaluate an expression and display the answer in the Command Window Use Auto Indent Selection to indent the selected text according to MATLAB syntax Debug Menu The Debug menu provides an interface to the graphical M file Debugger See Chapter 3 for information about debugging within MATLAB Tools Menu The Tools menu provides access to several dialog boxes that allow you to control existing features and to define new features for use with the E ditor Debugger Run Initially present on the Tools menu Run saves all files and runs the current file It is an example of the type of command you can create and add to the Tools menu using the Customize menu item 2 41 2 MATLAB W orking Environment 2 42 Customize Use Customize to create your own custom tool that can run any MATLAB command Customize Tools Menu Ea Tools Menu Contents Z ae a Remove Move Up Menu Text BR un MATLAB Expression stFite gt T
431. uations 8 12 The input arguments are ia String containing the name of the file that describes the system of ODEs tspan Vector specifying the interval of integration F or a two element vectortspan t0 tfinal thesolver integrates fromt0 to tfinal Fortspan vectors with morethan two elements the solver returns solutions at the given time points as described below Note thatt0 gt tfinal is allowed y0 Vector of initial conditions for the problem The output arguments are T Column vector of time points y Solution array Each row in Y corresponds to the solution at a time returned in the corresponding row of T Obtaining Solutions at Specific Time Points To obtain solutions at specific time pointst0 t1 tfinal specifytspan asa vector of the desired times The time values must bein order either increasing or decreasing Specifying these time points in thet span vector does not affect the internal time steps that the solver uses to traverse the interval fromtspan 1 to tspan end and has little effect on the efficiency of computation All solvers in the MATLAB ODE suite obtain output values by means of continuous extensions of the basic formulas Although a solver does not necessarily step precisely to a time point specified in t s pan the solutions produced at the specified time points are of the same order of accuracy as the solutions computed at the internal time points O DE Solvers Specifying Solver Options
432. uce data from other programs into MATLAB using several methods The best method for importing data depends on the amount and format of the data Method Enter data as an explicit list of elements Create data in an M file When to Use If you have a small amount of data it is easy to type the data explicitly using brackets This method is awkward for larger amounts of data because you cant edit your input if you make a mistake but must correct it using assignment statements See Getting Started with MATLAB for more information on this technique Use a text editor to create an M file that enters the data as an explicit list of elements This method is useful when the data is not already in digital form and must be entered anyway Although similar to the first method this method has the advantage of allowing you to use your text editor to change the data and to fix mistakes You can then just rerun the M file to re enter the data 2 26 Data Import Export Load data from an ASCII data file Read data using fopen fread and MATLAB s file I O functions Use a specialized file reader function for application specific formats An ASCII data file stores the data in ASCII form with each row having the same number of values and terminating with new lines carriage returns with spaces separating the numbers You can edit ASCII data files using a normal text editor You can read ASCII data files dire
433. uces this output Descriptor XYZ Date 17 Nov 1998 Type stock Current Value 2500 00 Number of shares 100 Share price 25 00 Here is the stock di spl ay method function display s DISPLAY s Display a stock object display s asset stg sprintf Number of shares g nShare price 3 2f n S num_shares s share price disp stg First the parent asset object is passed to the asset di sp ay method to display its fields MATLAB calls the asset di spl ay method because the input argument is an asset object The stock object s fields are displayed in a similar way using a formatted text string Note that if you did not implement a stock class disp ay method MATLAB would call the asset di sp ay method This would work but would display only the descriptor date type and current value 14 53 14 MATLAB Classes and O bjects Example The Portfolio Container Aggregation is the containment of one class by another class The basic relationship is each contained class is a part of the container class For example consider a financial portfolio class as a container for a set of assets stocks bonds savings etc Once the individual assets are grouped they can be analyzed and useful information can be returned The contained objects are not accessible directly but only via the portfolio class methods See Example Assets and Asset Subclasses on page 14 37 for information about the asse
