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MLAB Beginner`s Guide Manual

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1. The arithmetic notational rules and the corresponding numerical calculations are described in this chart Overflow or underflow may occur SE1 SE2 sum SE1 SE2 is computed SE1 negative SE1 is computed SE1 SE2 difference SE1 SE2 is computed SE1 SE2 product SE1 SE2 is computed SE1 SE2 quotient SE1 SE2 is computed division by zero may occur SE1 SE2 Either or may be used Let a SE1 and b SE2 b gt 0 a axa a b times b integer b 0 a 1 fora 40 b lt 0 Wa b integer b integer exp b log a for a gt 0 0 fora 0 b gt 0 SE1 equals SE1 Table 2 2 Scalar arithmetic notational rules For example the parenthesis rule applied to A 1 states that A 1 is a scalar expression Moreover 1 05 A 1 is a scalar expression by the rule SE1 SE2 In general parentheses are used for intermediate scalar expressions rather than for the final expression The following trigonometric functions may be used to form scalar expressions SIN SE1 sine of angle SE1 measured in radians COS SE1 cosine of angle SE1 measured in radians TAN SE1 tangent of angle SE1 measured in radians SIND SE1 sine of angle SE1 measured in degrees COSD SE1 cosine of angle SE1 measured in degrees TAND SE1 tangent of angle SE1 measured in degrees All of the functions above are defined everywhere but are meaningless for arguments of large magnitude Tabl
2. nF Move forward n characters nA Set incremental angle of n degrees R Reset all parameters Cx Control character corresponding to x Lx Lower case character corresponding to x Y Toggle vertical bars between characters on off Z Toggle underlining on off no Use color number n M Carriage return and line feed E Go to after rightmost character nIx Draw x for a total of n 1 times nX Draw centered over the n th preceding character Table 4 11 String modification command codes 4 7 EXAMINING AND MODIFYING CURVE AND WINDOW DATA ITEMS Displaying individual curves axes titles and windows can be controlled by MLAB BLANK and UNBLANK commands as described in Table 4 12 If a curve or axis is blanked then none of its parts curve matrix label matrix line type and point type are shown on the display although the curve data item continues to exist The BLANK command puts one or more curves in the blanked condition The UNBLANK command removes one or more curves from the blanked condition so that they again appear on the display The blanked or nonblanked status of each curve is independent of the VIEW and UNVIEW commands A BLANK or UNBLANK command may refer to a window data item This is equivalent to applying the BLANK or UNBLANK command to the entire list of curves and titles currently in that window object Suppose window W1 has been blanked in order to w
3. NORMRAN returns a pseudo random number from a series of such numbers If SE1 0 then the next value along the current series is returned If SE1 0 then a new series is started which will be a function of SE1 SE2 and SE3 are optional values specifying the mean and variance respectively of the normally distributed random numbers If not supplied SE2 defaults to zero and SE3 defaults to 1 Table 2 8 Normal random number operator 41 1 41 2 013 042 1 42 2 Q23 so that for example the element in row 1 and col j of A for appropriate integers i and j would be denoted by aj 1 In MLAB parentheses or equivalently brackets are used to denote matrix elements so that M 1 1 and M 1 1 both denote the first element of the first row of a matrix M For example DAY 5 2 or DAY 5 2 denotes the entry in row 5 and column 2 of a matrix DAY and A I 1 J or A I 1 J denotes the entry in row 1 1 and column J of an MLAB matrix A if I and J are scalar identifiers Table 2 9 summarizes construction of matrix entries in MLAB The construction and manipulation of MLAB matrices will be considered in Chapter 3 A list of all of the identifiers defined as scalars in an MLAB session can be obtained by giving the command TYPE SCALARS 22 In general a scalar expression describing any matrix entry can be constructed using an appropriate matrix identifier M and scalar expressions SE1 and SE2 to specify the row and c
4. displays the title string T prompt string array S initial string array fields H specifier of which initial string array is active X field width specifier Y and waits for user to fill in fields returns a string array containing the final string array fields WREAD SE1 Wname SE2 reads m SE1 ordered pairs specified by mouse clicks in window Wname with n SE2 specifying the mode returns a 2 column m row matrix 183 3 MLAB Commands for Input and Output e SAVE D1 D2 Dn IN filename saves or replaces data items D1 D2 Dn in a disk file named filename SAV e SAVE IN filename saves all currently defined data items in a disk file named filename SAV e SAVE D1 D2 Dn saves D1 D2 Dn in disk files named D1 SAV D2 SAV Dn SAV e USE filenamel filename2 filenamen e DELETE Di D2 Dn deletes MLAB data items D1 D2 Dn e DELETE M ROW ME1 deletes matrix rows e PRINT Di D2 Dn IN filename similar to the TYPE command but creates a character disk file named filename LST e PLOT creates a ps or 1j picture file e DO filename executes a DO character file of MLAB commands on a disk file e DO filename ext executes a character file of MLAB commands on the disk file filename ext e DO CON reads MLAB commands from console e DO CON same as DO CON e TYPE text e IF SE1 THEN commandgroup1 ELSE commandgroup2 4 MLAB session record on
5. 232 x m 2 11 136 The process shown above can be independently applied to n data points each obtained by re peated measurements The result is an n by 2 array of averaged data points and an n vector of corresponding weights estimating measurement precision If the variance o of the dependent variable y is proportional to the size of the corresponding independent variable x 0 c x for i 1 2 n the weight associated with the data point xi yi should be chosen as 1 c a wi fori 1 2 n On the other hand if the variance of the dependent variable y is inversely proportional to the size of the corresponding independent variable x 0 c x for i 1 2 n the weight associated with the data point x yi should be chosen as w Chi fori 1 2 n The reciprocal of variance method above is only one of the possible ways for choosing data point weights In MLAB one can select any weighted sum of squared error criterion that is appropriate for the data and model That is each data point can be given any fixed positive weight chosen by the user Of course the sum of squared error criterion is a special case obtained by setting all weights equal to one 7 3 EWT OPERATOR MLAB has an operator called EWT for estimated weights and shown in Table 7 1 which takes a data matrix M as input and produces an associated vector of weight values as output These weight values are comput
6. Check that the DO files are not too long Long files are hard to understand and debug 2 Tests for MLAB Identifier Misuse a Check that all MLAB data items required in a DO file have been created b Check that the same MLAB identifier has not been used for more than one purpose and that the identifier is not an MLAB reserved word see the latest edition of the MLAB Reference Manual c Check that MLAB identifiers have been deleted where necessary to free them for sub sequent use as unknown identifiers 3 Tests for Logical Flow Errors a Check whether conditions are improper or tests are reversed in IF THEN ELSE com mands or in the IF THEN ELSE scalar expressions and whether any needed operations have been omitted from the THEN or ELSE clauses b If the DO file is intended to be used repeatedly within the same MLAB session make sure that it works repeatedly not just once Check any commands preparing for the next use of this DO file for possible omissions and errors 4 General Error Correcting Methods a Examine error messages and output for clues to DO file malfunction For example b DIVISION BY ZERO or ASSIGNING MATRIX TO SCALAR suggest particular types of er rors Examine MLAB data items and graphical displays for clues in a DO file malfunction A curve not displayed may not exist may be blanked or may be outside the user rectangle or off the display screen Here a TYPE command fo
7. Note that in this introductory example MLAB commands are typed directly This interactive mode of operation is useful when one time exploratory computations are to be done and this is the way the examples in the next few chapters will be presented Generally however it is most convenient to construct a do file of MLAB commands with the text editor facility in MLAB under DOS or MSWindows or with a parallel running editor program in other environments You can repetitively execute such a do file and revise it to construct a useful reusable script Do files are discussed in Chapter 5 but you may wish to gain practice with this mode of operation immediately It is highly recommended that you use do files the added convenience obtained is well worth the small effort needed to learn to construct them 1 2 SUMMARY This chapter has shown a typical simple use of MLAB to make numerical calculations manipulate matrices of numbers fit curves to data prepare graphical displays of the results and save currently 13 defined variables for use in a subsequent MLAB session The full MLAB system can analyze much more complicated models including models defined by systems of ordinary differential equations It also has capabilities for numerical calculation approaching those of computer programming languages and generally far more convenient 14 Chapter 2 NUMERICAL CALCULATION IN MLAB The basic methods for calculation of numerical value
8. f FIT C K Y TO YD inal parameter values value error dependency parameter 1 2915176399 0 0283143124 0 818706264 C 0 4399303423 0 0015101591 0 818706264 K 4 iterations CONVERGED best weighted sum of squares 3 377390e 06 weighted root mean square error 6 125892e 04 weighted deviation fraction 3 940699e 03 R squared 9 999310e 01 The Legendre s equation model was set up for an MLAB FIT command just as it would be set up for numerical evaluation using an INTEGRATE operator The parameters C and K in the differential 298 equation system represent the unknown values of c and y 0 respectively and C and K were set equal to 1 as initial guesses The FIT command is used for a differential equation model similarly to its previously defined use for functional forms As shown in the example a differential equations system FIT command can adjust parameters appearing in differential equation FUNCTION commands in INITIAL commands or in both If one of the system conditions has the form INITIAL Y A B then B can be fit but not A As usual the FIT command above changes the values of the fitted parameters C and K to new values hopefully optimum found by the search process As usual the fitted system can be numerically solved by the INTEGRATE expression Appropriate MLAB dialog for continuing the above example is shown below TYPE C K C 1 29151764 K 439930342 YC POINTS Y 0 5 05 TYPE YC YD YC
9. nj j 1 2 m the weighted sum of squared error criterion S De Wil yi f tin Til2 Zip 721 wie yi2 f i21 Li22 Lia air yo Wim Yim Timi Tim2 iml is used as a measure of the agreement between the m models and corres ponding data Table 7 3 Joint models weighted sum of squared error criterion 7 6 THE CONSTRAINTS COMMAND Suppose that a weight has been selected for each data point obtained from an experiment For a completely specified model the weighted sum of squared error is just a number However the mathematical model will often contain parameters with unknown values If so then different values assigned to the parameters may yield different values for the sum of squared error criterion That is the weighted sum of squared error becomes a function of the parameters with unknown values For example consider the best straight line fit of the four data points xi yi with assigned weigths w i 1 2 3 4 shown below wi 6 for x1 y1 0 4 1 2 wz 8 for x2 y2 0 7 14 238 w3 9 for x23 y3 0 9 1 7 iy 8 for x4 ya 1 1 2 0 The model is the functional form of the straight line y a b x with unknown y intercept a and slope b Hence the weighted sum of squared error is S a b 6 1 2 a 5 4 1 8 1 4 a b 7 9 1 7 a b 9 8 2 0 a b 1 1 That is the sum of squared error criter
10. 7 5 1 7 5 1 5 7 1 5 and 7 1 Then came another marker row 100 0 and the cube s outline was completed by 7 1 5 7 35 1 9 7 8 1 9 7 8 1 4 and 7 5 1 The last three rows of the 15 by 2 curve matrix were a marker row 100 0 and the final edge 7 5 1 5 to 7 8 1 9 LINETYPE 6 MARKER can also be used with parametric drawing techniques as in Figure 4 5 c to draw several contours or curved lines on the display with a single DRAW command The complete form of the DRAW command shown in Table 4 7 is obtained by adding PTSIZE PTFONT LABEL LABELSIZE LABELFONT OFFSET PLACE ANGLE FORMAT and COLOR clauses to the short form of the DRAW command The complete form of the DRAW command is DRAW C ME1 POINTTYPE ptyp PTSIZE SE1 UNITS PTFONT SE2 LINETYPE ltyp LABEL ME2 LABELSIZE SE2 UNITS LABELFONT SE3 OFFSET SE4 SE5 UNITS PLACE A B FORMAT SE6 SE7 SE8 SE9 SE10 SE11 ANGLE SE12 COLOR col IN W Table 4 7 DRAW command complete form The complete DRAW command provides additional control over the point type and line type used for the curve matrix beyond that provided by the short form of the DRAW command It also allows some or all of the points in the curve matrix to be labelled with numbers The PTSIZE clause determines the size of the point symbols drawn at each x y coordinate in the curve matrix If the PTSIZE clause is omitted the default point type size is 0 02 units of the window s horiz
11. YN referring to the unknown functions A differential equation system has been defined as described above the INTEGRATE expression to solve it numerically is INTEGRATE Y1 T Y2 T Yn T ME1 Of course the list of n derivatives Yj T for j 1 2 n specifies the system of differential equations and initial conditions The matrix expression ME1 specifies a column vector of values for the independent variable T Much greater accuracy will be achieved if the successive values in ME1 move continually away from SEO that is if SEO lt al lt a2 lt lt am or SEO gt al gt a2 gt gt am where ME1 LIST a1 a2 am The result of the INTEGRATE operator is an m by 2n 1 matrix Column 1 is the given vector ME1 The remaining 2n columns are the numerical values for the functions and derivatives corresponding to the independent variable values in column 1 In each row the values of Yj and Yj T for the argument in column 1 are entered in columns 27 and 27 1 respectively for j 1 2 n Table 8 2 Solving first order differential equations systems continued 311 Before issuing a FIT command for a differential equation system the appropriate system must be defined in MLAB This is done as for an INTEGRATE operator by means of the FUNCTION and INITIAL commands FUNCTION Yj T T SEj INITIAL Yj SEO SEn j for j 1 2 n Differential equation FIT commands are disting
12. dv Ou dz dt Ox dt Ou dy Oy dt So the MLAB differentiator has applied the rule correctly with reference to U X and U Y by name and symbolic evaluation of X T as 2 for X T 2 T and Y T as 1 for Y T T It is also possible to define a function that makes reference to a derivative by name as in FUNCTION W Z U X Z C Z MLAB will attempt to differentiate a SUM expression term by term For example an erroneous expression is obtained for the coefficients as shown below FUNCTION P X TYPE P X FUNCTION P X X SUM T 0 N A I 1 X 1 SUM I 0 N 1 X I 1 A I 1 Then P X 0 is incorrectly computed However this can be corrected by separating some of the terms as shown in the following dialog FUNCTION P X TYPE P X FUNCTION P X X A 1 A 2 X SUM I 2 N ACI 1 X7I A 2 SUM I 2 N 1 X I 1 A 1 1 This is true in many cases an incorrect or unsatisfactory symbolic derivative can be avoided by carefully forming the function definition Derivative functions may be explicitly defined with function statements as shown in Table 2 16 This is how differential equations are specified It is also useful to define forward difference or centered difference derivatives by explicit function statements in some cases 2 9 THE TYPE COMMAND It is clear from the previous examples that the TYPE command enables the MLAB user to see the current values of scalar or matrix identifiers or to ob
13. 122 DRAW CX OFFSET 05 05 LABELSIZE 075 LABELFONT 29 IN W VIEW Note that the LABEL clause need not be repeated because the labels 1 5 are already associated with the curve data item CX The clause OFFSET 05 05 is used to specify the x and y displacements in horizontal frame fraction units between each label and the corresponding curve matrix point In Figure 4 8 b for example each label was printed 05 horizontal frame fraction units to the right and 05 horizontal frame fraction units below the corresponding point The clause LABELSIZE 075 specifies that the label digits are to be 075 horizontal frame fraction units high The clause LABELFONT 29 specifies that the label digits are taken from font table number 29 The label matrix can contain arbitrary numbers and may have fewer rows than the curve matrix For example the labelled curve of Figure 4 8 c was drawn by the MLAB commands N LIST 62 24 1 amp 1 5 2 DRAW CX LABEL N OFFSET 0 0 PLACE CENTER TOP LABELSIZE 05 IN W VIEW The LABEL clause with the 2 column label matrix N causes the number 62 to be printed at the first point in the curve matrix 24 to be printed at the third point in the curve matrix and 1 to be printed at the fifth point in the curve matrix The PLACE clause causes the top center of the label to be put at the corresponding data matrix point The LABELSIZE clause resets the size of the font characters to 05 horizontal frame fr
14. If ME1 1 1 MAXPOS then ME1 1 1 will be used as a marker value and thus several figures can be contained in ME1 and shaded simultaneously If two or more figures overlap then the symmetric difference is taken for shading purposes SE1 and SE2 have default values of 10 and 0 respectively Table 6 3 FILL matrix operator 6 4 SURFACE CONTOURS It is often the case that data will be collected over a range of values of an independent variable We have seen many MLAB operators designed to manipulate and display such information However 1t is sometimes the case that there are two independent variables that is the data would have 203 to be displayed in three dimensions While MLAB does include facilities for true 3 dimensional graphics it is often easier to analyze a 2 dimensional representation of your data in the form of a contour map A contour map assumes that you have no more than one data value for each x y pair and draws a surface using variably dashed lines indicating different levels The MLAB CONTOUR operator is used to develop such a contour map The CONTOUR operator takes as input two matrices The first must be a 3 column matrix of data values whose first two columns specify a rectangular grid of points on the x y plane and whose third column contains z coordinate data values The second matrix must be a column vector of z coordinate values whose levels are to be included in the contour map The result of this
15. dell EXERCDES o asas a a dr a a a Se e MAKING PICTURES WITH MLAB 4 1 GRAPHICAL DISPLAY CONTROL COMMANDS 4 2 WINDOW CURVE AND TITLE DATA OBJECTS ae THE DRAW COMMAND opos Eee ra ee ee Se a a 14 THE AXIS COMMAND 3 22 oe op eh Ree RADAR SEEKS A ERS SH le THE TITLE COMMAND o 0 45 4 4284 Fe 4e bp Ad 4 6 TITLE MODIFICATION COMMANDS oc sacet4 eek oh Ge eRe YE oS 4 7 EXAMINING AND MODIFYING CURVE AND WINDOW DATA ITEMS 4 8 SIMPLIFYING MLAB GRAPHICS 2 4 4 24 4bR Eve bee eee eS 49 SUMMARY ace 24 hoa ie Bee A AR we Oy gee ae eee Bee BO 4 10 EXERCDES oo ves be eS Ge ale eee Se ae PES Saw ae a INPUT AND OUTPUT FOR MLAB el PILES ee 6 tee Be ee NN iii 96 96 97 105 126 128 134 138 142 147 148 154 52 THE SAVE AND USE COMMANDS cos 2k ede ew or reses 160 5 3 THE READ READON KREAD AND KSREAD FUNCTIONS 163 94 THE PRINT COMMAND osos a ta ah ake dd 165 5 5 THEPLOTDEV AND PLOT COMMANDS o o o o 166 Sa THE DO COMMAND er a aa 6 ed 168 5 7 THE IF THEN ELSE COMMAND 2 266848 rs Ca Dew 171 5 8 THE MENUCHOICE GETSTRINGS and WREAD COMMANDS 177 So SUMMARY 6 220 w a a A A Ee REESE SAS ARA 183 ita AA 184 FURTHER MATRIX OPERATORS 189 Gl MATRIX COMBINATION lt sur e a A 189 6 2 MODIFYING DATA FOR ANALYSIS AND DISPLAY 194 6 3 SHADING AREAS OF A POLYGON CURVE 200 Gal SUR BA OLN TOUR oo o
16. wy 6 for 1 31 8 3 2 wz for x2 y2 1 4 1 85 w3 1 for x3 y3 2 1 1 8 wa 1 for z4 ya 3 3 1 1 we fb for 5 y5 4 1 7 Then the weighted sum of squared error S is given by 5 S J wiu Fa i 1 6 3 2 2 628 1 1 85 1 992 1 1 8 1 432 1 1 1 928 75 7 912 0 4152 Figure 7 2 shows the change obtained in the analysis of Figure 7 1 Shaded areas have been added to the five squares which geometrically represent the weighted squared error terms Each shaded area represents the fraction of the entire square s area given by the weight for the corresponding data point The weighted sum of squared error is the sum of the shaded areas A data point may have a weight greater than 1 It is also possible to assign zero weight to a data point which then has no effect on the weighted sum of squared error criterion Negative weights should not be used Note that multiplying each weight by a positive constant C is the same as multiplying the weighted sum of squared error criterion by C So only the relative sizes of the weights are important for selecting best fitting models However the reciprocal of variance weights described next have some desirable statistical properties In statistics if data are obtained with measurements of unequal precision the variances standard deviations squared of the measurements are not all equal When measurements
17. 2 n 4 1 p for all integers n where p is a positive constant Hint Adapt MOD INT T P 2 d 2 Dia wj 2 using an indexed sum and a column vector W with W 1 wi W 2 wa W 3 w3 W 4 wa and W 5 ws e The function h x defined by 0 lt 0 ne a 0 lt x lt c Cc TG where c is a constant The function where r nt even y Y 0 dx and c and k are constants 3 What are the arguments and parameters of each of the functions of exercise 2 4 Start a new MLAB session a Recover the data saved in the Chapter 1 tutorial as indicated in the following MLAB command 45 USE TUTOR1 b Evaluate the expressions of exercise 1 by TYPE commands first assigning arbitrary values to the variables occurring in these scalar expressions c Numerically evaluate F A M 7 2 using the scalar A and matrix M recovered from the Chapter 1 tutorial and function F defined in exercise 2 a Test all MLAB functions defined in exercise 2 by selecting arbitrary argument and parameter values To set up the indexed sum coefficients of exercise 2 d use the MLAB command W LIST 1 2 0 1 3 d Exit from MLAB and print a copy of MLAB LOG 46 Chapter 3 COMPUTATION WITH MATRICES IN MLAB It has been shown how by using scalar expressions symbolic descriptions of numbers and scalar assignment commands the MLAB user can perform scalar calculations and display the results through TYPE commands Sim
18. 4 9 4 10 4 11 4 12 4 13 4 14 4 15 4 16 4 17 4 18 4 19 6 1 6 2 6 3 6 4 6 5 6 6 6 7 6 8 Samples of point types and line types a 107 Examples of smooth curve draWiOg e 113 Examples of stick figure drawing e 116 Example of 7 parameter LINETYPE with values 1 1 1 1 0 05 0 120 Examples of point labeling o e o 124 Figure 4 2 after the user rectangle is redefined o 127 Smrtne texi A Sa he A ak eee ge OE aw A 130 Examples of a multiple line text 2 2 ee 132 Changing character size and position 1 e o oso 135 Changing character font tables lt s a e oa r cemoe raa ona ona o ea ranas 136 Changing character height width and angle a a 137 Part of a parabola drawn in the default window a a 144 Additional parts of parabolas drawn in the default window 145 Fiye practice drawit S cc 2 anoa i eee a EL es 150 Practice drawing using squares and triangles a a 151 Practice drawing using font tables ooa a a a 152 Shading a graph using MESH c e cocoa dosos o macs t pa a a eRe ee a a 190 Graphing in polar coordinates e 191 CDF step function graph s s ccs ma aoaie moa Syg dta poa ee EE Da 193 Toterp lating data ea ad ew a E a a al e aai A da 195 Smoothing datas c cons roda 5 eras aaa R
19. 69 77 80 80 83 89 9 1 03 1 05 1 04 1 15 1 18 1 1 2 1 26 1 34 1 41 1 38 1 33 1 33 1 29 1 36 The sum of squared error criterion all weights equal to one will be used and no linear constraints 262 35 30 25 20 15 10 JO 25 20 15 10 5 0 5 10 15 20 25 30 C Figure 7 7 Sum of squared error vs the parameter c on c will be assumed The curve fitting problem here is to slide the graph of h x to the left or right by appropriately choosing c until the portion of the graph between 1 and 1 best approximates the data minimizing the sum of squared error Since there is only one parameter to be fit here the sum of squared error S versus the parameter c can be graphed Part of this graph is shown in Figure 7 7 It is clear from the figure that c 1 is close to the correct value for minimizing S So this initial estimate can be used in the curve fit The MLAB commands and responses appear in the dialog below FUNCTION H X FUNCTION F X C 1 FIT C F TOM inal parameter values value error dependency parameter 0 5443525223 0 0706875861 o C 1 C0SH X 6 X X 8 X 18 X 5 COSH 2 X 5 H X C Hh xXx XX 263 1 14 0 78 0 06 10 6 2 2 6 10 Figure 7 8 Graph of f x for c 709 3 iterations CONVERGED best weighted sum of squares 3 119495e 01 weighted root mean square error 1 248899e 01 weighted deviation fraction 4 697038e 02 R squared 7 373404e 01 So the procedure
20. 83 38 gt VaI yey Figure A 1 ASCII character numbers 0 to 15 in Hershey fonts 1 to 5 asci1 16 E N WO N O NJ Has 24 25 26 28 29 30 31 Font g ll E KS FA PAPA R2 R 2 200 NOOI A WINER NOOIF UME UNUAU ll See CY IUD lt TT lt lt lt LIX LIT Lu S Il tyro tal z 4 LO El H gt eee HINA H I Ry A A V IA V NM OM Io gt i lt lt 1 lt Figure A 2 ASCII characters 16 to 31 in Hershey fonts 1 to 5 asci 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 8 gt Font SS SS SS O O US oe o o o s A A 8 o Gu E EB a A A bt lt db tb loi ik nn a X ao N n N A a a A A N A an n A A A A A A A a a na dX m amp A A N gt e te e tee ttt KKK Xx HK HH KK XXX KXHKXKX KH HH X Oe Ol e HHH aN OST NS NO O o o a a E AR A A AA ee T Bs a a ee ee SK es X 5025083 RRO g at BIN SNS ISI IAS ARR Dw OSOS ANO AA s OLEATE ALO 2 oD 1 DANS Sed SESE SRE HSE SRS SHS SS Se te FD se cot og SEE ROR me 2 See amp eB Se ke oe OS amp O amp ce eg j a o z la aL ey ea 2 el a A a e e O mn 8 8 8 Sat td Be i me ee E A l S a S e l a e e O e m m m O ANUMTILDUORDODNDO ANN TTD O NM AND TT 10 W Mm 09 00 H AH AA AH AH A A A NY OY AY OY NY AU ANN CUO 0M MO Figure A 3 ASCII characters 32
21. At each iteration X I is replaced by F X I for I from 1 to NROWS X and then J and the vector X are typed out That is J and the Jth iterate of F applied to the original vector X are typed out in each of N iterations for a specified scalar N The general form of the compound FOR command is FOR X ME1 DO command1 command2 3 commandn Note that semicolons are used to separate the MLAB commands With this FOR command the entire list of commands is repeatedly executed once for each scalar value assigned to the index variable X as specified by the column vector ME1 Table 3 33 FOR command compound form If an operation can be executed either by a suitable matrix expression or by a FOR command the FOR command should be avoided Evaluation of a matrix expression in MLAB is almost always more efficient than execution of an equivalent FOR command 90 3 10 SUMMARY In the following let SE1 SE2 denote arbitrary scalar expressions let ME1 ME2 denote arbitrary matrix expressions let U denote an unknown identifier let M denote a matrix identifier let F denote a function identifier defined by a FUNCTION command or a builtin function with m arguments X1 X2 Xm and DF denote an ordinary or partial derivative F X1 X2 gt Xn 1 Matrix Dimension Operations are a NROWS ME1 b NCOLS ME1 2 Matrix Expression Formation Rules are a Matrix identifiers are matrix expressio
22. INITIAL Y X 0O INITIAL Y X 0 0 INITIAL Y 0 0 M INTEGRATE Y X 0 1 25 TYPE M 2 XK XA XA XX XX M a 5 by 7 matrix 1 0 2 1 0 2 0 0 2 0 25 2 18786625 506025425 526059631 2 18786625 6 47793736E 2 526059631 3 0 5 2 25656185 5 55285646E 2 1 08396538 2 25656185 265676269 1 08396538 4 0 75 2 22508839 279225662 1 64592704 2 22508839 607080371 1 64592704 5 1 2 13893893 340972748 2 19178847 2 13893893 1 08725238 2 19178847 The resultant matrix M has 7 columns The values held in each column can be determined by typing TYPE ODESTR to which MLAB responds ODESTR X Y X X PPPX PX PPX Y YPX 291 As before ODESTR is a string variable that holds the names of the columns in the matrix returned by the INTEGRATE operator In this case column 1 of matrix M contains values of the independent variable X column 6 contains the corresponding values of the function Y columns 4 and 7 contain values of the first derivative of the function Y columns 2 and 5 contain values of the second derivative of the function Y and column 3 contains values of the third derivative of the function Y If only values of the unknown function Y are required the ON operator can be used instead of the INTEGRATE operator if only values of the independent variable and the corresponding values of the unknown function Y are required the POINTS operator can be used instead of the INTEGRATE operator The INTE
23. If the command SAVE IN filename is given in one session then the command USE filename given in a subsequent session allows the user to use the items saved in the earlier session An important use is to allow resumption of inter rupted work at a later time The complete filename for the disk file is obtained by deleting any characters after the first eight and adding a period and the extension Table 5 2 USE command e The computer disk file created by a SAVE command can not be changed by subsequent assignment FUNCTION DRAW or other MLAB commands that modify data items in computer memory A snapshot of the data items is created at the moment the SAVE command is executed Another SAVE command must be given if it is desired to save modified data items that were previously saved with different values e A SAVE command can cause a previously created file to be deleted before the new list of 161 items is saved However the user is first warned by the message Save file TUTOR1 SAV already exists R overwrite keep existing file The user can then respond R to replace the old file or to cancel the SAVE command leaving the old file unchanged The USE command can destroy MLAB data items currently in computer memory by replacing them Note the following dialog Y 1 10 USE TUTOR1 Using W CS1 DAY B A M diskuse an object with an existing name is being used R rename O overwrite If O is ty
24. It will evaluate the second expression with the value of the first expression substituted for each occurrence of the given identifier For example the function defined by f x sin x 2 exp a 2 r 2 could be easily defined with the MLAB statement FUNCTION F X EVAL A X X 2 SIN A EXP A A While the EVAL operator is not frequently used explicitly it arises automatically in calculating the symbolic derivative of the ROOT operator Like the INTEGRAL PRODUCT and SUM operators ROOT and EVAL may only be used in function definitions 36 The general form of the ROOT and EVAL operators are given by ROOT X SE1 SE2 SE3 EVAL X SE1 SE2 The ROOT operator returns a value for X which is in the interval SE1 SE2 and which is a root of SE3 If no such root exists then the value 1 7976 x 10 is returned If several such roots exist then one of them is chosen according to the internal algorithm used This will not necessarily be the smallest or the largest of these roots The EVAL operator evaluates SE2 with all occurrences of X replaced with SE1 This arises automatically in the derivative of ROOT The ROOT and EVAL operators may only be used in function bodies Table 2 14 ROOT and EVAL operators 2 8 DERIVATIVES OF FUNCTIONS In calculus the concept of the derivative of a function and the corresponding notations d Z o f dx are introduced for functions y f x of one variable Subseque
25. Lipschitz condition violations The numerical algorithm becomes less accurate near these points and the requested per step tolerance may be unobtainable When derivative functions with singularities are known to be present as for example when a step function appears then DISASTERSW 1 is often the best option ODERPT is a scalar which controls intermediate screen output from the differential equation solver It may take the following values e the value 0 means no itermediate reporting 296 e the value 1 means report independent variable value step size and method at each step e the value 2 means report the independent variable value step size method and derivative values at each step e the value 3 means report only the summary information after solving is complete This includes the number of steps taken and the number of derivative function evaluations and Jacobian evaluations This reporting is useful to identify regions of stiffness in a differential equation system The default value of ODERPT is zero During the INTEGRATE command execution the user may request information about the progress of the solution process by striking the R key for Report In such reports the independent variable is referred to as T and the solution of the differential equation is referred to as Y A second R will toggle reporting off again The user may also quit the INTEGRATE process by typing Q for Quit
26. Termination of input occurs either because 1 m rows of numbers each of length n have been entered in the matrix or 2 no more numbers are available from the input source e g CTRL Z or Escape was detected for device CON the end of the computer disk file was reached If termination of input occurs because of case 2 above then the current row of the matrix is completed with zero entries if necessary and no additional rows of the matrix are included If SE1 and SE2 are omitted from the READ matrix expression then the default values of 1000 rows and 1 column are assumed Table 5 3 READ matrix expression general form applied to the same file continues where the previous READON operation left off In this way the numbers in a file may be processed piecemeal If when executing READON F X Y fewer than Xx Y numbers remain in the file then only a FLOOR k Y Y matrix is returned where k is the number of values remaining in the file the last row is padded with 0 values as necessary If X lt 0 the file F is immediately closed and a 0 row null matrix is returned The MLAB operator KREAD constructs a scalar or matrix by reading one or more numbers from the keyboard There are four options for the arguments no arguments a single string argument S or one or two scalar arguments X and Y the argument X and the optional argument Y specify the number of rows and columns of the matrix result so that X Y is the number of
27. Then by sorting on column 1 of matrix N the matrix R is obtained In matrix R note that blocks of successive rows with the same entry in column 1 correspond to increasing sequences in column 2 That is 5 8 is an increasing sequence in column 2 for both 1 entries in column 1 and 3 4 is an increasing sequence in column 2 for both 7 entries in column 1 Lexicographic order may be obtained more directly by SORT ME1 0 A matrix may be sorted so that column k occurs in descending order by SORT ME1 k The general form of the SORT matrix expression is given by SORT ME1 SE1 Let ME1 a j an m by n matrix expression Suppose SE1 is a positive integer k k lt n Then the result is an m by n matrix obtained by per muting the rows of ME1 so that 1 column amp of the result has an increasing sequence of matrix entries and 2 row order is not changed for rows with the same kth component An error message appears if INT SE1 is not a valid column of ME1 Table 3 21 SORT matrix operator Other useful matrix editing operators are GETDIAG COMPRESS ROTATE MAXROW and MINROW which are described in Table 3 22 GETDIAG returns a 1 column matrix consisting of the largest main diagonal of a matrix COMPRESS returns a copy of a matrix with all rows removed in which an element in a specified column is zero ROTATE returns a copy of a matrix with its rows rotated 71 down a given number of indices MAXROW returns th
28. a then the result is an m by 1 column vector d such that d is the Euclidean length of a 1 di2 Qin for i lt m Table 3 25 Vector LENGTH operator In matrix algebra the product of an m by p matrix A and a p by n matrix B is an m by n matrix C given as follows p Cik Y aijbjk j 1 where A aij B bij and C ci That is the number of columns of A equals the number of rows of B and each entry of C AB is formed from a row of A and column of B by the inner product operation for vectors In MLAB matrix products are indicated by an asterisk as for scalar products The matrix product operation is described in Table 3 26 For example observe the following MLAB dialog TT TYPE A B A a 1 by 2 matrix B a 2 by 3 matrix e w o Pp a 1 by 3 matrix 1 5 2 3 If the first matrix argument has too many columns or the second matrix argument has too many rows an error message is printed The general form of the matrix product operation is the following ME1 ME2 If ME1 is an m by c matrix expression a and ME2 is an c by n matrix expression b then the result is an m by n matrix cik with C Cik Dj 1 lijbjk This is the usual matrix product Table 3 26 Matrix product operation In MLAB one can indicate repeated products A A A of a given square matrix A by raising A to a scalar integer power The matrix power operation is d
29. and A Fo Go K9 K 254 In this model Bo Fo and Go are constants The easiest and worst way to enter this model in MLAB is to enter it as it is written above with empty argument list functions for R S A and d This method is very inefficient because it requires that the functions for R S A and d be evaluated several times for a single computation of B t A far better way is to define a hierarchy of functions used cooperatively to evaluate B Each level must perform some of the calculation required for B For example we could notice that in the definitions of B above the terms involving R BO and S BO appear in the same fashion in both the numerator and the denominator We could start therefore with FCT F1 A B S R S A R B A B Then F1 must be called with values for A and B To simplify this slightly more we define FCT F2 R S X F1 R BO S BO X S R The next step then is to specify values for S and R which are given above and to specify a value for X This can be done with the definition FCT F3 A D T F2 A D 2 A D 2 EXP D K1T This leaves A and D the only intermediate values to be evaluated Since D is defined in terms of A this can be done with the definition FCT F4 A T F3 A SQRT A A 4 FO GO T Thus through these four definitions we can now define B t as FCT B T F4 F0 G0 K2 K1 T Each of these four functions need be evaluated once during the evaluation of B
30. giving the following paired data icu 832 22 49 1 41 85 13 y3 2 1 1 8 raya 83 L1 t5 y5 4 1 7 Assume that the quadratic polynomial f z 3 7 1 5 2 2 4 models this data The sum S of the terms yi a is the sum of squared errors for the agreement between the model and the data where y is the data value at x and f x is the predicted value at x 226 5 S Y lu fail i 1 3 2 2 628 1 85 1 992 1 8 1 432 1 1 928 7 912 0 5578 If a vertical line segment is drawn from a data point to the function graph of y f x then the length of the segment is the absolute value of the difference between the data value and the value predicted by the model The squared error equals the area of a square with this segment as a side That is the vertical segment has endpoints x y and x f x with length y f 2 for each i 1 2 3 4 5 In Figure 7 1 is plotted part of the function y f x showing the five data points as small triangles For each data point the vertical line segment between the point and the function graph and the square which has this segment as its left side are shown Thus the sum of squared error S equals the sum of the areas of the five squares shown in Figure 7 1 This picture shows some elementary properties of sum of squared error estimates Clearly the sum of squared error is never a negative number and it is n
31. r 52 333DJ gt 335382 222 33 83383233 M v 35353933 POE as E A es rae conovuNnonanbobnnn a mon OA asa A EAS A A bo Da AE To TUCO OOOO OO SS STOOD SHE O SH Oe oTo aFnotaqan BaAAArkKKEERARarie hron ht 10009022 ANO AUAN OO A A A A A SGA NY NY UY NY NY AIC 27 Figure A 8 ASCII characters 112 to 127 in Hershey fonts 1 to 34 322 ASCII 32 o E E S is KKK E ES DO E EN A A ER INN a yyy BBs Y oF oF Y OY oY ge de de de WELW Ve XXw UWUWARNAAAKHAAHHATM He SHS SH te He He Se HERES 49 ASCII ue Oe He Or IE OHO Que NAKA KAXKAANNAAAN lon Wan ad Wn nd a a my VVVVVVVVVVV VV en eN ON ary DSOADDAADADODO o 00 00 0 0 o0 Y 20 09 0000 00 vo SERNSAENNOENRNNNO OWSDWOCEOSOWDNDNO LOLI DO MINH AN LO LO LO LO o o SES SES St 1 O19 MEN EAM HOON MOI CD NA NNN ANN ANANA AAA A A R enn el el e o0o002o0 00000 ozy S NM 10 0 M5 OWA v4 vA vA v ha A O Figure A 9 ASCII characters 32 to 63 in TrueType fonts 2 to 14 65 ASCII 64 A 2000009900000 ZZZZZZZZZZZZZ SES NAAIISESSZ H H HN NNNHNN PERRA ARAS AR SS le RPP nt sonsa H HN pu ey ee ee pe ca AORTA OVULYVUVUOUUOOSL Fy fu Fy Ey Gy fay E ELL Foo E a a RQ U U U po AAgRmAAgRgoOAgaA lt OVUOVUOULLOOOCO XK CO Am AO AO A ARO a cala LAURA LZATAA vvv ve O O OOGG Oa NMF M AOAN oe et e et od l l nt Fo
32. 0 01 cosh 0 99 cosh 1 The expression POINTS COSH 0 1 0 01 describes a 101 by 2 matrix The first column is the column vector of arguments 0 0 01 etc The second column now gives the function values cosh 0 cosh 01 etc Note that the rows of this two column matrix may be regarded as x y coordinates of points on the graph of the function y cosh x Again recall the MLAB commands 6 15 05 POINTS Y M ss in Chapter 1 These commands assign the 181 by 2 matrix POINTS Y 6 15 05 to M Here Y was a defined function given by FUNCTION Y X A EXP B X which describes an exponential curve with parameters A and B The values of A and B were first set arbitrarily and then changed by a curve fitting command to describe the best fitting curve for the data DAY in Chapter 1 The matrix M may be regarded as a list of 181 points x y lying on this exponential curve with the consecutive x coordinates 6 00 6 05 15 00 Although the ON and POINTS operations are most often used with functions of one variable and arithmetic sequence colon column vectors they can be used with built in or defined functions of n variables provided that the matrix expression for the list of arguments specifies an n column matrix For example observe the following MLAB dialog FUNCTION F X Y 2 Y X TYPE B B a 2 by 4 matrix 1 1 2 3 4 84 2 5 6 T 8 TYPE F ON B COL 1 2 a 2 by 1 matrix 1 4 2 2 4 TYPE
33. 0004 142 4 15 The WINDOW statement with ADJUST clause o e 143 S L SAVE SOM cera a eae WS 161 o AER 161 5 3 READ matrix expression general form o 000 eee eee 164 Bl PRINT POIS o a he ow koe a Rd A ae id a he A 167 SE PLOT gomand i socio rana A a dee ced was 169 26 DM Cosi lt ses as ese die ke koa ee g Goa p Bee ee ee ee 170 of Conditional command 2 24652 aos 65 a a ot a a ee eee as 172 Oo Conerali ra ot MENUCHUICE oo coa 6005 8 wa a Ye ee ee ae ee 179 5 9 General form of GETSTRINGS 2 2 4 6 5 be ee tab ee eee ee es 187 5 10 The WREAD command 556 aaa soa du dta poa nd doa aha ga kpda 188 6 1 Matrix Combination Operators ea 0 ee ee 194 6 2 INTERPOLATE SMOOTH BARGRAPH and HISTO matrix operators 200 Geo FILL maik OPEL e a ee de a ra A A a 203 GA CONTOUR matrix operator lt o cp saraa ra boarn apoa a 208 63 ELIGEN matrik OPeradoL so s s 4 05 x oa fd eRe a a a a 213 329 6 6 6 7 6 8 6 9 del Tea 73 7 4 7 9 7 6 it 7 8 TA 7 10 8 1 8 2 8 3 8 4 Complex number operators lt coa a aora daoa be a e ee 214 SVD matrix Operador oe ro wa ee ee OG a BAS Re we a a 215 Covariance and correlation matrix operators e a 217 Discrete Fourier transform operators e a 222 ENT ODO Of cos oka wa Gee e A la a A a 234 Weighted sum of squared error criterion o oo ooe a e e ee 237 Joint models we
34. 0892392386 0 2894782089 o Cc 7 iterations CONVERGED best weighted sum of squares 2 035792e 00 weighted root mean square error 3 190448e 01 weighted deviation fraction 2 695622e 01 Here the process converged to the incorrect value c 3 089239 Examination of Figure 7 7 shows that this value corresponds to a local minimum of the sum of squared error S In terms of the latitude and longitude analogy the bottom of the valley that contained the starting point was found but another deeper valley elsewhere was not detected Thus as shown in Figure 7 10 the right peak of h was fit to the data without detecting that the left peak of h gives a better fit The next initial estimate again finds this local minimum C 10 FIT C F TOM final parameter values value error dependency parameter 3 0887892741 0 2897531343 o Cc 10 iterations CONVERGED best weighted sum of squares 2 035827e 00 weighted root mean square error 3 190476e 01 weighted deviation fraction 2 695820e 01 268 0 78 0 06 Figure 7 10 Graph of f x for c 3 09 269 In the above case the search process overshot the correct solution That is the algorithm at c 10 happened to generate a trial near the local minimum at c 3 09 before it generated trials sufficiently close to the true minimum at c 709 The search process then went astray converging to the local minimum at c 3 09 The last example shows that linear constrain
35. 1 1 for I 1 2 NROWS M So the INTEGRATE operator allows numerical computation of both the function Y and its derivative Y X which were implicitly defined by the differential equation and initial condition above The differential equation and initial condition given above have a precise analytical solution as follows 282 l r we age This function and its derivative in MLAB are evaluated as follows FUNCTION F X 1 1 X 4x EXP X TYPE 0 5 amp F ON 0 5 F X ON 0 5 a 6 by 3 matrix 1 0 5 5 os 1 1 97151776 1 72151776 3 2 874674466 652452244 4 3 449148273 261648273 5 4 273262556 113262556 6 5 193618455 5 47295658E 2 Note that the matrix M above generated by the differential equation solver is in close but not exact agreement with the theoretical result The maximum error is less than 0002 The accuracy of the INTEGRATE command s numerical solution may be controlled by setting the control variable ERRFAC which will be discussed later 8 2 COUPLED FIRST ORDER DIFFERENTIAL EQUATIONS SOLVED IN MLAB Frequently a mathematical model will involve a system of several interrelated first order differen tial equations and initial conditions for the corresponding unknown functions The MLAB user can also generate numerical solutions for such systems with the INTEGRATE operator Consider a simple example from idealized enzyme kinetics Suppose one molecule of an enzyme E combin