434. unction squares the input value Given two inputs it adds them together Here s a more advanced examplethat finds the first token in a character string A token is a set of characters delimited by whitespace or some other character Given one input the function assumes a default delimiter of whitespace given two it lets you specify another delimiter if desired It also allows for two possible output argument lists Functions function token remainder strtok string delimiters Function requires at leat if nargin lt 1 error Not enough input arguments end one input token remainder en length string if len return end If one input use white if nargin 1 space delimiter delimiters 9 13 32 White space characters end wee r while any string i delimiters Determine where non gt r Lo return end delimiter characters end begin 7 start i while any string i delimiters i i i 41 Find where token ends cde ihe Sal ei oteak ani L end finish i 1 token string start finish For two output arguments if nargout 2 count characters after first remainder string finish 1 end delimiter remainder end Note strtok isa MATLAB M filein thestrfun directory Note that the order in which output arguments appear in the function declaration line is important The argument that the function returns in most cases app
435. unctiont ext read reads string and numeric data froma fileinto MATLAB variables using the conversion specifiers you indicate Conversion specifiers denote for example the length of the field and the data format t ext read is most useful for files with a known and uniform format such as comma or tab delimited files but you can also use it for reading free format files For example the filemydata dat is Sally Typel 12 34 45 Yes Toread mydata dat as a free format file use the conversion format names types x y answer textread mydata dat s s f d S 1 Data Import Export where s reads a whitespace separated string f reads a floating point value and d reads a signed integer MATLAB returns names Sally types Typel X 12 34000000000000 y 45 answer Yes See all of the allowable delimiters on the reference page for textr ead Ex changing Data Files Between Platforms It s sometimes necessary to work with MATLAB implementations on different computer systems or to transmit MATLAB applications to users on other systems MATLAB applications consist of M files containing functions and scripts and MAT files containing binary data Both types of files can be transported directly between different computers e M files consist of ordinary text They are machine independent While different platforms terminate lines with various combinations of CR carriage return and LF line
436. unique however it is likely to be so 15 6 Binary Files Binary Files This section explains how to read from or write to binary files Function Purpose fread Read binary data from file fwrite Write binary data to file Reading Binary Files Thef read function reads all or part of a binary file as specified by a file identifier and stores it in a matrix In its simplest form it reads an entire file and interprets each byte of input as the next element of the matrix F or example the following code reads the data from a file named ni ckel dat into matrix A fid fopen nickel dat r A fread fid Toecho the data to the screen after reading it usechar to display the contents of A as characters transposing the data so it displays horizontally disp char A Thechar function causes MATLAB tointerpret the contents of A as characters instead of as numbers TransposingA displays it in its more natural horizontal format Controlling the Number of Values Read fread accepts an optional second argument that controls the number of values read if unspecified the default is the entire file For example this statement reads the first 100 data values of the file specified by f i d into the column vector A A fread fid 100 Replacing thenumber 100 with the matrix dimensions 10 10 readsthesame 100 elements into a 10 by 10 array 15 7 13 File O 15 8 Controlling the Data Type o