36. 10 optional window identifier Y specifies the MLAB window from which points are to be read If Y is omitted the default 2D window W is used optional scalar integer Z specifies the mode for selecting points If Z is omitted or equal 0 points are selected by positioning the mouse cursor at a desired point on the graph and pressing and releasing the mouse button If Z 1 then the current mouse cursor coordinates are displayed at the lower left corner of the image rectangle and points are selected as for Z 0 If Z 2 then points are selected by the mouse positions obtained while the mouse button is pressed and the mouse is dragged across the window If Z 3 then the current mouse cursor coordinates are displayed as for Z 1 and recorded as for Z 2 The user can terminate WREAD before X points have been selected by striking the ESCape key WREAD returns a 2 column matrix of coordinates Table 5 10 The WREAD command 188 Chapter 6 FURTHER MATRIX OPERATORS Many operators have been given that enable an MLAB user to define functions compute scalar and matrix quantities and aid in displaying results graphically This section will explain several other MLAB operators which while they are matrix operators are designed mainly for use in developing graphs and pictures Also explained in this section are other matrix operations which have many diverse uses including picture drawing 6 1 MATRIX COMBINATION MLAB has two
37. 127 AXES 141 axes 138 AXIS 126 128 148 axis 98 126 axis color 127 axis identifiers 128 axis label angle 127 axis label font 127 axis label format 127 axis label matrix 141 axis label offset 127 axis label place 127 axis label size 127 axis labels 127 axis matrix 126 141 axis names 128 bar graph 199 BARGRAPH 196 199 binomial coefficients 94 BLACK 125 BLANK 138 139 148 BLUE 125 Boolean AND 29 Boolean NOT 29 Boolean OR 29 BOTTOM 122 133 134 143 boundary value differential equation 300 bracket 22 BROWN 125 canonical form of a system of differential equa tions 210 canonical form of differential equations 293 carriage returns in titles 131 case 2 CASESW 2 CDF 83 192 CDIV 213 221 CENTER 122 133 chain rule 39 CHARTREUSE 125 chemical kinetics 237 247 254 283 chemical reaction kinetics 237 247 254 chemical reaction rates 283 choosing data point weights 233 CIRCLE 106 clipping 129 COL matrix assignment command 64 COL matrix operator 62 colon 54 COLOR 117 125 127 129 133 134 color designation by R G B triples 125 COLORN 125 colors 125 COLORX 125 column concatenation 64 column replication operator 67 command prompt 2 comments in do files 174 comparison expressions 30 38 complex division 213 221 338 complex exponentiation 213 complex multiplication 213 complex number operators 21
38. 2 5 3 4 3 2 5 4 3 2 4 5 1 1 5 Note the way in which we negated the second column of M in order to rank the matrix in descending order The general form of the CDF and RANKORDER operators are given by CDF ME1 ME2 RANKORDER ME1 The CDF operator for NCOLS ME1 NCOLS ME2 k will return a matrix whose last column gives the fraction of rows of ME1 whose values are all less than or equal to the corresponding values of ME2 The RANKORDER operator sorts ME1 on its last column and adds a column containing the rank values The rank values are 1 NROWS ME1 except that duplicate values are assigned the average of their row indices Table 3 29 The CDF and RANKORDER operators general forms 3 7 TABULATION OF A FUNCTION It has been shown that a defined function or built in function of n arguments can be evaluated at a given n tuple of real numbers in MLAB However the MLAB user often wants to evaluate a function numerically at many points in order to study its properties in some region The MLAB matrix operations ON and POINTS permit the evaluation of functions at many points in 83 one operation For example suppose that a table of numerical values of the hyperbolic cosine cosh x in the interval from 0 to 1 is desired and that computation at every step of size 0 01 is sufficient The expression COSH ON 0 1 0 01 describes a column vector of length 101 containing the numerical values cosh 0 cosh
39. 3 TYPE D V XK XA XX D a 5 by 2 matrix 1 0 8 3 2 2 1 4 1 85 3 2 1 1 8 4 3 3 1 1 5 4 1 0 7 V a 5 by 1 matrix 6 OP WNEF Orrro 75 Table 7 7 describes preparation of data for curve fitting operations In general suppose that m weighted data points il i2 Tin yi Of n 1 coordinates each with associated weight w i 1 2 m are to be curve fitted to a functional form Y f tiiti Cin Dig Od x Bp The m by n 1 data matrix and m by 1 column vector of weights are set up as MLAB matrices with ith rows respectively Ei E Zin Yil and wi for i 1 2 m That is the i data matrix row contains in the first n columns the t data point with the n independent variables in some fixed order and in the n 1 column the dependent variable As shown the it coordinate of the weight vector is the weight for the i data point Table 7 7 Data preparation for curve fitting operations 243 7 9 THE FIT COMMAND Suppose the parameters a b and c of the functional form f x a b 2 c 2 are to be fit to the data of Figure 7 2 Furthermore assume the following linear constraints f 0 gt 1 and f 1 lt 1 It is relatively easy to define functional forms in MLAB using FUNCTION statements To set up a quadratic polynomial model for the data of Figure 7 1 and Figure 7 1 for example the following MLAB commands can be used FUNCTION F X A B X Cx
40. 4 The MLAB dialog is shown below FUNCTION Y X X Y X x EXP Y X 1 INITIAL Y 0 3 INITIAL Y X 0 K K 1 FIT K Y TO LIST 1 4 inal parameter values value error dependency parameter 0 6498070118 0 Oo K 5 iterations CONVERGED best weighted sum of squares 0 000000e 00 weighted root mean square error 0 000000e 00 weighted deviation fraction 0 000000e 00 R squared 1 000000e 00 H A XX If the INTEGRATE operator is used to numerically evaluate the above system with y x 0 649804 the boundary value condition y 1 4 is satisfied 8 7 THE FIT COMMAND FOR DIFFERENTIAL EQUATIONS The FIT command can be used for curve fitting systems of differential equations Since the differ ential equation solver is invoked by such FIT commands the values of METHOD JACSW DISASTERSW MANDSW ODERPT and ERRFAC control the operation of the differential equation solver while exe cuting within the FIT command Furthermore the characters Q and R can be used here just as they are used during the evaluation of an INTEGRATE operator A differential equations FIT command may have several clauses referring to a system of differential equations in some clauses and to functional forms in others It is convenient but not required that the data matrices in the different clauses all have the same first column For example consider a particle that moves on the curve 301 y t 4 a Rees 4 1 5 k with a
41. 4 4 Note that point types corresponding to convex shapes can be generated by negating the point type values Also note in Figure 4 4 that with DASHED linetypes dashes are continued across curve matrix points That is dash lengths are preserved even if the resulting dashes are broken lines It is important to note that the curve matrix of a curve is completely separate from that matrix which is used to define it the defining matrix is copied to establish the curve matrix In the above example changing the matrix DAY has no effect on the curve CA Drawing with line type 6 MARKER combines features of drawing with line types 1 SOLID and 5 ALTERNATE The curve matrix begins with a dummy row identifying a marker value in column 1 The curve is then drawn as for line type 1 except that another dummy marker row is inserted between any two points where the user wants to lift the pen For example the matrix M in the following MLAB dialog TYPE M M a 12 by 2 matrix OANDOAARWNEFE e pi o m O o m m N N WWORPRNODORrFRWDAHFEHFO E N w 106 Point types PT O means no points wo 1i 2 3 5 6 7 E 9 0 11 42 19 14 45 a t AOw OX OF Line types LT O means no lines LT 6 means interrupted solid lines LT 7 means interrupted variable dashed lines LT 9 means dotted line with marker skipping LT 10 means dashed line with marker skipping LT 11 means marker skipping using spline curve LT 12 means variable dashed Jine
42. ASCII eo LEVES E SEA LFI 44 26224244845 5 Pp Manan AS A A A M N N Y NNNNNNNNNN A A A gt A A e AAA Bt Bel gt lt DK OK OK x e eREBBESESSS50 DO gt DD gt gt gt gt gt gt gt gt o ODDpDODDPHLHDD Ip EA RRAEARARREHERRRE NNANNNNHNHHNNNHnW Mm Me eM reo a aaa AOOO O ONWAA AAA OOo OE t O E 12 3 4 Fon Figure A 10 ASCII characters 64 to 95 in TrueType fonts 2 to 14 98 99 100 101 102 103 104 105 106 107 108 109 110 111 97 ASCII 96 Fo 0000000982800000 GEgecqgadserceces EESEEBBERREEEE HA NN A A A MM e A dl za MN NY ee e gt cl 4 H om om m w ee m n O o cl ASA a MOD Don oN so MODDD gt H H SH y a Ge ee HS OR ROR EOR ENSEN OR ROR ES DTUDUITESIODOD O CVOVHVOLHYLOOOOR 222 02013020020 CO VUOAOZIESSOCOADUEAS P m i v vs se Mou u a a WOoOrowon GN q O do Bo 114 115 116 117 118 119 120 121 122 123 124 125 126 127 113 ASCII 112 2122911122 21 A O pp a E n MM SN pj tp ip NNNNNNWNNNNNNS Pale ale a i et gt gt XM p be BRK KOK KD z o gt r gt or gt gt prn gt gt gt gt bEB 3323 3533330 PPPV ohms qm 2 o PP nunOUranr r unn nao OMYNoe nina CHOP oS TSOTOooD AMAWAAAgeoOmoQeR S TLA e E NMOrFMowoOnrR OMA DA oa ds fh i a A A O Figure A 11 ASCII characters 96 to 127 in TrueType fonts 2 to 14 325 Appendix B LIST OF TABLES List of Tables Ld al 2 2 2 3 2
43. COS BxT so that H has argument T scalar parameters A and B and four parameters in a 2 by 2 matrix C If matrix names are used in the parameter list of an MLAB model two restrictions must be ob served First linear constraints in which any matrix element parameter occurs are not permitted Second all the matrix element parameters will be curve fitted individual matrix element param eters are not permitted in the parameter list of a FIT command For example note the MLAB error obtained by trying to fit the matrix element parameter in the function H above A 1 B 1 C SHAPE 2 2 1 4 D SHAPE 10 2 1 20 FIT A B C 1 1 H TO D Error fitop an expression is of the wrong type The general problem of curve fitting a functional form is shown in Table 7 9 256 7 12 LINEAR MODELS There is a certain special class of mathematical models called linear models for which the curve fitting problem can be solved by a general formula The MLAB user should know when the model is linear and when it is not because nonlinear models require more careful treatment in many cases Furthermore some of the output of the MLAB FIT command pertains only to linear models These output values are also computed for nonlinear models but should be regarded as only rough guides in that case Consider the functional form f z a e b e sinh z with independent variable x and parameters a and b When numerical values are substitute
44. FOR command The median of an odd number 2 7 1 of scalars is the middle element of the list of given scalars if they are put in increasing order i e it is the n 1 in the ordered list Construct MLAB command s computing the median of the entries of a column vector S assuming that NROWS S is odd Suppose that an m by n matrix D is regarded as a list of m n dimensional row vectors Construct MLAB command s by which an m by n matrix ND can be created where ND is a list of m n dimensional unit row vectors vectors of Euclidean length 1 and each row vector of ND is a positive scalar multiple of the corresponding row vector of D Assume that no rows of D are all zeros Suppose that you have a matrix M containing points along a curve Use the LOOKUP and INTEGRAL operators to find the area under the curve contained in M Start an MLAB session Test the answers for exercises 1 9 by first assigning numerical values to the scalars and matrices that were not specified Keep a log of the session and have it printed 95 Chapter 4 MAKING PICTURES WITH MLAB This chapter describes the detailed principles of MLAB graphics and MLAB graphics commands For a simpler description of quick and automatic MLAB graphics refer to the book MLAB Graphics Examples Only two dimensional pictures will be considered here MLAB facilities for making images of curves and surfaces in three dimensional space will be discussed in Chapter 9 of a futu
45. K S S 1 X EXP A X 2 A 1 T 2 TYPE F 3 4 L0G T 3 223 481907 Note that the TYPE command for F returns the current function definition In order to numerically evaluate a reference to F the parameters K and A must be scalar identifiers that is must have numerical values This is not true for the arguments S and X which could be known or unknown MLAB identifiers As shown above the scalar expressions to be substituted for the argument list identifiers are supplied in parentheses following the function identifier For example the command TYPE F U 1 GG has exactly the same effect as the command TYPE Kx U 1 72 1 GG EXP A GG for scalar identifiers U and GG The formal argument list identifiers S and X are used only to interpret references to the function F and any actual data items named S or X are not damaged by using these names as formal arguments For example if the function Y is defined by FUNCTION Y X A EXP B X 24 The general form of the FUNCTION command is given by FUNCTION F A1 A2 An SEO Here F is either an unknown MLAB identifier or a function identifier according to whether one is defining a new function or redefining a previously existing function The next part of the command A1 A2 An is called the formal argument list and consists of n for n gt 0 distinct MLAB identifiers usually unknown identifiers which are en closed in parent
46. K and N may be different during different runs of the experimental procedure the user should be requested to assign values for K and N during execution of the DO file 9 Create files representing the DO files designed in exercises 6 7 and 8 above Run MLAB and test your answers to the previous exercises wherever it is feasible 186 The general form of the GETSTRINGS function is GETSTRINGS T S H X Y C L11 1 1 gt where T is a required string defining the title for the form if no title is desired set T S is a required 1 dimensional string array containing descriptive text associated with each text entry field These are the prompts that will appear to the left of each editable text string in the form These strings must be short enough so that the length of the prompt combined with the length of the text entry field is less than or equal to 78 characters H is an optional string array containing the initial contents of the editable text string fields to be filled in the form if omitted the initial text string fields are empty If H has fewer elements than the string array arguments the initial values of the text entry fields with no corresponding H element default to empty strings If H has more elements than S then only the first NROWS S elements of H are used X is an optional integer scalar specifying which of the text string fields to be filled in is active i e to where keyboard keys will be d
47. M LINETYPE NONE POINTTYPE CROSSPT DRAW SMOOTH M LINETYPE DASHED VIEW The resulting picture shown in Figure 6 5 demonstrates the effect that SMOOTH has on noisy data This can be useful whenever such data noise is likely and could have a disrupting effect on calculations Combined use of the SMOOTH and INTERPOLATE operators can take data points reduce noise and deduce intermediate points that lie along a smooth curve of the data This resulting matrix could be used as input for a curve fitting process explained later or used for graphically displaying data However care should be used to ensure that spurious results are avoided The BARGRAPH operator can also be used in preparing pictures BARGRAPH will take a 2 column matrix of x y coordinates sorted on column 1 as input and will return the points of a step function graph corresponding to the matrix Unlike the step function graph in Figure 6 3 the BARGRAPH operator will place the actual data points at the center of the steps not at the corners Histograms are often presented in this form as will be seen later A step function graph is made using the following commands RESET M POINTS COS 0 10 5 DRAW M LINE 2 DRAW BARGRAPH M WINDOW O TO 10 1 TO 1 VIEW XK XA XA XX XX The result of these commands is shown in Figure 6 6 196 70 56 42 2B 14 2 8 4 6 6 4 Figure 6 5 Smoothing data 197 9 2 10 0 2 Figure 6 6 Ste
48. MLAB will print a colon as a prompt on each continuation line The last line of the continued command should be followed by a carriage return as usual Beware you should end each line being continued with a blank just before the backslash to avoid unintended run on text on the next line You can also type several MLAB commands on the same line if you separate them by semicolons If you type an MLAB command incorrectly you will receive an error message DOS MSWindows and Macintosh OS7 8 9 versions of MLAB invoke a built in line editor so that you can edit the errant command On Linux Unix and Macintosh OSX systems the error is usually corrected by retyping the command correctly If typographical errors are noticed before a carriage return is typed correct them by striking the BACKSPACE key until the text cursor has backed up to the first error then continue to type from there On DOS and MSWindows holding down the SHIFT and CTRL keys and simultaneously striking the BREAK key will interrupt MLAB execution print the message Command in progress aborted Safest Action type EXIT then restart MLAB Riskier but often worthwhile Save your data objects SAVE IN MYFILE then EXIT restart MLAB and restore the objects USE MYFILE and return to the MLAB command prompt On Linux Unix and Macintosh OSX systems holding down the CONTROL key and simultane ously striking the C key i e 7C CONTROL C interrupts MLAB execution and
49. POINTS F B COL 1 4 a 2 by 3 matrix 1 1 4 8 2 5 8 3 2 Symbolic derivatives can be evaluated by using the ON and POINTS matrix operations For example consider the following MLAB dialog FUNCTION G X X EXP X TYPE G DIFF X POINTS G DIFF X 0 1 2 FUNCTION G X X EXP X EXP X x X a 6 by 2 matrix 1 654984602 402192028 219524654 8 98657928E 2 0 RROOOOO OnFN OonFRWNF The ON and POINTS matrix operations may also be used to evaluate symbolic partial derivatives of MLAB functions of several variables The MAPPLY operator is a useful extension of the ON operator A typical expression is MAPPLY F M where F is a function of one two or three arguments and Mis a matrix An NROWS M by NCOLS M matrix is returned whose i entry is F M i j if F is a function of one argument F i j if F is a function of two arguments and F i j M i j if F is a function of three arguments The general form of the ON POINTS and MAPPLY operators are shown in Table 3 30 For example suppose that B is a 10 by 4 matrix of positive numbers and a 10 by 4 matrix B1 is to be constructed such that the entries of B1 are the base 10 logarithms of the corresponding entries of B 85 The general descriptions of the ON POINTS and MAPPLY matrix operations are F ON ME1 Here F must be the identifier of a function f of n arguments either defined by a FUNCTION command or a built in function e g SIN
50. ROW SE1 SE2 SEn e M ROW SE1 SE2 SEn e U COL ME1 ME2 e M COL ME1 ME2 e U COL SE1 ME1 ME1 ME1 92 e M COL SE1 ME1 e U COL SE1 SE2 SEn ME1 e M COL SE1 SE2 SEn ME1 b TYPE command for matrix expressions e TYPE ME1 ME2 MEn c FOR command e FOR X ME1 DO command e FOR X ME1 DO command1 command2 3 commandn 3 11 EXERCISES 1 Suppose that a 7 by 5 matrix of data is recorded on paper and an MLAB matrix named N containing this data is to be created a What MLAB command should be given so that N can be typed b Suppose that the fourteenth number is accidentally mistyped while typing in the 35 matrix entries of N so that the corresponding matrix entry holds an incorrect value What MLAB command changes the incorrect entry to its correct value say 65 3 Suppose that a repetition of the 27 number on the list of 35 matrix entries is ac cidentally typed without noticing the error so that all matrix entries following the twenty seventh are not in the correct matrix positions for N Construct one or more MLAB commands by which N could be corrected without the necessity of retyping all 35 entries a wa Suppose that the first column of N should be the arithmetic sequence of the seven numbers 12 24 36 48 60 72 84 Construct N by an MLAB command or commands so that the entries in the first column need not be typed individually 2 Let A la a
51. SE1 defined for all SE1 GAUSSF SE1 normal probability function Fa JZ at for x SE1 defined for all SE1 Table 2 6 Commonly used special functions MLAB is based upon an algorithm for uniform 0 1 random numbers derived from Efficient and Portable Combined Random Number Generators by Pierre l Ecuyer in Communications of the ACM 31 1988 742 749 Often it is desirable to generate normal 0 1 pseudo random numbers having mean 0 and variance 1 Such normal pseudo random numbers may be generated with the NORMRAN function defined in MLAB The NORMRAN operator is summarized in Table 2 8 Normal pseudo random numbers are often used in generating error in simulated data MLAB provides random number generators for more than 20 different distribution functions See the MLAB Reference Manual or type HELP RANDOM at the MLAB prompt for a complete listing In mathematical text the elements of a matrix are often denoted by the use of subscripts For example the entries of a 2 row by 3 column matrix A may be labelled in the following way 21 The RAN operator has the form RAN SE1 RAN returns a pseudo random number from a series of such numbers If SE1 0 then the next value along the current series is returned If SE1 0 then a new series is started which will be a function of SE1 Table 2 7 Random number operator The NORMRAN operator has the form NORMRAN SE1 SE2 SE3
52. WMATCH The result of these commands is shown in Figure 6 8 The matrix M contains the points of a circle of radius one and this circle is shaded with 20 fill lines at an angle of 45 degrees The FILL operator can also use marker skipping similar to that of linetype MARKER in the DRAW command See Section 4 3 for a description of linetype MARKER To use this feature element 1 1 of the input matrix must specify a marker value of MAXPOS 1 7 x 10 8 This marker value can then be placed in column 1 of any row of the matrix to signal the end of one polygon and the start of another If polygons are contained inside one another or are overlapping then the symmetric difference of the two is used as is shown in the following dialog RESET 201 5 6 4 2 2 8 1 4 0 Figure 6 9 Shading several polygons M2 COL 1 KREAD 24 115511 123322 146644 178877 M2 COL 2 READ 24 022662 033553011441 022662 FUNCTION F X IF X lt O THEN MAXPOS ELSE X M2 MAPPLY F M2 DRAW M2 LINETYPE MARKER DRAW FILL M2 50 135 LINETYPE ALTERNATE WINDOW O TO 9 O TO 7 VIEW Note how the MAPPLY operator is used to change all elements of the matrix M that are less than 0 to MAXPOS The result of this dialog is shown in Figure 6 9 Note that the fill line matrix was still drawn with linetype ALTERNATE while the figure had to be drawn with linetype MARKER to take the marker skipping into account The FILL operator takes the value MAXPO
53. WORLD means user rectangle coordinate system of the window containing the curve If the units are omitted the coordinates are assumed to be in frame fraction units The clause SHIFT SE6 positions the text string shifted from the point SE4 SE5 The shifting is along a line in the direction implied by the angle SE2 if SE2 gt O and in the opposite direction if SE2 lt 0 The shifting line starts at the point SE4 SE5 and extends to the point of intersection of this line and the nearest extended boundary of the frame rectangle 133 Text characters may be printed anywhere in the frame rectangle they need not lie within the image rectangle of the window containing the curve However any characters in the text string lying partly or wholly outside the window frame will not be shown The clause COLOR col specifies that color number INT col is used if col is a number col may also be a color name as listed in Table 4 8 The clause PLACE A B specifies which part of the box that encloses the text string will be placed at the position determined by the AT and SHIFT clauses A may be either LEFT RIGHT or CENTER and B may be either TOP CENTER or BOTTOM The tables in Appendix A show all of the characters available in each of the MLAB fonts 4 6 TITLE MODIFICATION COMMANDS As was mentioned in the preceding section MLAB has special coded commands which may be embedded in strings These string modification commands enable the user to change t
54. X Y CVI X Y MN X MN Y FUNCTION CVI X Y XM YM SUM I 1 NROWS X X 1I XM Y 1I YM NROWS Y 1 FUNCTION SD X SQRT COVAR X X FUNCTION COR X Y COVAR X Y SD X SD Y H 1 1 COVAR D COL 1 D COL 1 H 2 2 COVAR D COL 2 D COL 2 H 1 2 COVAR D COL 1 D COL 2 H 2 1 H 1 2 TYPE H H a 2 by 2 matrix 1 1 67393876 645592141 2 645592141 1 24619036 H 1 1 COR D COL 1 D COL 1 H 2 2 COR D COL 2 D COL 2 H 1 2 COR D COL 1 D COL 2 H 2 1 H 1 2 TYPE H H a 2 by 2 matrix Led 446988405 2 446988405 1 216 The convenience of having COV and CORR predefined is clear from the complexity of defining these functions The general forms of the COV and CORR matrix operators are COV ME1 CORR ME1 If ME1 is a matrix of n tuple observations COV ME1 is the sample covariance matrix of this data and CORR ME1 is the sample correla tion matrix Table 6 8 Covariance and correlation matrix operators 6 8 DISCRETE FOURIER TRANSFORMS The Fourier series of a real valued periodic function x t of period p is a sum of oscillations which reconstitutes the function z x t Y M h p cos 27 h p t o h p h 0 where M s is the amplitude function of x and s is the phase function of x Both M and are real valued functions which are defined to be 0 except possibly at the points 0 1 p 2 p Note these points are the frequencies in cycles per
55. a system of n differential equations is to be solved y a hj amp Y1 Y2 Una yj u 25 J 1 2 e N for a sequence of values a1 a2 4 specified in an INTEGRATE matrix expression for the independent variable x Call the matrix produced by applying the INTEGRATE operator the result matrix The differential equation solver generates a potentially very large matrix called the solution matrix which is similar to the result matrix Like the result matrix the solution matrix has 2n 1 columns corresponding to the independent variable and n pairs consisting of values for an unknown function and its derivative The solution matrix is generated one row at a time and generation of a row is called a step The first column of the solution matrix corresponds to a sequence of values bi ba bt for the independent variable x beginning at x u The interval length b bj 1 is called the step size for the jt step 1 lt j lt t It is important to note that the vector of independent variable values specified in an INTEGRATE matrix expression generally has little to do with the step sizes used in solving the differential equation system These are determined automatically by the characteristics of the system being solved and the optional control settings on the differential equation solver In particular the step sizes are often different for different steps Typically there are many more rows in the solution matrix than in t
56. an m vector of weights ME2 g FIT P1 P2 Pp clausel clause2 clausek Multiple clause curve fitting all clauses must be of the form Z TO ME1 WITH WEIGHT ME2 Z T TO ME1 WITH WEIGHT ME2 or F TO ME1 WITH WEIGHT ME2 all data matri ces must have the same first column If the clause WITH WEIGHT ME2 is omitted all elements of the m vector of weights will be set equal to one Typing R with no RETURN during FIT overrides LSQRPT setting typing Q with no RETURN during any FIT terminates command at the end of the current iteration h TYPE commands e TYPE csi e TYPE Z initial condition e TYPE Z T differential equation i Control variables These may be typed before a differential equation solution command e METHOD SE1 Specifies the integration algorithm MIXED 0 means mixed ADAMS 1 means Adams method GEAR2 2 means Gear Shrager method GEAR 3 means Gear method Default method is MIXED 0 e ERRFAC SE2 Error tolerance used for stepsize control Default error tolerance is 001 e ODERPT SE3 Specifies the contents of intermediate reports 305 the value 0 means no intermediate reporting the value 1 means report independent variable value step size and method at each step the value 2 means report independent variable value step size method and the derivative values at each step the value 3 means report only the summary information
57. analysis of the chick embryo data for our purposes However we would like to redraw our present picture as in Figure 1 3 which is more satisfactory for publication or for a permanent record The curve can be rescaled and the titles added by typing the commands shown below On Linux Unix with X Windows Macintosh and MSWindows systems each graphics command will cause the part of the picture being modified to be displayed WINDOW O TO 20 O TO 3 DRAW CB POINTS Y 0 20 1 TITLE CC1 DRY WEIGHTS OF CHICK EMBRYOS AT 0 3 3 WORLD SHIFT 35 TITLE CC2 FROM AGES 6 TO 15 DAYS AT 0 3 15 WORLD SHIFT 4 BOTTOM TITLE Ages in Days LEFT TITLE Weight in Grams TITLE CF Y 0051 EXP 3342 X AT 16 5 1 5 WORLD ANGLE 180 PI ATAN2 Y 18 Y 16 5 3 20 SIZE 013 VIEW Oe He KR He After these commands Figure 1 3 is displayed on the screen The first command dimensions the MLAB window in the x y plane so that it covers x in the interval of 0 to 20 and y in the interval of 0 to 3 The fitted curve CB is then redrawn for x in the interval 0 to 20 The TITLE commands are used to place the picture labels Note that picture labels may use upper and or lower case characters and be written in different sizes and at different angles Also note the method used to calculate the angle for the title CF and how this calculation took the user rectangle size into account The identifier PI has a value approximating the transc
58. and or repetitive computations Our introductory problem concerns the exponential law y a e which describes some simple growth phenomena This relation is illustrated by the dry weights of chick embryos from ages 6 to 15 days recorded in Table 1 1 Ages in Days x Dry Weights in Grams y 6 111 n 078 8 095 9 126 10 243 11 180 12 202 13 387 14 465 15 831 Table 1 1 Simulated chick embryo weight data Our objective is to make a graph showing these data points together with the exponential curve best approximating the variation in dry weights with age First enter the data by following the MLAB dialog below In general lines beginning with asterisks contain MLAB commands which you are to enter other lines are the responses which you should see appear as a result of typing the indicated commands DAY 6 15 DAY COL 2 LIST 111 078 095 126 243 180 202 387 465 831 TYPE DAY The computer response to these commands is to create a 10 row by 2 column matrix rectangular array of numbers called DAY and to type its contents as follows DAY a 10 by 2 matrix 1 6 Os TIt 2 7 0 078 3 8 0 095 4 9 0 126 5 10 0 243 6 11 0 180 7 12 0 202 8 13 0 387 9 14 0 465 10 15 0 831 The first number in each line indicates the row number of the matrix The numbers in the column to the right of the row numbers are the ages of the embryos The numbers in the right most column are the corresponding dry
59. are SUM X SE1 SE2 SE3 permitted in functions only PRODUCT X SE1 SE2 SE3 permitted in functions only m Definite integral is INTEGRAL X SE1 SE2 SE3 permitted in functions only n The root operator is ROOT X SE1 SE2 SE3 permitted in functions only e EVAL X SE1 SE2 permitted in functions only o Reference to the symbolic function derivative of a function F with respect to a symbol X is e F X or F DIFF X 4 MLAB Commands for Numerical Calculation are a Scalar assignment command creates or changes a scalar eu SE1 or e X SEl b FUNCTION command creates or redefines a function e FUNCTION U A1 A2 An SEl or e FUNCTION F A1 A2 An SEL c TYPE commands for scalar expressions e TYPE SE1 SE2 5En or e TYPE F or e TYPE F A1 A2 An 44 2 11 EXERCISES 1 Convert the following mathematical expressions to appropriate MLAB scalar expressions a cos 3u tanh 3v e b arctan c y c 1 d c log p to 9999999 where z denotes the largest integer not greater than z 0 d max x1 x2 the larger of x1 and x2 2 Construct MLAB FUNCTION commands to define the following functions a f x y B loge f sin t dt where B is a constant b g z U f z 1 where U is a constant and f is defined in part a c A periodic square wave function s t of period 2 p defined by st 5 2 n 1 p lt t lt 2 n p 3 2 n p lt t lt
60. are fixed parameters 3 Initial estimates for fitted parameters bo1 bo2 bop 4 Fixed parameter values Co1 C02 Cog 5 Constraint set each constraint is one of the forms r bi r2 b2 Tp bp lt ro r bi r2 b2 Tp bp fg r bi r2 b2 Tp bp gt To 6 Weighted sum of squared error brida dp XO we yi F Wainer in A standard MLAB curve fitting procedure for functional forms follows with optional steps indi cated by an asterisk 1 Create MLAB matrices via USE DO or matrix assignment commands as follows a An m by n 1 data matrix b An m vector of data point weights 273 2 Create an MLAB function via a USE DO or FUNCTION command that represents the func tional form f Parameters may be represented by scalars or matrix identifiers 3 Create an MLAB constraint set via a USE DO or CONSTRAINTS command containing all the linear constraints for the model Matrix element parameters can not appear in MLAB linear constraints 4 Create MLAB scalars or matrices via USE DO or scalar or matrix assignment commands corresponding to all initial estimates for fitted parameters and the known values for all fixed parameters of f For nonlinear models the initial estimates should be generated as carefully as possible It is often desirable to evaluate the weighted sum of squared error for a variety of initial assignments to the fitted parameters before choosin
61. are independent 230 Figure 7 2 Example of weighted sum of squared error terms 231 but have different variances the reciprocals of the variances are used as weights for corresponding data points Thus the expected values of the weighted squared error terms Wi yi fa are all equal to one Some examples of the reciprocal of variance method for assigning weights follow Suppose k independent measurements y1 Y2 Yg are made of a dependent variable y at a par ticular value of an independent variable x The mean value 1 m 7 EN Yi k 4 i 1 is usually taken as the data value for y It is well known that V m 0 k is the variance of the mean m where is the standard deviation of the individual measurements Thus the weight associated with the data point x m should be chosen as 1 W lt SS V m o If is unknown w k may be taken as the weight associated with the data point x m since o is a constant In this case however the expected value of the squared error term is unknown For example if 11 21 10 98 11 06 11 25 11 18 are k 5 repeated measurements for the value of a dependent variable y at the value x 2 of an independent variable the mean value m given here by m 11 21 10 98 11 06 11 25 11 18 5 11 136 is taken as the data value for the repeated measurements Since o is unknown the value 5 is taken as the weight associated with the data point
62. are the real and imaginary parts of the kt eigenvector of A The kt eigenvector vz corresponds to the kt eigenvalue Az The relation Av Axux holds for k 1 2 n Eigenvalue eigenvector analysis is useful in the study of stability of differential equation systems Given a differential equation system in canonical form dy F i dt 1 Y1 Y2 Yn dya F a dt 2 Y1 Y2 Yn 4 Fy Gis Y2 lt Yn 0 which has an equilibrium solution y y9 y2 whereat di 0 the stability of that equilibrium is determined by the signs of the real parts of the eigenvalues of the Jacobian matrix OF OF OF ou o gt Dk OF OF2 OF Oy y2 Oyn OF Fn OF n Oy y2 Oyn where each of the indicated partial derivatives is evaluated at the point of equilibrium y y9 y Furthermore if an equilibrium is just barely unstable by this criterion e g there is only a single eigenvalue with positive real part then the direction of the trajectory of the dynamical system in the vicinity of the equilibrium is parallel to the eigenvector of the eigenvalue with positive real part passing through the initial point close to y y9 y Finally if there is a single pair of complex eigenvalues with positive real part then the trajectory will lie in the plane formed by the corresponding eigenvectors and will be a spiral whose rate of rotation is proportional to the imaginary part of the eigenvalue pair and whose rate
63. as B then A Band A B B are equivalent matrix expressions For a column vector B A B is equivalent to the matrix expression A B NCOLS A As previously noted this is the same as multiplying each row of A by the corresponding component of B The general form of the element by element matrix product is given by ME1 ME2 Let ME1 be an m by n matrix expression aij and ME2 be an m by n matrix expression bj The result c j is an m by n matrix where m is the larger of m and m and n is the larger of n and n Then Cij Qij bij where aij is defined for j gt n by successively subtracting n from j until the result is between 1 and n and bj is defined for j gt n by successively subtracting n from j until the result is between 1 and n Table 3 28 Element by element matrix product operation Similarly the matrix transpose and element by element matrix product operations can be used to multiply each column of a matrix component by component with a given row vector 3 6 RANKORDER AND DISTRIBUTION OF VALUES IN A MATRIX Given data in a matrix M it is often desired to calculate the empirical distribution or the numerical ranking of the data values in M One method of calculating the density function or histogram of data given in a matrix is given in Section 6 2 The empirical cumulative distribution of the data can be easily calculated using the MLAB CDF
64. as the curve name It is recommended that the user supply curve names when developing a new picture The curve matrix can not be omitted from a DRAW command that creates a new curve If the window clause is omitted from a DRAW command creating a new curve then IN W is assumed W is an MLAB window with special properties which is supplied automatically by the MLAB system in response to a DRAW command requiring it Use of W will be discussed in more detail later Finally the default clauses POINTTYPE 0 no points and LINETYPE 1 solid lines will be supplied if necessary for a DRAW command creating a new curve For example the command DRAW M COL 1 3 POINTTYPE SQUARE will create in window W a curve data item CO assuming CO was unknown with curve matrix M COL 1 3 using squares at the points connected by solid lines The window clause of the DRAW command includes either an unknown identifier or a window identifier If the identifier has not been previously defined the DRAW command will create a new window data item with default user frame and image rectangles and add the curve matrix to it If the identifier has been previously defined as a window the curve matrix is simply added to it If the window clause is not supplied in a DRAW command the curve matrix is added to a window named W which is created if it does not exist A common MLAB user error is to omit the window clause of the DRAW command while adding curves to a window no
65. avoiding unsupported assumptions that are based on the properties of a particular set of input test data Errors of this kind frequently occur when constructing pictures 4 If a DO file is to be used repeatedly during the MLAB session it may be necessary to modify data items before exiting from the DO file Design the necessary commands so that repeated executions of the DO file will work correctly 5 It is usually helpful to add comments or typed messages to a DO file so that its principles can be quickly recalled after a lapse of time It is particularly useful to state the name of the file in the first line of the file itself 6 It is often convenient to use RESET at the beginning of a do file It is also useful to use the assignment command ECHODO 3 this causes the do file to echo to the MLAB command window when it is executed When a large DO file is first constructed it is likely to have a number of errors that produce error messages or unintended results Some standard methods for discovering and correcting such errors are described below 174 1 Tests for Syntax Errors y O d E Ns Se Check that every DO file command and comment has a semi colon Check that all parentheses brackets and quotation marks are in proper pairs Check that scalar and matrix operations have the proper number order and type of arguments that matrix dimensions are correct and that MLAB commands have correct syntax
66. causes MLAB to print the message Do you really want to quit MLAB y YES NO If you strike the y key the MLAB session will be terminated otherwise MLAB prints the message Resume calculation or return to top level t TOP LEVEL RESUME If you strike the t key MLAB prints the message Command in progress aborted Safest Action type EXIT then restart MLAB Riskier but often worthwhile Save your data objects SAVE IN MYFILE then EXIT restart MLAB and restore the objects USE MYFILE and returns to the prompt otherwise MLAB resumes the interrupted calculation Note you can also suspend an MLAB session and return to the bash shell prompt by typing Z CONTROL Z After typing any bash shell commands you can return to the suspended MLAB session by typing fg On Macintoshes with OS7 8 9 or OSX MLAB can be forced to quit by holding down the Option and Apple keys and simultaneously striking the ESC key The Macintosh Finder will then display a dialog box with a list of currently running programs select MLAB in the list and click FORCE QUIT button to terminate MLAB This manual focuses on explaining and presenting examples for MLAB statements and operators Generally these examples are given in the context of keyboard input In practice you will find that constructing modifying and using scripts of MLAB commands called do files is the most common and convenient way of using MLAB particularly for complex
67. computer disk file MLAB LOG 5 10 EXERCISES 1 Suppose that one wants to create an 80 row by 5 column MLAB matrix M with the first column representing days 1 to 80 of an experiment and the other columns representing measured quantities as follows col 2 a col 3 h col 4 c col5 d for days i 1 2 80 Describe MLAB commands for creating M for each of the following circumstances a The numbers are written in a table in a laboratory notebook 184 b A text file has been created which has the numbers Q1 42 080 bj ba Dgo C1 C2 C80 d d2 dgo in successive lines of text c A text file contains 80 lines of text each line containing for i 1 2 80 the data i day number aj bi ci di in the given order 2 Suppose the contents of a computer disk file have been forgotten How can the contents of this file be determined in each of the following cases Also which of the files can be modified and how can it be done in each case The file is EX1 DO and is known to be an MLAB DO file The file is MAR SAV and is known to have been created by an MLAB SAVE command c The file is T2 LST The file is mlabp0 ps The file is DATA5 and is known to be numeric data which was read into MLAB using a READ matrix expression f The file is A DAT and nothing else is known about it 3 Assume that M and N are large two column matrices A DRAW command is execut
68. converged quickly to a value for c which is reasonable from examination of Figure 7 8 In Figure 7 8 the graph of f x for c 709 and the plotted data points clustered near the function graph for 1 lt x lt 1 are shown If S versus c had not been graphed to allow a good initial estimate for c to be chosen the desired result would not necessarily have been obtained Next are the curve fitting responses that would be obtained from several other initial estimates for c Observe the MLAB dialog 264 C 500 FIT C F TOM matherr overflow error in cosh arg 9 399497e 84 return value 1 797693e 308 matherr overflow error in cosh arg 3 759799e 84 return value 1 797693e 308 matherr overflow error in cosh arg 9 399497e 84 return value 1 797693e 308 matherr overflow error in cosh arg 3 759799e 84 return value 1 797693e 308 matherr overflow error in cosh arg 9 399497e 84 return value 1 797693e 308 matherr overflow error in cosh arg 3 759799e 84 return value 1 797693e 308 final parameter values value error dependency parameter 500 2 142282373e 84 o Cc 1 iterations CONVERGED best weighted sum of squares 2 620320e 01 weighted root mean square error 1 144622e 00 weighted deviation fraction 1 000000e 00 For c 500 the function graph of f x from 1 to 1 is very near the z axis and the computed values are identically zero because of exponent underflow Hence the curve fitting pr
69. dynamically control the size font and relative position of the various noncode characters in the string These are explained in the next section The title T is put in the list of titles of window W If the window clause is omitted then an existing title T remains in the same window and a new title T is put into the default window W The clause SIZE SE1 UNITS specifies character size and spacing for the text string in frame fraction image fraction or user coordinates as previously described If no units are specified SE1 is assumed to be in horizontal frame fraction units The clause ANGLE SE2 specifies that the text is to be displayed at an angle of SE2 degrees where 0 degrees is left to right horizontal text positive angles are measured counter clockwise from zero and negative angles are measured clockwise from zero The clause FONT SE3 specifies that the character table or type style in font table number INT SE3 is to be used The table number must be an integer between 1 and 34 inclusive to select a Hershey font or between 2 and 14 inclusive to select a TrueType font Recall TrueType fonts are not supported on Linux Unix with X Windows or DOS systems The character tables for these fonts are shown in Appendix A The clause AT SE4 SE5 UNITS specifies that the first character of the string is printed at the point SE4 SE5 in the specified units FFRACT means frame fraction units IFRACT means image fraction units and
70. each coordinate x of V corresponds to x m d in VN for m the mean average of the elements of V and d the unbiased standard deviation of elements of V b To compute VB Y ON V where each coordinate of V is normalized in VB so that the smallest element of V corresponds to 0 in VB and the largest element of V corresponds to 100 in VB c To compute VA Y ON V where coordinates of V are temperatures in Fahrenheit and the corresponding coordinates in VA are to be temperatures in centigrade 8 Suppose a 1000 row by 2 column MLAB matrix D containing the results of 10 different experiments has been created each result being a 100 by 2 matrix of time intensity mea surements That is the first experiment corresponds to D ROW 1 100 the second experiment to D ROW 101 200 and so on a Design a DO file LOOK DO such that the execution of E 1 and of DO LOOK displays an intensity vs time graph of the first experiment curve matrix D ROW 1 100 in default window W on a graphical display the second execution of DO LOOK displays the graph of the second experiment D ROW 101 200 and so on for subsequent executions of DO LOOK b Choose a method that overrides the consecutive sequence by modifying E so that any experiment in a series can be displayed or skipped as desired c Change this DO file to account for K experiments each corresponding to an N row by 2 column matrix in a K N row by 2 column matrix D Because the variables
71. error can also give helpful information about the correctness of the curve fit it should not be much larger than the noise in the data adjusted by the weights that were used 270 7 15 INITIAL PARAMETER ESTIMATES FOR NONLINEAR MODELS As noted the unknown parameters for nonlinear curve fitting problems should be estimated as accurately as possible If such estimates can not be obtained curve fits starting from several different initial estimates for the fitted parameters can be tried Also producing a graphical display of the weighted sum of squared error versus various selections of values for the parameters as in Figure 7 7 if possible may help to find the best curve fit It also may help to compute the weighted sum of squared error at certain points throughout the region of interest As an example suppose that parameters A B and C in an MLAB functional form FUNCTION F T A EXP B T SIN C T are to be fit to an m by 2 data matrix D with an m vector of weights V Assume that high and low estimates for A B and C are known say 6 lt A lt 9 106 lt B lt 1 0210 lt C lt l The weighted sum of squared error S A B C is to be evaluated for various triples of values satisfy ing these constraints The triple that minimizes S is then used as initial estimates for curve fitting the parameters A B and C It is decided to search uniformly evaluating S for 120 triples corre sponding to the 4 values 6 9 for A the 5
72. example if you have a matrix M containing values of a quantity corresponding to a range of times you may wish to see the value of the quantity at a certain time without typing out the entire matrix and looking through it manually The MLAB LOOKUP operator described in Table 3 31 serves just this purpose Given a 2 column matrix of key values and corresponding data values the LOOKUP operator will return the data value corresponding to any given key value 86 Using the matrix DAY from Chapter 1 have MLAB type out the dry weights corresponding to day 11 and day 8 by entering the following dialog USE TUTOR1 Using W CS1 DAY B A M Y TYPE LOOKUP DAY 11 LOOKUP DAY 8 18 095 In the above dialog LOOKUP s were done on key values that existed in the matrix As the following dialog shows the LOOKUP operator will perform linear interpolation if necessary M LIST 1 2 4 9 10 amp 1 5 TYPE LOOKUP M 4 3 TYPE LOOKUP M 5 LOOKUP M 8 3 2 3 8 The LOOKUP operator will perform linear interpolation only if the given key value is not found in the first column of the data matrix However the INTERPOLATE operator explained in Section 6 2 provides the method of cubic spline interpolation if this is necessary M LIST 1 5 3 2 4 amp 1 5 TYPE LOOKUP M 1 LOOKUP M 2 LOOKUP M 3 LOOKUP M 4 LOOKUP M 5 25 5 l onere As this dialog shows the LOOKUP operator will not perform pr
73. file names for PostScript plot files have an mlabp prefix followed by a number and a ps extension The file names for LaserJet plot files have an mlabp prefix followed by a number and an 1j extension Plot files are created and accessed in the directory specified by FILEDIR Print files mlab 1st 1st MLAB can be directed to create print files by using the PRINT command Print files have default name mlab 1st if no name is specified and moreover subsequent print operations append to this file A print file has the default extension 1st if the file name with no extension is given Alternately the full print file name can be given in quotes Print files are created and accessed in the directory specified by FILEDIR Print files are text files which may be used in any way desired including physically printing them when you wish In order to print any file you can type the MLAB command PRINT FILE lt filename gt or you may use the facilities of your operating environment to send the file to the printer This can be difficult in some cases Read files dat The READ operator in MLAB can be used to read numbers from a text file Such text files are accessed in the directory specified by FILEDIR Do files do Do files are text files containing MLAB commands thus they are executable scripts A do file z do is executed by typing DO z Such do files are accessed in the directory specified by control string DOFILEDIR If DOFILEDIR
74. for each of the curve fitting parameters 5 the 3 element column vector DEPVALS was assigned the dependencies for each of the curve fitting parameters and 6 the 3 by 3 matrix COVP was assigned the covariance matrix There is a pre defined scalar in MLAB called LSQRPT that controls what information is to be reported during and at the conclusion of the processing of a FIT command The default value 246 of LSQRPT is 8 which means that the MLAB curve fitting process is quiet final parameter values lagrange multipliers and a statistical summary are printed at the conclusion of the FIT process If LSQRPT is 0 the MLAB curve fitting process is silent nothing will be printed to the screen during or at the completion of the curve fitting process For a more complete description of LSQRPT see The MLAB Reference Manual If the FIT command is used some output regarding the progress MLAB is making in reducing the weighted least sum of squares error can be obtained by typing the letter R without carriage return during execution of the FIT command R stands for Report This will toggle an MLAB switch which when ON will print out information about the fitting process Typing Q without carriage return terminates any FIT command at the end of the iteration currently being computed Q stands for Quit In MLAB multiple clauses can be used in a FIT command to compute overall estimates of the
75. form of the MLAB FIT command for curve fitting to func tional forms are given by FIT X1 X2 Xn clausel clause2 clausek CONSTRAINTS csi where clausej has the form Fj TO ME2 j 1 WITH WEIGHT ME2 j for j dy 225 wee Ke Here Fj describes a functional form used for curve fitting which is an MLAB function The list X1 X2 Xn contains MLAB scalar or matrix identifiers Each scalar identifier is a parameter to be fitted and all elements of each matrix identifier are also parameters to be fitted The values of these scalars and matrix elements when the FIT command is executed are the initial estimates for these parameters New values will be given to these scalars and matrix elements in each iteration of the curve fitting process The matrices ME2 j 1 and ME2 j are the data ma trices and associated weight vectors respectively for the curve fitting process If a clause WITH WEIGHT ME2 j is omitted then weights equal to one are used The word WEIGHT may be abbreviated by WT csi is an optional MLAB constraint set for linear constraints on the fitted parameters If there are no constraints the CONSTRAINTS csi clause may be omitted In the FIT command each fit clause has an associated weighted sum of squared error criterion For the given functional forms and data the iterative search process modifies the listed parame ters scalars and matrix elements to seek the minimum sum of the k weighted sum of squared er
76. graph of y x in the same range So Figure 4 5 b can be drawn by the commands UNVIEW DELETE G 112 a b Figure 4 5 Examples of smooth curve drawing 113 c WINDOW 1 TO 1 3 TO 3 IN W FUNCTION Y1 X 0 25 SQRT 1 X X M POINTS Y1 1 1 02 DRAW EA M IN W M COL 2 M COL 2 DRAW EB M INW VIEW The above technique is not suitable for the curve of Figure 4 5 c which twists and turns frequently Curves of this type are often best drawn by a parametric representation That is a pair of functions x s and y s are defined so that the desired curve consists of the points x s y s for all s in a certain range of values say from c to d In that case the point x s y s traces the curve as s increases from c to d Here an arithmetic sequence matrix c d e is set up to obtain a sufficient number of s values and then the F ON M matrix expression is used twice to generate the curve matrix from the parametric definition The curve of Figure 4 5 c has the parametric description als sin 8 s y s sin s for s in the range from 0 to m The MLAB commands to draw it were UNVIEW DELETE EA EB WINDOW 3 TO 3 O TO 1 INW FUNCTION X S 0 25 SIN 8 S FUNCTION Y S SIN S M 0 P1 02 N COL 1 X ON M N COL 2 Y ON M DRAW C N IN W VIEW Now consider the situation in which straight strokes are required but are not arranged in any en
77. hee eae a aa 49 KREAD expression Simple form lt o co sardea 06 0 4444 2 oe be ee eke as 50 READ expression console form a o coa maaa aa eo i a e aa a e a a 52 READ expression vector form lt lt s 2 2 2 ee ndana ap ia 52 Matrix entry assipnment comni nd o s os 64 a a aa a a a eR a a 55 Arithmetic sequence matrix expression e a eee 56 Fixed number of members arithmetic sequence matrix expression 57 Matrik transpose WOTAMION aos 6 26 nb kw eB a E 59 LIST expression general form oeoo a 59 SHAPE expression general form oa sa ac aa esama e e 61 ROW DALE Operators 6 6 eG eG REE EEA Be SHAE Ce RAG 62 COL matrix OESTE lt a a ee ee Sk Deh EL ee eee A 63 HOW matrik assignment command ocaso ee es ae a ek ee a 65 COL matrix assignment command lt s s s o ee e a e a a p 66 3 17 3 18 3 19 3 20 3 21 3 22 3 23 3 24 3 25 3 26 3 27 3 28 3 29 3 30 3 31 3 32 3 33 4 1 4 2 4 3 4 4 4 5 4 6 4 7 Matrix joining Operators cesos aa a a 68 Matrix row replication operator o o 68 Matrix column replication operator e e o 68 Matrix joining and repeating operator evaluation 69 BURT matrix operat sisi as a A OR ED Re EEE 71 Some useful matrix editing operators e e 72 Matrix expressions with scalar matrix arguments o 75 Matrix expressions with matrix ar
78. id UM Shy ommeiz7DS 1181233357373 ay aS Sea 7 y Es A ao Le H G terrtirir re ere role aes da EEE OMUNHOOUOUOULEL_HOOOARAHNYVOGFH X 000000 Lob eo eo eeeoe eR BHRoOWO 4 Ht om AL Lt bbdd ooo DA m mk o y a a y U OARANADADADA lt IA lt IBRARARNRAADH lt lt o0MAAQQA OVOOLLUODODODODXxXA AXxX gt XDOLDONVDOLOD ses OODOOOO MAOMQMMANAMANANAAAARRRDOE mamma QQ TGLITCIA lt ALALIAL IAE RARUA On LALAN Laya OOOO 00000000000 o LL ODANMFMOWR ONOANMFTAMOWOR ONOANM THOMA 100 MM OM AAA AAA AAA A NU NY OY ONU AY NU NU NY NU AU ON MUDA oH mM Figure A 5 ASCII characters 64 to 79 in Hershey fonts 1 to 34 319 asc1180 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 ted a mid T OULD ARO EEE ETA ECEE MD ADA AA oo SS PAI MEE ARTE EM Eee A RR E DD ee ee eae ae NeNNNNNNNNNNSNNSNNNN RN ODO gt Stl NNNNNN a AI U VWXYZ PDA DS NNO MPD 1 Da A AO AAA KOK PK DL S P AAA RR RK DS xO a KK A OK BS 333RI SFE S3ECCCOORERERIESE NODO OO Sa SS SAR DSABAPM n AD gt gt gt SD SIDO PD DID se Hb DE SRESDS gt P0333333 REKEFERBREBERE RE ERRRRDDBARDE u OFrOoOrFrFEKAR FontP Q R S AMNNNNNNNNNWNANWANIN QSROOONLDHANDNDAHHDH EARL ERES RBAA gt RENO TNOBMOCCCOCOPO SO OOCCSEONAN LAO H ooao00 a Kj UM Pg RS QqgdaMmatmamcHhohQQWAageseade YOLUN ODA A A AH SADA TATA NY AY N
79. implied when Y alone is written so Y alone is sufficient The MLAB INITIAL command is used to specify the initial condition y 0 5 in the obvious way shown The function Y is now implicitly known to MLAB by means of the differential equation for Y X and the initial condition for Y As usual the TYPE command will describe the associated data items as shown in the dialog TYPE Y Y X INITIAL Y O 5 FUNCTION Y X X Y X X 1 72 The response is essentially a repeat of the INITIAL and FUNCTION commands Note that Y is an MLAB data item of data type initial it is by this means that MLAB knows the corresponding function Y is defined by a system of one or more differential equations Only the MLAB INITIAL command can be used to create or modify data items of initial type and these data items are used only to specify initial conditions for systems of differential equations The implicitly defined function Y can be evaluated numerically by any one of several methods The first method is to simply evaluate Y as if it were an explicitly defined function For example M Y ON 1 5 TYPE M a 6 by 1 matrix 5 1 97155402 874696273 449159263 273267675 193620648 OORUN The second method uses the POINTS operator and returns a two column matrix with the indepen dent variable in column 1 and the solution to the differential equation in column 2 For example observe the following 281 M POINTS Y 0 5 TY
80. in Section 7 1 Table 7 2 Weighted sum of squared error criterion X t Ao k1 k1 ka gt e kt ete B t Apg 1 e x t derived from the consecutive kinetic reaction ki ko A gt X gt B contain the common parameter k Suppose data measurements 1 2 n Of the function X t with associated weights v1 v2 Un and data measurements b1 ba bm of the function B t with associated weights w1 wa Wm are obtained Although it is possible to model data measurements T1 T2 Zn by the function X t and to model data measurements b1 b2 bm by the function B t separately it is usually preferable to model these data to X t and B t simultaneously There is a weighted sum of squared error criterion Si a xi X t i 1 associated with the model X t and another weighted sum of squared error criterion So Su b Bt i 1 237 associated with the model B t The weighted sum of squared error criterion for the joint models is then given as S S1 S2 a xi X t Lay b B t i 1 i 1 Table 7 3 shows the joint models weighted sum of squared error criterion In the general case suppose that the dependent variables y1 Y2 Ym are functions fi Ei 2 cp Jol Ci 2 20 Bp my fm T1 T2 oh Ep respectively of p independent variables x1 2 p For data points Zijl Lij2 Zijp Yij and assigned weights wij i 1 2
81. is empty then do files are accessed in the direc tory specified by FILEDIR Do files provide the usual way in which you control MLAB to develop a computation or do repetitive tasks MLAB control files MLAB HLP gxfonts 0 etc MLAB consists of an executable file and a number of associated control files These files are 159 found in the MLAB executable directory and possibly in certain ancillary directories These directories are fixed at the time MLAB is installed 5 2 THE SAVE AND USE COMMANDS You can save data objects on the disk for later reuse Such files are called SAVE files they are created by the MLAB SAVE command The SAVE command is described in Table 5 1 To execute the SAVE command of MLAB the user supplies 1 a list of MLAB identifiers separated by commas for currently defined data items and 2 a filename for a computer disk file to be created or modified For example recall the command SAVE Y M A B DAY CS1 W IN TUTOR1 from the Tutorial in Chapter 1 which created a computer disk file named TUTOR1 DAT This file contained the function definition for Y the matrices M and DAY the scalars A and B the set of constraints CS1 and the window W that were defined during the session When the window W is saved all its curves titles and axes are automatically saved also However curves titles and axes can only be saved by saving the window containing them To recover these data items in a subseque
82. matrix element to correspond to each fitted and each fixed parameter The fitted para meters must be assigned initial estimates bo1 bo2 bop and the fixed parameters must be assigned numerical values co1 C02 Cog by appropriate scalar or matrix assignment or input statements Table 7 9 The general problem of curve fitting a functional form In general let y f 1 2 Zn b1 b2 bp be a functional form for a dependent variable y independent variables x1 9 Y para meters b1 b2 bp with unknown values and possibly other para meters with fixed values Then f determines a linear model with respect to b1 b2 bp if and only if For any numbers 701 02 Zon assigned to the arguments of f there exist numbers 11 12 Tp depending upon 21 102 Ton and any fixed parameters such that f 01 02 Zon b1 b2 bp ro ribi r2b2 ss roda for all numerical values assigned to b1 ba bp Table 7 10 Linear model functional form 279 Chapter 8 SYSTEMS OF DIFFERENTIAL EQUATIONS IN MLAB MLAB has a built in differential equation solver which permits computation of numerical solu tions for systems of differential equations This chapter discusses 1 conventions for setting up differential equation systems in MLAB 2 the matrix valued operator INTEGRATE which generates numerical solutions 3 the variables METHOD ERRFAC ODERPT MAN
83. numbers to be read KREAD is a convenient way to read data from within a do file If there are no arguments a single number entered at the keyboard is read and returned as a scalar value If when a single number is requested the user enters no number but simply presses the ENTER RETURN key then the value MAXPOS is returned Testing for MAXPOS is a convenient way to determine that some default input value should be used 164 If there is a string argument S MLAB will print the string to the MLAB command window and wait for the single number to be entered followed by the RETURN ENTER key This is convenient for dialog directed input Otherwise this case is identical to the no argument case If X is given and Y is omitted then X numbers are read and a 1 column matrix with X rows is returned If both X and Y are given then X Y numbers are read and a matrix with X rows and Y columns is returned You must enter at least X Y numbers The numbers may be typed in free format separated by spaces tabs or commas Input lines are terminated by pressing the RETURN ENTER key If the Escape key is typed in an input line the current row of the matrix is completed with zero entries and no more rows are added to the matrix Excess numbers are ignored Thus KREAD will read in a fixed number of numbers i e if you specify X 8 and Y 2 you must type in 16 numbers unless the Escape key is struck To read in an indeterminate number of numbers READ
84. of convolution as a matrix multiplication of the form r G f where r and f denote the vectors r 1 r 2 and f 1 f 2 and g1 0 0 0 o 99 am 0 0 9 3 g 2 g0 0 is a lower triangular matrix In both cases we write r f x g using x to denote convolution Convolution arises in various situations One common application is in processes described by probability distributions If X and Y are random variables with density functions f and g respectively then the random variable X Y has the density function r f g This rather abstract situation has numerous concrete realizations For example suppose that the random 220 variable Y is the background level of some substance in a population while the random variable X is the added amount of the same substance as a result of some treatment When the substance is measured in the population the detected level will be a random variable Z X Y with density function r If the background density function g is known the treatment density function f can be found by deconvolution i e solving the above matrix equation to yield f G te Since G is triangular the solution is easier than in the general case i 1 f t r O al i 1 9Q for1 lt is lt n j l This operation can be done directly in MLAB using the DECONV operator but it may be done more efficiently by taking advantage of the discrete Fourier transform operators If the MLAB vecto
85. operator The CDF operator described in Table 3 29 takes a matrix of values M and a matrix of values A over which to compute the values of the distribution function of the data It will return a matrix whose first column contains the values of A and whose second column contains the number of values from M which are less than or equal to the corresponding value in A divided by the total number of values in M While graphing this distribution matrix will be left to Section 6 1 we can look at the CDF operator in the following dialog M LIST 7 1 5 2 7 5 8 1 3 7 TYPE CDF M 1 10 81 w ja o by 2 matrix COOANOOBRWNEH FOAN OOF WNE RRPROOOOOOO ODDOFA AWN E 0 TYPE CDF M SORT M a 10 by 2 matrix ooon AUNE 0 Y YX 0d00uNRe RA PPOO0O0OO0OO0OO0OOOO ODMOMDAAWNND E The MLAB RANKORDER operator described in Table 3 29 will take as input a matrix M sort it on its last column and add an additional column beyond the last column which contains the rank values For rows of M with identical last column values the rank values in the result matrix will be averaged This is shown in the following dialog M 1 5 LIST 1 4 2 5 3 TYPE RANKORDER M a 5 by 3 matrix ORPWNF PHO OW Fe OP WNEF oORPWNF M 1 5 LIST 1 3 2 3 4 82 TYPE RANKORDER M RANKORDER M COL 1 amp M COL 2 a 5 by 3 matrix NON N sNOaere 5O0uNR Gi 0 w o a a a 5 by 3 matrix 1 5 4 1 2 2 3
86. operator is a 2 column matrix whose rows appear in groups headed by marker rows Each marker row contains MAXPOS in the first column and a value V between zero and one in the second column The following rows in the group are points on the contour level line corresponding to the particular group Each value V is determined as l lmin lmax where I is the level value of the curve lmin is the minimum level value and lmaz is the maximum level value to the height of the level of the particular group This matrix format corresponds to that of linetype MARKER VMARKER SMARKER or SVMARKER which are explained below The following MLAB commands will draw a contour map of the function f x y y cos x y shown in Figure 6 10 RESET FUNCTION F X Y X Y Y COS X Y M POINTS F CROSS 3 3 3 3 C CONTOUR M 27 27 DRAW C LINETYPE VMARKER WINDOW ADJUST WMATCH VIEW Often however the data we want to analyze are not taken over a grid of x y values but are scattered randomly in space In this case it is necessary to use the INTERPOLATE operator to interpolate values over a grid of values and it is often beneficial to SMOOTH the data beforehand This technique is shown in the following command sequence RESET DA COL 1 LIST 2 2 1 5 1 5 1 1 5 5 0 0 5 5 1 1 1 5 1 5 2 2 DA COL 2 LIST 2 2 1 1 5 1 5 2 1 5 1 0 2 2 1 5 2 1 5 1 5 204 INS V NG WE SAS AW uw o
87. or a graph strike any key on the keyboard to continue Selecting the second option causes MLAB to terminate the STARTUP DO script file and wait for a command Selecting the third option causes MLAB to print a list of suggestions on how to proceed with MLAB Striking the RETURN ENTER key or clicking the CONTINUE button causes MLAB to execute the do file MLABEX DO which was described previously Note if you edit the file STARTUP DAT in the directory where the MLAB application resides type a number i e 1 in the file and save the file then when the script STARTUP DO is executed in future MLAB sessions MLAB will simply go to top level For the purpose of this tutorial select the second option Go to top level MLAB That will print an asterisk on the last line of the MLAB command window This asterisk is MLAB s prompt for the user to enter an MLAB command It may matter whether you type upper or lower case characters for MLAB commands the MLAB system interprets non system variables in the case typed unless the variable CASESW is 0 In this tutorial type all characters in either upper case or lower case In general a carriage return is to be typed at the end of any complete line of input to MLAB On some computer keyboards the carriage return key is marked ENTER on others it is marked RETURN If an MLAB command is too long to type on a single line you can type a backslash to continue the command on the next line
88. printer devices For some computers systems the color number in Table 4 8 may not correspond to the listed color For example COLOR 11 does not appear to be pink on MSWindows with a video graphics adapter set to 16 color mode The color numbers listed for MSWindows MLAB in Table 4 8 are color numbers for a computer with a video graphics adapter set to display 256 or more colors the color numbers will be different for a computer with a video graphics adpater in 16 color mode The clause COLOR COLORX r g b can be used to obtain the color number closest to the color defined by the combination of red green and blue intensity specifications given by values r g and b respectively r g and b must be numbers in the interval 0 1 For example COLOR COLORX 0 0 1 would result in blue and COLOR COLORX 1 1 1 would result in white The COLORN function maps the integers 1 2 3 15 to the colors in Table 4 8 This function is convenient when using a FOR loop to draw different color curves For example the following MLAB commands will draw 6 different exponential curves each in a different color 125 FCT F X EXP A X FOR I 1 6 DO A 5 I DRAW POINTS F 0 10 100 COLOR COLORN T VIEW Of course you can always define your own vector of color numbers and use an index into the vector to select your own sequence colors 4 4 THE AXIS COMMAND The XAXIS YAXIS and AXIS commands are used to create axis data objects The general
89. problems The INTERPOLATE operator will take a matrix of data points and return a matrix containing selected points on a smooth cubic spline curve passed through these points This is shown in the following command sequence DELETE W FUNCTION F X SIN X LOG X M POINTS F LIST 1 2 4 5 18 19 DRAW M LINETYPE NONE POINTTYPE CROSSPT DRAW INTERPOLATE M 0 20 25 DRAW POINTS F 0 20 25 LINETYPE DASHED VIEW The result of these commands shown in Figure 6 4 shows the data points as plus signs the interpolated curve as a solid line and the actual function curve as a dashed line It can be plainly seen in this figure that the INTERPOLATE operator provides good estimates when approximating 194 Figure 6 4 Interpolating data 195 values close to existing data points but may not behave as you wish when interpolating far from other points The INTERPOLATE operator also applies to surfaces as will be seen in Section 6 4 The SMOOTH operator is designed to eliminate oscillations from a curve matrix by modifying the last column based on the surrounding data This can be done as preparation for curve fitting which will be explained in Chapter 7 for other mathematical analysis or for drawing or plotting The effects of the SMOOTH operator can be seen visually by entering the following commands RESET FUNCTION F X X X 4 X 5 M POINTS F 1 10 DRAW M M 3 2 M 3 2 10 M 7 2 M 7 2 7 DRAW
90. spond to one or more pairs of coordinates in an MLAB graphics window that are specified with the mouse pointing device The WREAD command is described in Table 5 10 Given a scalar X an MLAB window name Y and a scalar Z the command M WREAD X Y Z 181 causes MLAB to show the window corresponding to window identifier specified by Y on the display screen as if a VIEW command were given and wait for the user to specify X ordered pairs with the mouse pointing device using the method specified by integer argument Z Once the required number of points have been stored or earlier if the user strikes the ESCape key the MLAB graphics window is removed from the screen and the 2 column matrix of specified points is returned If Z is omitted or equal to 0 then an ordered pair is selected by positioning the mouse cursor at a desired point in the window Y and then pressing and releasing the mouse button WREAD changes the pixel value at each selected point to the color WHITE The world coordinates corresponding to the mouse position are then stored as a row of the output matrix in this case matrix M The user may terminate WREAD before X ordered pairs have been selected by striking the ESCape key on the keyboard If Z is equal to 1 WREAD will dynamically display the world coordinates of the current mouse position at the bottom left corner of the image rectangle Ordered pairs are then selected as for Z 0 by positioning the mouse cursor at a desire
91. supplied in all DRAW commands with no LABELFONT clause The OFFSET clause specifies where the lower left corner of a box enclosing the label will be placed with respect to the corresponding curve matrix point If omitted the offset is 0 0 this means the lower left corner of the label box is placed at the corresponding curve matrix point If included the OFFSET clause requires two numbers separated by a comma and enclosed in parentheses the first number specifies the horizontal displacement of the lower left corner of the label with respect to the curve matrix point the second number specifies the vertical displacement of the lower left corner of the label with respect to the curve matrix point These numbers may be positive or negative If no units are specified these numbers are assumed to be in horizontal frame fraction units alternatively including IFRACT will cause the offsets to be interpreted in units of horizontal image fraction units Including WORLD will cause the offsets to be interpreted in units of horizontal user coordinates For example the clause OFFSET 5 5 WORLD would put the lower left corner of the box enclosing a label at a point 5 user z coordinates to the left and 5 user x coordinates above the corresponding curve matrix point 121 The PLACE clause takes two words separated by a comma and enclosed in parentheses and specifies a point on the box surrounding the label which should be placed at the offset to
92. the curve fitting command invokes system atic search methods to seek new parameter values that give better agreement between the data and the values predicted by the least squares criterion discussed below In Chapter 1 the mathematical model was an exponential relationship between the age X and the dry weight Y of chick embyos This was expressed by the exponential function defined by the MLAB command FUNCTION Y X A EXP B X with two parameters A and B The curve fitting operation FIT A B Y TO DAY 225 changed the parameters A and B systematically until the exponential model was in close agreement with the experimental data of the matrix DAY The equation of the exponential function with parameters A and B is an example of a functional form Curve fitting to various other functional forms will be considered next 7 1 SUM OF SQUARED ERROR CRITERION FOR CURVE FITTING Usually the experimental data contains random measurement errors which make it impossible for every data point to agree exactly with the value predicted by a model It is convenient to have a computable quantity for measuring the agreement between the data and the predicted model values Because of an extensive mathematical and statistical theory for the method of least squares the sum of squared error is the most frequently used measure As an example suppose that an experimental value y has been measured at various values of an independent variable x
93. the numbers 300 0 300 600 900 1200 and 1500 respectively As a result of the WINDOW command which redefined the user rectangle Y1 was changed so that it consisted of the 7 points 1 5 300 1 5 350 1 5 400 1 5 450 1 5 500 1 5 550 and 1 5 600 labelled with the numbers 300 350 400 450 500 550 and 600 respectively The axis command begins with either XAXIS YAXIS or AXIS If XAXIS is used labels and points for the resulting axis data item will re scale when the user rectangle s x coordinates are changed by 127 The complete form of the AXIS command is lt X Y gt AXIS A ME1 POINTTYPE ptyp PTSIZE SE1 UNITS PTFONT SE2 LINETYPE ltyp LABEL ME2 LABELSIZE SE2 UNITS LABELFONT SE3 OFFSET SE4 SE5 UNITS PLACE A B FORMAT SE6 SE7 SE8 SE9 SE10 SE11 ANGLE SE12 COLOR col IN W Table 4 9 AXIS command complete form subsequent WINDOW commands If YAXIS is used labels and points for the resulting axis data item will re scale when the user rectangle s y coordinates are changed by subsequent WINDOW commands If AXIS is used only the points not the labels for the resulting axis data item will re scale when the user rectangle s x and or y coordinates are changed by subsequent WINDOW commands If the axis name A is omitted from an XAXIS command the default name W1 XAXIS is provided where W1 is the window name specified in the IN clause If the axis name is omitted from an YAXIS comma
94. to the left image border 106 solid line extending to the right image border tick mark 106 solid line extending to the top image border 106 solid line with marker skipping line type 106 117 204 solid line with marker skipping using spline curves line type 106 solid line type 106 solving a third order differential equation 290 solving boundary value differential equation 300 solving differential equations 280 281 solving first order differential equations sys tems 290 solving linear equations 93 solving systems of first order differential equa tions 283 SORT 70 87 SORT matrix operator 71 sorting matrices 70 SOSQ 246 special cases of differentiation 38 special functions 19 square point type 106 110 square root 19 SQUARES 106 110 stability of differential equation systems 210 STAR 106 star point type 106 starting MLAB 1 startup do file 158 STARTUP DAT 2 STARTUP DO 2 STDEST 246 step function 192 196 step functions 26 step size 293 step specified sequences 54 STEPGRAPH 192 stick figure drawing 115 stiff systems of differential equations 295 stopping MLAB 176 string 15 345 string addition 131 string array 15 string modification command codes 136 submatrices 62 subtraction 18 SUM 33 34 sum of squared error criterion for curve fitting 226 sunspot cycle periodicities 219 surface 15 surface contours 203 SVD 213 215 S
95. two pictures shown in the Tutorial of Chapter 1 covered the entire MLAB picture As in those examples the contents of a frame rectangle typically consists of an image of the user rectangle with some surrounding area in which titles or axis labels appear A frame rectangle is usually specified in screen fraction units Screen fraction units are numbers which range from 0 to 1 and specify a fraction of the MLAB picture s extent in the horizontal x direction or the vertical y direction For example the frame rectangle for a window that occupies the full MLAB picture would be specified by O TO 1 O TO 1 in screen fraction units Note that 0 0 in screen fraction units corresponds to the lower left corner of the MLAB picture A frame rectangle for a window object that occupies the upper right quarter of the MLAB picture would be specified by 5 TO 1 5 TO 1 in screen fraction units The image rectangle of a window object describes the location of the graphical image of the user clipping rectangle within the frame rectangle If the usable part of the frame rectangle is regarded as an aligned rectangle with origin 0 0 in its lower left corner then the horizontal and vertical extents of the image rectangle may be specified in frame fraction units The image rectangle of a window object can describe the entire usable frame rectangle or can be any aligned rectangle which lies within the usable frame For example a window with image rectangle 5 TO 1 O TO 1 w
96. values 1 06 1 02 01 for B and the 6 values 0 1 02 for C MLAB dialog for this process follows FUNCTION F T A EXP B T SIN Cx T FUNCTION G T A B C A EXP B T SIN C T FUNCTION S A B C SUM J 1 NROWS D V J D J 2 G D J 1 A B C 2 DIN 6 1 02 01 DO 1 02 DO N N 1 M ROW N I amp J amp K 120 XA XA X 1 0 0 22 7 a 4 by 3 matrix 1 6 1 06 0 2 6 1 06 0 02 3 6 1 05 0 4 9 1 02 0 1 271 M POINTS S M M SORT M 4 TYPE M ROW 1 a 1 by 4 matrix 1 8 1 02 0 1 846954933 A 8 B 1 02 C 1 FIT A B C F TO D WITH WEIGHT V final parameter values value error dependency parameter 4 6795085893 1 5539811937 0 9144956732 A 0 5947033424 0 3363179636 0 9631001287 B 0 1087617401 0 1816061286 0 8749637384 C 4 iterations CONVERGED best weighted sum of squares 2 153034e 01 weighted root mean square error 3 281032e 01 weighted deviation fraction 1 117593e 01 R squared 9 348442e 01 Note that the asterisks and colons appearing at the beginning of command lines are printed by MLAB and should not be typed by the user The second and third function definitions above express the weighted sum of squared error S A B C as an MLAB function After setting N to 0 the next FOR command generates a 120 by 3 MLAB matrix M which lists all triples such that the first column contains test values for A in 6 9 the second column contai
97. with marker skipping using splines ie K ue y i Figure 4 4 Samples of point types and line types 107 Description Point types Symbol 0 NONE 1 VBAR 2 CROSSPT 3 TRIANGLE 4 SQUARES 5 STAR 6 HBAR 7 OCTAGON 8 XPT 9 LTICK 10 RTICK 11 UTICK 12 DTICK 13 CIRCLE 14 ARROW 15 DOTPT 16 DIAMOND 17 XERRBAR 18 YERRBAR 19 XLOGTICK 20 YLOGTICK 21 NBAR 22 TBAR 23 RBAND 24 LBAND 25 UBAND 26 DBAND 27 DRBAND 28 DLBAND 29 DUBAND 30 DDBAND 31 ARROWTIP on no points default vertical bars plus signs triangles squares stars horizontal bars eight sided figure X shaped symbol leftward tick mark rightward tick mark upward tick mark downward tick mark variable resolution circle vector field arrow a small dot a diamond symbol horizontal error bar vertical error bar horizontal logarithmically spaced ticks vertical logarithmically spaced ticks normal bar tangent bar solid line extending to the right image margin solid line extending to the left image border solid line extending to the top image border solid line extending to the bottom image border dotted line extending to the right image border dotted line extending to the left image border dotted line extending to the top image border dotted line extending to the bottom image border directed arrow the character c in a specified font could be a curve matrix for a curve to be drawn with LT MARKER T
98. with y 0 1 and y2 0 3 for t i 10 using the Adams method the Gear method the Gear Shrager method and the combined Adams Gear method Compare the results 3 Recall the fundamental theorem of the calculus dH 2 f x if He f t dx for any continuous function f So the definite integral as a function of the upper limit is the unique solution of the differential equation system dH a f x Hig 0 307 Evaluate the definite integral ff 1 t dt numerically by using the INTEGRATE operator and the INTEGRAL operator described in Chapter 2 and compare the results Given the system of differential equations in two unknown functions f t and g t df dg o 273 PELO 28 0 dg dg ae IO Bt 4 t 0 with initial conditions f 1 3 1 g 1 2 d 1 a Numerically solve this system for t 1 2 1 b Numerically solve the above system with boundary conditions f 1 3 1 g 1 2 g 2 6 for t 1 2 1 A dose of a radioactive labelled drug is injected in the blood stream of a rat and enters the bile The amounts of the drug are measured in both the blood and the bile over a period of ten minutes Because a delay of d minutes is needed for the bile to pass through a catheter the amount of the drug measured in the bile at time t is actually the amount at time t k If all measurements are given in terms of percent of the initial dose a simple model for the incorporation of the dru
99. 0 matrix D dij such that di i for i lt 10 and dij 0 for i j Hint Use a FOR command Write MLAB commands to compute a 5 by 1 column vector M such that M I is the average mean value of the 7 numbers in column I of the matrix N in exercise 1 for I lt 5 Then write MLAB commands to divide each element of N by the mean value of the column of N to which that element belongs Hint Use the and operations Let a function f be given by f x t sin x t cos x where t is constant Write MLAB commands that will create a 21 by 2 matrix P which lists x y coordinates for 21 points on the graph of y f x with the x coordinates going from 1 to 1 in steps of size 1 10 Create a similar matrix PD for the derivative df x dx of f x The binomial coefficient C m n is defined for integers m gt n gt 0 by the formula m n m n where 0 1 A 5 by 5 matrix of binomial coefficients with C 1 7 1 in row and column 7 is given as eee Re Ae Q0NF O aOwnr oo rere COO oO H oo co oc 10 11 It can be proved that C m n C m 1 n 1 C m 1 n for m gt n gt 0 and clearly C m 0 C m m 1 for all m Construct a 10 by 10 matrix BC of binomial coefficients extending the above matrix with BC I J C I 1 J 1 for J J lt 10 Hint Examine the effects of V LIST 1 followed by repeated execution of the command V V 0 V and then use a suitable
100. 10 approximately 3 Scalar Expression Formation Rules are a Scalar constants and scalar identifiers are scalar expressions b Scalar arithmetic operations are e SEL SE2 e SE1 SE2 e SE1 e SE1 SE2 e SE1 SE2 42 e SE1 SE2 or SE1 SE2 c Order of operations is specified by parentheses or by the hierarchy of operations as follows e Evaluation of functions e Exponentiation or e Multiplication and division from left to right e Addition and subtraction from left to right d Trigonometric functions include e SIN SE1 e COS SE1 e TAN SE1 e SIND SE1 e COSD SE1 e TAND SE1 e Inverse trigonometric functions include e ASIN SE1 e ACOS SE1 e ATAN SE1 e ATAN2 SE1 SE2 f Logarithmic and exponential functions include e LOG SE1 e LOG10 SE1 e EXP SE1 e SINH SE1 e COSH SE1 e TANH SE1 g Special functions include e ABS SE1 e SQRT SE1 e INT SE1 e MOD SE1 SE2 e LOGGAMMA SE1 e ERF SE1 e GAUSS SE1 h Matrix entries are written e M SE1 SE2 43 e M SE1 i The value of a user defined function at the actual arguments SE1 SE2 SEn is F SE1 SE2 SEn See FUNCTION command 3 Scalar comparison operators are SE1 lt SEQ SE1 lt SE2 SE1 gt SEQ SE1 gt SE2 SE2 SE2 SE1 NOT SE2 k Logical operators are NOT SE1 SE1 AND SE2 SE1 OR SE2 1 Indexed sum and product operators
101. 107 is computed during a scalar expression evaluation computation continues with zero replacing the scalar value causing underflow The MLAB user should be aware of the possibility of numerical inaccuracy and should exercise due caution in the choice of scalar expressions for computation Theoretically every MLAB scalar expression can be constructed step by step beginning with scalar identifiers and scalar constants and then applying the notational rules in some order In practice correct MLAB expressions can be written directly by using the familiar algebraic nota tion Whenever a notational rule is introduced the numerical operation it describes must also be known Let SE1 SE2 etc indicate both the names and the numerical values of the scalar expres sions in this discussion We may now present some of the more common scalar operators available in MLAB Table 2 2 summarizes the scalar arithmetic notational rules Table 2 3 summarizes the common trigonometric functions in MLAB Table 2 4 summarizes the common inverse trigono metric functions in MLAB Table 2 5 summarizes the logarithmic exponential and hyperbolic trigonometric functions in MLAB and Table 2 6 summarizes some commonly used special func tions In general this book does not discuss all the operators and functions available in MLAB please refer to the MLAB Reference Manual for a complete definition of every operator and function in MLAB 18
102. 1712 B 4 iterations CONVERGED best weighted sum of squares 3 465262e 02 weighted root mean square error 6 581472e 02 weighted deviation fraction 1 251451e 01 R squared 9 297622e 01 In addition to other information the response specifies best fitting parameter values of 0050847413 for the parameter A and 334188183 for the parameter B In fact A and B have been changed to these new values as you can see by typing the MLAB command TYPE A B and observing the output gt i 5 08474129E 3 B 334188183 In order to draw the best fitting curve we create a new matrix of x y coordinates by typing in the folowing MLAB commands 6 15 05 POINTS Y M ss This creates M as a two column matrix which can be regarded as a list of 181 points in the a y plane which lie on the best fitting exponential curve Redisplay Figure 1 1 with the fitted exponential curve by typing the MLAB commands DRAW CB M VIEW This causes a curve segment called CB that is part of the best fitting exponential curve defined by the points in M to be drawn in the default window on the screen The result is shown in Figure 1 2 0 9 0 72 0 54 0 26 Figure 1 2 Data points and best fitting exponential curve Weight in Grams DRY WEIGHTS OF CHICK EMBRYOS FROM AGES 6 TO 15 DAYS 0 6 Ages in Days Figure 1 3 Best fitting exponential curve suited for publication 10 Suppose that this is sufficient
103. 2 SE1 SE2 1 if SE1 SE2 0 if SE1 SE2 SE1 gt SE2 1 if SE1 gt SE2 0 if SE1 lt SE2 SE1 lt SE2 1 if SE1 lt SE2 0 if SE1 gt SE2 SE1 NOT SE2 1 if SE1 SE2 0 if SE1 SE2 Logical scalar expressions values NOT SE1 1 if SE1 0 0 if SE1 40 SE1 AND SE2 1 if SE1 4 0 and SE2 40 0 if SE1 0 or if SE2 0 SE1 OR SE2 1 if SE1 0 or if SE2 40 0 if SE1 0 and SE2 0 Conditional expression values IF SE1 THEN SE2 ELSE SE3 SE2 if SE1 0 SES if SE1 0 Table 2 11 Comparison logical conditional expressions 2 5 RECURSIVE FUNCTIONS AND THE TYPEOUT OPER ATOR There are a few situations in which functions are most easily defined recursively that is where evaluation of the function involves evaluating the function at a different value An example of this is the factorial function n mathematically defined as the product n n n 1 n 2 8 2 1 The value of 4 is thus 4 4 3 2 1 24 The easiest way of defining the factorial function is 1 forx 1 FACTORIAL z E FACTORIAL x 1 fora gt 1 31 This form of function definition is possible in MLAB with the statement FUNCTION FACTORIAL X IF X 1 THEN 1 ELSE X FACTORIAL X 1 Note however the computation specified is very wasteful when we want to tabulate FACTORIAL on a range of successive integers Another example of a function that is most often defined recursively is the function which when called with the integer n