437. ur with C 2 4 ans ans ans Cell Arrays Similarly you can create a new numeric array using the statement B C 1 C 2 C 4 You can also use content indexing on the left side of an assignment to create a new cell array where each cell represents a separate output argument D 1 2 eig B D 3x3 double 3x3 double You can display the actual eigenvalues and eigenvectors usingD 1 andD 2 Note Thevarargin andvarargout arguments allow you to specify variable numbers of input and output arguments for MATLAB functions that you create Both varargin andvarargout are cell arrays allowing them to hold various sizes and kinds of MATLAB data See Chapter 10 for details Applying Functions and Operators Use indexing to apply functions and operators to the contents of cells For example use content indexing to call a function with the contents of a single cell as an argument Atl ToS 2 2 348 A 1 2 randn 3 3 A 1 3 1 5 B sum A 1 1 B 13 27 13 Structures and Cell Arrays 13 28 To apply a function to several cells of a non nested cell array use a loop for i 1l length A M i sum A 1 i end Organizing Data in Cell Arrays Cell arrays are useful for organizing data that consists of different sizes or kinds of data Cell arrays are better than structures for applications where e You need to access multiple fields of data with one statement e You want to access
438. ure where each point on the graph represents the location of a nonzero array element 9 13 9 Sparse Matrices 9 14 For example spy west 0479 0 so gt X ya NUE 100F D A 150 T K X 200l lt Si A 1 250 afl gt as N 300 aN 3 350 m J Sam BI cal ae Ka i Jl ve an 450 ad A gt 0 100 200 300 400 nz 1887 The find Function and Sparse Matrices For any matrix full or sparse thef i nd function returns the indices and values of nonzero elements Its syntax is i j s find S fi nd returns the row indices of nonzero values in vector i the column indices in vector j and the nonzero values themselves in the vector s The example below uses f i nd to locate the indices and values of the nonzeros in a sparse matrix Thes parse function uses thef i nd output together with the size of the matrix to recreate the matrix i j s find S m n size S S sparse i j s m n Example Adjacency Matrices and Graphs Example Adjacency Matrices and Graphs The formal mathematical definition of a graph is a set of points or nodes with specified connections between them An economic model for example is a graph with different industries as the nodes and direct economic ties as the connections The computer software industry is connected to the computer hardwareindustry which in turn is connected tothe semiconductor industry and so on This definition of a g
439. urns a string containing the class name of a For example class pi class hello cl ass p return double char pol ynom Use thewhos command to see what objects are in the MATLAB workspace whos Name Size Bytes Class p 1x1 156 polynom object The display Method MATLAB calls a method named di spl ay whenever an object is the result of a statement that is not terminated by a semicolon For example creating the variablea which is a double calls MATLAB s di spl ay method for doubles gt gt a 5 a 5 You should define adi sp ay method so MATLAB can display values on the command line when referencing objects from your class In many classes di spl ay can simply print the variable name and then use thechar converter method to print the contents or value of the variable since MATLAB displays output as strings You must define thechar method to convert the object s data toa character string Designing User Classes in MATLAB Examples of display Methods See the following sections for examples of di sp ay methods e The Polynom display Method on page 14 27 e The Asset display Method on page 14 44 e The Stock display Method on page 14 52 e The Portfolio display Method on page 14 57 Accessing Object Data Y ou need to write methods for your class that provide access to an object s data Accessor methods can use a variety of approaches but all methods that change
440. urrent directory Later you can view the saved results using your Web browser 3 25 3 Debugger and Profiler 3 26 Another way to save results is with thei nfo profile command which displays a structure containing the profiler results Save this structure so that later you can generate and view the profile report using profreport info Example Using Structure of Profiler Results The profiler results are stored in a structure that you can view or access This example illustrates how you can view the results 1 Run the profiler for code that computes the Lotka Volterra predator prey population model profile on detail builtin history t y ode23 lotka 0 2 20 20 2 Toview the structure containing profiler results type stats profile info MATLAB returns stats FunctionTable 28x1 struct FunctionHistory 2x774 double ClockPrecision 0 00999999999840 3 You can view and access the contents of the structure For example type stats FunctionTable MATLAB displays the Funct i onTabl e structure ans 28x1 struct array with fields FunctionName Mf i eName Type NumCalls Total Ti me Total RecursiveTime Children Parents ExecutedLi nes M File Profiler example stats FunctionTable 2 MATLAB returns the second element in the structure ans FunctionName Mf il eName Type NumCalls Total Ti me Total RecursiveTi me Children Parents ExecutedLi nes Save
441. ursor on the line and selecting Clear Breakpoint from the Debug menu Or alternatively right click to bring up the context menu and choose Clear Breakpoint Continue executing the M file by selecting Continue from the Debug menu Open thes qsum m file and set a breakpoint at line 4 to check both the loop indices and the computations that take place inside the loop Run variance again still using the vector v as input Execution pauses at line 4 of sqsum If MATLAB Debugger we look at thevari ance function in the Editor Debugger we see the call to sqsum indicated by an arrow with a vertical line through it MATLAB Editor Debugger C WINDOWS Desktop variance m J File Edit View Debug window Help pisla le Sle ae Aae suck Pome a function y variance x mu sum x length x tot sqsum x mu y tot length x 1 Evaluate the loop index i K gt gt i 1 Then select Single Step from the Debug menu to execute the next line Evaluate the variablet ot K gt gt tot tot 4 Select Single Step again s qs um only goes through thef or loop once for ij 1 length mu Theloop only iterates until the length of mu a scalar rather than the length of x the input vector Select Quit Debugging from the Debug menu to end the M file execution 3 9 3 Debugger and Profiler 3 10 Toseeif changingt ot toits expected value produces the correct answer clear the breakpoint from s qs um and
442. usewhos to see what variables are now in the workspace whos To check the value of mu K gt gt mu mu MATLAB Debugger Usedbst ep tostep onelinein the function When execution stops again before line 4 check the value of t ot to see if the mean removed squared sum calculation matches the expected value K gt gt dbstep 4 y tot length x 1 K gt gt tot tot 4 It appears the problem may bein thes qs um function Changing Workspace Context Usedbup anddbdown to move between function workspaces and the base workspace To step up from thevari ance function and see the base workspace contents use dbup whos Thetest variablev as well as any other variables you may have created shows up in the listing To step back down to the vari ance workspace use dbdown Displaying an M File with Line Numbers Without leaving keyboard mode usedbt ype to views qsum Set breakpoints to check both the loop indices and the computations that take place inside the loop K gt gt dbtype sqsum 1 function tot sqsum x mu 2 tot 0 3 for i 1l length mu 4 tot tot x i mu 2 5 end K gt gt dbstop sqsum 4 K gt gt dbstop sqsum 5 3 13 3 Debugger and Profiler 3 14 Viewing the Function Call Stack and Continuing Execution Return from keyboard mode using dbqui t At the MATLAB prompt type dbclear variance to clear the breakpoints from thevari ance function while retaining the new
443. utive lines These two examples require the same amount of memory result function2 functionl input 99 result functionl input 99 result function2 result Variables and Memory Memory is allocated for variables whenever the left hand side variable in an assignment does not exist The statement x 10 allocates memory but the statement x 10 1 does not allocate memory if the 10th element of x exists 10 67 10 M File Programming 10 68 To conserve memory e Avoid creating large temporary variables and clear temporary variables when they are no longer needed e Avoid using the same variables as inputs and outputs to a function They will be copied by reference For example y fun x y is not preferred becausey is both an input and an output variable e Set variables equal to the empty matrix to free memory or clear them using clear variable_name e Reuse variables as much as possible Global Variables Declaring variables asgl obal merely puts a flag in a symbol table It does not use any more memory than defining nonglobal variables Consider the following example global a a 5 Now there is one copy of a stored in the MATLAB workspace Typing clear a removes a from the MATLAB workspace but it still exists in the gl obal workspace clear global a removes a from the global workspace PC Specific Topics e There are no functions implemented to manipulate the way MATLAB hand
444. ve specified x the string returned is a syntactically correct MATLAB expression which you can then evaluate 14 25 14 MATLAB Classes and O bjects Hereis pol ynom char m function s char p POLYNOM CHAR CHAR p is the string representation of p c if all p c 0 5 0 else d length p c 1 s for a p c if a 0 if isempty s if a gt 0 s s else s s a ys a end end ifa 1 d s s numstr a if d gt 0 Soe bs SEn end end if d gt 2 s s x int2str d elseif d 1 E see E end end d d 1 end end Evaluating the Output If you create the polynom object p p polynom 1 0 2 5 14 26 Example A Polynomial C lass and then call thechar method on p char p MATLAB produces the result ans X 3 2 x 5 The valuereturned bychar isastring that you can pass toeval once you have defined a scalar value for x For example x 3 eval char p ans 16 See The Polynom subsref Method on page 14 28 for a better method to evaluate the polynomial The Polynom display Method Here is pol ynom display m This method relies on the char method to produce a string representation of the polynomial which is then displayed on the screen This method produces output that is the same as standard MATLAB output That is the variable name is displayed followed by an equal sign then a blank line then a new line with the va