104. 2 SE3 TO SE4 As before the notations SE1 SE2 etc denote MLAB scalar expressions and their corresponding numerical values The phrase above specifies an aligned rectangle with horizontal x range SE1 to SE2 and vertical y range SE3 to SE4 For example 4 TO 8 10 TO 40 is the aligned rectangle shown in Figure 4 1 Each window data object has three aligned rectangles associated with it the user clipping rect angle the frame rectangle and the image rectangle The user clipping rectangle of a window 98 object is specified so that the x axis and y axis ranges correspond to the units of measurement appropriate to the user s application For example suppose the user wishes to graph counts of radioactive samples in the range 100 to 100 000 measured every four hours for two days in a certain experiment Then O TO 48 O TO 100000 would be an appropriate description of the user clipping rectangle corresponding to a range 0 to 48 measured in hours along the z axis and a range 0 to 100 000 measured in counts along the y axis Alternatively the base 10 logarithms of the radioactive counts in the range 2 to 5 might be the desired graph An appropriate user clipping rectangle would then be O TO 48 2 TO 5 The user clipping rectangles for different window objects are completely independent The frame rectangle of a window object defines what portion of the MLAB picture will contain the window object The frame rectangles of the windows in the
105. 27 label 101 label matrix 141 LABELFONT 117 121 127 labeling points on curves 123 LABELM 141 LABELSIZE 117 121 127 Lagrange multiplier 252 law of mass action 284 LBAND 106 LDASH 106 LEFT 122 133 134 143 leftward tick mark 106 Legendre s equation 298 LENGTH 77 line feeds in titles 131 line type 101 106 118 linear constraints 239 linear differential equation 280 linear interpolation 86 87 linear models 239 257 259 LINETYPE 5 106 118 Lissajous figures 114 LIST 5 49 53 59 LJ2L 168 LJ2P 168 log file 13 158 logarithm transformed data 229 logarithmic functions 19 logical expressions 30 38 long commands 3 long dash line type 106 LOOKUP 86 87 95 LSQRPT 247 250 252 LT 106 118 LTICK 106 128 magnitude of numbers 16 making pictures 96 MANDSW 280 294 301 MAPPLY 85 86 202 MARKER 106 117 204 marker skipping 202 mathematical models 225 matrix 15 47 matrix arithmetic operations 73 342 matrix assignment statement 47 matrix column replication operator 67 94 matrix combination operators 192 matrix compression 72 matrix diagonals 213 matrix element assignment 53 matrix elements 22 26 matrix elements as curve fitting parameters 255 matrix entry assignment command 54 matrix expansion by cyclic extension 61 matrix expansion by zeroes 54 63 matrix exponentiation 78 matrix expressions with both scalar and ma trix argumen
106. 3 complex numbers 208 COMPRESS 72 computation with matrices 47 conditional command 172 conditional expressions 30 38 connect every other point line type 106 114 190 200 constant matrix 69 constraint set 15 CONSTRAINTS 7 238 240 245 250 constraints 239 252 constraints on curve fitting parameters 238 construction of matrix entries 22 CONTOUR 204 208 contour graphs 203 contour maps 204 208 contour splines 208 control variables for curve fitting 251 control file 159 controlling the differential equation solver 293 CONVERGED 252 convolution 220 coordinate axes 126 CORR 215 217 correlation 215 217 cosine 19 COV 215 217 covariance 215 217 COVP 246 CPOW 213 CPROD 213 CROSS 189 192 cross point type 106 crosshatching 203 CROSSPT 106 cubic polynomial 112 cubic spline interpolation 87 cubic splines 194 208 cumulative distribution 81 83 192 curly braces 89 DBAND 106 current directory 155 DDASH 106 curve 15 98 138 DDBAND 106 curve color 117 125 DECONV 221 curve identifiers 110 deconvolution 220 221 curve label angle 117 123 default axes 146 curve label font 117 121 default frame rectangle 100 curve label format 117 122 default image rectangle 100 curve label matrix 141 default user clipping rectangle 100 curve label offset 117 121 default window 144 146 curve label place 117 122 defining functions 24 2
107. 4 2 5 2 6 aot 2 8 2 9 Simulated chick embryo weight data o ooa a aa a 4 Scalar assienment command ooro oie 2 4 eo a a aa 16 Scalar arithmetic notational rules o a 19 Trigonometrie Tmetions o ce o a o Bee E A a A a a me a 19 Inverse trigonometric functions e e a 20 Logarithmic exponential and hyperbolic trigonometric functions 20 Commonly used special functions e 21 Random number operator secs aoo a b a peg doa dona e a aoa i b e a E 22 Normal random number operator lt cs asasena sar a A a 22 Construction of matrix Ediles 2 0 so aod os ea eR a a a a eaa ee a 23 2 10 2 11 2 12 2 13 2 14 2 15 2 16 2 17 3 1 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 3 10 aL 3 12 3 13 3 14 3 15 3 16 FUNCTION command oe e ceos 46 5 HEGRE eae ee ee ee ee ee RE So 25 Comparison logical conditional expressions 2 2 0 0 ee eee 31 TYPEDUT scalar Operatit s s scs sacs cds ee wR a bo 33 SUM PRODUCT and INTEGRAL operators soc s oee da od ketara ee ee 35 ROOT and EVAL Operators lt e p aaroo ea baci eR ee EH 37 Special cases of differentiation e 38 Denning MLAB iderveativess 2 500 054 6 eee aa A 41 TYPE c Mminamd s nba be Pear bee we ee ee ee Se ee A A 42 General forms of matrix assignment command o 48 Matrix dimension Operador ee ee ee e a 49 LIST expression simple form 2 0 2 4 6 5
108. 5 POINTTYPE O LINETYPE 1 DRAW C2 M ROW 7 9 POINTTYPE O LINETYPE 1 DRAW C3 M ROW 11 12 POINTTYPE O LINETYPE 1 Clearly the first piece draws a triangle and the second and third pieces form a letter F within the triangle 109 Drawing with line type 7 VMARKER is similar to drawing with line type 6 MARKER except that dashed lines with variable lengths instead of solid lines are drawn The second column in a marker row contains the dash length in inches to be used For example if M 1 2 125 and M 6 2 M 10 2 25 in M COL 2 of the matrix M above and LT 7 is specified the triangle will be drawn with dashed lines of lengths 1 8 inch and the letter F within the triangle will be drawn with dashed lines of lengths 1 4 inch It is possible to omit certain parts of the short DRAW command To modify an existing curve only the curve name and the clauses specifying the desired changes are needed For example the command DRAW CA IN W LINETYPE 1 will move CA to window W and change its line type but will leave its curve matrix and point type unchanged Similarly the curve matrix could be modified while leaving the window point type line type unchanged and so on If a new curve item is to be created default values will be supplied for omitted parts If the curve identifier is omitted it is assumed that a new curve is to be created and the first unknown identifier in the list CO C1 C2 C3 etc will be used
109. 5 T TO 875 T in screen fraction units where S and T are the width and height respectively of the frame in screen fraction units Alternatively suppose PIC is an existing window data item Then e the WINDOW command above changes the user clipping rectangle of PIC to the specified value 1 TO 5 0 TO 100 leaving the frame rectangle and image rectangle unchanged e the FRAME command changes the frame rectangle of PIC to the specified value 0 TO 5 0 TO 1 in screen fraction units leaving the user clipping rectangle and the image rectangle unchanged e the IMAGE command above changes the image rectangle of PIC to the aligned rectangle 1 TO 9 1 TO 9 leaving the user clipping rectangle and frame rectangle unchanged 100 The simple forms of the WINDOW FRAME and IMAGE commands are WINDOW SE1 TO SE2 SE3 TO SE4 IN wi FRAME SE1 TO SE2 SE3 TO SE4 IN wi IMAGE SE1 TO SE2 SE3 TO SE4 IN wi where wi is either a window identifier or unknown The clause IN wi is optional If the IN clause is not included the command will apply to the default window named Y For a window wi the user clipping rectangle frame rectangle or image rectangle is changed to the given specification SE1 TO SE2 SE3 TO SE4 For wi unknown wi becomes a window with no curves and the specified user clipping rectangle frame rectangle or image rectangle The WINDOW command creates a window with default frame rectangle O TO 1 O TO 1 and default image re
110. 5 TO 1 O TO 1 VIEW XK XA XK XA XA XX XA XA XA XX XX D In the FIT command above the clause X TO DX tells MLAB to find the sum of squared error 5 associated with X the clause B TO DB tells MLAB to find the sum of squared error S2 associated with B and the sum of the two criteria S 53 is used to find the agreement between the predicted and observed values of both X and B Because the values of X T are much greater than the values of B T the sum S1 53 is dominated by the term S1 Consequently the function B T shown in Figure 7 4 is a poor fit to the observed data The weighted sum of squared error criterion is appropriate in this case Observe the following dialog that uses the MLAB operator EWT in the clauses of the FIT command to estimate weight vectors corresponding to the matrices DX and DB and displays Figure 7 5 UNVIEW BLANK XFIT1 BFIT1 Ki 05 K2 0001 FIT K1 K2 X TO DX WITH WEIGHT EWT DX B TO DB WITH WEIGHT EWT DB inal parameter values value error dependency parameter 0 0445653323 0 0059939972 0 8353122888 Ki 0 0001048492 5 260479277e 06 0 8353122888 K2 3 iterations CONVERGED best weighted sum of squares 3 199382e 01 weighted root mean square error 1 333205e 00 weighted deviation fraction 5 176151e 02 H Xx XA 248 X T EKT Figure 7 4 Fit of X t and B t using unit weights 249 10 1 10 0 c c x m 1 6 2 0 2 0 20 40 60 Go 100 0 20 vo 60 Go 100 T T Fi
111. 55 curve label size 117 121 definite integral 33 curve labels 117 119 delay term 288 curve matrix 101 103 141 DELETE 5 curve names 110 density function 81 curve fit with EWT estimated weights 250 DEPVALS 246 curve fit with unit weights 248 derivative 281 curve fitting 7 225 244 derivatives of functions 37 40 85 curve fitting a differential equation 298 descending sort of matrix 71 curve fitting a functional form 241 DFT 221 curve fitting control variables 251 274 DIAG 89 curve fitting data preparation 241 diagonals of a matrix 213 curve fitting differential equation models 301 DIAMOND 106 curve fitting differential equations 298 301 diamond symbol point type 106 302 DIFF 38 curve fitting functions of more than 1 vari differential equation error handling 296 able 236 differential equation models 298 curve fitting joint models of several functional differential equation solver controls 293 forms 236 differential equation solver reports 296 297 curve fitting matrix elements 255 differential equation solver s error tolerance curve fitting models 253 296 curve fitting parameter constraints 238 differential equations 280 curve fitting reports 247 differential equations with a delay term 288 curve fitting sum of squared error criterion 226 directories 154 CURVEM 141 DISASTERSW 280 296 301 CURVES 141 discrete convolution 220 cyclic extension of matrices 61 63 64 75 81
112. 66009 490854 148559 447956 717571 800983 617647 9296 040238 9584832 0243578 8869249 3059407 9440997 6689501 4281439 200874 9788642 7588146 of MLAB commands 10 2653884 9 96067162 9 83889007 1000 350 695183 132 761927 51 7556015 20 4071213 8 08225647 3 20070866 1 26136837 489211958 181375622 5 84876871E 2 13211049 3666009 22490854 56148559 69447956 74717571 76800983 17617647 179296 78040238 oR Cola oes ol oes oki oes okie cmon o U INTEGRATE S T C T P T 0 10 1 WINDOW O TO 10 8000 TO 10000 286 5 geno E y p j 600 Y zo Es 8000 y 2 a 20 o o 2 E a 20 TIME ITI TIME ITI 10 a m z Bn 40 P o a 2 t 10 TIME ITI Figure 8 1 Plots of S t C t and P t DRAW U COL 1 2 LEFT TITLE SUBSTRATE S BOTTOM TITLE TIME T FRAME O TO 5 5 TO 1 Wi W DRAW U COL 1 4 WINDOW O TO 10 O TO 1000 LEFT TITLE ENZYME SUBSTRATE COMPLEX C BOTTOM TITLE TIME T FRAME 5 TO 1 5 TO 1 12 W DRAW U COL 1 6 WINDOW O TO 10 O TO 100 LEFT TITLE PRODUCT MATERIAL P BOTTOM TITLE TIME T FRAME 25 TO 75 O TO 5 VIEW Of course new values could be assigned to the scalars K1 K2 K3 EO and SO and the INTEGRATE 287 operator could be used to obtain new numerical solutions as often as required Observe that the INTEGRATE operator again has the differential eq
113. 7 3 4 2 2 TYPE MA COL 3 MB a 2 by 4 matrix te 2 4 2 2 2 3 4 2 2 For example suppose an experiment continues for three weeks with one daily measurement taken from each of 50 experimental animals Assume that one has entered three 50 by 7 matrices W1 W2 and W3 of data recorded from the first second and third weeks of the experiment Then the MLAB command E 1 21 amp W1 amp W2 amp W3 will create a 51 by 21 matrix E which has days 1 to 21 of the experiment numbered across the top row and has the full 21 day responses of the 50 experimental animals in rows 2 through 51 There is a matrix repeating expression denoted by double carets which is related to the ampersand amp matrix expression This is described in Table 3 18 For example MA 773 is equivalent toMA amp MA amp MA That is the specified number of copies of the given matrix are joined successively from top to bottom of the resulting matrix To join copies of a matrix from left to right to form a larger matrix the column wise replication operator described in Table 3 19 can be used For example the matrix expression MA 77 gt 4 is equivalent to MA MA amp MA MA The matrix joining and repeating operations can also act on scalar expressions which are then treated like column or row vectors as appropriate with all components equal to the given scalar 67 Let ME1 be an m by n matrix expressio
114. A ROW 1 4 3 l Q has the same effect as the three commands A ROW 1 C ROW 1 A ROW 4 C ROW 2 A ROW 3 C ROW 3 Observe that only three rows of C were actually used above If C was a 1 by 7 matrix A ROW 1 4 3 C would substitute C for each of rows 1 4 and 3 There are two MLAB matrix operations denoted by ampersand amp and ampersand apostrophe amp gt which allow one to join two matrices together to form a larger matrix These operators are described in Table 3 17 Roughly speaking the ampersand operation forms the larger matrix by placing the second matrix argument below the first while the ampersand apostrophe operation forms the larger matrix by placing second matrix argument to the right of the first These oper ations are called row concatenation amp and column concatenation amp Matrix expansion by cyclic extension is used if necessary to fill any unspecified entries in the resulting matrix Observe the following MLAB dialog 64 TYPE MA 1 1 2 T The general forms of the ROW matrix assignment commands are M ROW ME1 ME2 M ROW SE1 ME2 M ROW SE1 SE2 SEn ME2 Here M is either an MLAB matrix identifier or an unknown identifier If M is unknown the command is executed just as if M was the 1 by 1 matrix 0 Because SE1 abbreviates LIST SE1 and because SE1 SE2 SEn abbreviates LIST SE1 SE2 SEn in the second and third forms of the matri
115. AB will be automatically be supplied with the do command to execute the selected do file In either case when MLAB is run from the NeXTStep Workspace your current directory will be your home directory as determined when you logged in Beware if your home directory could not be found at login time the system will have assigned the top level root directory as your home directory Also when MLAB is run from the NeXTStep workspace a front end MLAB interface process is run which in turn communicates with an MLAB session process You may establish further MLAB sessions by clicking the Session gt New menu item In a DOS system the MLAB executable directory is specified during the installation process it is C MLAB if MLAB was installed in the default location Your home directory is the same as the current directory To run MLAB you should cd change directory to the working directory you desire and then run MLAB by typing mlab assuming the DOS PATH environment variable is properly set For example if you create a directory called C MLAB WORK cd to that directory and type mlab then C MLAB WORK is the current directory MLAB accesses and creates files Each such file must be in a specific directory There are MLAB control strings whose values determine where some files will be Other files are assumed to be in certain fixed directories MLAB commands may be typed in the screen or the text window Currently aside
116. AD operator offers another method for entering scalars into a matrix For the moment only the two READ expressions used for typing in a matrix at the keyboard will be considered The first form of the READ expression summarized in Table 3 5 is used to enter an array matrix of numbers A typical MLAB dialog using the console READ expression for typing in a matrix is shown below MAT READ CON 2 4 Type lt CONTROL gt Z after last number 5 35 6 1 2 3E 3 45 1 1 72 Z TYPE MAT MAT a 2 by 4 matrix 5 35 6 1 0 0023 45 Ld 72 0 0 ol The keyboard entry form of the READ expression is READ CON SE1 SE2 The word CON is short for console The expressions SE1 and SE2 are positive integers usually constants If m SE1 and n SE2 the READ expression will create an m by n matrix m rows and n columns If necessary SE1 and SE2 are made into inte gers by applying the INT function When a console type of READ expression is in a matrix assignment command or other MLAB command the command s execution is halted and the following message appears TYPE lt CONTROL gt Z AFTER LAST NUMBER Scalar constants may now be typed in to fill the m by n matrix named proceeding from left to right across each row and giving each complete row from the first to the last Carriage returns are typed wherever con venient It is not necessary to type rows as separate lines of input If mn scalar constants are entered to completely fill the
117. AXIS command is shown in Table 4 9 Axis data objects are similar to curve data objects in that they include a curve matrix linetype pointtype and label matrix The use of XAXIS and YAXIS is demonstrated by the MLAB commands that made the axes in Figure 4 2 XAXIS X1 2 3 amp 300 PT UTICK LABEL 2 3 LABELSIZE 015 FFRACT OFFSET 01 025 IN W YAXIS Y1 2 amp 300 1500 300 PT RTICK LABEL 300 1500 300 LABELSIZE 015 FFRACT OFFSET 09 01 IN W VIEW Just as the DRAW command specifies a curve matrix point type line type label matrix color etc for a curve so the AXIS commands define these elements for a curve to be used as an axis The curve matrix for the axis named X1 includes the six points 2 300 1 300 0 300 1 300 2 300 and 3 300 Since the line type is not specified the default SOLID 1 is used The point type is an upwards directed tick mark UTICK Each point in the curve matrix is also labelled with a number from the label matrix which consists of the numbers 2 1 0 1 2 and 3 The height of each number is 015 horizontal frame fraction units Each label appears 01 frame fraction units to the left and 025 frame fraction units below each point in the curve matrix Since a FORMAT clause was not included the label numbers are formatted by FORMAT 3 9 0 0 2 0 There are two important differences between a curve resulting from a DRAW command and an axis resulting from an A
118. CT PLACE LEFT BOTTOM FORMAT 3 5 0 0 2 0 ANGLE 0 COLOR 1 e Title creation and modification e TITLE T text IN W SIZE SE1 UNITS ANGLE SE2 FONT SE3 AT SE4 SE5 UNITS SHIFT SE6 COLOR SE7 PLACE A B New title defaults name TO T1 etc default window W SIZE 02 FFRACT ANGLE 0 FONT 5 AT 0 0 FFRACT SHIFT 0 COLOR 1 PLACE LEFT BOTTOM Curve and window blanking e BLANK D1 D2 Dn e UNBLANK Di D2 Dn g TYPE command for windows curves axes or titles e TYPE W TYPE C TYPE A or TYPE T h Renaming windows or curves U WU C U AjorU T i Recovering curve and label matrices from curves and axes e CURVEM C or LABELM C e CURVEM A or LABELM A 3 Special facilities e default window W Automatically supplied by MLAB in response to a DRAW or TITLE command requir ing it Frame and labelled x axis and y axis are included Automatic rewindowing is performed when DRAW commands introduce new curves 4 10 EXERCISES 1 Start MLAB and reproduce Figure 4 2 as follows a Create a window W by using the WINDOW FRAME and IMAGE commands b Create the x axis with labelled tick marks the y axis with labelled tick marks and the rectangle bordering the image rectangle c Draw the three straight line segments between adjacent data points and label the points as shown 2 Rename the window W in exercise 1 with the new name F2 and blank F2 148 3 Suppose HL is a scalar
119. DIVISION BY ZERO does not occur if this command is executed in a be typed nonzero The general form of the conditional command is IF SE1 THEN command group1 ELSE command group2 If SE1 NOT 0 then the commands following the THEN clause are executed If SE1 O then the commands following the ELSE clause are executed This agrees with the convention that true is represented by 1 and false is represented by 0 for arithmetic comparison and logical con nectives considered as scalar operations The specifications for command group1 and command group2 are 1 nothing 2 a single MLAB command without semicolon or 3 a se quence of n MLAB commands in parentheses separated by semicolons of the form command1 command2 commandn The ELSE clause including command group2 may be omitted Table 5 7 Conditional command THEN TYPE ERROR X IS 0 ELSE V 1 X DO file since only expressions in the executed clause are evaluated The next IF K lt 3 This command increases K by one when K lt 35 and sets K to zero and causes the message K example shows an ELSE clause with two MLAB commands 5 THEN K K 1 ELSE K 0 TYPE K RESET TO 0 RESET TO 0 to be typed when K gt 35 In MLAB the THEN clause can not be omitted from a conditional statement but the ELSE clause can be omitted However the MLAB commands within either clause can be omitted For example T 0 THEN X T 0 THEN X
120. DSW JACSW and DISASTERSW which are used to control the computation process and 4 curve fitting with models specified by systems of differential equations 8 1 A FIRST ORDER DIFFERENTIAL EQUATION SOLVED IN MLAB Consider the linear differential equation _dy x _ Ha eee y 1 This is a first order differential equation because only the first derivative of the unknown function y appears There are infinitely many functions y y x that satisfy this equation but an initial condition e g y 0 5 can be specified to select a unique solution from the set of all solutions of the differential equation In effect that solution of the differential equation whose graph passes through the particular point 0 5 in the plane is specified by the initial condition y 0 5 To set up the above differential equation and initial condition in MLAB use the following com mands 280 FUNCTION Y X X Y X X 1 72 INITIAL Y 0 5 The differential equation is represented by the FUNCTION command which defines Y X as a function with argument X and the appropriate defining scalar expression That is the differential equation is treated as a function definition for the derivative y 1 in terms of the unknown function y x and other expressions involving x Observe that y is written as if it were a scalar in the expression Y X X 1 72 defining Y X One could also use the more explicit expression Y X X X 1 72 but Y X is
121. E G COL 1 NROWS G 1 G COL 2 S MESH G R DEL S ROW NROWS S DRAW S H CDF M 0 1 05 DRAW H LINETYPE NONE POINTTYPE VBAR VIEW XK XA XA A XA XA XA XX KX The method used to draw the step function by these commands can be easily understood by considering that the bottom endpoint of each vertical line which is every other point in the matrix has the x coordinate of the next point and the y coordinate of the preceding point These steps are all combined in the function STEPGRAPH The CROSS operator also combines two matrices by column concatenating the rows of the first matrix with the rows of the second in every possible combination If the first and second input matrices are m by n and ma by na matrices respectively then the resulting matrix will have m ma rows and n na columns The CROSS operator is often used to specify a grid of points on the plane for example CROSS 5 5 5 5 specifies an 11 by 11 grid of points with the origin at the center spaced at unit intervals This could be used to analyze a function of two variables as will be discusssed in Section 6 4 It could also be used to list and manipulate values for a function of several variables as will be done in Section 7 15 One of the most common uses of the CROSS operator is to generate a rectangular grid to use in generating contour plots as shown in section 4 later in this chapter Finally the CROSS of two lists each stored as a 1 column
122. E O In this example there is a compound condition obtained by joining the two conditions T gt 0 and T lt 180 by the Boolean operator AND That is the THEN clause is used only if both conditions are true MLAB also has the Boolean operators OR inclusive and NOT logical negation Arithmetic comparisons and Boolean operators are just scalar expressions under the convention that true is represented by 1 and false is represented by 0 The graph of y h a in Figure 2 3 corresponds to the function l z if0 lt a lt 1 Ma 4 l x if1 lt x lt 2 1 otherwise One way to define h x in MLAB is by using an auxiliary function K as shown below FUNCTION H X IF X gt 0 AND X lt 2 THEN K X ELSE 1 FUNCTION K X IF X lt 1 THEN 1 X ELSE 1 X 29 Note that it does not matter what values are taken by K X except when X is between 0 and 2 if K is used only to help evaluate H Another more efficient way to define h x in MLAB is using nested conditional expressions as in the statement FUNCTION H X IF X gt O AND X lt 2 THEN IF X lt 1 THEN 1 X ELSE X 1 ELSE 1 Functions of several variables can be defined by conditional scalar expressions in a similar way For example the MLAB command FUNCTION Z X Y IF X X Y Y lt 1 THEN SQRT 1 X X Y Y ELSE 0 defines a function of two variables whose graph is obtained from the x y plane by replacing the unit disk by a hemisphere Any two scalar expres
123. E2 ME1 and ME2 are 2 column matrices whose first columns contain the real parts and second columns contain imaginary parts of a list of complex numbers If ME1 and ME2 do not have the same number of rows the shorter of the matrices is cyclically extended column wise The result of CPROD is a 2 column matrix containing the real and imaginary parts of the complex product of corresponding rows of ME1 and ME2 The result of CDIV is a 2 column matrix containing the real and imaginary parts of the complex quotient of corresponding rows of ME1 and ME2 The result of CPOW is a 2 column matrix containing the real and imaginary parts of the power Table 6 6 Complex number operators and the degree to which the light is attenuated in passing through the sample One source of such spectra is the titration of a mixture of acids and bases monitored by changes in the optical spectrum of the mixture A titration is a common chemical analysis procedure in which a specimen is subjected to the sequential addition of equal amounts of some chemical in this case hydrogen ion Following each addition a measurement is made of some property of the specimen The data takes the form of a matrix whose columns are the measured spectra at some hydrogen ion concentration and whose rows are the absorbance at successive points through the titration at fixed wavelength Applying the SVD operator to this matrix we obtain the matrices U P and V Exam
124. ELSE O THEN ELSE X X 1 X 1 X 1 172 are equivalent conditional commands As an example of the use of these techniques consider the quadratic equation a 12 b 2 e 0 There are two roots of this equation which are both real or which form a complex conjugate pair according to whether the discriminant d b 4 a c is positive or negative The DO file QUAD DO shown below solves any such quadratic equation QUAD DO TYPE SET CF LIST A B C FOR ROOTS OF A X 2 B X C 0 DO CON DSCR CF 2 72 4x CF 1 CF 3 IF DSCR gt O THEN S CF 2 SQRT DSCR 2 CF 1 T CF 2 SQRT DSCR 2x CF 1 TYPE REAL ROOTS S AND T ELSE S CF 2 2 CF 1 T SQRT DSCR 2 CF 1 TYPE COMPLEX CONJUGATE ROOTS S Ti AND S Ti TYPE S T Lines 1 2 request that the user enter the desired coefficients a b and cinto a 3 by 1 matrix CF In line 3 the discriminant is computed as DSCR Lines 4 10 contain a single large conditional com mand which computes S and T correctly in the two possible cases and issues the correct message Finally S and T are typed at line 11 Note that all statements are separated by semicolons An example of MLAB dialog using QUAD DO is given below DO QUAD SET CF LIST A B C FOR ROOTS OF A X 2 B X C CF LIST 1 5 6 REAL ROOTS S AND T S 3 r 2 DO QUAD SET CF LIST A B C FOR ROOTS OF A X 2 Bx X C CF LIST 2 7 23 ll o Il 173 COMPLEX CONJUGATE RO
125. EO SE1 and SE2 denote scalar expressions ME1 and ME2 denote matrix expressions T denote a scalar identifier P1 P2 Pp denote scalar or matrix identifiers and clause1 clause2 clausek denote clauses that refer to MLAB defined functional forms or unknown functions or derivatives of unknown functions in one differential equations system 1 MLAB Commands for Systems of Differential Equations are a FUNCTION Z T T SE1 b INITIAL Z SEO SE2 c INTEGRATE Y1 T Y2 T Yn T ME1 Typing R with no RETURN during a numerical solution of a system of differ ential equations causes a report on the current status of the computation to be typed typing Q with no RETURN causes the computation to be stopped and zeros used for values not computed d CONSTRAINTS csi Constraint terms N1 S Ni S and S N1 are separated by or to make expressions two expressions are separated by lt or gt to make a constraint e g A 4 2 B C gt 1 304 e FIT P1 P2 Pp Z TO ME1 WITH WEIGHT ME2 Fits a parameter list P1 P2 Pp of scalar or matrix identifiers of an unknown function Z in a differential equation system to an m by 2 data matrix ME1 with an m vector of weights ME2 f FIT P1 P2 Pp Z T TO ME1 WITH WEIGHT ME2 Fits a parameter list P1 P2 Pp of scalar or matrix identifiers of a derivative Z T in a differential equation system to an m by 2 data matrix ME1 with
126. EXP etc and ME1 must be a matrix of n columns If ME1 is an m by n matrix a then the result is an m vector with i component f aj1 aj2 Qin for 1 2 m POINTS F ME1 Equivalent to ME1 F ON ME1 DF ON ME1 Equivalent to F ON ME1 except that the symbolic derivative F DIFF X1 DIFF X2 DIFF Xk specified by DF is evaluated POINTS DF ME1 Equivalent to ME1 amp DF ON ME1 MAPPLY F ME1 F must be a function of one two or three arguments If ME1 is an m by n matrix the result is an m by n matrix whose i j entry is F M i j ifFisa function of one argument F i j if F is a function of two arguments or F i j M i j if F is a function of three arguments Table 3 30 ON POINTS and MAPPLY matrix operators The expression LOG10 ON B does not work because LOG10 is a function of one argument and B has more than one column However the computation can be performed with B1 MAPPLY LOG10 B Note that the result of an ON operation is always a column vector but the result of a POINTS operation is a matrix of n 1 columns for a function or derivative of n variables while the result of an MAPPLY operation has both the row and column dimensions of the matrix to which the function was applied 3 8 LINEAR INTERPOLATION There are many times when you may wish to search through a matrix of data to find the data value corresponding to a given measurement or key value For
127. Englewood Cliffs NJ 1987 6 Nordsieck Arnold On numerical integration of ordinary differential equations Math of Comp 16 1962 22 49 7 Shrager Richard I Numerical integration of ordinary differential equations some methods with variable stepsize and order unpublished 1973 8 Tu Kai Wen Stability and convergence of general multi step and multi value methods with variable stepsize Dept of Computer Science Report No uiucds r 72 526 Univ of Illinois Urbana Champaign 1974 335 Appendix E INDEX The index begins on the next page Words and symbols used in MLAB commands appear in BOLD TYPEWRITER FONT 336 Index amp 64 amp gt 64 37 58 133 281 O 22 48 2 18 79 18 18 131 1s 18 174 0 Om ty 32 8 lt 30 lt 30 16 30 gt 30 gt 30 22 23 67 gt Ls 75 94 222 67 _ 16 i 88 active constraints 252 ADAMS 295 Adams Method 294 addition 18 ADJUST 142 203 aligned rectangle 98 146 ALTERNATE 106 114 190 200 alternating long and short dash line type 106 337 ampersand 64 ampersand apostrophe 64 AND 29 30 ANGLE 117 123 127 129 133 apostrophe 37 58 133 281 AQUA 125 arithmetic sequence 84 ARROW 106 arrow point type 106 ARROWTIP 106 ASCII 131 aspect ratio 103 143 203 assignment statement 7 16 47 asterisk apostrophe 79 AT 129 133 automatic scaling of axes
128. F I J IF J lt I THEN 1 ELSE O M MAPPLY F SHAPE 5 5 Although the FOR command is useful it is slow and should be avoided when possible Often there are matrix operators that can be used in place of FOR commands In general the simple FOR command has the form FOR X ME1 DO command An underscore _ may replace the equal sign The matrix expression ME1 is usually a column vector if NCOLS ME1 gt 1 then LIST ME1 is used The repeated MLAB command usually uses the index variable X in some scalar expression but this is not required Table 3 32 FOR command simple form The matrix expression need not be an arithmetic sequence The forms shown below are other examples of acceptable FOR commands FOR V LIST 2 1 0 3 3 2E4 DO command FOR KK B COL 3 DO command The repeated command can itself be a FOR command for example 89 FOR I 1 10 DO FOR J 1 10 DO A I J I J 1 In addition to the above simple form the FOR command can also be used to specify repetitive execution of a list of MLAB commands as described in Table 3 33 For example suppose F is an MLAB function via some FUNCTION command and X is a k by 1 column vector of numbers 11 72 Tk The functional values F a F Ple FP 2 F N xi obtained by iterated composition of the function F can be evaluated for alli lt k by the compound FOR command FOR J 1 N DO X F ON X TYPE J TYPE X
129. G Y G KA G T FUNCTION G X X Y A EXP X KA Y EXP X KA Y A X FUNCTION G Y X Y KA Y Y EXP X KA Y A X FUNCTION G KA X Y 1 Y EXP X KA Y A X FUNCTION G T X Y 0 Note that the variables of differentiation X Y KA and T in the four examples above may be either arguments or parameters of the function to be differentiated or may even not occur in the function The symbolic differentiation operation with respect to a given variable treats all other arguments or parameters of the function as scalar constants in accordance with the usual rules of calculus Multiple differentiations iterate the above process as shown in the example of a second order partial derivative below FUNCTION P1 X1 X2 CxX1 X1 X2 TYPE P1 X1 C FUNCTION P1 X1 C X1 X2 2 X1x X2 TYPE P1 X1 C 3 5 30 Observe that the partial derivative of P1 with respect to X1 and C was computed and numerically evaluated correctly If the scalar expression defining a function contains references to other defined functions then the chain rule will be applied to compute derivatives of this function The resulting expression will define the derivative in terms of the derivatives of other functions For example observe the following MLAB dialog LOG X Y FUNCTION U X Y U 2 T T FUNCTION V T TYPE V T FUNCTION V T T U T 2 T T 2 U X 2 T T U Y 2x T T 39 The chain rule for u t u x t y t is given by
130. GA KS BS be Se eae 197 Step function graph of a CULOS ocios ocnisas isa 198 Histogram of random daba ooo we a ee a a a 199 Shading In A tirela cda a a a 201 6 9 Shading several polygons c accs a 48 2 2 eses 202 O20 A contour map o URL eds he sd a A 205 6 11 A contour map of interpolated data e 206 6 12 A contour map of interpolated smoothed data 207 6 13 A contour map of interpolated data with spline curve contours 209 6 14 A solution to the van der Pol equation 2 0200004 212 6 15 Average monthly sunspot number top and its Fourier transform amplitude bottom 219 6 16 Shading around text s 4 66 eRe oe aa p a ag ee eee ES EE ae ee 224 7 1 Example of sum of squared error terms soosoo 228 7 2 Example of weighted sum of squared error terms e e 231 7 3 Illustration of EWT generated weights o e e 235 TA Wit of Ad and Bid using unit weights lt aoo ek ee wee ee dS cassan 249 78 Fit of A t and Bit using the EWT operator x 4 420 4 eke eee eeu ds 250 TO Grophof fie end Ais fer ea Z se ha eee eae ke Sa ee EEE So 262 7 7 Sum of squared error vs the parameter C 2 0 ee e 263 Ta Tp OO ior es IW os eee Ga iaa 264 70 Gap Fie gor eee o circa oes ee eA EE a e 266 T10 Graph ol fie Bee KI e te th Oh eR RES Se ERAS AA 269 BI Piste tr Bit Gh Sd Pike yace cc kd ek ee de em REE eee EE 287 8 2 Experimental and
131. GRATE operator returned a 7 column matrix because MLAB solves the third order differ ential equation by transforming it into three first order differential equations It introduces new unknown functions for the lower order derivatives of y say 2 1 y 1 and w x y x Then the above differential equation is equivalent to the system _Y dx dz dx dw dx with initial conditions The same resultant matrix M would have been obtained if the following MLAB commands were given FUNCTION Y X X Z FUNCTION Z X X W FUNCTION W X X Z SEC X INITIAL Y 0 0 INITIAL Z 0 0 INITIAL W 0 2 M INTEGRATE W X Z X Y X 0 1 2 XX XA XA XA XA XA 292 It is always possible to transform a given system of differential equations in canonical form i e with the highest order derivative appearing linearly into a first order system 8 4 CONTROLLING THE DIFFERENTIAL EQUATION SOLVER The special scalar quantities METHOD ERRFAC ODERPT MANDSW JACSW and DISASTERSW are used to control the MLAB procedure for the numerical solution of systems of differential equations These parameters control the options for the solution method the tolerance level the reporting of intermediate output interpolation or no interpolation Jacobian matrix calculation method and the error handling method To explain these options it is helpful to first describe briefly the operation of the differential equation solver Suppose that
132. ITIAL Y 0 1 M INTEGRATE Y T 0 3 5 297 8 5 AN EXAMPLE OF CURVE FITTING A DIFFERENTIAL EQUATION Like other mathematical models systems of differential equations may contain scalar parameters with unknown values The MLAB FIT command can then be used to search for parameter values that minimize the weighted sum of squared error between functions defined by the differential equation model and the experimental data Models defined by systems of differential equations should be treated like nonlinear functional form models The problems given in Section 7 14 of Chapter 7 may also arise for differential equation models As an example of the FIT command applied to a differential equation system consider Legendre s equation cy 2a ae a dx 1 22 Assume an MLAB matrix YD exists that represents experimental measurements of a function y x approximately equal to a solution of Legendre s equation The initial condition y 0 0 is known but only rough estimates of y 0 and the parameter c are available The following MLAB dialog adjusts the parameters c and y 0 so that the corresponding Legendre s equation solution is close to the experimental measurements YD COL 1 0 5 05 YD COL 2 LIST 0O 02179195 04390796 06505254 08783066 1095252 1308729 1501906 1703068 1905134 2105629 FUNCTION Y X X 2 X Y X C C 1 Y 1 X X INITIAL Y 0 0 INITIAL Y X 0 K C 1 K 1
133. It is beyond the scope of this Beginner s Guide to describe the methodology and mathematical theory associated with the differential equation solver Further discussion and references may be found in the bibliography As previously noted the tolerance level only controls the error of individual steps and it is possible that much larger errors may accumulate over many steps Unfortunately the analysis of numerical and algorithmic error in solving a system of differential equations involves difficult mathematical problems of the stability or instability of such systems The MLAB algorithms have no provision for computing overall estimates of error occurring during the numerical solution of differential equation systems Perhaps the user will have available an error analysis for the system under consideration If not varying the tolerance level perhaps dividing by 10 and recomputing one or more times can be tried If there are large discrepancies between the numerical solutions at different tolerance levels one should conclude that there is too much instability to rely upon any numerical solution Other obvious warnings are given by MLAB messages for inability to maintain the tolerance level singularities arithmetic overflow division by zero exponentiation underflow or overflow and so on during the numerical solution of the system as illustrated in the MLAB dialog FUNCTION Y T T 1 IF T lt 1 THEN 5 ELSE IF T lt 2 THEN 5 ELSE 0 IN
134. J E o 1 B Figure 6 10 A contour map of f z y 205 Roe XA XA A X gt gt XK XA XA XA XX k 1 2 0 4 0 4 1 2 S NN AN NN D A 3 a e 2 6667 1 6 563332 533234 1 6 2 6667 Figure 6 11 A contour map of interpolated data 1 2 1 DA COL 3 LIST 7 04 8 6 2 1 2 5 1 3 3 55 13 48 1 1 3 5 5 3 22 4 3 2 35 8 96 2 74 DAI INTERPOLATE DA CROSS 2 2 5 2 2 5 DRAW C1 CONTOUR DAI 9 9 5 LINETYPE VMARKER DRAW C2 DA COL 1 2 LINETYPE NONE POINTTYPE CROSSPT WINDOW ADJUST WMATCH VIEW DELETE W DAS SMOOTH DA DAI INTERPOLATE DAS CROSS 2 2 5 2 2 5 DRAW C1 CONTOUR DAI 9 9 5 LINETYPE VMARKER DRAW C2 DA COL 1 2 LINETYPE NONE POINTTYPE CROSSPT WINDOW ADJUST WMATCH VIEW 1 2 AA NN 2 6667 Figure 6 12 A contour map of interpolated smoothed data 207 The resulting contour maps are shown in Figure 6 11 and Figure 6 12 Note the regularity gained by smoothing the data in the second example Notice also that the interpolation yielded much greater detail in the areas near which the original data points were located You can see this more easily by entering the command DRAW C2 COLOR 2 after drawing Figure 6 11 The contours in the contour map shown in Figure 6 11 can also be smoothed by using the SVMARKER line type The SVMARKER line type draws variable dashed lines with marker skipping using spline curves The flatness of the spli
135. ME2 in clause F TO ME1 WITH WEIGHT ME2 should be a column vector with the same number of rows as ME1 The clauses may be abbreviated by omitting the weight specification WITH WEIGHT ME2 the default weight vector is a column of ones The word WEIGHT may be abbreviated to WT as usual The FIT command for a system of differential equations causes an itera tive search for the values of the fitted parameters that minimizes the weighted sum of squared error just as for FIT commands using functional forms The typed response to a FIT command for a system of differential equations is the same as for a FIT command for curve fitting a functional form Table 8 4 FIT command differential equations systems continued 313 Appendix A FONT TABLES The tables on the following pages show all of the printable characters in each of the thirty four Hershey fonts specified by font numbers equal to postitive integers 1 2 34 and in each of the thirteen TrueType fonts specified by font numbers equal to the negative integers 2 3 14 Note TrueType fonts are supported by MSWindows and Macintosh displays and PSL PSP PSCL and PSCP PostScript printer devices However they are not supported by Linux with X Windows displays or LJ2L and LJ2P Hewlett Packard printer devices 314 8 9 10 11 12 13 14 J lt O nO GD w w ate a B lt 0 70 o JM J NOOIF UME Y
136. MLAB Beginner s Guide by Daniel Kerner Revision Date January 17 2014 Civilized Software Inc 12109 Heritage Park Circle Silver Spring MD 20906 USA Tel 301 962 3711 Email csi civilized com URL http www civilized com aT IN NM p AN AN l ue M ji INN XN N Y ih Na Y SN MI NANNININ jo LO Je y NE NES MORE NS ae tis PREFACE MLAB Beginner s Guide is an updated version of the book of the same name written by George A Hutchinson and George Atta and revised by Gary Knott and Bruce Krulwich in the early 1980 s at the National Institutes of Health Division of Computer Research and Technology Bethesda MD The original edition of this book provided an introduction to using the MLAB mathematical modeling program on a DEC PDP 10 mainframe computer This book provides an introduction to using MLAB on Microsoft DOS Windows MSWindows Apple Macintosh OS7 8 9 X and Linux with X Windows computers Chapter and section names in the Table of Contents figure numbers in the text and List of Figures table numbers in the text and List of Tables and page numbers in the index appear in the color blue and provide clickable links to the respective content If the reader should find errors in this text please send an email to kerner civilized com with a description of the error so that corrections can be made to fu
137. MLAB commands to create data items or change existing data items of the twelve possible types by referring to identifiers that one has selected 15 MLAB scalars and matrix entries are real numbers with a precision of approximately sixteen digits A nonzero MLAB scalar or matrix entry must have a magnitude approximately in the range 2 225 x 107808 to 1 798 x 109 MLAB constants may be written in ordinary decimal notation for example 43 76 12 or 003 Scientific notation may also be used where powers of 10 are preceeded by the capital letter E as in the following examples 6 3E23 for 6 3 x 10 2E 5 for 2 x 107 The number following the E must be an integer between 308 and 308 Scalar constants are the simplest type of scalar expressions A scalar expression may be roughly described as an algebraic or symbolic description of a number The MLAB user constructs scalar expressions to describe the numerical calculations that are to be performed via MLAB 2 2 SCALAR ASSIGNMENT STATEMENTS AND SCALAR EXPRESSIONS The MLAB assignment command is the most important method for making identifiers correspond to scalars or matrices in MLAB For example the MLAB assignment commands A 003 B 4 in the tutorial section of Chapter 1 caused the unknown identifiers A and B to become scalar identifiers with the values 003 and 4 respectively Blanks to the left or right of the equal sign have no effect and are optional The scala
138. N MLAB 280 8 1 A FIRST ORDER DIFFERENTIAL EQUATION SOLVED IN MLAB 280 8 2 COUPLED FIRST ORDER DIFFERENTIAL EQUATIONS SOLVED IN MLAB 283 8 3 HIGHER ORDER DIFFERENTIAL EQUATION SOLVING IN MLAB 290 8 4 CONTROLLING THE DIFFERENTIAL EQUATION SOLVER 293 8 5 AN EXAMPLE OF CURVE FITTING A DIFFERENTIAL EQUATION 298 8 6 AN EXAMPLE OF A BOUNDARY VALUE DIFFERENTIAL EQUATION 300 8 7 THE FIT COMMAND FOR DIFFERENTIAL EQUATIONS 301 NA SUMMARY siseses rya ee Ad be Saw ee a p a 302 8 9 EXERCISED 24 2p eg rdsu die g a Sea a RE bee SE a 307 A FONT TABLES B LIST OF TABLES C LIST OF FIGURES D BIBLIOGRAPHY E INDEX vi 314 326 331 335 336 Chapter 1 A TUTORIAL MLAB SESSION In this first chapter we will go through a simple MLAB session from beginning to end In order to show many different aspects of MLAB only brief explanations of commands are given MLAB can run on many different types of computers Initially of course you must carry out the appropriate steps needed to install MLAB on your computer These steps are detailed in the installation instructions provided with the MLAB distribution 1 1 THE TUTORIAL If the operating system on your computer has a graphical user interface i e MSWindows or Macintosh double click on the MLAB icon on the Desktop to begin the MLAB session If the operating system on your computer has a command line interface Linux Unix or DOS type the co
139. O 1 0 TO 5 IN W VIEW W W1 Figure 6 15 shows the result Notice that the data consists of 1020 t x t pairs and 1020 is not a power of 2 Therefore the matrix returned by REALDFT has 257 rows where 257 is 1 2l 0821920 1_ To convert the frequencies in column 1 of the matrix returned by REALDFT to periods elements of that column 219 were replaced by their reciprocals However to avoid a zero divide the first row of the REALDFT result was deleted The amplitude for that 0 frequency component represents the average value about which the function oscillates i e its offset and has a value of 59 2 The REALDFT operator also applies to samples of periodic functions of two arguments When deconvolution or transfer function calculation is the motive for using the Fourier transform the complex valued form is desirable The complex valued discrete Fourier transform and its inverse are produced by the MLAB function DFT and IDFT respectively DFT operates upon a matrix M which specifies n NROWS M equally spaced sample values of a complex valued function f The result is a 3 column matrix which is the complex valued Fourier transform Convolution is an operation on a pair of functions f and g which yields a third function r defined by r x Fg zdz When f and g are defined only on the positive integers we have the discrete convolution Me ll m r i gt fG gli j 1 fori gt 1 J In this case we may think
140. OTS S Ti AND S Ti as 1 75 T 2 90473751 In DO files any comments can be put between quotation marks and followed by a semicolon These comments have no effect upon the execution of the DO file and are not displayed Comments may also be bracketed by and as in the C programming language This section concludes with some general advice on designing and testing large DO files The methods below are not intended to be exact rules but guides for a coherent approach To design DO files 1 Keep in mind the DO file objective and design MLAB data items scalars matrices windows and curves which will be required to generate the wanted numerical results or pictures 2 Analyze the data items such as scalars MLAB functions data matrices and partially completed windows which will be required as input in order to create the final MLAB data items that are wanted Design a strategy for input to the DO file whether data items are to be created before executing the DO file or during execution by means of DO CON commands and whether saved data items such as windows are to be obtained by USE commands assignment commands or other MLAB input commands 3 Design the computational and output procedures that is the MLAB commands and ex pressions which will compute the desired MLAB data items using the input data items Use conditional commands IF THEN ELSE commands where required to treat alternatives Develop expressions in full generality
141. PE M M a 6 by 2 matrix 1 0 5 2 1 1 97155402 3 2 874696273 4 3 449159263 5 4 273267675 6 5 193620648 A third method uses the INTEGRATE operator to invoke the differential equation solver to produce a numerical solution Note that INTEGRATE should not be confused with the scalar operator INTEGRAL described in Section 2 6 The use of INTEGRATE is shown in the following dialog M INTEGRATE Y X 0 5 TYPE M M a 6 by 3 matrix 1 0 5 5 2 1 1 97155402 1 72155777 3 2 874696273 652474248 4 3 449159263 261659281 5 4 273267675 113267675 6 5 193620648 5 47317594E 2 The first argument s to INTEGRATE is are the name s of the differential equation s in the system In this example the single name Y X specifies the single corresponding differential equation The last argument is a column vector of values for the independent variable X Column 1 of the matrix M obtained from the INTEGRATE operator contains this column vector which here is 0 5 Of course one would usually solve for more than the six values in this example in order to obtain an acceptable graph of the function y x Column 2 of the matrix obtained from the INTEGRATE operator above contains the corresponding values for Y and column 3 contains the corresponding values for Y X That is M COL 2 is like Y ON 0 5 and M COL 3 is like Y X ON 0 5 More precisely M I 2 is the value of Y M I 1 and M 1 3 is the value of Y X M
142. PRINT mat will create the file Users csi mat 1st 157 e on Macintosh OS7 8 9 systems if FILEDIR Disk MLAB for Macintosh kinetics then the MLAB command PRINT mat will create the file Disk MLAB for Macintosh kinetics mat lst If FILEDIR is the empty string the current directory is used Note the special pattern may not be used on Linux Unix or Macintosh OSX systems when defining FILEDIR MLAB has no understanding of this shorthand notation Also note in the examples above the character used to separate directory names and filenames differs according to the system on which MLAB is running On DOS and MSWindows systems the character is backslash on Linux Unix and Macintosh OSX the character is slash and on Macintosh OS7 8 9 the character is colon If the MLAB control string DOFILEDIR is set then it will be used instead of FILEDIR to determine the directory where MLAB do files will be sought This does not affect the automatic execution of the start up do file mlabinit do The various types of files accessed or created by MLAB are discussed below in more detail e Log files mlab log mlab0 log When MLAB is run it creates a log file in your current directory This log file is named MLAB LOG During the MLAB session all MLAB commands that are typed in and all MLAB responses that are displayed are entered into this disk file However the pictures on the graphical display are not entered into the log file These
143. RAL operators for numerical calculation in MLAB For example suppose f n log n is to be computed by means of the equivalent indexed sum do log j j l Then this function can be defined and evaluated as shown in the following MLAB dialog FUNCTION F N SUM J 1 N LOG J TYPE EXP F 6 F 40 720 110 32064 The function f n log n can also be computed by means of the equivalent indexed product is log 5 gl This function can be defined and evaluated as FUNCTION F N LOG PRODUCT J 1 N J 33 Redefining F TYPE EXP F 6 F 40 720 110 32064 Similarly the integral I x exp t dt 0 can be defined and evaluated as shown in the following MLAB dialog FUNCTION I X INTEGRAL T 0 X EXP T T TYPE I 2 I 1 5 I 1 882076325 109359866 The SUM PRODUCT and INTEGRAL operations summarized in Table 2 13 are permitted only in FUNCTION commands although they may be used indirectly in assignment commands or other MLAB commands by reference to functions in which they appear For example suppose a x and b x are functions of one variable f x y is a function of two variables and A B and F are the corresponding function identifiers defined by MLAB FUNCTION commands Then the definitions b n Y f n j a n b n IT fa j a n Q 2 Il Il where a n and b n are integers and the definition b x h x f o Oud can be express
144. S in element 1 1 to indicate marker skipping Notice also that the FILL operator assumes that you are drawing closed polygons and will add a final additional side to close each polygon if necessary 202 With close scrutiny the reader will notice that the angle of the fill lines in Figure 6 9 are not exactly 135 degrees This is because the FILL operator when calculating the fill line endpoints does not know in what window the fill lines will be drawn and thus does not know the relation between the image and user window coordinates The angle at which the fill lines are drawn is perfect only when the user window is a square or is in proportion with a user specified image window Use the command WINDOW ADJUST WMATCH to scale the window axes so that a unit of length in the x direction equals a unit of length in the y direction A crosshatching or grid effect can be accomplished using fill lines at right angles to each other Note that the angle adjustments described in the preceding paragraph are necessary to have this done properly The general form of the FILL operator is shown in Table 6 3 The general form of the FILL matrix operator is given by FILL ME1 SE1 SE2 where ME1 is a matrix containing the polygons to be shaded SE1 is the number of fill lines to be generated and SE2 is the angle in degrees of the fill lines The matrix returned contains pairs of fill line endpoints and should be drawn using linetype ALTERNATE
145. TION F X 10 EXP X SIN X POINTS F 0 10 2 M 0 NROWS M CRAN ON N RAN ON N COL 2 M COL 2 E 4 DRAW G DRAW M LINETYPE NONE POINTTYPE FUNCTION R X SQRT 1 X N R ON EWT M G COL 2 G COL 2 N DRAW G LINETYPE DASHED G COL 2 G COL 2 N N DRAW G LINETYPE DASHED VIEW 2z mZ QZ XA XA XA A XX XA XA XA XA A k The operator RAN in the above command sequence generates a uniform pseudo random number strictly between zero and one The simulated data dots the function F solid line and the function F plus and minus standard deviations obtained from EWT generated weights dashed lines are plotted in Figure 7 3 A look at this figure suggests that the EWT generated weights are quite reasonable 234 11 5 8 3 2 0 6 2 Figure 7 3 Illustration of EWT generated weights 235 7 4 FUNCTIONAL FORMS OF SEVERAL VARIABLES The examples given above use functions of one variable to model data consisting of two variables one independent and the other dependent It is also possible to model data consisting of n 1 variables n independent and 1 dependent by a function of n variables For example suppose three variables x y and v have been measured in each repetition of an experiment It is predicted that v should equal the average x y 2 of the independent variables x and y The model is then v f x y 1 y 2 Suppose the data xi yi vi and assigned we
146. VMARKER 106 204 208 systems of first order differential equations 283 tabulating function values 83 tangent 19 tangent bar point type 106 TBAR 106 terminating an MLAB session 13 third order differential equation 290 tick marks 106 126 128 TITLE 11 128 133 143 148 title 15 98 title angle 129 133 title color 129 133 134 title font 129 133 title identifiers 131 title modifications 134 title names 131 title offset 133 title place 133 134 title position 129 133 title size 129 133 TITLES 141 titles 128 138 titles for graphs 128 TO 99 100 TOLSOS 251 TOP 122 133 134 143 TPOLAR 191 transfer function calculation 220 transforming a system of differential equations to canonical form 293 TRIANGLE 106 triangle point type 106 trigonometric functions 19 trigonometric model 258 TrueType fonts 326 TURQUOISE 125 tutorial 1 TYPE 5 22 24 26 40 72 141 148 166 170 281 TYPEOUT 31 32 typing out matrices 59 73 UBAND 106 UNBLANK 138 139 148 underflow 18 uniform 0 1 random numbers 20 unit aspect ratio 143 203 unknown identifier 15 UNVIEW 7 96 138 147 upward tick mark 106 USE 13 160 168 user clipping rectangle 98 103 129 143 146 user rectangle 98 103 129 143 146 UTICK 106 128 van der Pol oscillator 210 variable dashed line with marker skipping line type 106 110 204 variable dashed line with m
147. XIS command The first difference is that the points line segments and labels of a curve that extend beyond the user rectangle of a window are not visible in the window s image rectangle However the points line segments and labels of an axis can appear anywhere in the window s frame rectangle The second difference is that if a window s user rectangle is redefined a curve data item is not changed its curve and label matrices remain the same But an axis data item is changed when a window s user rectangle is redefined the axis s curve matrix is rescaled and if the axis is an x or y axis its label matrices are rescaled so that they are appropriate for the new user rectangle For example if the user rectangle for window W shown in Figure 4 2 is changed by the command 126 Figure 4 9 Figure 4 2 after the user rectangle is redefined WINDOW 1 5 TO 5 300 TO 600 IN W Figure 4 9 results Note that the labels of curve C and the line segments connecting the points of curve C have not changed in the user coordinate system however since the rectangle in the user coordinate system that is visible in the window s image has changed parts of curve C have been clipped and are no longer visible Also note that the labels and curve matrix points on the z axis and y axis have changed For example the axis Y1 in Figure 4 2 consisted of the 7 points 2 300 2 0 2 300 2 600 2 900 2 1200 and 2 1500 labelled with
148. Xx X A 3 7 B 1 5 C 2 Thus the quadratic polynomial f z 3 7 1 5 2 2 4 graphed in Figure 7 1 and Figure 7 2 is now defined as F in MLAB Notice however that param eters A B and C are used for the coefficients of F rather than simply defining F with constant coefficients So F can now be regarded as defining the functional form of the quadratic polynomial f Then the functional form is the main definition for the model and the scalar values assigned to A B and C may now be regarded as initial estimates for the coefficients Since f 0 a and f 1 b 2c the following MLAB dialog can be used to set up these conditions as linear constraints in a constraint set named Z1 CONSTRAINTS Z1 A gt 1 B 2xC lt 1 TYPE Z1 constraints Z1 A gt 1 B 2 C lt 1 The MLAB definition of the desired model has now been completed To initiate the curve fitting process using the data prepared in Section 7 8 the FIT command of MLAB is used as shown in the following dialog 244 FIT A B C F TO D WITH WEIGHT V CONSTRAINTS Z1 final parameter values value error dependency parameter 3 8935241474 0 9319706332 0 9655163403 A 1 4034884111 0 8666083211 0 9942652507 B 0 1583819796 0 1711160115 0 9874324868 C 2 iterations CONVERGED best weighted sum of squares 2 605776e 01 weighted root mean square error 3 609554e 01 weighted deviation fraction 1 256144e 01 R squared 9 158419e 01 no active con
149. Y NY AIC ar Figure A 6 ASCII characters 80 to 95 in Hershey fonts 1 to 34 320 905540000000000000073 95000 000000 CRERCZAOCACEAEZAS2LeLE eee KM ceeces sesertesG Chess nxnkEScceees IC EEEEESE A d Wee et ee ee ey aa ee Lvi sx Nxs MY NY LPL RR wie YF AN o Nea mea A Ds SA A As dro a A lt gt a 0 m TIN DOR ee NNS Ae Asci 96 97 98 99 100101 102103104 105106 107 108109110111 Font b c d e q J m m CectclTGcGe gre eee ec eee poe A oe S Leeces o MY 00 WDMWDA ASADO AA y 00500000 us gol Gg yb AE AAA IAS VHOUNOWODYDOYOOLINHHVOHOS dH4ee Ou OLODO Dio uT LTAT OTT TMS oo BTT BTV Vaas Ho oO BDUBBBYB OX UROQOVOVDOX amp X ROOCO OO UL me 000000 Am oO VAMALe2MOAMVWwMwOo0O8 naD O PASA AAA SSmaLCOLBeomoatongdsass snd sox x GHNE pe ea a ce ea eee ees H ORRIN OANM DwWUNONO NUN DwIUNDONDO NM ANO TT 100 MO 03433393 AA 9 A NY NY NY OY NY NANNY OU Ao od oa AMO Figure A 7 ASCII characters 96 to 111 in Hershey fonts 1 to 34 321 ascid12113114115116117118119120121122123124125126 127 Fontp q lt O lt gt 4 l lene EEES AEREA AEREA AAA A A A 5 SS As A A a A A Na NN NNNNNNNNAa As N NN Rit Aho ANN gt TL DH RAR RAS SSS DDD mas DAL Gl SR mRNA Kae X X XXH x MO la BBR BM we a De E pE e X X 2332313 E gt FE R033333231388323H L013 323223 e Sie eS pa 333445950 thliossdann S
150. a 11 by 2 matrix 1 0 0 2 0 05 021987713 3 0 1 4 39223598E 2 4 0 15 6 57496541E 2 5 0 2 8 74127844E 2 6 0 25 108850927 7 0 3 129997448 8 0 35 150777641 9 0 4 171105703 10 0 45 190880623 11 0 5 209980237 YD a 11 by 2 matrix 1 0 0 2 0 05 0 02179195 3 0 1 0 04390796 4 0 15 0 06505254 Br O22 0 08783066 6 0 25 0 1095252 7 0 3 0 1308729 8 0 35 0 1501906 9 0 4 0 1703068 10 0 45 0 1905134 11 0 5 0 2105629 299 D 22 0 176 0 132 0 088 0 044 Figure 8 2 Experimental and fitted values of Legendre s equation DRAW YC DRAW YD LT NONE PT CIRCLE BOTTOM TITLE X LEFT TITLE Y VIEW XX XA XX XX Figure 8 2 and a comparison of the second columns of YD with YC show that there is close agreement between the data and the fitted model 8 6 AN EXAMPLE OF A BOUNDARY VALUE DIFFEREN TIAL EQUATION Numerical solutions of boundary value problems can be generated by curve fitting parameters in the INITIAL commands This is often called the shooting method For example consider the differential equation problem 300 dy dy aY ay Y_ 1 dx dx e with the boundary conditions y 0 3 and y 1 4 One initial condition is y 0 3 but the other initial condition y 0 is not available However there is a condition for y 1 To solve numerically a parametric initial condition can be set up say y 0 k and then k can be fit so that y passes through the point 1
151. a linear constraint determines a part of the boundary of the region of permissible assignments of numerical values to the fitted parameters and a constraint is active if the currently assigned values are located near this part of the boundary In particular constraint equations are always active but constraint inequalities may be active or not during different iterations The Lagrange multiplier of a constraint is zero if a constraint is inactive and will usually be nonzero when the constraint is active If a set of linear constraints has been specified for a FIT command then the output for each iteration and for the final summary will contain a Lagrange multiplier for each constraint given in the same order as in the constraint set list If the Lagrange multiplier is x for a constraint equation r by t ro bo p bp ro then the sign of x indicates the direction in which the search process would move if the equation were deleted from the model Specifically x gt O ifthe search process would require increased ro x lt 0 ifthe search process would require decreased ro 252 Because the Lagrange multiplier is not dimensionless the magnitude of this quantity is not a simple estimate of the influence of the corresponding constraint on the search process If a FIT command terminates with the final result NOT CONVERGED then the parameter values after the final iteration can be used as the initial values for another FIT command That
152. action units Figure 4 8 d was drawn by the command DRAW CX FORMAT 2 5 0 3 0 1 LABELSIZE 1 IN W VIEW Here the FORMAT clause changes the format of the labels and the LABELSIZE clause increases their size The clause ANGLE SE12 specifies the angle in degrees of the baseline on which the label is drawn with respect to the x axis For example ANGLE 90 would cause the label to be printed on a line parallel to the y axis and directed toward the top of the window 123 y la 0 10001 d 1 2 200E0 4 a 12400E0 5 A b c Figure 4 8 Examples of point labeling Color name Linux with X Windows DOS Color No MSWindows Macintosh Color No WHITE if 1 BLACK 0 0 GREY 43 2185 RED 49 3841 GREEN 13 241 YELLOW 61 4081 BLUE 4 16 VIOLET 36 2351 PURPLE 36 2351 ORANGE 57 4001 PINK 59 4029 AQUA 32 1006 CHARTREUSE 45 2289 BROWN 37 2115 ROSE 55 3947 TURQUOISE 31 1245 Table 4 8 Color names and numbers The clause COLOR col specifies the color of line type point type and label given in the DRAW command for color graphics displays and plotting devices col may be a number or a name of a color The color numbers and corresponding color names are listed in Table 4 8 If a COLOR clause is not included in a DRAW command the default color is 1 i e WHITE on Linux Unix with X Windows MSWindows Macintosh OS X and DOS systems the default color is 0 i e BLACK on Macintosh OS7 8 9 and
153. ade into MLAB scalars before curve fitting For example the commands C1 1 C2 1 A 07 B 2 16 would accomplish the second step of the model setup for the functional form F above by assigning numerical values to all the parameters of F Similarly P must be made an MLAB scalar before curve fitting the LP model above Failure to assign numerical values to parameters in a functional form is a common error Some guesses must be made no matter how ill prepared one is to do so For formulating models having several functional forms to be fit simultaneously the first step is accomplished by transcribing each functional form into an MLAB FUNCTION command For example the model in Section 7 5 is set up as shown in the commands FUNCTION X T AO K1 K2 K1 EXP K1 T EXP K24T FUNCTION B T AOx 1 0 EXP K1 T X T with the argument symbol T unknown parameters K1 and K2 and the fixed parameter AO The second step is accomplished by assigning appropriate numerical values to AO K1 and K2 For example AO 200 Ki 05 K2 0001 The third step is accomplished if necessary by creating a constraint set For example CONSTRAINTS CS K1 gt 0 K2 gt O Using the correct strategy in defining complex functional forms is very important Take for example the equation B t S R Bo R S Bo e Ex R Bo S Bo eo HE where R A d 2 S A d 2 d 42 4 Fy Go 2
154. adjustment codes WNICE WNICE WNICE WNICE frame color WHITE 0 NO FRAMEBOX frame in screen fraction units width height 1 x 0 to 1 y 0 to 1 frame in screen inches x 0 to 4 y 0 to 3 image color WHITE 0 IMAGEBOX color BLACK 1 image in frame inches x 0 7 to 3 7 y 0 375 to 2 625 image in frame fraction units x 0 175 to 0 925 y 0 125 to 0 875 image in screen fraction units x 0 175 to 0 925 y 0 125 to 0 875 image in screen inches x 0 7 to 3 7 y 0 375 to 2 625 CURVES E D C AXES W XAXIS W YAXIS TITLES T DRAW and TITLE commands can be used to add and modify curves and titles in the default window W just as for any other window The user frame or image rectangle of W can also be changed by the WINDOW FRAME or IMAGE commands Indeed a user can define W as desired the MLAB supplied default window is only used if no window called W is present 146 4 9 SUMMARY In the following let SE1 SE2 denote arbitrary scalar expressions ME1 ME2 denote arbitrary matrix expressions U denote an unknown identifier C denote a curve identifier T denote a title identifier A denote an axis identifier and W denote a window identifier 1 MLAB two dimensional graphical data types a window user rectangle frame rectangle image rectangle framebox imagebox list of curves list of axes list of titles b curve curve matrix label matrix pointtype linetype c axis curve matrix label matrix pointt
155. after solving is com plete This includes the number of steps taken and the number of derivative function evaluations and Jacobian evaluations The default value is 0 e MANDSW SE4 Specifies how result matrix is obtained from solution matrix the value 0 means interpolate to get result matrix from solution matrix a value 0 means adjust step size so that requested values of independent variable are included in the solution matrix The default value is 0 e JACSW SE5 Specifies how partial derivatives for the Jacobian matrix are to be computed NUMERICAL 0 means that numerical partial derivatives are used SYMBOLIC 1 means that symbolic partial derivatives are used CONSTJAC 2 means that partial derivatives are computed numerically only once The default value is 0 e DISASTERSW SE6 Specifies how the integrator should continue if truncation or round off errors are too large the value 2 means do not report errors and increase tolerance indefinitely as required the value 1 means do not report errors and increase tolerance as required If the internal value of ERRFAC becomes greater than 1 2 then terminate the integration the value 0 means report errors and increase tolerance as required If the inter nal value of ERRFAC becomes greater than 1 2 then terminate the integration the value 1 means report errors and increase tolerance five times If it would be necessar
156. ain sensitive differential equations although the user suffers a penalty of increased computer memory requirements and usually also an increase in computation time MANDSW is FALSE 0 unless specified otherwise by the user No known method for solving systems of differential equations will work well for all problems In MLAB there are four available methods which provide state of the art performance with respect to a wide variety of problems encountered in applications e Adams Method e Gear Method Gear Shrager Method mixed Adams Gear Combination Method The value of the control variable METHOD determines which method is to be used The Adams method which is selected by setting METHOD equal to ADAMS ADAMS is the value 1 performs well for a wide variety of problems but poorly for systems of stiff differential equations Stiff systems tend to involve widely varying magnitudes for the system derivatives For example in a situation where a toxin is being released very slowly from fat into the bloodstream in a period of weeks while 294 electrolyte balances in the blood are changing from minute to minute the model is very likely to lead to a stiff system The Gear method is an algorithm appropriate for the situation of stiff differential equations MLAB has two variants of the Gear method implemented in the differential equation solver In the first variant which is selected by setting METHOD equal to GEAR 3 at each step a cubi
157. al discrete Fourier transform of the interpolated function values and returns a matrix with 3 columns and nn 2 1 rows Note that if n is not a power of 2 some information may be lost in the process of resampling zx For example given the samples of average monthly relative sunspot numbers from 1900 to 1985 available from Appendix 1 of S Lawrence Marple Jr s Digital Spectral Analysis with Ap plications Englewood Cliffs New Jersey Prentice Hall Inc 1987 pp 474 475 in a file called SLM DAT the following MLAB dialog draws the data and the amplitude of its Fourier transform versus period so that one can see that the primary sunspot cycle is about 10 or 11 years for that interval RESET M READ SLM DAT 1020 2 M SORT M 1 TYPE M ROW 1 5 XX XA a 5 by 2 matrix 1900 9 4 1900 08333 13 6 1900 16667 8 6 1900 25 16 1900 33333 15 2 OPWNF P REALDFT M NROWS P 257 TYPE P ROW 1 5 a 5 by 3 matrix 0 59 252449 0 Le 1 2 17651765E 2 23 0381617 369664928 218 260 208 156 AMANDA 1900 1918 1936 1954 1972 1990 55 44 33 22 11 Figure 6 15 Average monthly sunspot number top and its Fourier transform amplitude bottom 3 023530353 2 52949674 1 05383481 4 3 52955295E 2 10 9995289 495864914 5 047060706 11 8243895 1 31285923 DELETE P ROW 1 P COL 1 1 P COL 1 DRAW M Wi W FRAME O TO 1 5 TO 1 IN W1 DRAW P COL 1 2 FRAME O T
158. al to that of DFT but returns the in verse discrete Fourier transform so IDFT DFT ME1 ME1 REALDFT ME1 takes a matrix holding values of a real valued func tion F and computes the amplitude and phase spectrum of F according to the formula given at the beginning of this section Table 6 9 Discrete Fourier transform operators Matrix smoothing is a SMOOTH ME1 A step function of a matrix is a HISTO ME1 The fill lines to shade a matrix are a FILL ME1 SE1 SE2 A contour map of a surface matrix is a CONTOUR ME1 ME2 Eigenvalues and eigenvectors operator 222 a EIGEN ME1 8 Complex number arithmetic is a CPROD ME1 ME2 b CDIV ME1 ME2 c CPOW ME1 ME2 9 Singular value decomposition of a matrix is a SVD ME1 10 Discrete Fourier transforms are a DFT ME1 b IDFT ME1 c REALDFT ME1 6 10 EXERCISES 1 How well can interpolation and smoothing imitate curve fitting Find the answer to this question by drawing the data from Section 1 1 along with a smoothed interpolated curve Try this twice the first time by interpolating and then smoothing and the second time by smoothing and then interpolating 2 Integral calculus teaches the concept of an integral by approximating the area under a curve using Riemann sums How close does a step function drawn with the HISTO operator approximate integration Using a sine wave ranging from 0 to 7 calculate the area un
159. am of your initials using a matrix constructed from the given matrices Make any assumptions needed about the MLAB window to be used for the display Hint Be careful that the matrix constructed for LINETYPE 6 drawing has consistent marker values Use the string modification commands listed in Table 4 11 to draw the following equations in an MLAB graphics window a sin 0 b y aoz aj c f 3x dx a Draw cos x versus x at 100 points with x in the interval 0 7 4 Define the window so that it has unit aspect ratio Draw tan x versus x at 100 points in the same window What are the coordinates of the point where the two curves intersect Use the ROOT operator to refine the precision of the coordinates of the point of intersection 153 Chapter 5 INPUT AND OUTPUT FOR MLAB Digital computers perform computations by means of a central processor which can access com puter memory A typical computer task involves input operations computational operations and output operations Input operations occur when data or commands are transmitted to the com puter memory In the computational process commands in computer memory are interpreted operations are performed on data in computer memory and the results are placed in computer memory Then output operations transmit these results from computer memory to the computer display or to some other computer output or storage device Much of MLAB input and output is directly betwee
160. amp amp ME1 n fold repeti tion For any number SE1 the expression is equivalent to ME1 77 INT SE1 Table 3 19 Matrix column replication operator 68 value See Table 3 20 The matrix dimensions are computed so that the resulting operation makes sense For example observe the following MLAB dialog TYPE 6 amp 2 3 amp 1 a 2 by 3 matrix 2 1 3 1 1 6 2 6 TYPE 1774 a 1 by 4 matrix 1 1 1 1 1 TYPE 9 amp LIST 3 2 4 a 2 by 3 matrix 1 9 9 9 22 3 2 4 The following general forms of matrix joining and matrix repeating operations are evaluated as indicated SE1 amp SE2 a 2 by 1 column vector with the first element equal to SE1 and the second element equal to SE2 SE1 amp SE2 a 1 by 2 row vector with the first element equal to SE1 and the second element equal to SE2 SE1 SE2 an n by 1 column vector with all elements equal to SE1 where n INT SE2 gt 1 SE1 amp ME1 equivalent to SE1 NCOLS ME1 ME1 ME1 amp SE1 equivalent to ME1 amp SE1 NCOLS ME1 SE1 amp ME1 equivalent to SE1 NROWS ME1 ME1 ME1 amp SE1 equivalent to ME1 gt SE1 NROWS ME1 Scalars are treated like constant row vectors for ampersands and like constant column vectors for ampersand apostrophes Table 3 20 Matrix joining and repeating operator evaluation A constant mat
161. and TITLE QT Here is a quotation mark ASCII 34 FONT 5 AT 2 1 43 WORLD IN W will cause the title QT to have the text Here is a quotation mark Continuing a title onto a second line also requires special treatment It requires that the title string contain an embedded control character for the line feed and carriage return at the end of the first line The MLAB commands DELETE W WINDOW O TO 10 O TO 10 IN W DRAW SHAPE 2 2 LIST 0 6 10 7 IN W LT DASHED DRAW SHAPE 2 2 LIST 0 7 10 8 IN W TITLE TD Experimental solid line MControl dashed line AT 1 5 WORLD IN W VIEW XA XA XA XX X 131 Experimental solid line Control dashed line Figure 4 11 Examples of a multiple line text 132 provide an example of multiple line titles The two lines of text corresponding to the title TD are shown in Figure 4 11 Note that the apostrophe and the letter M in the TITLE command s quoted string were interpreted as embedded control characters and caused the characters immediately following the M to be positioned below the first character of the title s string The general form of the TITLE command is TITLE T text IN W SIZE SE1 UNITS ANGLE SE2 FONT SE3 AT SE4 SE5 UNITS SHIFT SE6 COLOR col PLACE A B Table 4 10 TITLE command The text string is typed between quotation marks as shown above It can also be coded that is have special codes in it that
162. and W is a window with default user frame and image rectangles and the following MLAB commands are executed WINDOW ADJUST WFIX TITLE HL AT 6 4 WORLD DRAW LIST 7 4 LABEL LIST HL LABELSIZE 02 FFRACT What would be the effect on the displayed picture 4 Let a function F be given by F x log x k Run MLAB and give commands to accomplish the following objectives e Express F as an MLAB function with a parameter k e For parameter values k 1 2 3 4 5 draw the graph of F from 0 to 4 in the default window W Below each curve at the value x 2 label the curve with the inscription k and the corresponding value of k for that curve e Rewindow the default window W so that the x axis and y axis labels are integers but the five curves remain visible 5 In Figure 4 17 are shown some geometric figures drawn by MLAB From the following descriptions design sequences of MLAB commands to create the required curve matrices and DRAW commands to display the curves A window with user rectangle 10 TO 10 10 TO 10 should be used After designing the commands run MLAB and test to see whether each figure is obtained a The regular hexagon in the upper left corner has center at 7 7 leftmost vertex 9 7 and rightmost vertex 5 7 b The rectangular grid in the upper right corner has horizontal lines for y 4 10 and vertical lines for x 4 10 c The triangle of circles in the lower right co
163. apter 2 and Chapter 3 as well as an understanding of the analytic geometry applying to the desired figures Consider smooth curves in the plane The simplest such curves are the graphs of functions which can be defined by MLAB FUNCTION statements Suppose F X is a defined function of one variable and its graph in a certain range from c to d is to be examined The normal procedure would be to set up a matrix M by the arithmetic sequence operation c d e where e is a fraction of d c chosen so that NROWS M is of moderate size say between 25 and 200 If a DRAW command with the curve matrix POINTS F M and default types POINTTYPE 0 LINETYPE 1 is set up then the eye sees the result as a smooth curve of the graph of F X from c to d The actual drawing of course is obtained by connecting many short line segments end to end Examples are shown in Figure 4 5 In Figure 4 5 a the graph of a cubic polynomial y x 7x 14 8 in the range from 0 to 6 was constructed in the window object W by the MLAB commands WINDOW O TO 6 10 TO 40 IN W NO IMAGEBOX IN W FUNCTION P X X73 7 X72 14 X 8 DRAW G POINTS P 0 6 1 IN W VIEW XK XA XA In Figure 4 5 b there is an ellipse defined by the equation z 16y 1 This figure was drawn by two applications of the function graph method Observe that the upper half of the ellipse is the function graph of y 2 5 1 2 Aj in the range from 1 to 1 and the lower half is the
164. are to be created using the PLOT command After the MLAB command EXIT is given to end the session the disk file MLAB LOG can be printed or used as desired One common use is to edit the log file to construct a do file for further use Any previously existing file named MLAB LOG will be renamed to MLABO LOG any file named MLABO LOG will first be shifted to be MLAB1 LOG any file named MLAB1 LOG will first be shifted to be MLAB2 LOG any file named MLAB2 LOG that is replaced by MLAB1 LOG will be deleted e Startup do files mlabinit do HOME mlabrc In Linux Unix NeXTStep Unix and Macinosh OSX if the do file mlabrc is present in your home directory it will be interpreted when MLAB is first run Subsequently in Linux Unix NeXTStep Unix Macinosh OSX and in MSWindows Macintosh OS7 8 9 or DOS if the do file called mlabinit do is present in your current directory it will be interpreted when MLAB is run e Save files sav 158 MLAB can be directed to create binary files holding MLAB data objects scalars matrices windows functions initials strings via the SAVE command These files are called save files and have the extension sav The data objects stored in a save file can be read into MLAB by means of the USE command Save files are created and accessed in the directory specified by FILEDIR Plot files mlabp ps mlabp 1j MLAB can be directed to create PostScript and LaserJet plot files by using the PLOT com mand The
165. arker skipping us ing splines line type 106 204 208 variable resolution circle point type 106 VBAR 106 192 vector field arrow point type 106 vector length operator 77 vertical bar point type 106 192 vertical error bar point type 106 vertical logarithmically spaced tick marks 106 VIEW 5 96 138 147 VIOLET 125 VMARKER 106 110 204 W 144 146 346 WEIGHT 245 weighted root mean square error 252 weighted sum of absolute values of errors 250 weighted sum of squared error criterion 229 236 238 248 WEIGHTS 233 250 weights 230 232 WEXPAND 143 WFIX 143 WFLOAT 143 WHITE 125 WINDOW 11 100 101 142 143 147 203 window 15 98 window names 101 window 3D 15 WINDOWS 139 windows 138 WITH 245 WMATCH 143 203 WNICE 143 WORLD 117 121 133 world units 117 121 WREAD 177 181 183 WSHRINK 143 WSLACK 143 X point type 106 XAXIS 126 128 XERRBAR 106 XLOGTICK 106 XPT 106 YAXIS 126 128 YELLOW 125 YERRBAR 106 YLOGTICK 106 zeroes of a function 36
166. as ME1 was The endpoints of ME1 and SMOOTH ME1 are always the same The BARGRAPH operator will take a 2 column matrix of x y coordinate pairs sorted in increasing order on column 1 and return a 2 column matrix of x y coordinates corresponding to the step graph of the input matrix The HISTO operator will return the points of the step function approxi mation of ME1 with the points of ME1 at the center of the steps Table 6 2 INTERPOLATE SMOOTH BARGRAPH and HISTO matrix operators on curves and not curves themselves and can thus be combined with each other and with other matrix operators in order to achieve a desired result The general forms of the SMOOTH BARGRAPH and HISTO matrix operators are shown in Table 6 2 6 3 SHADING AREAS OF A POLYGON CURVE In Figure 6 1 we saw how to use the MESH operator to shade the area between two curves This method can usually only be used to draw vertical lines In general the MLAB FILL operator may be used to fill in a polygon with fill lines It takes as input a matrix of points the number of fill lines to generate and the angle at which to draw the fill lines The result is a matrix containing the endpoints of the fill lines which can then be drawn using linetype ALTERNATE The following commands show this 200 Figure 6 8 Shading in a circle RESET M COSD ON 0 360 SIND ON 0 360 DRAW M DRAW FILL M 20 45 LINETYPE ALTERNATE WINDOW ADJUST
167. as about other data types previously discussed A list of defined window objects can be obtained by the command TYPE WINDOWS 139 a typical MLAB response would be currently defined WINDOWS W For a window identifier W the command TYPE W produces a response having the following information the list of curves axes and titles in W and the user rectangle frame rectangle and image rectangle of W The format of the response follows window W UNBLANKED priority 0 window clipping limits world units wxmin 1 wxmax 10 wymin 1 wymax 10 adjustment codes WNICE WNICE WNICE WNICE frame color WHITE 0 NO FRAMEBOX frame in screen fraction units width height 1 x 0 to 1 y 0 to 1 frame in screen inches x 0 to 4 y 0 to 3 image color WHITE 0 IMAGEBOX color BLACK 1 image in frame inches x 0 7 to 3 7 y 0 375 to 2 625 image in frame fraction units x 0 175 to 0 925 y 0 125 to 0 875 image in screen fraction units x 0 175 to 0 925 y 0 125 to 0 875 image in screen inches x 0 7 to 3 7 y 0 375 to 2 625 CURVES CO AXES W XAXIS W YAXIS TITLES T1 The curves axes and titles are always listed in the order in which they were created from last to first Note that the user rectangle frame rectangle and image rectangle are clearly defined The following MLAB dialog shows how lists of the currently defined curves axes and titles may be obtained TYPE CURVES currently defined CURVES C2 in Y W YAXIS
168. as much as 10 percent of the minimum shrink extent to obtain round numbers on axes WMATCH or 15 means use the larger of minimum zx or y shrink extent to achieve unit aspect ratio Table 4 15 The WINDOW statement with ADJUST clause Another labor saving device to precede the TITLE command with one of the words TOP BOTTOM LEFT or RIGHT This will automatically place the title string at the specified position in the window s frame For example TOP TITLE T The MLAB Learning Curve BOTTOM TITLE B time LEFT TITLE L MLAB Fluency 143 2 5 1 4 0 3 1 9 Figure 4 15 Part of a parabola drawn in the default window would place the title T in the center of the margin above the image rectangle the title B in the center of the margin below the image rectangle the title L in the center of the margin to the left of the image rectangle written at an angle of 90 degrees A third technique is to make use of the special window called W For example the MLAB commands RESET FUNCTION G X 3 X X 8 X 3 DRAW C POINTS G 0 2 02 VIEW result in Figure 4 15 showing a parabola Adding the commands UNVIEW 144 Parabo las Figure 4 16 Additional parts of parabolas drawn in the default window FUNCTION H X G X DRAW D POINTS H 0 2 02 LT 2 DRAW E POINTS G 3 3 02 TOP TITLE T Parabolas VIEW XX XA XA XX results in Figure 4 16 Note that the DRAW co
169. as the first row of DM DM COL 20 1 1 denotes the matrix obtained from DM by reversing the order of the columns from the twentieth to the first A simpler use for ROW and COL is to choose a particular row or column of a matrix That is DM ROW 4 and DM COL 1 are the corresponding 1 row by 20 column vector a row vector and 6 row by 1 column vector a column vector respectively In general ROW matrix operations have one of the following forms ME1 ROW ME2 For an arbitrary m by n matrix ME1 and a k by 1 column vector ME2 of integers 71 72 Jx with all 7 between 1 and m the result is a k by n matrix with i row equal to row j of ME1 for t 2 inks All entries of ME2 are made into integer values by applying INT if necessary and an error message appears if a nonexistent row of ME1 is indicated Use with NCOLS ME2 gt 1 is equivalent to ME1 ROW LIST ME2 In other words elements are taken from ME2 row by row to use as row numbers of ME1 ME1 ROW SE1 equivalent to ME1 ROW LIST SE1 ME1 ROW SE1 SEn equivalent to ME1 ROW LIST SE1 SEn Table 3 13 ROW matrix operators In addition to the simple matrix assignment commands and matrix entry assignment commands previously described there are additional matrix assignment commands based on ROW or COL specifications as summarized in Table 3 15 and Table 3 16 That is one or more given rows of a matrix can be directly replaced leaving the other r
170. atios are generally different for the x and y coordinates so that figures may be foreshortened or extended for convenient examination For example the image rectangle of Figure 4 2 is a square but it corresponds to a user rectangle which is 360 times as high as it is wide 5 units on the z axis vs 1800 units on the y axis A precise rule for conversion of a point x y in the user rectangle to its corresponding coordinates x yi in inches on the MLAB picture can be given Assume that the MLAB picture dimensions and user frame and image rectangles are MLAB picture c inches wide d inches high user rectangle a toc b to d in user coordinates oe a frame rectangle a toc b tod in screen fraction units image rectangle af to c b to d in frame fraction units as shown in Figure 4 3 Since the inches per unit ratios are c a c a c a c a for the x axis and d b d b d b d b for the y axis the proportionality rule corresponds to the formulas a gt c u atv et xi e a a c a LE 103 MLAB PICTURE 0 031 FRAME RECTANGLE c d USER eT RECTANGLE USER COORDINATES Figure 4 3 Illustration of MLAB user rectangle frame and image rectangles 104 That is x is a linear function of x and similarly y is a linear function of y Recall that WINDOW FRAME and IMAGE commands ca
171. be a set of points scattered throughout a picture or a curved line drawn in the picture or a sequence of straight line segments connected end to end or numbers to be drawn at specified locations in the picture A title data object may contain some text labeling coordinate axes or a legend explaining what is drawn in a graph A window consists of lists of curves axes and titles plus certain values controlling the positioning and display of the curves and the window in the MLAB picture Be careful not to confuse the MLAB window data type with the MLAB picture that appears as a result of the VIEW command Frequently the MLAB user makes into an individual picture a single window data object that is being defined However it is possible 97 60 ALIGNED RECTANGLE 50 X RANGE 4 TD B Y RANGE 10 TD 40 40 30 HEIGHT 30 20 WIDTH 1 2 10 LDWER LEFT CDRNER AT 4 10 Figure 4 1 The aligned rectangle with x in 4 8 and y in 10 40 to make a picture in which several MLAB window data objects are displayed simultaneously and the window objects may overlap on the viewing screen if the user requests it The positioning and display of curves axes titles and windows are specified by giving aligned rectangles An aligned rectangle in the x y plane is a rectangle with two sides parallel to the x axis and two sides parallel to the y axis In MLAB such rectangles are specified by four numbers using the phrase SE1 TO SE
172. be used as the curve matrix for CA 3 a clause specifying the point type of the curve CA 4 a clause specifying the line type of the curve CA 5 and the clause IN W to specify that CA will belong to the list of curves of the window W The point type clause specifies what is to be drawn at each of the n points of the plane specified by the n by 2 curve matrix For example the clause POINTTYPE X would specify that the capital 105 letter X is to be drawn at each of the n curve matrix points POINTTYPE may be abbreviated as PT The specification POINTTYPE TRIANGLE of the example asks for small triangles Alternatively a point type number may be given instead of a name PT 3 POINTTYPE 3 PT TRIANGLE and POINTTYPE TRIANGLE all have the same result Similarly the line type clause specifies what is to be drawn at each of the n 1 lines between adjacent curve matrix points It requires the word LINETYPE which may be abbreviated LT and a number or a line type name LT 1 LINETYPE 1 LT SOLID and LINETYPE SOLID each would cause solid line segments to be drawn as in Figure 4 2 The clause LINETYPE NONE of the example in Chapter 1 specifies that nothing is to be drawn at the lines associated with the curve matrix DAY That is the specification POINTTYPE TRIANGLE LINETYPE NONE causes scattered triangles to be drawn in the picture The possible point types and line types are listed in Table 4 3 Table 4 4 and Table 4 5 and shown in Figure