445. y 3 y 2 y 1 1 This ODE file must accept the argumentst andy although it does not have to use them Here the vector dy must be a column vector Apply a solver function to the problem The general calling syntax for the ODE solvers is T Y solver F tspan yO wheresol ver is a solver function likeode45 Theinput arguments are F String containing the ODE file name tspan Vector of time values where t0 tf inal causes the solver to integrate fromt0 tot final y0 Column vector of initial conditions at the initial timet 0 For example to usetheode45 solver to find a solution of the sample IVP on the time interval 0 1 the calling sequence is T Y ode45 F 0 1 0 1 1 Each row in solution array Y corresponds toa time returned in column vector T Also in the case of the sample IVP Y 1 isthe solution 2 isthe derivative of the solution and Y 3 is the second derivative of the solution Representing Problems Representing Problems This section describes how to represent ordinary differential equations as systems for the MATLAB ODE solvers The MATLAB ODE solvers are designed to handle ordinary differential equations These are differential equations containing one or more derivatives of a dependent variable y with respect to a single independent variable t usually referred to as time The derivative of y with respect to t is denoted as Y the second derivative as Y and so on
446. y editing This is essentially the same array editing facility as the one provided for Windows 2 57 2 MATLAB W orking Environment platforms See the section Microsoft Windows Handbook for a discussion of this tool File Edit View Debug Tools Window Help Ready 8 59 AM Preferences An optimization option in the Xdef aul ts file starts the environment tools but does not display them until you explicitly ask for them If you want to turn this feature off set the mat ab preload DE variable to Of f The default is on The environment tools will not display well on 16 bit X displays If you are using such a display turn the environment tools off in your Xdefaults file matlab builtlnEditor Off matl ab graphical Debugger Off matlab preloadl DE Off License Management Tools Although your system administrator has probably taken care of the details of installing and configuring MATLAB s license manager some background information is helpful for you to know This section contains some of the same UN IX Handbook information provided for the system administrator F or complete details see the section by the same name in the MATLAB Installation Guide for UNIX On UNIX platforms MATLAB uses a license manager called FLEXIm to manage the per computer or per user licensing FLEXIm manages per user licenses with a key system Each time a user invokes MATLAB the license
447. y is measured at several different values of time t to produce the following observations t y 0 0 0 82 0 3 0 72 0 8 0 63 1 1 0 60 1 6 0 55 2 3 0 50 This data can be entered into MATLAB with the statements E gh abe R E 3 y 82 72 63 60 55 50 4 15 4 Matrices and Linear Algebra 4 16 It is believed that the data can be modeled with a decaying exponential function y t C Ces This equation says that the vector y should be approximated by a linear combination of two other vectors one the constant vector containing all ones and the other the vector with components et The unknown coefficients c and can be computed by doing a least squares fit which minimizes the sum of the squares of the deviations of the data from the model There are six equations in two unknowns represented by the 6 by 2 matrix E ones size t exp t E 1 0000 1 0000 1 0000 0 7408 1 0000 0 4493 1 0000 0 3329 1 0000 0 2019 1 0000 0 1003 The least squares solution is found with the backslash operator c Ely 0 4760 0 3413 In other words the least squares fit to the data is y t 0 4760 0 3413 The following statements evaluate the model at regularly spaced increments in t and then plot the result together with the original data T 0 0 1 2 5 Y ones size T exp T c plot T Y t y 0 Solving Linear Equations You can see that E c is not exactly equal toy but that the diff