173. below READ CON 3 2 TYPE lt CONTROL gt Z AFTER LAST NUMBER 1 2 3 4 5 6 Z a 3 by 2 matrix WNr ooo nue ooo OPN Suppose a number is mistyped while entering a matrix using a LIST KREAD or READ expression If the error is noticed immediately the user can backspace until the incorrect entry is deleted and continue by typing the correct number However if the error is not noticed immediately the matrix input operation can be completed with one or more incorrect matrix entries Then scalar values can be assigned to any incorrect entries For example if TA is a 4 row by 7 column matrix which is correct except that TA 3 6 is 1 84 instead of the desired value 1 48 then the matrix entry assignment command TA 3 6 1 48 corrects the error while leaving all other entries of TA unchanged An additional feature of this matrix entry assignment command is that it can enlarge the matrix dimensions For example observe the following MLAB dialog A LIST 1 7 TYPE A A a 2 by 1 matrix 53 TYPE A A a 2 by 3 matrix Note that the matrix dimensions are enlarged sufficiently so that the operation makes sense Also note that zeros are supplied for matrix entries that are not otherwise determined This principle called matriz expansion by zeroes is also used for many other MLAB matrix operations A matrix can be created by a matrix entry assignment command if an unknown identifier is given again using matrix expansio
174. c o ae BM BR 203 6 5 EIGENVALUES EIGENVECTORS AND COMPLEX NUMBERS 208 6 6 SINGULAR VALUE DECOMPOSITION OF A MATRIX 213 6 7 COVARIANCE AND CORRELATION cocos cios ee ct be eee yeas 215 68 DISCRETE FOURIER TRANSFORMS lt 4 660 hea we Baw eee eee a 217 09 SUMMARY oon eo iepa eS OE Ke ada ree EAE 221 G10 EXERO SES covers EE Eee ee ee eee Le Re eS Bo 223 CURVE FITTING IN MLAB 225 7 1 SUM OF SQUARED ERROR CRITERION FOR CURVE FITTING 226 7 2 WEIGHTED SUM OF SQUARED ERROR CRITERION 229 La ENT OPERADOR oa stat RA a O eee eee Sea A 233 7 4 FUNCTIONAL FORMS OF SEVERAL VARIABLES 236 7 5 JOINT MODELS OF SEVERAL FUNCTIONAL FORMS 236 7 6 THE CONSTRAINTS COMMAND cg ce wank wade ae beens eee e 238 7 7 GENERAL PROBLEM OF CURVE FITTING A FUNCTIONAL FORM 241 7 8 PREPARING DATA FOR CURVE FITTING OPERATIONS 241 TO THEFITCOMMAND os ce caca ba ee Ara 244 7 10 THE FIT COMMAND CONTROL VARIABLES aaa 251 7 11 FORMULATING FUNCTIONAL FORM MODELS 253 T12 LINEAR MODELS bmo coe cdas aaa E aa a ia a 257 7 13 STATISTICAL CONSIDERATIONS FOR CURVE FITTING 259 7 14 CURVE FITTING SEARCH STRAIBGIES i4 6 heehee SES eRe ew 260 7 15 INITIAL PARAMETER ESTIMATES FOR NONLINEAR MODELS 271 VAG SUMMARY e A Oe eR eee CAS A A e See 273 Teall BXEROUDES 20 3 eas das ES Se ate wee eee ee SS Oe ES Ge 275 SYSTEMS OF DIFFERENTIAL EQUATIONS I
175. c polynomial approximation to the solution is used In the second variant which is selected by setting METHOD equal to GEAR2 2 a quadratic approximation to the solution is used at each step MLAB also can combine the Adams and the Gear methods It makes internal tests of the stiffness of the differential equations system during the numerical computations alternating between the two methods according to which method is indicated as most favorable This method is used if METHOD is set equal to MIXED 0 This is the default method that MLAB uses if the user does not specify another method When solving systems of differential equations Gear s method and its variants require estimates of a Jacobian matrix these are partial derivatives of one unknown function with respect to the independent variable and the other unknown functions The MLAB variable JACSW is used to determine how the Jacobian matrix is to be computed It may take the following values e 0 means that numerical partial derivatives are to be used e 1 means that symbolic partial derivatives are to be used e 2 means that the partial derivatives are to be computed only once numerically The default value of JACSW is 1 so that symbolic partial derivatives are computed unless the user specifies otherwise It is usually more accurate and faster to use symbolic partial derivatives in computing the Jacobian matrix used with Gear s method However for large and complicated diffe
176. cal computation Some examples of linear constraints on parameters b1 b2 and b3 are given in Table 7 4 both in a simplifed form that MLAB will accept and in an equivalent combination of the forms described above 239 b1 gt 0 1 6 0 b2 0 63 gt 0 1 3 lt bo lt 1 4 0 6 1 b2 0 b3 gt 1 3 0 6 1 b2 0 b3 lt 1 4 by gt b3 1 b 0 b2 1 b3 gt 0 by 2 b3 0 bj 1 b2 2 b3 0 2 bi gt b3 1 2 bi 0 b2 1 b3 gt 1 b b2 b3 1 1 b 1 b94 1 b63 1 by b3 2 b2 b1 4 3 75 b1 25 b2 5 b3 3 Table 7 4 Examples of linear constraint forms The curve fitting process in MLAB may be restricted by specifying a system of linear constraints on the parameters to be fitted If a system of linear constraints is given then the search for a minimum of S will observe these constraints The curve fitting method is designed to avoid violating constraints and beyond that to converge to a constrained minimum of S If a model involves linear constraints on the parameters then an MLAB constraint set must be created to specify them MLAB conventions for describing linear constraints are less restricted than the aforesaid forms The general form of the CONSTRAINTS command is shown in Table 7 5 For example it is permitted to write K 4 to describe the term 25 K and to write A 2 B to describe the expression 1 A 2 B Any inequality or equation of such expres
177. common parameters in joint models of functional forms For example recall the equations X t Ao ki ki ka e 4D e t B t Ag L eA X t used in the example of Section 7 5 Suppose that the data measurements of the function X t are entered in the MLAB matrix DX and the data measurements of the function B t are entered in the MLAB matrix DB as shown in the following dialog DX COL 1 10 100 10 DX COL 2 LIST 66 4 141 150 8 174 6 207 1 155 7 207 2 215 6 220 3 188 6 DB COL 1 10 100 10 DB COL 2 LIST 02 23 24 42 59 67 1 04 1 22 1 47 1 68 Then the following dialog curve fits the functions X T and B T simultaneously and displays Figure 7 4 FUNCTION X T AO K1 K2 K1 EXP K1 T EXP K2 T FUNCTION B T AO 1 EXP K1T X T AO 200 K1 05 K2 0001 FIT K1 K2 X TO DX B TO DB final parameter values value error dependency parameter 0 0511870446 0 00511935 0 0898001789 Ki 0 000161445 0 0003352155 0 0898001789 K2 247 2 iterations CONVERGED best weighted sum of squares 3 213720e 03 weighted root mean square error 1 336189e 01 weighted deviation fraction 8 383730e 02 squared 9 808347e 01 DRAW XFIT1 POINTS X 0 100 DRAW DX POINTTYPE CROSSPT LINETYPE NONE LEFT TITLE X T BOTTOM TITLE T FRAME 0 TO 5 0 TO 1 Wi W DRAW BFIT1 POINTS B 0 100 DRAW DB POINTTYPE CROSSPT LINETYPE NONE LEFT TITLE B T BOTTOM TITLE T FRAME
178. ctangle 125 TO 875 125 TO 875 The FRAME command creates a window with default user rectangle O TO 10 O TO 10 and default image rectangle 125 S TO 875 S 125 T TO 875 T where S is the width of the frame rectangle and T is the height of the frame rectangle The IMAGE command creates a window with default user rectangle O TO 10 O TO 10 and default frame rectangle O TO 1 O TO 1 Table 4 2 WINDOW FRAME and IMAGE commands Curve data objects contain a considerable amount of information divided basically into a curve matrix part a point type part a line type part and a label part The MLAB graphics command DRAW is used to specify some or all of these parts in a curve A curve matrix usually has two columns corresponding to x and y coordinates That is a curve matrix is an n row by 2 column matrix which is part of a curve data object and which is regarded as a list of n points in the z y plane For example the matrix 9 1200 1 1 350 2 5 650 0 200 considered as a curve matrix is regarded as a list of the four points 9 1200 1 1 350 2 5 650 and 0 200 in the plane An n row by 2 column curve matrix also has n 1 line segments associated with it the line segments going between adjacent points on the list The matrix above for example has three straight line segments associated with it The first goes from 9 1200 to 1 1 350 the second goes from 1 1 350 to 2 5 650 and the last g
179. d differential equation modeling features of MLAB are organized around the MLAB concept of a defined function Suppose that a functional form F is specified in mathematical text by the equation 1 F s 1 K s e This defines F to be a function of two real variables s and x called arguments which can be computed by means of the given expression defining F the right side of the equation above The variables K and a which appear in the expression defining F but are not arguments of F are called parameters of F In fact the equation above describes an infinite family of functions of two variables since different numerical values for the parameters will lead to different functions 23 Before F can be numerically evaluated at specified points e g F 1 2 F 3 5 0 etc numerical values for K and a must be specified The MLAB FUNCTION command is equivalent to the above mathematical technique for defining a function in parametric form For example the definition of F could be expressed by the MLAB command FUNCTION F S X K S S 1 X EXP A X Note that the FUNCTION command is a straightforward translation of the mathematical text The meaning is also very similar In particular after executing the above FUNCTION command the function F can be referenced in subsequent scalar expressions For example observe the MLAB dialog below FUNCTION F S X K x S S 1 X EXP A X K 2 A 0 TYPE F 11 F 2 2 FUNCTION F S X
180. d for the argument x of f the pattern 3679a 2 718b 1 175 a b f 5 1 649a 6065b 5211 2 718a 3679b 1 175 and so on is established In each case the value of f is a sum of terms as follows Fx r1 a ra b r3 for certain numbers r 72 and r3 depending upon x Now the expression ri a r2 b r3 for numbers 71 72 and rg is called a linear function of the parameters a and b This indicates a sum of one or more terms each summand being either a number or the product of a number and a parameter Since the functional form f above has this property it is said to be a linear model with respect to the parameters a and b Note that a linear model can involve nonlinear functions such 257 as sinh x and exponential functions So long as such functions do not involve fitted parameters they will reduce to numbers when numbers are substituted for the arguments Consider the trigonometric model h z y s sin a x t cos b y of two independent variables z and y and four parameters s t a and b Then h 1 1 s sin a t cos b is not a linear function of s t a and b because there are no numbers T1 2 13 T4 T5 such that for all s t a and b h 1 1 r1 s r2 t r3 a r4 b rs Therefore h is a nonlinear model for the parameters s t a and b However a and b might be given fixed values say a 2 and b 15 derived from scientific theory or data Then h x y would be linear i
181. d point in the window and then pressing and releasing the mouse button If Z is equal to 2 WREAD starts to read and store world coordinates of the mouse cursor position when the mouse button is pressed and continues to read and store world coordinates of the mouse cursor position as the mouse is dragged i e the mouse is moved while the mouse button is down across the window In this mode WREAD stops reading mouse coordinates when the mouse button is released or X coordinate pairs have been stored If Z is 3 both of the features for Z equal 1 and 2 are enabled If argument Y is omitted the default 2D window W is displayed and points are specified in that window If argument X is omitted it defaults to a value of 10 The following MLAB commands demonstrate how the mouse can be employed to specify the location of a title DRAW 1 5 M WREAD 1 W 2 TITLE LINE SEGMENT AT M 1 1 M 1 2 WORLD IN W VIEW XA The first command establishes a curve in the default window W The second command causes MLAB to display the default window and wait for the user to click the mouse button The current mouse cursor coordinates will be displayed at the lower left corner of the image rectangle of the default window Once the user clicks the mouse button the picture is removed from the screen 182 and the mouse coordinates are returned in the matrix M Then the title containing the text LINE SEGMENT is established at the world coordina
182. d the PRINT command 5 5 THE PLOTDEV AND PLOT COMMANDS In Chapter 4 techniques were discussed by which pictures can be generated and displayed using MLAB The MLAB user can make permanent records of the displayed pictures if there is a PostScript or Hewlett Packard LaserJet printer accessible to the computer MLAB provides the PLOT command described in Table 5 5 for making a disk file containing data necessary to reproduce the currently displayed picture The output file can contain either PostScript language commands which are ASCII commands developed by Adobe Inc that can be edited embedded in documents or displayed with various other programs such as Ghostscript 166 The general form of the PRINT command is PRINT Di D2 Dn IN filename where each of D1 D2 Dn is an MLAB identifier scalar array defined function constraint set initial curve window surface or 3D window a scalar or matrix expression or a derivative of a function The PRINT command creates a computer disk file named filename LST with filename truncated to eight characters if necessary containing alphabetical and numerical characters suitable for printing The data has the same format as would be obtained from a TYPE command ap plied to the same list of identifiers and expressions The clause IN filename can be omitted in which case the data items are appended to the end of the file named MLAB LST An MLAB file can be ph
183. d to end pattern Examples are shown in Figure 4 6 In Figure 4 6 a the clause LINETYPE 5 ALTERNATE is used to lift the pen between strokes That requires a 12 row by 2 column curve matrix in order to draw the 6 strokes The following dialog shows how Figure 4 6 a can be drawn 114 UNVIEW WINDOW 1 TO 5 O TO 4 INW DELETE C MD SHAPE 12 2 LIST 0 1 0 3 1 4 3 4 4 3 4 1 3 0 1 0 0 2 4 2 2 0 2 4 TYPE MD MD a 12 by 2 matrix 1 0 1 230 3 Seid 4 4 3 4 5 4 3 6 4 1 7 3 0 8 1 0 9 0 2 10 4 2 11 2 0 12 2 4 DRAW CD MD LINETYPE 5 IN W VIEW The clause LINETYPE 5 ALTERNATE can also be used to draw dashed lines or curves with the user controlling the lengths of the dashes In Figure 4 6 b LINETYPE 6 MARKER was used to do stick figure drawing This curve was specified by the following MLAB commands UNVIEW DELETE CD WINDOW 6 7 TO 8 1 1 TO 1 9 IN W MC SHAPE 15 2 LIST 100 0 7 1 7 5 1 7 5 1 5 7 1 5 7 1 100 0 7 1 5 7 35 1 9 7 8 1 9 7 8 1 4 7 5 1 100 0 7 5 1 5 7 8 1 9 TYPE MC MC a 15 by 2 matrix 1 100 0 2 7 1 3 7 5 1 4 7 5 1 5 5 T 1 5 6 7 1 7 100 0 115 a b Figure 4 6 Examples of stick figure drawing 116 8 9 10 11 12 13 14 15 o 0 aooo o w a O Oo NNENNNNN a RRORR RR DRAW CC MC LINETYPE 6 IN W VIEW The curve matrix MC consists of a marker row 100 0 followed by the cube s front face given by 7 1
184. der this curve with the INTEGRAL operator and approximate it using the HISTO operator along with either a FOR loop or a SUM expression function 3 Draw a circle of radius 1 5 centered at the origin with a letter A at the point 1 0 Shade this with the FILL operator as shown in Figure 6 16 leaving space around the letter 4 Given the function defined by f x y x cos x y suppose that you wanted to find the roots of this function for x between 1 and 1 and y between 0 and 5 How would you use the contour operator to do this Hint draw a contour map with only one level Are you getting an exact answer 223 Chapter 7 CURVE FITTING IN MLAB There are several motives for fitting curves to data The investigator may want to 1 compare data with the values predicted by an appropriate mathematical model 2 test a mathematical model relating variables for example a growth curve that has been proposed from previous research or a growth curve that has been derived from mathematical analysis of a mechanism by which the variables are connected or 3 estimate parameters of a model that have physical meaning for example the dissociation rate constants for enzymatic reactions The formulas describing the mathematical model will often contain parameters having uncertain values The curve fitting facility of MLAB can be used to search for parameter values that may improve upon user supplied initial estimates That is
185. discrete Fourier transform operators 221 discrete Fourier transforms 217 DASHED 106 display a form and return user responses 177 dashed line with marker skipping line type 179 106 display a menu and return user choice 177 DASHMARK 106 division 18 data types 15 41 339 division by zero 30 DLBAND 106 DO 168 169 174 do file 4 154 159 168 169 174 DOFILEDIR 159 dot point type 106 DOTMARK 106 DOTPT 106 DOTTED 106 dotted line extending to the bottom image border 106 dotted line extending to the left image border 106 dotted line extending to the right image bor der 106 dotted line extending to the top image border 106 dotted line with marker skipping line type 106 dotted line type 106 downward tick mark 106 DRAW 5 11 101 105 147 DRAW command complete form 117 DRAW command short form 110 drawing functions of 1 variable 112 DRBAND 106 DTICK 106 128 DUBAND 106 DW SAV 146 dynamical systems 210 editing matrix elements 53 efficient function definitions 255 EIGEN 208 211 eigenvalues 208 211 eigenvectors 208 211 element by element matrix product 79 81 ellipse 112 embedded string control characters 131 134 entering numbers in a column vector 49 entering numbers into a matrix 49 enzyme kinetics 283 enzyme substrate complex 284 340 ERRFAC 280 283 296 301 error function 19 error tolerance level in solving differential e
186. down and striking the Break key Then the currently executing MLAB command is suspended and the message Command in progress aborted Safest Action type EXIT then restart MLAB Riskier but often worthwhile Save your data objects SAVE IN MYFILE then EXIT restart MLAB and restore the objects USE MYFILE is printed On Macintosh OS7 8 9 and OSX systems MLAB can be interrupted at any time by pressing the Apple key and the Option key and simultaneously striking the ESC key A dialog window then appears offering the option of terminating the MLAB session On all systems the MLAB FIT INTEGRATE MINIMIZE and POINTS operations can be interrupted and stopped by striking the letter Q MLAB will then return to the command prompt 176 5 8 THE MENUCHOICE GETSTRINGS and WREAD COM MANDS The MLAB functions MENUCHOICE GETSTRINGS and WREAD can be used to provide a graphical user interface to do files on Linux Unix with X Windows DOS MSWindows and Macintosh systems The MLAB function MENUCHOICE creates a menu from which the user can select one of a number of options GETSTRINGS creates a dialog window in which the user can fill in a form and WREAD can be used to store a 2 column matrix containing coordinates specified by mouse clicks in an MLAB window The function MENUCHOICE is described in Table 5 8 It takes 4 arguments the last three are optional MENUCHOICE T S X C displays a menu in a dialog window with a title and a li
187. e 19 partial derivatives 39 85 210 343 particle motion on a parabola 301 path 154 PI 11 pictures 96 piece wise defined functions 26 PINK 125 PLACE 117 122 197 133 134 PLOT 11 166 168 plot file 11 159 166 PLOTDEV 168 point type 101 106 point type font 117 118 127 point type size 117 127 POINTS 8 11 48 84 86 191 281 292 POINTTYPE 5 106 polar graphs 190 positioning titles 131 power spectrum 217 precedence of arithmetic operations 18 precision 16 preparing data for curve fitting 241 PRINT 165 166 print file 154 159 165 PRODUCT 33 34 prompt 2 PSCL 168 PSCP 168 PSL 168 PSP 168 PT 106 PTFONT 117 118 127 PTSIZE 117 127 PURPLE 125 quadratic equation 173 quadratic model 226 244 quadratic polynomial 226 quit curve fitting 247 quit differential equation solving 297 quitting MLAB 13 quotation marks 129 131 174 radioactive count data 227 RAN 234 random numbers 20 ranking data 82 83 RANKORDER 82 83 rate equations 237 247 254 RBAND 106 READ 49 51 53 163 read mouse clicks specifying coordinates 177 read file 159 163 READON 163 164 real discrete Fourier transform 218 REALDFT 218 221 reciprocal of variance method 232 rectangle 98 recursive functions 31 RED 125 redimensioning matrices 60 relational operators 38 renaming windows 142 reports on curve fitting process 247 repo
188. e 2 3 Trigonometric functions 19 The following inverse trigonometric functions may be used to form scalar expressions ASIN SE1 arcsine of SE1 measured in radians from 1 2 to 7 2 defined for SE1 from 1 to 1 ACOS SE1 arccosine of SE1 measured in radians from 0 to m defined for SE1 from 1 to 1 ATAN SE1 arctangent of SE1 measured in radians from 1 2 to 7 2 defined for all SE1 ATAN2 SE1 SE2 arctangent of SE1 SE2 defined as the angle z measured in radians 7 lt z lt Tr formed from the positive x axis and the ray from the origin passing through the point SE1 SE2 in the x y plane ATAN2 0 0 equals 0 by convention Table 2 4 Inverse trigonometric functions The following logarithmic and exponential functions may be used to form scalar expressions LOG SE1 natural logarithm of SE1 defined for SE1 gt 0 LOG10 SE1 base 10 logarithm of SE1 defined for SE1 gt 0 EXP SE1 exponential function exp x for z SE1 defined for 709 lt SE1 lt 709 overflow for larger values underflow for smaller values SINH SE1 hyperbolic sine of x SE1 exp x exp x 2 defined for 709 lt SE1 lt 709 overflow otherwise COSH SE1 hyperbolic cosine of x SE1 exp x exp x 2 defined for 709 lt SE1 lt 709 overflow otherwise TANH SE1 hyperbolic tangent of x SE1 exp x exp 2 exp x exp lt def
189. e do not know the correct values of the parameters a and b of the exponential curve that best approximates the data we assign initial values for later modification You must guess these values Type the following MLAB commands FUNCTION Y X A EXP Bx X A 003 B 4 Of course other initial guesses for the parameters A and B in the above MLAB commands could have been used In MLAB the parameters of a model may have constraints For example we may want to specify that the parameters A and B of the exponential curve approximating the data must have positive values Use the MLAB command CONSTRAINTS CS1 A gt 0 B gt O to create a set of constraints The constraints A gt O and B gt O are now named that is they are members of a set of constraints called CS1 We are now prepared to fit the exponential curve to our data the matrix DAY That is we want to estimate the parameters A and B such that the corresponding curve passes as near as possible through the data points For the moment we will not describe the precise method of measuring nearness of a curve to given points called the least squares method First give the MLAB command FIT A B Y TO DAY The response to this command is a computation of the parameter values corresponding to the best fitting exponential curve final parameter values value error dependency parameter 0 0050847413 0 0029804036 0 9897411712 A 0 334188183 0 042492703 0 989741
190. e g A 4 2 B C gt 1 b Functional Form Curve fitting Commands e FIT B1 B2 Bp F TO ME1 WITH WEIGHT ME2 This command fits a parameter list B1 B2 Bp of scalar or matrix identifiers of 274 a functional form F of n arguments to an m by n 1 data matrix ME1 with an m vector of weights ME2 e FIT B1 B2 Bp clause1 clause2 clausek This command does multiple clause curve fitting All clauses must be of the form F TO ME1 WITH WEIGHT ME2 If the clause WITH WEIGHT ME2 is omitted all elements of the m vector of weights will be set equal to 1 Typing R with no CR during any FIT overrides the current LSQRPT setting and toggles verbose printout about the fitting process typing Q with no CR during any FIT terminates command at the end of the current iteration c TYPE Command for Constraint Sets e TYPE csi This command displays the constraint set d Functional form linear models e For any numerical values 01 02 Zon there exist numbers ro 71 72 Tp Such that for all b1 b2 bp icone L025 5 LON bi ba fea bp T0 11 by 19 ba le Tp bp 7 17 EXERCISES 1 Suppose that the functional form model f t sin a t e is to be modelled to an m by 2 data matrix M Assume that the variance of the dependent variable is proportional to the size of the independent variable and that 7 lt a lt 1 2 and 01 lt k lt 04 Design MLAB commands to a b se
191. e later measurements would have negligible effect on the sum of squared error criterion For example a model with 2 error at t 1 and 5 error at t 10 might have a larger sum of squared error than a model with 1 error at t 1 and 100 error at t 10 This undesirable effect can sometimes be avoided by making a mathematical transformation of the measured values to transform the expected errors to approximately equal size For example suppose a user wanted to fit an exponential washout model to the radioactive count data using the formula 227 Figure 7 1 Example of sum of squared error terms where y is the predicted radioactive count at time t for certain constants a and b One commonly used method is to take logarithms of the model and data using the formula log y log a e7 log a b t So the user can fit the transformed data points ti log yi for i 1 2 n to a straight line on a graph of t versus log y The slope will equal b and the y intercept will equal log a If the error of each original data point is roughly proportional to the size of that data point then the errors of the transformed data will be roughly equal in magnitude For example suppose the measured values at times t 1 and t 2 are both 10 larger than the predicted values Then the vertical distances on the log plot are as follows log 1 1 a e log a e7 log 1 1 log 1 1 a e72 log a e7 log 1 1 Tha
192. e plot device is an HP Graphics Language device then the first unused name in the sequence mlabpO 1j mlabp1 1j etc is created containing the current picture PLOT can be executed without displaying the picture Table 5 5 PLOT command Issuing the command DO TEST causes all MLAB commands in the files TEST DO TESTA DO and TESTB DO to be executed It is also possible for a DO file to contain several DO commands inter spersed with other MLAB commands It is the user s responsibility to avoid an infinite circle of DO file references The DO command may be executed repeatedly by using a FOR command For example FOR J 1 100 DO DO RUN will execute the MLAB commands in RUN DO a hundred times with J equal to 1 2 100 consecutively Of course scalar expressions containing J may occur in MLAB commands that 169 In general the DO command has one of the three forms DO filename DO filename ext DO STR Here the ext is optional The first form refers to a file having extension DO in the user s disk area Note that the extension is not typed in the DO command It is valid to put DO commands into DO files and to chain together several such files so long as an infinite circularity is avoided Ifa serious error condition occurs during execution of a DO file then exe cution of all DO files is terminated immediately and an error message will appear Certain error conditions cause warning messages to be ty
193. e row index of the maximum of all values in a specified column Similarly MINROW returns the row index of the minimum of all values in a specified column Let ME1 a j an m by n matrix expression The following forms of matrix editing operations are evaluated as indicated GETDIAG ME1 Let k be the smaller of m and n The result is a k by 1 column vector d such that di ai i 1 2 k That is the result is the largest main diagonal of ME1 COMPRESS ME1 SE1 Let SE1 be a positive integer k k lt n and m be the number of elements in column k of ME1 not equal to zero The result is an m by n matrix which is a copy of the rows of ME1 in which the k element aj is not 0 ROTATE ME1 SE1 Let SE1 be an integer k The re sult is an m by n matrix whose rows are the rows of ME1 rotated down k indices That is the result is the matrix M ROW NROWS ME1 1 k NROWS ME1 1 NROWS ME1 k If k gt m then the rows of ME1 are rotated down k mod m rows If k lt 0 then the rows of ME1 are rotated up k rows MAXROW ME1 SE1 Let SE1 be a positive integer k k lt n The result is an integer r 1 lt r lt m such that ark gt Qik for 1 2 m That is the result is the row index r for which a is the maxi mum for all values in column k In case of ties the result is the least row index MINROW ME1 SE1 Let SE1 be a positive integer k k lt n The result is an integer r 1 lt r lt m
194. e shown in Figure 4 13 The nT command changes the font table used to font number n This is used to draw the a f and 7 characters Double apostrophes are used to have an apostrophe drawn in the string instead of it being interpreted as a string modification command The commands TITLE Ti 33W 2H M 2W 66H L 2W 66H A 2W 66H B SIZE 025 TITLE T2 20AThis is a circle ANGLE 90 SIZE 025 VIEW produce the picture shown in Figure 4 14 The nH command and the nW command change character height and width respectively by a scale factor of n In the example the height is initially set to twice the usual and the width is initially set to one third the usual The height and width are then scaled by factors of two thirds and two respectively between each of the four letters The nA command however need only be used once and will cause an angle change of n degrees between each of the following characters This character rotation can be turned off by the 0A command 136 Figure 4 14 Changing character height width and angle 137 String modification commands listed below are embedded in the text string of a TITLE command nS Multiply character size by n nW Multiply character width by n nH Multiply character height by n nT User character font table n nU Move up n characters nD Move down n characters nB Move back n characters
195. ed by the MLAB commands 34 The general forms of these FUNCTION command scalar expressions are shown below SUM X SE1 SE2 SE3 PRODUCT X SE1 SE2 SE3 INTEGRAL X SE1 SE2 SE3 Here X denotes an MLAB identifier used formally as the index variable of the sum or product or as the variable of integration That is X may be an unknown identifier when the SUM PRODUCT or INTEGRAL expression is evaluated and any data item with the same name as the identifier X is irrelevant to the evaluation of the sum or integral In particular X should not be an argument of the function defined using the SUM PRODUCT or INTEGRAL expression The scalar expressions SE1 and SE2 should not contain the identifier X For a SUM expression SE1 and SE2 are the lower and upper limits re spectively of the indexed sum or product For an INTEGRAL expression SE1 and SE2 are the lower and upper integration bounds respectively for the definite integral The last scalar expression SE3 will usually contain the identifier X It gives the numerical description of the terms being summed for the SUM operation multiplied for the PRODUCT operation and of the function being integrated for the INTEGRAL operation Table 2 13 SUM PRODUCT and INTEGRAL operators FUNCTION G N SUM J A N B N F N J FUNCTION P N PRODUCT J A N B N F N J FUNCTION H X INTEGRAL CY A X B X F X Y Note that a particular sum product or i
196. ed subject to certain assumptions and are sometimes inappropriate although they are often quite suitable The input data matrix M must be a 2 or 3 column matrix with column 1 in ascending order The standard deviation of each data point is estimated by the difference of the data from a smoothed form of the data Thus the error is assumed to be similar to white noise The reciprocal of the squares of these standard deviations estimates form the resulting weight vector The use of EWT generated weights is sometimes not appropriate where the data points are too few in number or where wide gaps appear in a low frequency disturbance Nevertheless EWT generated 233 The general form of the EWT operator is given by EWT ME1 If ME1 is an n by 2 or an n by 3 matrix a with the elements of the first column in ascending order the result is an n by 1 column vector w such that w are estimated weights associated with data points aj a 2 or ail d a aig for i 1 2 lt 1 If NCOLS ME1 gt 3 an error message appears Table 7 1 EWT operator weights are often useful In particular the problem of unit differences when fitting several curves simultaneously described in Section 7 9 can be overcome by using EWT generated weights since the weight vector entries are computed based on the actual units of data For example consider the simulated data matrix M computed as shown in the following MLAB commands FUNC
197. ed to remind the user what is required here say an assignment statement for the scalar K You can also direct the input of values within a do file For example you might program K KREAD Enter the value K in the file READK DO Then typing DO READK causes MLAB to print Enter the value K and wait for the user to type a number Then one might enter 2E 4 so that the computation continues with this value assigned to K 5 7 THE IF THEN ELSE COMMAND There is another MLAB command the conditional or IF THEN ELSE command described in Table 5 7 which can be used to control execution of MLAB commands It is generally useful in DO files This command is similar to the IF THEN ELSE scalar expression described in Chapter 2 However the THEN and ELSE clauses for the conditional command contain MLAB commands or parenthesized lists of MLAB commands not scalar expressions There is little point to using the conditional command when typing in MLAB commands since one can simply find out whether the condition is true or false and then type in the MLAB commands in the THEN or ELSE clause as appropriate In a DO file however one must provide for both alternatives in advance and so this form is useful Consider the example 171 IF X 0 which is a proper conditional command This command causes the message ERROR X IS 0 to when X is zero and causes the assignment statement V 1 X to be executed when X is The error condition
198. ed to the right edge of the window so that the right edge of the entry area is uniform 180 eL 4 This layout is identical to the layout specified by L 1 except that the prompt strings are right justified in the prompt area The following MLAB commands demonstrate how GETSTRINGS might be used to display a form for obtaining map coordinates and set the variables LAT and LON to the values filled in the form TTL Enter the coordinates of a point PR 1 Latitude 90 lt South lt 0 0 lt North lt 90 PR 2 Longitude 180 lt West lt 0 0 lt East lt 180 DF 1 39 DF 2 74 STR1 GETSTRINGS TTL PR DF STR LAT STR1 1 DO STR STR LON STR1 2 DO STR In the last four lines after the GETSTRINGS command has been executed the string STR is defined and then executed with a DO statement to convert the responses in string array STR to scalar values and store the scalar values in the variables LAT and LON On Linux Unix with X Windows MSWindows and Macintosh systems the form would something like this Enter a the coordinates of a point Latitude 90 lt South lt 0 0 lt North lt 90 39 lt Longitude 180 lt West lt 0 0 lt East lt 180 74 Continue On DOS systems the form would not have a CONTINUE button and the currently active text field would be drawn in reverse video The MLAB operator WREAD X Y Z returns a 2 column matrix where rows of the matrix corre
199. ed which creates a curve C with a curve matrix which is a copy of M When another DRAW command is executed to create a curve D with a curve matrix which is a copy of N the error message response is Error allocator failed num 1000001 size 8 This means there is insufficient memory available in MLAB What would be the quickest attempt to overcome this obstacle 4 Suppose that X is a 400 row by 12 column matrix which is to be printed Since it is too wide for the printer page it is to be printed as two 400 row by 6 collumn matrices the first matrix containing columns 1 6 and the second containing columns 7 12 Furthermore the row numbers 1 400 are to be printed as a seventh column for each of the two matrices Design a sequence of MLAB commands to achieve this 185 5 Suppose that one wants to draw Figure 4 18 in two colors The axes and labels are to be black and the function graph and shading red How could one do this using MLAB 6 Construct a DO file which operates on a matrix M to make separate graphs of the curves given by M COL 1 3 for J 2 3 NCOLS M Each picture is to be plotted and no picture is to appear on the graphical display Modify the DO file so that it can also put all the curves in the same graph 7 Construct DO files that normalize vectors that is compute Y ON V for a given vector V where FCT Y X A X B for suitable scalars A and B in the following cases a To compute VN Y ON V where
200. en the INTEGRATE operator is applied One sample calculation is given in the following MLAB dialog Ki 0001 K2 01 K3 01 EO 1000 SO 10000 U INTEGRATE S T C T P T 0 10 TYPE ODESTR ODESTR T S ST C CIT P PT TYPE U COL 1 3 U COL 1 4 5 U COL 1 6 7 XX XA XX a 11 by 3 matrix 10000 1000 9383 1931 356 827294 9152 31555 141 128528 9057 62222 60 98051 9014 5455 29 9686069 8991 60794 17 776736 8976 61348 12 9478844 8964 77087 11 0293782 ONOOFWNEFE NO oOPWNFO 285 9 8 895 10 9 894 11 10 893 a 11 by 3 matrix 0 FOWOANOAAPEPWNE FOANODOAFWNFO Rh a 11 by 3 matrix 0 3 5 11 PRRO0O0O0YJO00d4syuyN FOANOAOPWNFO w 00 Rh The string ODESTR is set whenever INTEGRATE is executed it contains the labels of the columns of the output matrix In this case the first column of the matrix U holds the values of the independent variable T the second column of U holds values of the function S the third column of U holds values of the function S T the fourth column of U holds values of the function C the fifth column of U holds values of the function C T the sixth column of U holds values of the function P and the seventh column of U holds values of the function P T Plots of S C and P versus T shown in Figure 8 1 may be produced as shown in the following set 613 836 922 956 969 974 976 977 977 O 978 4 18148 4 09154 4 20095 211049
201. endental number 3 141592654 The ATAN2 operator as well as the other commands and operators used here will be explained later In order to make a permanent record of this picture as a PostScript file type the MLAB command PLOT and observe the response PLOT file is mlabpO ps The response to the PLOT command gives the name of the plot file containing the saved picture which is mlabpO ps unless you have already created a file with that name If the plot filemlabp0O ps exists the first unused name in the sequence mlabp1 ps mlabp2 ps etc is selected To print the PostScript file on a printer supporting the PostScript language type the command 11 PRINT FILE mlabp0 ps to which MLAB responds file printed mlabp0 ps You could also generate a Hewlett Packard PCL printer language file of an MLAB picture or embed the PostScript language picture file in a document using any of several methods We will defer discussion of the required steps to Section 5 5 The dialog of the MLAB session can be scrolled for review from the MLAB prompt With DOS and MSWindows MLAB strike the up and down arrow keys and the Page Up and Page Down keys to scroll backwards and forwards If you position the text cursor on a line and strike the Insert key the line that the text cursor is on will be copied to the built in line editor so that you may edit the line that the text cursor was on and create a new MLAB command With Linux Unix and Mac
202. entry fields The widths of each text entry field is determined by the associated text entry width value as determined by the entry width argument processing The start of the text entry field immediately follows the prompt field on the line The visual result is that the prompts are all in a rectangular column and the text entry fields are all left justified against the prompt column while the right edge of the entry area will be ragged if the entry fields have different widths or uniform if all entry fields have the same width as in the case of a constant or scalar entry width argument eL 2 The prompt strings are left justified in the form The length of each prompt field is de termined by the length of the prompt string If all prompt strings have the same length the prompt area is rectangular otherwise the right edge of the prompt area is ragged The text entry fields start immediately after the associated prompt fields and the text entry field widths are determined by the entry width argument so in general both the left and right sides of the entry area are ragged eL 3 The prompt strings are left justified in the form The width of each prompt field is de termined by the length of the prompt string If all prompt strings have the same length the prompt area is rectangular otherwise the right edge of the prompt area is ragged The text entry fields start immediately after the associated prompt fields and the entry widths are extend
203. ents Any of the WINDOW FRAME or IMAGE commands above would create a window object named PIC which has no curves e If only the WINDOW command were given it would define PIC to be a window data item and set its user clipping rectangle to the specified rectangle 1 TO 5 O TO 1 the PIC frame rectangle would be set to the default rectangle O TO 1 O TO 1 in screen fraction units and the PIC image rectangle would be set to the default rectangle 125 TO 875 125 TO 875 in frame fraction units e If only the FRAME command were given it would define PIC to be a window data item and set its frame rectangle to the left half of the MLAB picture the user clipping rectangle of the window would be set to the default value O TO 10 O TO 10 and the image rectangle would be set to the value 0625 TO 4375 125 TO 875 in frame fraction units e If only the IMAGE command were given it would define PIC to be a window data item and set its image rectangle to 1 TO 9 1 TO 9 in frame fraction units the user clipping rectangle would be set to the default value O TO 10 O TO 10 and the frame rectangle would be set to the default value 0 TO 1 O TO 1 in screen fraction units The default user clipping rectangle is always O TO 10 0 TO 10 The default frame rectangle is always O TO 1 O TO 1 in screen fraction units The default image rectangle expressed in frame fraction units is always 125 TO 875 125 TO 875 which corresponds to 125 S TO 875 S 12
204. es MLAB will apply rules for symbolic differentiation of scalar expressions containing arithmetic operations builtin functions and so on However it is advisable to examine symbolic derivatives before using them because some special expressions may be differentiated in a way that is not desired This applies to non standard expressions involving IF THEN ELSE INT or relational operators as shown in Table 2 15 Special cases of differentiation follow with d A signifying the derivative of the expression A with respect to some variable d A B A B d A lt B A B E gt 3 A B d A B A B d A B A B w A B E SUM I A B d E d A AND B d a AND d B d A OR B d A OR d B d NOT A NOT d A d IF A THEN B ELSE C IF A THEN d B ELSE d C d ABS A IF A gt O THEN d A ELSE d A d INT A INT A A d MOD A B d INT INT A INT B Table 2 15 Special cases of differentiation MLAB derivative functions named by DIFF expressions can be used for numerical calculation in the usual way For example the derivative of the above mentioned function F could be evaluated as shown in the following dialog 38 A 2 TYPE F DIFF X PI 2 F X PI 2 2 2 The expression 2 SIN PI 2 was numerically evaluated as 2 The rules for partial derivatives in MLAB are quite similar Consider the MLAB dialog FUNCTION G X Y A X EXP X KA Y TYPE G X
205. es reversibly with one molecule of a substrate S to form an intermediate enzyme substrate complex C which reacts irreversibly to release a product material P and the original unmodified enzyme This process may be represented by the reaction ky k3 E S C gt P E ka 283 where ky k2 and kg are the rate constants of the reaction Let S t E t C t and P t be the concentrations of S E C and P respectively at time t Eo is the total concentration of enzyme whether free or coupled and Sy is the total concentration of substrate whether free enzyme bound or converted to the product Then So S t C t P t Assume the initial concentrations at time zero of the enzyme substrate complex C and the product material P are zero Thus C 0 0 P 0 0 E 0 Eo S O So The law of mass action states that the reaction rates are proportional to the product of the reactants where the rate constants k1 k2 and kg are the proportionality constants Thus the rate at which S will be converted to C is k S t E t and the rate at which S will be formed from C is k2 C t Hence the differential equation S 1 a ky S t E t kp C t ky gt S t gt En E CH describes how the concentration of S will change with time Similarly the rate at which C will be formed from E and S is k S t E t the rate at which C will be converted back to E and S is kacdotC t and the rate at which C will be conve
206. escribed in Table 3 27 For example note the following MLAB dialog TYPE A A a 2 by 2 matrix 78 TYPE 4 73 a 2 by 2 matrix 1 223 777 2 222 778 Note that TYPE A 3 is the same as TYPE A A A Raising a square matrix to the power zero yields an identity matrix of the same size and raising a nonsingular square matrix A to the 1 power yields the matrix inverse as shown in the continuation of the above dialog TYPE AO za 2 by 2 matrix 1 1 0 2 0 1 B A 1 TYPE B A B B a 2 by 2 matrix 8 2 1 0 0 7 2 0 0 3 za 2 by 2 matrix 1 1 2 2 22044605E 16 1 Note that A B is not a perfect identity matrix due to small rounding errors in computing the entries of B Nonsingular square matrices can also be raised to negative integer powers For example A 7 5 is the same as B 5 if B is the inverse matrix of A Matrices raised to integer powers are also defined for negative powers of singular square matrices and even for negative powers of non square matrices The Moore Penrose generalized inverse is used however further discussion of these cases will be omitted here In MLAB an element by element matrix product operation denoted by asterisk apostrophe is also available as described in Table 3 28 For matrices with the same dimensions the result is obtained by multiplying corresponding matrix entries pairwise If one argument is a column vector then the result is obtained by multipl
207. esti mates 259 If these assumptions are not satisfied then the error values are no more than rough guides If the reciprocal of variances method of assigning weights is used and the errors between the model and the data are samples from normal distributions with zero means then the weighted sum of squared error will have a chi square distribution The degrees of freedom will be equal to the number of data points minus the number of fitted parameters In particular the expectation of the weighted sum of squared error will be equal to the number of degrees of freedom Because of the limited scope of this Beginner s Guide only brief mention can be made of the statistical problems associated with curve fitting The discussion and references in the MLAB Reference Manual and MLAB Applications Manual give more detailed information about these concerns 7 14 CURVE FITTING SEARCH STRATEGIES The curve fitting search process in MLAB has been designed to work for most linear models and a wide variety of nonlinear models However the search methods will fail in some cases For linear models failure will be a result of numerical computation difficulties such as arithmetic overflow division by zero or unacceptable buildup of numerical error during computation The problems may be inherent in the model and data However they can be avoided in some cases by mathematical transformations of the model and data rescaling taking logarithms etc Si
208. except for the element in the lower right corner which equals SE3 The two forms M SE1 SE3 and U SE1 SES are usually used for column vectors and are equivalent to M SE1 1 SE3 and U SE1 1 SE3 If necessary SE1 and SE2 are made into integers by the INT function As usual the underscore _ may be used for the equal sign An error message is typed if a zero or negative row or column is specified Table 3 7 Matrix entry assignment command a a d a 2 d a 3 d a n 1 d with the jt term a j 1 d The MLAB colon expression a b d describes the column vector of the finite arithmetic sequence which goes from a to b in steps of size d The form a b may be used if d 1 Some examples are given below 55 2 6 2 equals LIST 2 0 2 4 6 e 1 6 1 equals LIST 1 0 1 2 3 4 5 6 e 2 01 2 08 0 03 equals LIST 2 01 2 04 2 07 e 0 10 0 05 describes a column vector of length 201 with components 0 0 05 0 1 9 9 9 95 10 0 195 terms omitted 11 25 describes a column vector of length 15 with components 11 12 13 25 respectively since d 1 is assumed e 1 10000 2 describes a column vector of all the odd integers less than 10000 Note that the sequence may be increasing d gt 0 or decreasing d lt 0 Also b determines the length n of the sequence by the agreement that b is the largest acceptable term if the sequence is increasing or is the smallest acceptable term if the sequence is dec
209. ext file that contains too few numbers to fill up the specified size of the matrix The complete form of the READ command is shown in Table 5 3 In that case rows are successively filled until all available numbers have been used any incomplete rows are filled with zeroes For example if ROD1 contains 145 numbers then the command in the previous paragraph would create a 73 by 2 matrix M14 with M14 73 2 equal to 0 This property of the READ matrix expression permits the user to overestimate the number of rows in the matrix if the exact number of rows of data are not known However the number of columns must be known exactly If the number of columns is not specified 1 is assumed The MLAB operator READON constructs a matrix with numbers read from a file It is identical to READ except that it leaves the file open after a portion of it has been read A subsequent READON 163 The general forms of the READ matrix expression are READ filename ext SE1 SE2 READ filename SE1 SE2 READ CON SE1 SE2 The first form is used for an arbitrary disk file The second form is used if the disk file name has the extension DAT The third form is used for keyboard input The expression is evaluated to an m by n matrix where 1 lt m lt SE1 and n SE2 The matrix entries must be MLAB constants in character form see Section 2 1 on the source specified Alphabetic and other non numeric characters except a period are treated like blanks
210. fitted values of Legendre s equation 300 A 1 ASCII character numbers 0 to 15 in Hershey fonts 1 to 5 315 A 2 ASCII characters 16 to 31 in Hershey fonts 1 to 5 316 A 3 ASCII characters 32 to 47 in Hershey fonts 1 to 34 317 333 A 4 ASCII characters 48 to 63 in Hershey fonts 1 to 34 318 A 5 ASCII characters 64 to 79 in Hershey fonts 1 to 34 319 A 6 ASCII characters 80 to 95 in Hershey fonts 1 to 34 320 A 7 ASCII characters 96 to 111 in Hershey fonts 1 to 34 321 A 8 ASCII characters 112 to 127 in Hershey fonts 1 to 34 322 A 9 ASCII characters 32 to 63 in TrueType fonts 2 to 14 323 A 10 ASCII characters 64 to 95 in TrueType fonts 2 to 14 324 A 11 ASCII characters 96 to 127 in TrueType fonts 2 to 14 325 334 Appendix D BIBLIOGRAPHY 1 PEcuyer Pierre Efficient And Portable Combined Random Number Generators Communi cations of the ACM 31 1988 742 749 2 Kerner Daniel MLAB Graphics Examples Civilized Software Silver Spring MD 2014 3 Knott Gary D and Kerner Daniel MLAB Reference Manual Civilized Software Silver Spring MD 2002 4 Knott Gary MLAB Applications Manual Civilized Software Silver Spring MD 1998 5 Marple S Lawrence Digital Spectral Analysis with Applications Prentice Hall Inc
211. from backspace no special line editing facilities are present so for example don t try to use arrow keys while typing commands When an error occurs however such as a mistyped command DOS MSWindows and Macintosh OS7 8 9 MLAB will invoke a special single line editor which allows you to correct the erroneous command Macintosh OSX and Linux Unix versions of MLAB do not yet have such a feature However you can use the up and down arrow keys to bring previously typed commands to the current command line and use the left and right arrow keys to move the text cursor in the current command line and delete or insert characters to edit the command For individual exploratory computations it is appropriate to directly type MLAB commands Usually however it is most convenient to construct a do file with the text editor facility in MLAB under DOS or MSWindows or with a parallel running editor program in other environments You can repetitively execute such a do file and revise it to construct a useful reusable script MLAB accesses or creates the following types of files which are discussed below 156 1 log files 2 startup do files 3 save files 4 plot files 5 print files 6 read files 7 do files and 8 MLAB control files Each computer disk file has a name of the form filename ext where filename is a user selected name In the case of DOS the filename is limited to eight characters beginning with a letter and f
212. g in the bile is dy dt dyz dt k y key t d with initial conditions y 0 100 and y2 0 0 where y t is the amount of the drug in the blood at time t and y2 t is the amount of the drug in the bile at time t a Curve fit d and k to the unknown functions y and ya using the following measurements Time Blood Bile 3 23 3 21 8 4 15 1 52 1 5 7 9 71 8 6 4 3 84 3 7 2 5 90 9 8 2 2 96 0 9 2 1 97 3 10 0 6 98 4 308 Use d 2 5 and k 5 as initial estimates of the parameters Use the Gear method with symbolic derivatives no interpolation and tolerance level of 003 b Draw graphs of the fitted curves and the experimental data 309 MLAB can solve a first order system of n differential equations n lt 1 with n initial conditions having the same argument that is the system is an implicit definition of n unknown functions y 1 yo x Yn 1 in an independent variable x given by means of n differential equations of the form dy hg CCE cg Via 2 haa yi a ya 2 Yn x dee hy z 2 vle y 2 together with n initial conditions one for each unknown function at a common value x u for the independent variable as follows yilu 21 yo u 22 Yn U Zn The functions h1 ho An are to be given by expressions which may involve the independent variable x the unknown functions y a the derivatives y x of the unknown functions and delay te
213. g ps file it always creates a new file with the first unused name in the sequence mlabp0 ps mlabp1 ps mlabp2 ps etc The MLAB response to a PLOT command shows the name of the computer disk file that was created For example the MLAB dialog PLOT PLOT file is mlabpO ps indicates that the computer disk file mlabpO ps was created containing data needed to display or plot the current picture on a PostScript printer If PLOTDEV is set to generate Hewlett Packard Graphics Language output i e PLOTDEV LJ2P or LJ2L then the name of the disk file created by a PLOT command is mlabp 1j where is 0 1 2 The PLOT command never modifies an existing 1j file it always creates a new file with the first unused name in the sequence mlabp0 1j mlabp1 1j mlabp2 1j etc Using the PLOT command it is even possible to do MLAB graphics blind without viewing the current picture However for elaborate plots such a procedure is likely to lead to errors unless it has been tested thoroughly in advance For simple plots the danger of requiring several iterations is small MLAB output ps and 1j files are not in a form suitable for subsequent input to MLAB The MLAB SAVE and USE commands may be used for window data items for that purpose In order to physically plot an MLAB plot file you must either type a PRINT FILE command or use the facilities of your operating system to send the plot file to your printer 5 6 THE DO COMMAND As previously de
214. g the initial estimates 5 Set MLAB FIT control variables LSQRPT MAXITER TOLSOS and FITNORM as necessary 6 Execute an MLAB FIT command directly or via DO command which specifies the list of fitted parameters functional form data matrix and weight vector The list of fitted parameters may contain matrix names in which case all elements of listed matrices are fitted parameters Multiple clauses each specifying a functional form and weighted data may be used in the FIT command 7 After examining the FIT command output modify the model continue curve fitting or prepare graphical displays as needed Using the above framework as a guide the MLAB user can often construct DO files and SAVE data items that will substantially speed up data analysis involving curve fitting of functional forms In the following let CR denote a carriage return F denote a function identifier csi denote a constraint set name sci and sc2 denote positive scalar constants si denote a scalar identifier SE1 denote a scalar expression ME1 and ME2 denote matrix expressions B1 B2 Bp denote scalar or matrix identifiers and clause1 clause2 clausek denote functional form clauses 1 MLAB Commands for Curve fitting a Constraint Set Command e CONSTRAINTS csi expressions Constraint terms sc si sc si and si sc are separated by or to make expres sions Two expressions are separated by lt lt gt or gt to make a constraint
215. g the result to the first The resulting equation and the second differential equation above yield the equivalent system dx dt dy dt 2 t t x 1 22 dx t pane 289 Then z t and y t can be numerically evaluated as shown in the following MLAB dialog FUNCTION X T T FUNCTION Y T T INITIAL X 0 0 INITIAL Y 0 0 M INTEGRATE X T Y T 0 5 TYPE M COL 1 3 M COL 1 4 5 2xT T X 1 T T X T X DIFF T XK XA XA XA XX a 6 by 3 matrix 0 0 585834027 707041658 1 10558751 357763487 1 36755442 189733946 1 51493649 114132694 1 60777408 7 54280607E 2 DO 0H4sS0Ne Munro a 6 by 3 matrix 0 0 0 1 585815653 1 29295578 2 2 21117613 1 82111252 3 4 10266227 1 93675472 4 6 05974405 1 97146672 5 8 03886791 1 aAmnRWNEF 98491439 The general techniques in MLAB for setting up a system of first order differential equations and obtaining numerical solutions of such systems by the INTEGRATE operator are given in Table 8 1 and Table 8 2 8 3 HIGHER ORDER DIFFERENTIAL EQUATION SOLVING IN MLAB Consider the third order differential equation or 290 y x y x sec x 0 with three initial conditions This is a third order differential equation because the third derivative of the unknown function y x is the highest order derivative appearing in the equation It can be solved numerically in MLAB with the following commands FUNCTION Y X X Y X SEC X
216. gned fixed values For data points i11 112 Ziln yij with associated weights wij i 1 2 nj j 1 2 m the weighted sum of squared error criterion is given by S bi bas bp Didi Wil Yi fii Ti Vial da bp 072 wiz yi2 f vi21 Vie Lian b1 ba bp HETT Wim Yim f Binns Dimas lt lt a Band Oty b23 bpl Note that S b1 b2 bp is a function of the parameters only because numerical data values are substituted for Y1 Y2 Ym and 1 T2 En when evaluating the expression for S The model may include linear constraints on the parameters 61 ba bp of the form described in Section 7 6 The curve fitting problem for this model and data is to find numerical values for the parameters b1 b2 bn which satisfy all the given linear constraints and minimize the weighted sum of squared error criterion S b1 b2 bn with respect to the permitted assignments It is possible that no such assignment exists Also multiple solutions may occur that is two or more different assignments of numerical values to the fitted parameters may yield the same minimum value for S Table 7 6 Curve fitting a functional form matrix D and column vector V corresponding to the data of Figure 7 2 can be created as shown in the following MLAB dialog TMP LIST 8 3 2 6 1 4 1 85 1 2 1 1 8 1 3 3 1 1 1 4 1 7 75 242 TMP SHAPE 5 3 TMP D TMP COL 1 2 V TMP COL
217. grow according to a proportionality constant k and the growth rate will diminish as the population approaches another constant p This model is defined as n t Tr n t k 1 7 n t and is defined in MLAB by the statements FUNCTION N T T K 1 N T R P N T INITIAL N 0 NO This is a perfectly legitimate MLAB ordinary differential equation model definition It may be integrated as shown by the following dialog NO 1 P 50 K 5 R 2 TYPE INTEGRATE N T 1 10 a 10 by 3 matrix de d 1 63231881 799836218 288 37 49 COOAONOOAFWN FOANA AFEWND E 2 66436437 4 33528933 6 99557018 11 17 26 1366652 3491089 1231128 3497516 5466991 O 59 4770903 1 304811 2 09612954 3 30946766 5 08077898 7 44985736 10 1292169 12 1545347 11 7793934 7 52399967 In this model MLAB uses NO as the value for N for any time before T 1 Next consider the two differential equations dx dt dy dt dy Oe dt a dx 7 dt The following MLAB dialog produces an error message because of the circularity of X T XX XA XA XX FUNCTION X T T FUNCTION Y T T INITIAL X 0 0 INITIAL Y 0 0 M INTEGRATE X T Y T 0 5 2 T T Y DIFF T T X T X DIFF T T Error stack space too low probable infinite recursion To avoid this circularity eliminate dy dt from the two differential equations above by multiplying the second equation by t and addin
218. guments 0 0200005 76 Vector LENGTH Operat 2 26 sae a ee ee ee ON Ge oe ee ors 77 Matriz product oprati ec e pratade a Se EER RR OE be AE eR GH 78 Matrix power operatio caca ka ee bok we a he ee ee ee 80 Element by element matrix product operation 2 2 0 00 ee eee 81 The CDF and RANKORDER operators general forms 2048 83 ON POINTS and MAPPLY matrix operators 2 se onrera tesar eeunaat 86 LOOKUP Matrix Operator o clack Go She ae eon ee Bo BR ee a ah as 88 FUR command simple form s s s s s sadaca eee ee de 89 FOR command compound forms cesa OR OR a ea eee ew 90 The VIEW and UNVIEW commands oaoa 97 WINDOW FRAME and IMAGE commands 2 000004 eee 101 a o ss e Re ea eee ee a Oe ee ee a 108 Solid filled point typ S s ee saa sas sco aam EERE A ER OR Re ees 109 A oe ne Gr Ge een Ge a ee Sk ae a Ree e e a ed 109 DRAW command short Terie nce ee eee eo ew we a ae Be EO ee a 111 DRAW command complete forni o s ad sonaa cee eee ee Ee 117 AS Color names and numbers s ccna sonore d aa a RES ES 125 AO AXIS command complete Torn oa see a lt 2 48 Fee baa hehe NA ead 128 A10 TITLE SOARES me ee A ae ee Se Re 133 4 11 String modification command codes 000002 eee ee 138 4 12 BLANK and UNBLANK commands coco e eee 139 4 13 CURVEM and LABELM operators osos 060480 5546 He a ee ee ee a 142 4 14 Window renaming assignment commands 2 0 0
219. gure 7 5 Fit of X t and B t using the EWT operator R squared 9 768205e 01 DRAW XFIT2 POINTS X 0 100 IN W1 DRAW BFIT2 POINTS B 0 100 IN W VIEW There is now considerable improvement in the fit of B T to the observed data as shown in Figure 7 5 Table 7 8 presents the general form of the MLAB FIT command Criteria other than the sum of squared error or weighted sum of squared error criteria are avail able in MLAB for curve fitting The MLAB commands FITNORM k FIT B1 B2 BP F TO D WITH WEIGHT W for k gt 0 will minimize the quantity 250 m S X wi yi Fu tia Lini bi bajes bp TA el with respect to parameters b1 b2 bp Of the function f When k 1 S is the weighted sum of the absolute values of the errors When k 2 S is the weighted sum of squared error criterion discussed in Section 7 7 7 10 THE FIT COMMAND CONTROL VARIABLES When an MLAB FIT command is used to curve fit a functional form a collection of predefined MLAB scalars control the curve fitting process The previous section described two of these control variables LSQRPT and FITNORM there are others These scalars have default values that may be changed by the user prior to the FIT command Another control variable is MAXITER MAXITER determines the maximum number of iterations to be performed That is the curve fitting process will terminate either because it converges or because the specified maximum n
220. he charac teristics of a string while in the process of drawing it These codes are embedded in the string of a TITLE command and are of the form nC where the apostrophe indicates the start of the code n is an optional real number argument and C is a command code These codes are listed in Table 4 11 and examples of many of them are given in the following commands WINDOW O TO 1 O TO 1 INW TITLE T1 f x 6U Zxty Z 3B 1Dx y AT 1 1 WORLD SIZE 06 IN W TITLE T2 d 4t 6U 582 25 6D 2 AT 1 5 WORLD SIZE 06 IN W VIEW This dialog produces the picture shown in Figure 4 12 The nU and nD commands cause following characters to be moved up or down respectively by n character heights The nZ command causes underlining to be toggled on or off which in this example is used to make the horizontal fraction bar The nB command causes the following n characters to appear exactly over the n preceding character In other words nB means backspace n characters The nS code will change the character size by a scale factor of n so the 5S command in the second string will halve the size and the 2S will double it to return it to its original size The commands TITLE T1 i5Ta 1T 26TI 1Tu dx TITLE T2 y 25Tkjjj 3B 1Tx y VIEW 134 Figure 4 12 Changing character size and position 135 Figure 4 13 Changing character font tables produce the pictur
221. he number in M 1 1 which is 100 is interpreted as an interruption marker for drawing the corresponding curve Since the marker occurs 3 times in M COL 1 at M 1 1 M 6 1 and M 10 1 the curve is drawn in 3 pieces corresponding to rows 2 through 5 7 through 9 and 11 through 12 of M rows 1 6 and 10 Table 4 3 Point types 108 Filled Point types Symbol Description 3 TRIANGLE solid filled triangles 4 SQUARES solid filled squares 5 STAR solid filled stars 7 OCTAGON solid filled eight sided figure 13 CIRCLE solid filled variable resolution circle 16 DIAMOND a solid filled diamond symbol Table 4 4 Solid filled point types Line types Symbol Description 0 NONE no lines 1 SOLID solid lines default 2 DASHED short dashes 3 LDASH long dashes 4 DDASH alternating long and short dashes 5 ALTERNATE connect every other point 6 MARKER solid line with marker skipping T VMARKER variable dashed line with marker skipping 8 DOTTED dotted line 9 DOTMARK dotted line with marker skipping 10 DASHMARK dashed line with marker skipping 11 SMARKER solid line with marker skipping using spline curves 12 SVMARKER variable dashed line with marker skipping using splines Table 4 5 Line types are not data points More precisely a new curve created by the MLAB command DRAW C M PT NONE LT MARKER could be replaced by three new curves drawn by the commands DRAW C1 M ROW 2
222. he result matrix since it is necessary to keep the step sizes small in order to control the error in each step of the computation The solution matrix is complete when the last specified value for x in the INTEGRATE matrix expression is passed that is when a lt b for increasing values b1 b2 by or ax lt bi for decreasing values b1 ba bt 293 The result matrix is then computed using the solution matrix and is returned as the value of the INTEGRATE operator The MLAB user has no access to the solution matrix it is discarded when the differential equation solver completes a numerical solution The control variable MANDSW determines how the result matrix is computed from the solution matrix If MANDSW is FALSE 0 larger step sizes will be used and each row of the result matrix is obtained by a cubic polynomial interpolation formula applied to the solution matrix That is for any 7 lt k there may exist j such that bj lt aj lt bj41 or b gt 0 gt bj 1 and so the result matrix row is not in the solution matrix If MANDSW is TRUE 0 then no interpolation is specified and each result matrix row is forced to belong to the solution matrix That is the step size is reduced where necessary so that for each i lt k there exists j gt 1 such that bj aj because bypassing the next specified value of the independent variable is not permitted More accurate results can be obtained with no interpolation for cert
223. he window W1 In Figure 4 15 for example the critical points 0 3 1 34 2 3332 and 2 1 lie on the boundary of the user rectangle which goes from 0 to 2 on the z axis and from 3 to 2 5 on the y axis If another curve is added to a window which has been defined with the ADJUST WNICE clause the user rectangle will be redefined if necessary so that all curves are within the user rectangle WNICE is one of seven options for the ADJUST clause in the WINDOW statement The other options are WFIX WSHRINK WEXPAND WFLOAT WSLACK and WMATCH Each of these options is explained in Table 4 15 An alternative form of the WINDOW command is WINDOW ADJUST p IN wil The clause IN wi is optional If omitted IN W is supplied p is either one of the words WFIX WSHRINK WEXPAND WFLOAT WSLACK WNICE or WMATCH or a corresponding number WFIX or 0 means neither expand nor shrink the user rectangle limits when adding or deleting curves WSHRINK or 1 means decrease the user rectangle limits if necessary when deleting curves from the window WEXPAND or 2 means increase the user rectangle limits if necessary when adding curves to the window WFLOAT or 3 means shrink or expand user rectangle limits if necessary to accomodate added or deleted curves WSLACK or 7 means increase user rectangle limits by 5 percent of the rectangle extent beyond that determined by minimum shrink extent WNICE or 11 means increase user rectangle limits by
224. heses and separated by commas A defined function may have a null argument list for example FUNCTION F SEO The final part SEO is the scalar expression defining F Usually the scalar expression SEO contains the formal argument list identifiers as well as other scalar identifiers The word FUNCTION may be abbreviated by FCT Only the equal sign is acceptable in a FUNCTION command the underscore _ can not be used The FUNCTION command creates or modifies a function data item with identifier F This data item which consists of a formal argument list and a defining scalar expression is stored in symbolic form An existing function data item can be referenced in subsequent scalar expressions If SE1 SE2 SEn is an actual argument list scalar expresssion list the value F SE1 SE2 SEn is obtained by evaluating scalar expres sion SEO which is obtained from the scalar expression SEO defining F If the formal argument list for Fis A1 A2 An then SEO is ob tained from SEO by substituting SEk for each occurrence of Ak in SEO k 1 2 n and then SEO is evaluated as usual Table 2 10 FUNCTION command as in the Chapter 1 tutorial and A 003 B 4 and C 3 are scalars then Y C 1 is computed in MLAB by the steps Y C 1 Y 3 1 Y 4 A EXP Bx4 003 EXP 4 4 003 EXP 1 6 0148591 A scalar expression defining a function may contain references to other defined function
225. ied are treated as scalar identifiers A list of all of the identifiers defined as functions in an MLAB session can be obtained by giving the command TYPE FUNCTIONS 2 4 RELATIONS AND DEFINITION OF FUNCTIONS BY CASES In mathematical applications functions are often defined by fitting together parts of several other functions In MLAB the conditional scalar expression IF THEN ELSE can be used to construct MLAB functions of this kind In Figure 2 1 for example the function y f x given _ fexp az ifx gt 0 He 1 otherwise is shown for a 3 This function can be defined and used as in the following MLAB dialog 26 1 25 0 75 0 5 0 25 0 4 0 2 0 8 Figure 2 1 Graph of y f x 27 1 1 05 0 83 0 61 0 29 10 30 70 110 150 190 Figure 2 2 Graph of y g t FUNCTION F X IF X gt O THEN EXP A X ELSE 1 A 3 TYPE F 2 F 1 1 4 97870684E 2 Observe that the condition X gt 0 following the word IF determines whether the THEN clause or the ELSE clause scalar expression is evaluated to obtain the numeric value of the conditional expression In Figure 2 2 the graph of y g t is given by t sin t if 0 lt t lt 180 degrees 0 otherwise This function can be defined by the MLAB command 28 1 05 0 83 0 61 0 29 0 05 0 37 0 79 1 21 1 63 2 05 Figure 2 3 Graph of y h x FUNCTION G T IF T gt O AND T lt 180 THEN SIND T ELS
226. ighted sum of squared error criterion osooso a a 238 Examples of linear constraint forms e a 240 CONSTRAINTS command i e roo oa doae a eR aa a a i A 241 Curve fitting a functional form lt o e oos sa sacre aa o ee 242 Data preparation for curve fitting operations a 243 FIT commands for a functional form lt se a sew e sco namma doe sa oa 278 The general problem of curve fitting a functional form o oo a a a a 279 Linear model functional formi s s s s s sas ea c 00 ponina a G e da ee 279 Solving first order differential equations systems a 310 Solving first order differential equations systems continued 311 FIT command differential equations systems 1 2 ee ee 312 FIT command differential equations systems continued 313 330 Appendix C LIST OF FIGURES List of Figures 1 1 1 2 1 3 2l 2 2 2 3 4 1 4 2 4 3 Plot of dry weights ve age io ee a 6 Data points and best fitting exponential curve o a 9 Best fitting exponential curve suited for publication 0 10 Cie OH FT ces hee a eee eee Re CRS ee ee Cee 27 Ci A 28 ph Gt PS MEL ciar ni A Sea SRS 29 The aligned rectangle with x in 4 8 and y in 10 40 98 Plot ol Gree TB ee Ge a A ee a we we a 102 Illustration of MLAB user rectangle frame and image rectangles 104 331 4 4 4 5 4 6 4 7 4 8
227. ights w i 1 2 3 4 are wi 4 for 21 y1 U1 S 1 0 2 0 2 3 wea 8 for x2 ya ve 1 1 1 8 1 4 w3 25 for 23 Y3 V3 1 2 1 6 1 3 wa 400 for xa ya va 1 3 1 4 1 2 The weighted sum of squared error S is given by S 4 2 3 1 0 2 0 2 8 1 4 1 1 1 8 2 25 1 3 1 2 1 6 2 400 1 2 1 3 1 4 2 11 83 In effect weighted areas of squares formed from line segments parallel to the v axis in three dimensional x y v space are summed Each line segment goes from 2 yi vi to i yi f i yi l for i 1 2 3 4 A vertical line segment was dropped from the data point to the graph of v f x y a surface in three dimensional space The weighted sum of squared error criterion is described in Table 7 2 7 5 JOINT MODELS OF SEVERAL FUNCTIONAL FORMS In certain experiments the user may have two or more sets of data to be modeled to several functional forms that contain some common parameters For example the models 236 In the general case suppose that a dependent variable y is a function F x1 22 Zn of n independent variables 21 12 Zn For data points 241 L 2 Tin Yi and assigned weights w i 1 2 m the weighted sum of squared error criterion S Xia wily Fzs1 Li2 ls is used as a measure of the agreement between model and data If n 1 and w 1for 1 2 m S is the sum of squared error criterion discussed
228. ilarly the MLAB user by using matrix expressions and matrix assignment commands can perform matrix calculations The general form the matrix assignment command is shown in Table 3 1 With the TYPE command the user can show all or part of a specified matrix or directly type out the matrix that corresponds to a specified matrix expression Because matrices have more structure than scalars the matrix operations and assignment com mands tend to be somewhat more complicated than scalar operations However matrix manip ulations give the MLAB user considerably greater power to perform calculations and generate pictures Mastery of the basic matrix operators and manipulation techniques is an important part of the MLAB user s training 3 1 MATRIX ASSIGNMENT STATEMENTS AND MATRIX DIMENSIONS The simplest matrix assignment commands for creating and changing MLAB matrices are similar to the corresponding scalar assignment commands In Chapter 1 the successive commands 6 15 05 POINTS Y M ss AT The matrix assignment command has the general forms where ME1 is a matrix valued expression unknown identifier ME1 matrix identifier ME1 As in scalar commands an underscore symbol _ may be substituted for the equal sign Table 3 1 General forms of matrix assignment command were matrix assignment commands of the first and second types respectively Here 6 15 05 and POINTS Y M are matrix expres
229. ill map the user clipping rectangle to the right half of the frame rectangle The border about an image rectangle of a window object is drawn automatically unless the user makes special provision not to do so That is a window data object will be displayed with the outline of the image rectangle even if it has an empty list of curves The curves in a window are always drawn within the image rectangle of that window in the MLAB picture There is an exception characters of titles and axes may lie outside the associated image rectangle The image rectangles of different windows are completely independent In particular several windows objects can be displayed together to form a picture with the corresponding image rectangles disjointed overlapping or with small rectangles inset in larger rectangles as needed The MLAB graphics command WINDOW can be used to create a window object or to modify the user clipping rectangle of an existing window object The MLAB command FRAME can be used to specify or modify the frame rectangle in a specific window Similarly the MLAB command IMAGE can be used to create a window data item or to modify the image rectangle of an existing window The WINDOW FRAME and IMAGE commands are summarized in Table 4 2 For example consider 99 the commands WINDOW 1 TO 5 O TO 100 IN PIC FRAME O TO 5 O TO 1 IN PIC IMAGE 1 TO 9 1 TO 9 IN PIC Suppose PIC is an unknown MLAB identifier in each of these statem
230. in W W XAXIS in W CO in W 140 C1 in W TYPE AXES currently defined AXES W YAXIS in W W XAXIS in W TYPE TITLES currently defined TITLES W TOP in W Information about a specific curve axis or title can be obtained using the TYPE statement as well For example if a curve item has the identifier CO the MLAB command TYPE CO responds with information as appropriate about the DRAW command clauses that are currently in effect for CO A typical response is curve CO UNBLANKED in window W curve data matrix 10 rows points pointtype 0 font 5 size 0 02 FFRACT color BLACK 1 lines linetype 1 color BLACK 1 no label matrix labels font 5 size 0 02 FFRACT color BLACK 1 format 3 5 0 0 2 0 angle 0 place left bottom xoffset 0 yoffset O FFRACT initial axis window 0 10 O 10 wclipsw 1 However the curve matrix and label matrix are not typed out The curve matrix and label matrix of a curve can be recovered by using the CURVEM and LABELM operators as described in Table 4 13 For example the command TYPE CURVEM CO LABELM CO would yield a complete description of CO s curve matrix and label matrix Thus it is not necessary to retain matrices which have been used to create curves or labels since such matrices can always be recovered from the corresponding curve data items As usual the CURVEM and LABELM operators can be used as parts of more complicated matrix expressi
231. ination of the magnitude of the successive singular values combined with tests of randomness in the columns of U and V provides sufficient evidence to estimate the number of titrating species with some confidence Only the largest singular values associated with non random columns of U and V as measured by autocorrelation are associated with identifiable titrating species Using the non random columns of V the data may be fit to a sum of Henderson Hasselbach titration functions by least squares methods using the squares of the singular values as weighting factors The Henderson Hasselbach function is a physically motivated mathematical model for the optical trace arising from the titration of a single acid or base See the chapter Analysis of Absorption Spectra 214 Titration Data in the MLAB Applications Manual The general form of the SVD matrix operator is SVD ME1 If ME1 is an m by n matrix A with m gt n then SVD ME1 is an m n 1 by n matrix H such that P DIAG H ROW 1 U H ROW 2 m 1 and V H ROW m 2 m n 2 where UU I VV I and UPV M If M is symmetric that is M M the singular values of M are the abso lute values of the eigenvalues of M and the columns of V are the eigen vectors In any case the product of the singular values is the absolute value of the determinant of M If ME1 is an m by n matrix with m lt n SVD ME1 is computed Table 6 7 SVD matrix ope
232. ined for all SE1 Table 2 5 Logarithmic exponential and hyperbolic trigonometric functions Random numbers are often useful in testing mathematical models Successive use of the MLAB RAN operator summarized in Table 2 7 will generate numbers in a pseudo random series Each use of RAN O will return the next value from the series in use while RAN n for n 0 will begin a new series which will be a function of n The user of random numbers should be aware that there is a theoretical difference between pseudo random and random numbers The algorithm by which pseudo random numbers are produced in 20 Certain commonly used special functions can be employed in scalar expressions as follows ABS SE1 absolute value of x SE1 defined for all SE1 SQRT SE1 square root of SE1 defined for SE1 gt 0 INT SE1 the largest integer y such that y lt SE1 defined for all SE1 Note INT 1 5 is 2 not 1 In mathematics texts sometimes denoted by x for real x MOD SE1 SE2 If SE1 and SE2 are positive integers of sixteen digits or less the value computed is the integer division remainder of SE1 divided by SE2 For example MOD 18 5 3 For any scalars SE1 and SE2 the expression is equivalent to SE1 if SE2 0 and to SE1 SE2 INT SE1 SE2 if SE2 40 LOGGAMMA SE1 natural logarithm of I SE1 defined for all SE1 except SE1 n for n 0 1 2 ERF SE1 error function Era for x
233. ing the corresponding matrix expressions Because some matrix operations have scalar arguments SE1 SE2 SEn will continue to be used to indicate scalar expressions and the number corre sponding to a scalar expression chosen by the MLAB user The matrix dimensions number of rows and number of columns can be obtained by the operations NROWS and NCOLS described in Table 3 2 Each of these operations defines a scalar expression with one matrix argument For example recall the 181 row by 2 column matrix M created and saved in the Chapter 1 Tutorial These dimensions can be checked by following the MLAB dialog USE TUTOR1 USING W CS1 DAY B A M Y TYPE NROWS M 48 181 TYPE NCOLS M 2 In general these dimension operations take any matrix expression argument as shown below NROWS ME1 is the number of rows of the matrix ME1 a scalar NCOLS ME1 is the number of columns of the matrix ME1 a scalar Table 3 2 Matrix dimension operators Like other scalar operations NROWS and NCOLS can be used as parts of more complicated scalar expressions For example if Mis a matrix then 2 NROWS M 1 is a scalar expression 3 2 ENTERING NUMBERS INTO A MATRIX Three matrix expressions LIST KREAD and READ can be used to enter numbers into matrices The LIST expression described in Table 3 3 can have matrix arguments but only the form having scalar arguments will be considered here The KREAD and READ ex
234. ing the contents of the updated text entry fields is returned For DOS systems the optional argument C is a scalar or a 1 or 2 dimensional array that specifies 179 the color attributes for each of the components of the text entry form in the following order the title text the prompt text the text entry field text the title box border and the prompt entry field box border If C is omitted altogether all background colors are black and all text and box border foreground colors are white If C is a scalar all background colors are the color defined by the value of that scalar If C is a 1 column array the background colors for each of the components are defined by the corresponding values of the array and the text and or box border foreground colors are white If C is a 2 column array the first column defines the background colors and the second column defines the text and box border foreground colors The optional argument L is a scalar that specifies the layout style of the tabular form L determines the columnar layout of the prompts and entry fields The width of the tabular form is determined by the overall maximum of the width of the title string and the maximum of the combined widths of the prompt and entry fields We have the following options e L 1 default The prompt strings are all left justified in the form The widths of all the prompts are the same and this width is the width of the entire form less the width of the text
235. ining the entries in the menu X is an optional scalar defining the menu option that is selected when the menu is first displayed on the screen If the user should strike the ESCape key at any time while the menu is displayed on the screen the option specified by argument X will again be selected If omitted the first top most option is selected If X is less than 1 no option is initially selected when the menu is displayed C is an optional vector defining the colors used to draw the title title background menu options and menu background C is only supported on DOS systems On Linux Unix with X Windows Macintosh and MSWindows systems the menu text and selected radio button are drawn in black on a white background and a Continue button appears under the menu If the argument T is provided as a string array then each element of the string array T will appear as a line of text above the menu MENUCHOICE returns a scalar integer corresponding to the number of the menu option selected by the user when the Continue button was clicked or the RETURN ENTER key was struck If argument S is not provided then only the text in argument T will appear in the menu window and the window will remain on screen until the Continue button is clicked or RETURN ENTER key is struck In this case MENUCHOICE returns the value 0 Table 5 8 General form of MENUCHOICE the form is removed from the display and a string array contain
236. into the curve fitting strategy Usually linear models will converge to the unique answer in one or two iterations starting from almost any initial values for the fitted parameters Although the following nonlinear curve fitting problem is somewhat artificial it serves to show many of the difficulties that can arise in curve fitting nonlinear models Consider a functional form f x with one parameter c based on a particular function h a as shown f z h c h a 1 cosh x 6 x x 8 2 18 2 5 cosh 2 2 5 261 Qe JO 25 20 15 10 5 0 5 10 15 20 25 30 Xx Figure 7 6 Graph of f x and h x for c 12 Part of the graph of the function A is shown as a solid line in Figure 7 6 The z axis is a horizontal asymptote to the graph of h in both directions h approaches O from below as x goes to plus infinity and h approaches 0 from above as x goes to minus infinity For particular values of c the graph of f is obtained by shifting the graph of h left or right For example at c 12 f x h w 12 so f 12 A 0 and so on Then the graph of f is obtained by shifting the graph of h twelve x axis units to the right This is shown as the dashed line in Figure 7 6 Similarly for c 1 4 the graph of f is obtained by displacing the graph of h 1 4 x axis units to the left The data for this example is assumed to be entered into an MLAB matrix M as shown in the following dialog M COL 1 1 1 1 M COL 2 LIST
237. intosh OSX MLAB move the elevator scroll box on the side of the bash or Darwin shell window up or down with the mouse pointing device to review the MLAB session history Striking the up and down arrow keys will bring previous MLAB commands of the current session to the command line When a previous command is brought to the current command line in this manner it can be edited by moving the text cursor anywhere in the command line with the left and right arrow keys and then typing or deleting characters The Insert key on the keyboard will toggle the keyboard between Insert and Overwrite modes With Macintosh OS7 8 9 MLAB move the elevator scroll box with the mouse pointing device to view MLAB commands and responses from earlier in the current MLAB session Descriptions of many MLAB functions can be obtained with the HELP command Type HELP at the MLAB prompt to get an overview of the help system or type HELP lt topic gt where lt topic gt is an MLAB statement or function name for information on the specific topic It is preferable however to use the MLAB Reference Manual to obtain the definitive description of the commands and functions provided by MLAB You are almost ready to complete this MLAB session and return control to the computer s oper ating system Let us suppose that you first wish to save the matrices the current picture and other data that you have defined so that they may be used in another MLAB session You can save
238. ion S a b is an explicit function of a and b depending upon the data and the functional form for the straight line In this case it can be shown that there are uniquely determined values for a and b such that S a b is a minimum These values for the y intercept a and slope b determine the regression line of y on x for this data In many cases a mathematical model will not be appropriate for all possible assignments of values to the unknown parameters For example it may be known that a certain parameter is positive or lies between certain known values Alternatively it may be known that one given parameter is larger than another or that one parameter is a constant multiple of another or that the sum of certain parameters is a known constant In MLAB any number of these conditions can be described by a system of linear constraints Given a list b1 b2 bn of parameters to be fitted each linear constraint for these parameters has one of the three forms ry by r2 b2 Tp dp lt ro r bi r2 b2 Tp bp 19 ry by r2 bo Tp bp gt TO where ro 11 Tp are specified scalar constants Note that lt and gt can be used but are replaced by lt and gt respectively when displaying the constraint set Theoretically equal quantities will often have slightly different computed values due to numerical error so this difference is usually unimportant when applying linear constraints during a numeri
239. irected when the form is first displayed If X is omitted the initial active text entry field defaults to 1 the top most field Y is an optional scalar or 1 dimensional array that specifies the width in characters of the text entry fields if Y is omitted all text entry fields will be 10 characters wide if Y is an integer scalar all text entry fields will be Y characters wide if Y is a 1 dimensional array the width of each text entry field is determined by the value of the corresponding element of Y In all cases if an initial value for a text entry field is defined by H and the width of the initial string is more than the width defined by Y then the width will be expanded to accomodate the initial string defined in H C is ignored on Linux Unix with X Windows Macintosh and MSWindows systems All background colors are white and all foreground text colors are black For DOS systems the optional argument C is a scalar or a 1 or 2 dimensional array that specifies the color attributes for each of the components of the text entry form as described in the text The optional argument L is a scalar equal to 1 2 3 or 4 that specifies the layout style of the tabular form as described in the text 187 Table 5 9 General form of GETSTRINGS The general form of the WREAD command is WREAD XL YL Z where scalar integer X specifies the number of points to be read If X is omitted the default value of X is
240. irectory which is the directory where the MLAB program is located 2 your home directory which on multi user computer systems is the directory you are in when you log on to the computer and 3 your current directory which is the directory you are in when you start the MLAB program In an MSWindows system the MLAB executable directory is specified during the installation process it is C MLAB if MLAB was installed in the default location The home directory is the same as the current directory The current directory can be determined by examining the properties of the MLAB application with the Program Manager which refers to the current directory as the working directory To run MLAB double click on the MLAB icon located in the Desktop folder or the MLAB program group It is not easy to choose your own current directory In a Macintosh system if you followed the installation directions the MLAB executable directory the home directory and the current directory are all the same namely the desktop folder entitled e MLAB for Macintosh OSX e MLAB for Power Macintosh for Macinsosh OS9 and e MLAB for Macintosh for Macintosh OS7 8 On Macintosh OSX systems MLAB is a terminal application that runs in a Darwin terminal window On Macintosh OS7 8 9 systems MLAB runs in its own command window To run MLAB locate the MLAB application icon in the MLAB directory and double click on it It may be helpful to locate the MLAB applicati