448. ynom 11 33 AM Z Note You cannot use the Editor on UNIX platforms to cut and paste to and from X Window Systems applications The Editor on UNIX platforms does not accept J apanese characters as input UN IX Handbook Setting the Default Editor The editing and debugging features are both set 0n by default when MATLAB is installed If you want to substitute a different editor such as Emacs or not use graphical debugging you can turn these tools Of f by setting the appropriate variablein your home Xdefaults file matl ab builtInEditor Off matl ab graphical Debugger Off Run xrdb merge home Xdefaults before starting MATLAB If you set the Editor Of f the option matl ab external EditorCommand EDITOR FILE amp controls what thee di t command does MATLAB substitutes EDI TOR with the name of your default editor and FI LE with the filename This option can be modified to any sort of command line you want Changing the Editor During a Session Toturn off the buil ti nEdi tor during a MATLAB session use system dependent builtinEditor off Thenedi t uses the editor defined for your UNIX EDI TOR environment variable Toturn the MATLAB Editor Debugger back on during the session use system dependent builtinEditor on You can include thes ystem dependent command in your startup m file See the mat abrc reference page for more information Saving Editor O ptions The MATLAB Editor st
449. yntax Using Braces The curly braces are cell array constructors just as square brackets are numeric array constructors Curly braces behave similarly to square brackets except that you can nest curly braces to denote nesting of cells see page 13 29 for details Curly braces use commas or spaces to indicate column breaks and semicolons to indicate row breaks between cells For example C 1 2 3 4 5 6 7 8 results in cell 1 1 cell 1 2 1 2 3 4 cell 2 1 cell 2 2 5 6 7 8 Use square brackets to concatenate cell arrays just as you do for numeric arrays Cell Arrays Preallocating Cell Arrays with the cell Function Thecel function allows you to preallocate empty cell arrays of the specified size For example this statement creates an empty 2 by 3 cell array B cell 2 3 Use assignment statements to fill the cells of B B 1 3 1 3 Obtaining Data from Cell Arrays You can obtain data from cell arrays and store the result as either a standard array or a new cell array This section discusses e Accessing cell contents using content indexing e Accessing a subset of cells using cell indexing Accessing Cell Contents Using Content Indexing Y ou can use content indexing on therright side of an assignment to access some or all of the data in a single cell Specify the variable to receive the cell contents on theleft side of the assignment Enclose the cell index
450. ys A cell can contain another cell array or even an array of cell arrays Cells that contain noncell data are called leaf cals You can use nested curly braces the cell function or direct assignment statements to create nested cell arrays You can then access and manipulate individual cells subarrays of cells or cell elements Building Nested Arrays with Nested Curly Braces You can nest pairs of curly braces to create a nested cell array For example clear A A 1 1 magic 5 A 1 2 5 2 8 7 3 0 6 7 3 Test 1 2 41 547i 17 A 5x5 double 2x2 cell Note that the right side of the assignment is enclosed in two sets of curly braces The first set represents cell 1 2 of cell array A Thesecond packages the 2 by 2 cell array inside the outer cell 13 29 13 Structures and Cell Arrays 13 30 Building Nested Arrays with the cell Function To nest cell arrays with thece function assign the output of cel toan existing cell 1 Create an empty 1 by 2 cell array A cell 1 2 2 Create a 2 by 2 cell array inside A 1 2 A 1 2 cell 2 2 3 Fill A including the nested array using assignments A 1 1 magic 5 A 1 2 1 1 5 2 8 7 3 0 6 7 3 A 1 2 1 2 Test 1 A 1 2 2 1 2 4i 547i A 1 2 2 2 cell 1 2 A 1 2 2 2 1 17 Note the use of curly braces until the final level of nested subscripts This is required because you need t
451. zes do not have to conform for every element in an array In the patient example thename fields can have different lengths thet est fields can be arrays of different sizes and so on Building Structure Arrays Using the struct Function You can preallocate an array of structures with thest ruct function Its basic form is str_array struct fieldl vall field2 val2 where the arguments are field names and their corresponding values A field value can bea single value represented by any MATLAB data construct or a cell array of values All field values in the argument list must be of the same scale single value or cell array You can use different methods for preallocating structure arrays These methods differ in the way in which the structure fields are initialized As an example consider the allocation of a 1 by 3 structure array weather with the structurefieldst emp andr ai nf all Threedifferent methods for allocating such an array are shown in this table 13 5 13 Structures and Cell Arrays 13 6 Method Syntax Initialization struct struct with r ep mat struct with cell array syntax weather 3 struct temp 72 rainfall 0 0 weather repmat struct temp 72 rainfall 0 0 1 3 weather struct temp 68 80 72 rainfall 0 2 0 4 0 0 weather 3 iSinitialized with the field values shown The fields for the other structur

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