241. is issuing a FIT command just like the previous one will continue the curve fitting process through additional iterations 7 11 FORMULATING FUNCTIONAL FORM MODELS To formulate functional form models in MLAB 1 MLAB functions which represent the functional forms must be created 2 numerical values to all parameters in the functional forms including both the parameters to be fitted and any other parameters must be assigned and 3 an MLAB constraint set must be created if the model involves linear constraints For simple models the first step is usually a straightforward translation of the mathematical formula describing the functional form into a suitable FUNCTION command For example the formula b x f z c e c e expressing the function f as a linear combination of two exponential functions could be transcribed into the MLAB command FUNCTION F X C1 EXP A X C2 EXP B X which sets up a functional form with argument X and parameters A B C1 and C2 The Lp norm in Euclidean three space Lol y 2 2P y 2h could be set up as a model by the MLAB command 253 FUNCTION LP X Y Z X P Y P Z P 1 P with arguments X Y Z and the single parameter P In these examples parameters in the functional form are represented by MLAB identifiers When the FUNCTION command is executed these MLAB identifiers may be unknown or correspond to any MLAB data However all the parameter identifiers must be m
242. is used for all items If rows 3 or 4 are omitted the information from rows 1 and 2 is used respectively If C is a scalar it defines the background color while all text and box borders are white If C is omitted the menu text title text menu box border and title box border all default to white and the backgrounds all default to black The following MLAB commands demonstrate how MENUCHOICE might be used to display a menu of mathematical distributions and conditionally define the function F according to the user s selection from the menu T Select a distribution 177 S 1 Normal S 2 Student s t s 3 Uniform S 4 Triangular S 5 Poisson X 3 M MENUCHOICE T S X IF M 1 THEN FCT F X GAUSSF X 1 0 ELSE IF M 2 THEN FCT F X STUTF X 2 0 ELSE IF M 3 THEN F X UNIF X 1 0 ELSE IF M 4 THEN F X TRIF X 0 1 ELSE IF M 5 THEN F X POISSF X 1 In Linux Unix with X Windows Macintosh and MSWindows systems the menu would look something like this Select a distribution o Normal Student s t Uniform Triangular ojo e O Poisson Continue The MLAB operator GETSTRINGS provides a form fill in mechanism for obtaining user input that is also useful in do files The general form of the GETSTRINGS function is summarized in Table 5 9 GETSTRINGS takes 6 arguments the last 4 are optional GETSTRINGS T S H X Y L displays a tabular form wi
243. ithmetic operations available for use in MLAB matrix expressions Arithmetic operations involving one scalar argument and one matrix argument allow adding a scalar to a matrix e subtracting a scalar and a matrix from each other multiplying a matrix by a scalar e dividing a matrix by a scalar Caution the operation of raising a matrix to a scalar power follows a different rule to be discussed later The forms of matrix arithmetic operations with scalar matrix arguments are given in Table 3 23 and examples are given in the following MLAB dialog TYPE V V a 2 by 3 matrix 11 1 7 0 73 2 6 2 4 U 6 TYPE U 2 V a 2 by 3 matrix te 2 10 3 2 9 5 1 TYPE 0 5 V a 2 by 3 matrix de 1 5 6 5 O 2 5 5 1 5 4 5 5 TYPE 2 V a 2 by 3 matrix ey 2 14 0 2 12 4 8 TYPE V COL 1 2 U 1 a 2 by 2 matrix ls 0 2 1 4 23 1 2 0 4 There are MLAB matrix expressions for the standard matrix arithmetic operations as shown in Table 3 24 The entry by entry negative of a matrix can be computed Also the sum or difference of a pair of matrices can be formed For example observe the following matrix dialog TYPE M1 M2 Mi a 2 by 3 matrix 1 1 5 2 2 3 0 M2 a 2 by 3 matrix de 1 4 12 2 3 6 2 74 Let SE1 and ME1 be respectively a scalar expression and an m by n matrix expression a in all operations below Assume the value of SE1 is x The general form
244. ition the text of the curve JX created above while leaving the text string window size angle font table number shift value and color unchanged If the title identifier is omitted from a TITLE command the new curve will be identified as TO T1 etc just as curves in DRAW commands are identified as CO C1 etc Since first attempts to position text are often unsat isfactory it is advisable to supply title identifiers for new text to facilitate further modifications Any clause can be omitted from a TITLE command creating a new curve except the text string itself As needed the following default clauses will be supplied for a new curve IN W AT 0 0 FFRACT SIZE 02 FFRACT ANGLE 0 FONT 5 SHIFT 0 COLOR 1 PLACE LEFT BOTTOM Note that the default SIZE is the same as the default LABELSIZE for point labels The ANGLE clause is frequently omitted as in most of the TITLE commands shown near the end of Chapter 1 Since quotation marks are used to delimit a string a text string that contains one or more quotation marks presents a special problem One can proceed by using the MLAB built in function ASCII and the addition operator to define the string ASCII X returns the character corresponding to the ASCII value of INT X where X is a number The operator appearing between two string variables results in a string that is the concatenation of the two string varibles Since the quote character is ASCII number 34 in font 5 the MLAB comm
245. ix identifier or an unknown identifier SE1 abbreviates LIST SE1 and SE1 SE2 SEn abbreviates LIST SE1 SE2 SEn Let ME1 denote a column vector if NCOLS ME1 gt 1 then LIST ME1 is used of positive integers j1 j2 jk representing column numbers of M If necessary the INT function is applied to the entries of ME1 to produce jj J2 jk If any ji is O or negative an error message appears Let M be an m by n matrix and ME2 be an m by n matrix expression Then the command M COL ME1 ME2 causes M to become an m by n matrix where n is the maximum of n and ji jo jk Again ifm gt morn gt ka warning message is printed that the dimensions of ME2 exceed the dimensions of the left hand side matrix Then if n lt n n n columns of zeros are added to M Next for each i lt k the i column of ME2 replaces column j of M with cyclic extension or truncation of the i column of ME2 as necessary If i gt n then n is subtracted from 7 until the remainder is between 1 and n That is the columns of ME2 are used cyclically to replace the columns of M as required Table 3 16 COL matrix assignment command MB a 1 by 3 matrix 1 4 2 2 TYPE MA amp MB a 3 by 3 matrix 1 1 1 2 2 6 A 3 66 3 4 2 2 TYPE MA ROW 2 1 COL 2 amp MB za 2 by 4 matrix 1 1 4 2 2 22 4 2 2 TYPE MA COL 1 amp MB a 3 by 3 matrix 1 1 1 1 2 f T
246. ix expansion by cyclic extension is applied when there are insufficient data for an explicit row and or column assignment Matrix expansion by zero extension is applied for implicit row and or column assignment This distinction will become clearer in the examples to follow Some typical uses of ROW and COL matrix assignment commands follow If A is a 20 by 5 matrix then A ROW 5 8 KREAD 4 5 replaces rows 5 6 7 and 8 of matrix A by a 4 by 5 matrix that is typed in at the keyboard If B is a column vector of length 18 A COL 3 B replaces the first 18 elements of column 3 of matrix A by column vector B A 19 3 gets B 1 and A 20 3 gets B 2 in accordance with the cyclic extension matrix expansion principle After these assignment commands A is still the same size but the command 63 A COL 7 B enlarges matrix A to a 20 by 7 matrix with column 6 all zeros and column 7 equal to column vector B plus B 1 and B 2 in rows 19 and 20 respectively Note column 6 was not explicitly assigned so the principle of matrix expansion by zero extension was used for column 6 Data was explicitly provided for column 7 so the principle of matrix expansion by cyclic extension was used for column 7 The command A ROW 3 6 11 19 A ROW 6 19 3 11 permutes rows 3 6 11 and 19 of matrix A while the command A COL 1 2 5 A COL 5 1 2 permutes columns 1 2 and 5 of matrix A If C is an n by 7 matrix for n gt 3
247. k for monochrome display and plotting devices In general the AT clause specifies the location in user rectangle coordinates for the bottom left corner of the box surrounding the first character in the string Note that this is the same as the AT clause for point labels The SHIFT clause allows the string to be shifted along the line running from the starting location of the string to the edge of the user rectangle at the specified angle For example the following commands will produce the picture shown in Figure 4 10 UNVIEW DELETE W WINDOW O TO 10 O TO 10 IN W TITLE Shifting AT 0 8 WORLD SHIFT 0 SIZE 05 IN W TITLE is so AT 0 5 WORLD SHIFT 5 SIZE 05 IN W TITLE nice AT 0 2 WORLD SHIFT 1 SIZE 05 IN W VIEW Note that unlike curves resulting from a DRAW command which are clipped to the user rectangle titles from a TITLE command are clipped to the frame rectangle Titles can extend beyond the image rectangle if the image rectangle is smaller than the frame rectangle Like the short DRAW command the TITLE command can often be abbreviated by omitting unnec essary clauses A TITLE command to modify an existing title need only supply the title identifier and all clauses containing new information Other clauses will remain as previously specified For example the MLAB command TITLE JX AT 7 21 WORLD 129 aia LING Figure 4 10 Shifting text strings 130 could be used to repos
248. l the apostrophe notation can be used to indicate matrix transpose in arbitrary matrix expressions ME1 If ME1 is an m by n matrix a j the result is the transposed n by m matrix a Of course it is sometimes necessary to use parentheses i e the form ME1 to indicate the transpose operation that is desired Table 3 10 Matrix transpose notation TYPE DAY a 2 by 10 matrix 1 6 7 8 9 10 11 12 13 14 15 2 0 111 0 078 0 095 0 126 0 243 0 18 0 202 0 387 0 465 0 831 Note that when a matrix has a large number of columns its rows may be broken between the end of one line and the beginning of the next in matrix output as shown above A method to avoid this annoyance will be given later As noted in the introduction to Section 3 2 the LIST expression may have matrix arguments as well as scalar arguments Table 3 11 summarizes the general form of the LIST operator The general LIST expression of n arguments n gt 1 has the form LIST E1 E2 En The expressions Ei may be scalar expressions SEi or matrix expres sions MEi i 1 2 n The result is a column vector of values If Ei is a scalar expression SEi the element of the vector is the value SEi If Ei is a matrix expression MEi the elements of the vector are the values of MEi taken row by row The value or values are concatenated on the result Table 3 11 LIST expression general form For example
249. l models 256 generating arithmetic sequences 54 GETDIAG 72 213 GETSTRINGS 177 179 181 graphical display control commands 96 graphics shortcuts 142 graphing data 5 graphs 96 GREEN 125 GREY 125 HBAR 106 HELP 12 Henderson Hasselbach titration functions 215 Hershey fonts 314 HISTO 199 histogram 81 199 home directory 155 horizontal bar point type 106 horizontal error bar point type 106 341 horizontal logarithmically spaced tick marks 106 hyperbolic trigonometric functions 19 identifier 15 IDFT 221 IF THEN ELSE 26 30 38 171 172 IFRACT 117 121 133 IMAGE 100 101 147 image fraction units 117 121 image rectangle 98 99 103 146 IMAGEBOX 105 147 IN 100 101 110 128 129 133 143 146 inactive constraints 252 indexed product 33 indexed sum 33 infinite recursion in systems of differential equa tions 289 INITIAL 281 300 initial condition 15 280 300 installing MLAB 154 installing MLAB 1 INTEGRAL 33 34 95 INTEGRATE 280 282 INTERPOLATE 87 194 199 204 interpolation 199 interrupting MLAB execution 4 176 inverse Fourier transform 221 inverse of a matrix 79 inverse trigonometric functions 19 Jacobian matrix 210 295 JACSW 280 295 301 joint models 238 247 254 joint models weighted sum of squared error cri terion 238 kinetics 237 247 254 KREAD 49 50 53 163 164 171 KSREAD 163 165 LABEL 117 119 1
250. lar MLAB identifier which is assigned different scalar values for the different executions of the repeated command Users familiar with traditional programming languages already know the similar concept of the loop To execute a similar sequence of MLAB commands repeatedly a general form for the MLAB command desired must first be constructed using an index variable In the above example of constructing M the general form of the command using the index variable J is M ROW J COL 1 J 1 Note that executing the above command four times with the index variable J assigned scalar values 1 2 3 4 and 5 respectively will create the matrix M wanted The full MLAB command for the loop creating M is 88 FOR J 1 5 DO M ROW J COL 1 J 1 This kind of repeated MLAB command is called a FOR command The simple form of the FOR command is described in Table 3 32 The part of the command between FOR and DO looks like a matrix assignment command associating the column vector 1 5 to the index variable J This is interpreted as specifying the scalar values to be assigned to the index variable J during consecutive repetitions of the command following DO The number of repetitions equals the length of the column vector 1 5 The MLAB commands appearing between the left and right curly braces are repeatedly executed with J taking on the values 1 2 3 4 and 5 Note that the matrix M can also be computed as M DIAG 1775 77 5 0 4 or FUNCTION
251. matrix is the collection of all pairs with a member from each list stored as a 2 column matrix The general forms of the MESH and CROSS operators are presented in Table 6 1 192 0 6 0 2 0 2 0 4 0 6 Figure 6 3 CDF step function graph 193 0 8 The general forms of the MESH and CROSS matrix expressions are MESH ME1 ME2 CROSS ME1 ME2 The MESH matrix operator will alternate the rows of ME1 and ME2 If ME1 is an m by n matrix and ME2 is an m by n matrix the result is an mx by n matrix where mx is two times the larger of m and m and nx is the larger of n and n ME1 and ME2 are expanded with zeros as necessary The CROSS matrix operator will pair up and column concatenate the rows of ME1 and ME2 in all possible combinations If ME1 is an m by n matrix and ME2 is a m by n matrix the result is an m by n matrix where m m m and n n w Table 6 1 Matrix Combination Operators 6 2 MODIFYING DATA FOR ANALYSIS AND DISPLAY Many problems may arise when trying to use experimental measurements for mathematical cal culations and modeling One problem is the irregularity of measurements that is data existing plentifully in some areas but not in others Another problem often found is that noisy measure ments have introduced spurious fluctuations which we wish to remove from the data MLAB has two operators INTERPOLATE and SMOOTH which are designed to combat these respective
252. matrix and are follow ed by a carriage return input is ended and the matrix assignment com mand or other MLAB command will proceed with the READ expression evaluated to the m by n matrix that was typed in Typing CONTROL Z Z or hitting the Escape key means that the typing of the numbers has been completed If fewer than mn numbers are entered the matrix row containing the final number is completed with zeros and no more rows are created Table 3 5 READ expression console form The column vector form of the READ expression is READ CON The method for entering numbers is the same as described above except that CONTROL Z Z is now the only way to terminate input If n numbers are entered up to a maximum of 1000 a column vector of the n given numbers is used for the READ CON expression s value Table 3 6 READ expression vector form The second form of the READ expression shown in Table 3 6 is used simply to enter a column vector of numbers 52 The READ expression has a third form that can be used to read matrix entries from a computer disk file That expression will be explained in Section 5 1 Because it is a common beginner error the user should note that LIST KREAD and READ are matrix expressions and they should be assigned to a matrix or unknown identifier Without an assignment LIST KREAD and READ will simply echo the input This is shown for the READ expression in the MLAB dialog
253. matrix operators MESH and CROSS which are designed to combine two matrices in ways that are difficult to do with just the matrix joining operators of Section 3 4 The MESH operator takes as input two matrices and returns a matrix containing successive rows of the two input matrices taken alternately The odd rows of the resulting matrix rows 1 3 5 will contain the rows of the first input matrix and the even rows of the resulting matrix rows 2 4 6 will contain the rows of the second input matrix The two input matrices will be cyclically extended as necessary so the resulting matrix will have 2m rows and n columns where m is the greater of the numbers of rows in the two input matrices and n is the greater of the numbers of columns of the two input matrices The following commands show two common uses of the MESH operator M1 POINTS SIN 0 10 25 M1 COL 2 M1 COL 2 2 M2 POINTS COS 0 10 25 DRAW M1 DRAW M2 DELETE M1 ROW 1 NROWS M1 2 189 2 2 1 4 0 6 0 2 lt 0 2 4 6 g 0 Figure 6 1 Shading a graph using MESH DELETE M2 ROW 1 NROWS M2 2 DRAW MESH M1 M2 LINETYPE ALTERNATE VIEW The result of these commands is shown in Figure 6 1 The MESH operator was used to combine the points on the two curves into one matrix containing the endpoints of the shading lines LINETYPE ALTERNATE then draws this matrix as desired The MESH operator can be used in many other ways to compute a ma
254. milarly curve fitting non linear models may lead to problems of numerical computation which may or may not be avoidable In addition theoretical difficulties can arise in curve fitting nonlinear models In this section the search strategies and some situations in which the curve fitting process converges inappropriately for nonlinear models will be considered To explain the methods used consider an analogy of a curve fitting process involving two parame ters with a problem involving the latitude and longitude of a certain region of the earth s surface Suppose that the altitude above sea level at any latitude and longitude equals the weighted sum of squared error for a model and data at that point Recall that the weighted sum of squared error is a function of the fitted parameters Imagine standing at some point within the region which corresponds to the initial values chosen for the latitude and longitude parameters An altimeter is available and the problem is to find the point of minimum altitude in the region that is the point nearest to sea level no point is below sea level The terrain near the current location can be examined but the entire region can not be seen There is one obvious strategy go downhill If the current location is on a hillside move in the direction that would lead to the most rapid loss of altitude in the immediate vicinity After moving a distance in this direction stop to assess the new lower altitude and
255. mmand mlab in response to the operating system prompt in a bash shell or DOS command window to begin the MLAB session A response similar to the following should appear on the display screen or in a new window MLAB Mathematical Modeling System Revision November 5 2013 Executing file usr local lib mlab mlab Copyright Civilized Software Inc 301 962 3711 email csi civilized com Web site WWW CIVILIZED COM Tue Nov 5 11 00 03 2013 Your current working directory is C Users csi mlab Use FILEDIR to reach any other directory is the command prompt This copy of MLAB belongs to csi Type exit to exit from MLAB or type another MLAB command In this Beginners Guide text that the user types at the keyboard and text that the computer produces itself are typed in BOLD TYPEWRITER FONT MLAB will then display a menu with the three options e Look at a list of MLAB demonstration do files e Go to top level MLAB e Display some tips on using MLAB This is done by executing the do file STARTUP DO Selecting the first option causes MLAB to execute the do file MLABEX DO in the EXAMPLES sub directory MLABEX DO displays a menu of example scripts many of which are explained in separate chapters of the MLAB Application Manual Selecting an example script from the menu will execute the script and then recursively executes the MLABEX DO script While executing an example script MLAB may pause occasionally to display instructions
256. mmands did not specify a window identifier with which to associate the curves C D and E and title T and yet e the user rectangle is always defined as the aligned rectangle that encloses all of the points in all of the curve matrices e the frame rectangle extends over the entire MLAB picture e the image rectangle has an even margin and its border is drawn 145 e labelled x and y axes with 6 labels and tick marks are supplied and they are relabelled appropriately when the user rectangle changes The reason for this automatic behavior is that MLAB provided the default clause IN W to the DRAW statement Then since W was an unknown identifier MLAB read the save file DW SAV for the window specifications Save files will be described in more detail in the next chapter Typically the default window object W saved in DW SAV was defined by the following MLAB commands WINDOW O TO 10 O TO 10 ADJUST WNICE IN W FRAME O TO 1 O TO 1 FFRACT IN W IMAGE 125 TO 875 125 TO 875 IN Y IMAGEBOX IN W XAXIS 0 10 6 0 LABEL 0 10 6 LABELSIZE 015 FFRACT OFFSET 01 025 FFRACT FORMAT 3 5 0 0 2 0 PT UTICK IN W YAXIS 0 0 10 6 LABEL 0 10 6 LABELSIZE 015 FFRACT OFFSET 09 01 FFRACT FORMAT 3 5 0 0 2 0 PT RTICK IN W The following MLAB dialog shows the status of the window object W when drawn as shown above TYPE W window W UNBLANKED priority 0 window clipping limits world units wxmin 0 wxmax 3 wymin 6 wymax 3
257. n a and ME2 be an m by n matrix expression b throughout In general the matrix joining operations are given as follows ME1 amp ME2 The result is an m m by n matrix cj where n is the larger of n and n The m rows of ME1 are followed by the m rows of ME2 with all rows extended to length n by cyclic replication of columns as necessary More precisely Cig 05 14 G n 1 mod n if i lt m and j lt n Cm i j eee coe ae mod n if i lt m and Jj lt n ME1 amp ME2 The result is an m by n n matrix cij where m is the larger of m and m The n columns of ME1 are followed by the n columns of ME2 with all columns extended to length m by cyclic extension of rows if necessary More precisely Cij U14 i m 1 mod m If i lt m and j lt n Cin j O14 itm 1 mod m j if i lt m and J lt n Table 3 17 Matrix joining operators In general the matrix row repeating operation is given by ME1 SE1 If SE1 is a positive integer n then this expression is equivalent to ME1 amp ME1 amp ME1 n fold repeti tion For any number SE1 the expression is equivalent to ME1 INT SE1 Table 3 18 Matrix row replication operator In general the matrix column repeating operation is given by ME1 77 SE1 If SE1 is a positive integer n then this expression is equivalent to ME1 amp ME1
258. n be used to modify the respective user frame and image rectangles in an existing window These commands will usually change the positions of curve matrix points in the MLAB picture even though the curve matrix itself contains the same numbers as before The dimensions of the MLAB picture c and d are fixed to the full screen dimensions for DOS PC MLAB they may be changed by resizing the MLAB picture on systems with windowing graphical user interfaces The border of the frame rectangle is not drawn in an MLAB picture unless the command FRAMEBOX command is given The border of the frame rectangle will be hidden if the command NO FRAMEBOX is given The border of the image rectangle is drawn in an MLAB picture unless the command NO IMAGEBOX is given The border of the image rectangle can be drawn in the MLAB picture by giving the command IMAGEBOX 4 33 THE DRAW COMMAND The short form of the DRAW command shown in Table 4 6 is primarily used to create a curve data object with a specified curve matrix part and to assign the curve to a specified window It can also be used to modify the curve matrix of an existing curve Consider the DRAW command DRAW CA DAY POINTTYPE TRIANGLE LINETYPE NONE IN W This is similar to the DRAW command given in the Tutorial of Chapter 1 There are five parts to this short DRAW command 1 an unknown MLAB identifier CA to be used as the name of the new curve data object 2 the two column matrix DAY to
259. n by zeroes For example an unknown identifier X becomes a matrix identifier in the following MLAB dialog TYPE X value unknown X 2 3 7 2 TYPE X X a 2 by 3 matrix 0 0 1 0 2 0 0 UN There are similar matrix entry assignment commands specifying only one index which are com monly used with column vectors For example V 5 1 changes V 5 to 1 if V has length 5 or more or first extends V to length 5 by supplying components with zero values if necessary Table 3 7 summarizes the matrix entry assignment command 3 3 GENERATING ARITHMETIC SEQUENCE VECTORS The colon is used in MLAB matrix expressions to denote column vectors of consecutive integers or more generally finite arithmetic sequences of real numbers An arithmetic sequence with n terms and difference d d 4 0 between successive terms has the form 54 Suppose 7 SE1 and j SE2 are positive integers for some scalar ex pressions SE1 and SE2 If M denotes an m by n MLAB matrix then M SE1 SE2 SE3 changes the matrix entry in row 7 and column j to SE3 ifi lt m and j lt n If i gt m or j gt n or both then M becomes a p by q matrix where p is the larger of m and i and q is the larger of n and j The element in row u column v of the enlarged matrix is unchanged ifu lt m u lt n SE3 ifu iandv j 0 otherwise If U denotes an unknown MLAB identifier then U SE1 SE2 SE3 creates an 2 by 7 matrix U with all elements equal to zero
260. n s and t because h 1 1 1987 s 9888 t h 4 1 1 0799 s 9864 t and so on So parameters that remain at fixed values may appear inside nonlinear functions used in linear models Again substitution of a fixed value reduces the nonlinear expression to a number For curve fitting in MLAB a functional form defines a linear model if it acts linearly with respect to the parameter list specified in the FIT command The model h above is nonlinear if s t a and b are fitted but is linear if s and t alone are fitted Note that curve fitting to a polynomial to a truncated Fourier series or to certain other forms of truncated series leads to a linear model If the functional form f appears as 258 fo x1 2 say bn bi AG LQ n b folti 2 ia bp Jol Gin sang re where the functions fo Fi f2 fp do not involve any of the fitted parameters by b2 bp then the model associated with f and b1 b2 bp is linear The general linear model functional form is shown in Table 7 10 7 13 STATISTICAL CONSIDERATIONS FOR CURVE FITTING If parameters in a functional form are curve fit to data it is natural to hope that they will be experimentally reproducible That is it is desirable that repeating the experiment and the curve fitting would lead to stable statistical results Good agreement between the functional form and the data as well as approximately the same values for the fitted parameters sh
261. n the user s keyboard and the computer s memory However other types of input and output are also useful either to save time and effort or to make best use of the computer s facilities In this chapter most of the commands available to the MLAB user for input and output operations are discussed 5 1 FILES MLAB creates and reads various files These files are indexed in various directories In operat ing systems with graphical user interface such as Macintosh and MSWindows systems directories are often referred to as folders For such systems substitute the word folder for directory in the discussion that follows MLAB is itself a file which resides in the MLAB directory When you installed MLAB you should have manually established the appropriate path information for the MLAB directory to allow the MLAB executable file to be located by the operating system when you request that MLAB be run MLAB mandatorily reads and writes certain files and optionally reads and writes certain other files as specified below Some of these files must reside in certain specific directories others may reside in arbitrary directories of your choice 154 It is usual to create at least one specific working directory where your MLAB related data files and do files are kept Generally this directory should be your current directory when you use MLAB In any event there are three directories of interest for running MLAB 1 the MLAB executable d
262. nd 165 PRINT FILE MDATA LST or employ another program or operating system service to physically print the produced file For example in DOS or an MSWindows Command prompt window we can type PRINT MDATA LST at the prompt to cause the contents of our file to be printed on the computer s printer In Linux Unix NeXTStep Unix and Macintosh OSX we can type lpr MDATA LST at the bash shell XTERM or Darwin terminal window prompt to print the file on the computer s printer Note that the MLAB and DOS PRINT commands are different and that the MLAB PRINT command does not itself initiate the printing of data the command PRINT FILE lt filename gt must typed in MLAB The MLAB PRINT command is often used to prepare large arrays or other large volumes of data for subsequent printing on the printer Another valuable use for the PRINT command is to create a disk file of character data for input to another computer program The MLAB TYPE command causes descriptive information to be listed in the MLAB command window when typing out certain data types For example when typing a scalar the scalar name and an equal sign are typed as well as the value of the scalar when typing a matrix the matrix name its dimensions and row numbers are typed as well as the values of matrix elements This descriptive information can be suppressed if the scalar NAMESW is set equal to 0 This suppression applies to the results of both the TYPE command an
263. nd the default name W1 YAXIS is provided where W1 is the window name specified in the IN clause If the axis name is omitted from an AXIS command then the first unique name in the series AO A1 A2 is used as the axis name Tick marks and bands are the usual point types for axes Point types LTICK 9 or RTICK 10 are typically used for tick marks on y axes UTICK 11 or DTICK 12 are typically used for tick marks on x axes NONE is the default point type for axes The PTSIZE PTFONT LINETYPE LABEL LABELSIZE LABELFONT OFFSET PLACE FORMAT ANGLE and COLOR clauses of the AXIS command are provided optionally as in the DRAW statement The window name defaults to W if the IN clause is not included 4 5 THE TITLE COMMAND The TITLE command shown in Table 4 10 is used to place text in an MLAB picture It defines or modifies a title data object which consists of a string and some information about where and how the string is to appear In computer science an arbitrary sequence of characters is called a string The characters are usually printable characters such as upper and lower case Latin letters digits punctuation marks and other symbols that can be typed An example of the TITLE command follows TITLE JX HEIGHT 30 SIZE 015 FFRACT ANGLE 90 FONT 5 PLACE BOTTOM LEFT AT 7 20 WORLD SHIFT 0 COLOR GREEN IN W 128 This command shows nine parts of the TITLE command Its effect would be to place the text
264. nd B bij be m by n matrices and suppose that t is a number Construct MLAB command s by which an m by n matrix C cij can be constructed so that cij aig iy t for alli lt m and j lt n 3 Suppose that real numbers z1 2 and x3 are to be determined by solving the system of linear equations below z 2 23 144 2 T 2 T3 63 LY 192 23 30 93 It is known that this system is equivalent to the matrix equation AX B where 1 2 1 z 144 A 2 0 2 X lxws B 6 Lali il T3 30 If A has an inverse matrix 47 it is known that X AT B is the unique solution for the matrix equation AX B Construct MLAB commands to create the matrices A and B set X equal to 47 B type X and finally type AX to test whether AX B is satisfied Construct MLAB commands to efficiently create the following matrices a A 10 by 10 matrix Z zij such that zi 0 and zij 2 if i j for all i j lt 10 b A 7 by 6 matrix K kij such that kij 1 for i j lt 2 kij 1 if 3 lt i lt 7 and 3 lt j lt 6 and kij 0 otherwise c A 25 by 3 matrix U where column 1 of U contains the integers 25 24 23 1 in descending sequence column 2 is given by U I 2 U I 1 I I for I lt 25 and column 3 is given by U I 3 I U I 1 U I 2 for I lt 25 d An 8 by 5 matrix CPT3 with every row equal to 6 4 1 3 7 2 1 1 1 Hint Use the repeat operator e A 10 by 1
265. ne curves is controlled by the value of the scalar FLATNESS the default value of FLATNESS is 1 In the limit as FLATNESS goes to 0 the spline curve becomes piecewise straight between the actual points of the contour DELETE W DRAW C1 CONTOUR DAI 9 9 5 LINETYPE SVMARKER DRAW C2 DA COL 1 2 LINETYPE NONE POINTTYPE CROSSPT WINDOW ADJUST WMATCH VIEW Figure 6 13 shows the result The CONTOUR operator is described in Table 6 4 The general form of the CONTOUR matrix operator is CONTOUR ME1 ME2 where ME1 is a 3 column matrix of 3 dimensional data values taken over a rectangular grid of x y coordinates and ME2 is a column vector of z coordinate levels to be shown on the resulting contour map The result is a matrix of points which will generate the desired contour map when drawn using linetype VMARKER Table 6 4 CONTOUR matrix operator 6 5 EIGENVALUES EIGENVECTORS AND COMPLEX NUM BERS The EIGEN operator described in Table 6 5 will compute the eigenvalues A1 A2 An and eigenvectors U1 V2 Un of an n by n matrix A EIGEN A is a 2 n 2 row by n column matrix 208 X 0 4 h n a 1 2 CLS 7 SAAS 63393 53223 1 6 2 6667 Figure 6 13 A contour map of interpolated data with spline curve contours 209 whose first two rows have as columns the real and imaginary parts of the n eigenvalues of A and whose 2 k 1 and 2 k 2 rows
266. new terrain Obviously the point of minimum altitude had not been found if the new location remains on the hillside That is the minimum point must be at least as low as its surrounding terrain a local minimum 260 If a relief map of the region were available the search process would be less crude and more likely of success For nonlinear models this might require evaluation point by point but for linear models a short cut is available It can be shown that the relief map for a linear model with two fitted parameters is a paraboloid bowl The vertical cross sections of the surface are parabolas This bowl has a unique point of lowest altitude Furthermore the location of the bottom of the bowl can be computed by analyzing the slope of the terrain near the current location So a second strategy is available assume the model is linear or nearly so and look at the latitude and longitude that the current terrain predicts is the bottom of the bowl Similar analysis is available for all linear models regardless of the number of fitted parameters The curve fitting routines in MLAB use a strategy that combines both of these approaches In each iteration one or more trial points are examined Some of these may be near the current location others near the predicted solution for a linear model and others may be intermediate between the two The first trial point encountered that has significantly lower altitude than the current location is cho
267. ns b Parentheses may be used to indicate order of operations c Matrix input operations are e LIST SE1 SE2 SEn e LIST ME1 ME2 MEn e KREAD SE1 SE2 e READ CON SE1 SE2 e READ CON d Finite arithmetic sequences are e SE1 5E2 SE3 e SE1 SE2 e SE1 SE2 SE3 e Matrix editing operations are e ME1 transpose e ME1 ROW ME2 row selection e ME1 COL ME2 column selection e ME1 amp ME2 above below concatenation e ME1 amp ME2 left right concatenation e ME1 SE1 repeated above below concatenation e ME1 77 SE1 repeated left right concatenation e SORT ME1 SE1 e GETDIAG ME1 e COMPRESS ME1 SE1 91 e ROTATE ME1 SE1 e MAXROW ME1 SE1 e MINROW ME1 SE1 Some operations also work with scalar arguments Matrix arithmetic operations are SE1 ME1 ME1 SE1 ME1 MEQ SE1 ME1 ME1 SE1 ME1 ME2 SE1 ME1 ME1 SE1 ME1 ME2 SE1 ME1 ME1 SE1 ME1 MEQ ME1 SE1 may replace ME1 ME2 ABS ME1 g Functions evaluated at matrix lists of arguments are e F ON ME1 POINTS F ME1 MAPPLY F ME1 DF ON ME1 POINTS DF ME1 MAPPLY DF ME1 3 MLAB Commands for Computation with Matrices are a Matrix assignment commands e U MEI e M ME1 e U SE1 SE2 SE3 e M SE1 SE2 SE3 e U SE1 SEQ e M SE1 SEQ e U ROW ME1 ME2 e M ROW ME1 ME2 e U ROW SE1 ME1 e M ROW SE1 ME1 e U
268. ns test values for B in 1 06 1 02 01 and the third column contains test values for C in 0 1 02 The first TYPE command shows the first second seventh and last rows of M The evaluation of the POINTS expression adds a fourth column to M which equals the weighted sum of squared error S A B C corresponding to the A B and C values in columns 1 2 and 3 The rows of M are then sorted so that the fourth column is in ascending order Since the first row of the sorted matrix is a minimum for S among these trials it is typed The values in columns 1 2 and 3 of the first row are then used as initial estimates for A B and C Note that the MLAB statement M POINTS S CROSS 6 9 CROSS 1 06 1 02 01 0 1 02 which uses the MLAB CROSS operator creates the matrix M above more easily and efficiently than using nested FOR statements The CROSS operator column concatenates each row of the first matrix 272 with every row of the second matrix If the columns of both matrices are in increasing order the rows of the resulting matrix appear in lexicographic order The Table 7 16 SUMMARY Assume the following 1 Data points n 1 coordinates 2 i2 Tin yi With weight w for i 1 2 m 2 Functional form yy fj 1 82 lt lt En b1 b2 y bp Ci C2 s lt Cy for j 1 2 m where the arguments 1 T2 En are the independent variables by b2 bp are parameters to be fit and C1 C2 Cq
269. nt MLAB session the MLAB USE command described in Table 5 2 is given with the filenames previously chosen as in the following dialog USE TUTOR1 Using W CS1 DAY B A M Y Note It is permissible to state an extension other than sav for the name of a SAVE file If the filename is delimited by double quotes the characters between the double quotes will be taken as the exact file name Some USE and SAVE command cautions should be noted e USE commands only work with files created by SAVE commands 160 The general forms of the SAVE command are SAVE D1 D2 Dn IN filename where D1 D2 Dn is a list of MLAB identifiers cannot be unknown curve or surface identifiers and filename is the name of a disk file This statement causes the values of the identifiers D1 D2 Dn to be saved in the specified file On a DOS computer all characters in the filename after the first eight are deleted In any case the extension DAT is added If the identifier list D1 D2 Dn is omitted then all currently defined data items are saved If the filename is omitted then each data item in the identifier list will be saved in its own file with the identifier names used as above for filenames Table 5 1 SAVE command The general forms of the USE command are USE filenamei filename2 filenamen where filenamel filename2 filenamen are names of disk files created by the SAVE command
270. ntegral can be evaluated by using a function with no arguments For example the MLAB commands FUNCTION SS SUM J 1 12 Jx JxJ S SSO assigns to S the scalar value of the sum of the first 12 cubes 35 2 7 EVALUATING AND FINDING ROOTS OF EXPRESSIONS Instead of finding the value of a function at a certain point it is often desirable to find the point at which the function assumes a certain value This is done in MLAB with the ROOT operator summarized in Table 2 14 which finds a point at which an expression equals zero The ROOT operator takes an identifier to use as the independent variable an expression containing that identifier and an interval of independent variable values in which to search for a root It returns a value for the independent variable within the given interval at which the expression equals zero For example to find the point at which the function SIN X equals 75 within the interval 1 57 1 57 the following dialog is used FUNCTION F A B ROOT X A B SIN X 75 TYPE F 1 57 1 57 848062079 Note that like the SUM PRODUCT and INTEGRAL operators the ROOT operator may only appear in function bodies The EVAL operator also summarized in Table 2 14 is used to compactly define functions that have subexpressions embedded repeatedly in them This operator takes an expression an identifier for which to substitute in this expression and an expression into which to make the substitution
271. ntly partial derivatives of functions of several variables are introduced with notations such as w or hy x y 2 Oy yl y for a function w h x y z of three variables for example There are also notations for multiple ordinary and partial derivatives such as dy n Pw a E i E gt or hezit yz etc These notations are not convenient for typing or computer printing and so the symbol apostro phe is used in MLAB to indicate ordinary or partial differentiation The identifier F X denotes the derivative of the function F with respect to the symbol X The symbol with respect to which 37 we are differentiating must be stated F X is the MLAB name of the function 4 Le but F is illegal The special identifier DIFF can be used in place of thus F DIFF X can be written in place of BAX This notation is used to specify various derivatives of MLAB functions which are defined by FUNCTION statements First consider some ordinary derivatives in the following MLAB dialog FCT F X A COS X TYPE F X F X X FUNCTION F X X SIN X A FUNCTION F X X X COS X xA Observe that F X denoting the first derivative aa and F X X denoting the second derivative a are also functions with argument X that is they are defined function data items Secondly note that MLAB has generated correct scalar expressions for derivatives of the trigonometric functions sin x and cos x In most cas
272. observe the following MLAB dialog TYPE R1 R2 Ri a 3 by 2 matrix 59 1 1 2 2 3 4 3 5 6 R2 a 2 by 3 matrix M LIST R1 35 R2 TYPE M M a 13 by 1 matrix OMONODOBPWNEFE hee WNROo Pree Oo Ne O Another operator that is useful for editing matrices is the SHAPE operator The SHAPE operator is used to convert a column vector into a matrix of any size as is shown in Table 3 12 and the following dialog TYPE SHAPE 2 4 1 8 a 2 by 4 matrix 1 1 2 3 4 2 5 6 7 8 M LIST 0 1 3 5 10 21 TYPE SHAPE 4 2 M a 4 by 2 matrix 1 0 1 60 5 0N Or w Oo N The SHAPE operator requires three arguments the number of rows and columns for the resultant matrix and a column vector of values As the dialog above shows if the resultant matrix requires more than the number of elements in the column vector the column vector is extended with as many copies of itself as are necessary to fill the resultant matrix This demonstrates a second prin ciple for supplying missing matrix elements in MLAB called matriz expansion by cyclic extension The general SHAPE expression has the form SHAPE SE1 SE2 ME1 If SE1 and SE2 are positive integers m and n respectively and ME1 is a column vector B of k rows then the result is an m by n matrix whose elements are the elements of ME1 assigned on a row by row basis More precisely if k gt m n the elements of the resulting matrix A are the element
273. ocess can not compute a direction of steepest descent because all values for c near 500 have the same sum of squared error S 26 20320 as seen in Figure 7 9 In terms of the latitude and longitude analogy at the beginning of this section there is no direction of steepest descent from a location in a level plateau A similar phenomenon might occur at a local maximum of the sum of squared error function but this is not very likely to happen The initial value c 30 also produces an inappropriate result as shown in the following dialog C 30 FIT C F TOM matherr overflow error in cosh arg 3 641769e 04 return value 1 797693e 308 matherr overflow error in cosh 265 0 78 y flix ABS c large Figure 7 9 Graph of f x for large c 266 arg 1 456707e 04 return value 1 797693e 308 matherr overflow error in cosh arg 3 641779e 04 return value 1 797693e 308 matherr overflow error in cosh arg 1 456711e 04 return value 1 797693e 308 matherr overflow error in cosh arg 3 641789e 04 return value 1 797693e 308 matherr overflow error in cosh arg 1 456715e 04 return value 1 797693e 308 final parameter values value error dependency parameter 36418 6856061939 8829 7637665699 2 220446049e 16 C 1 iterations CONVERGED best weighted sum of squares 2 620320e 01 weighted root mean square error 1 144622e 00 weighted deviation fraction 1 000000e 00 Since the initial value is
274. oes from 2 5 650 to 0 200 Figure 4 2 shows the four points and three lines associated with the matrix above 101 300 1 Figure 4 2 Plot of a curve matrix 102 Each curve belongs to exactly one window object and the user rectangle frame rectangle and image rectangle of that window together determine where curve matrix points will be positioned in the MLAB picture The n points of the curve matrix are given with the x and y coordinates corresponding to the window s user rectangle It is the user s responsibility to insure that all desired curve matrix points lie within or on the user rectangle Curve matrix points and lines lying outside the user rectangle will not be displayed and lines will be clipped if they cross the user rectangle boundary There is an exception to this rule the window object can be defined such that the user clipping rectangle is automatically recomputed to capture all of its curves This will be discussed later The dimensions of the MLAB picture and the frame and image rectangles of the window are used to compute the point positions in the MLAB picture Essentially each x y point in the user rectangle is displayed in proportionally the same position in the image rectangle That is the center of the user rectangle is displayed as the center of the image rectangle the lower left corner of the user rectangle is displayed as the lower left corner of the image rectangle and so on The inches per unit r
275. of departure from the vicinity of the equilibrium is proportional to the real part of the eigenvalue pair As an example consider the van der Pol oscillator which obeys the differential equations dyi _ dt Y2 dy 2 HO es 1 2 Ht y u 1 yj ya 210 which has an equilibrium at 0 0 The Jacobian matrix of this system evaluated at the equilib rium is En The following MLAB dialog shows the evaluation of the eigenvalues and eigenvectors of this Ja cobian matrix for u 1 J LIST 0 1 LIST 1 1 M EIGEN J TYPE M M a 6 by 2 matrix 1 0 5 0 5 2 866025404 866025404 3 353553391 707106781 4 612372436 0 5 353553391 707106781 6 612372436 0 Since the eigenvalues are complex with positive real part we conclude that the equilibrium 0 0 is unstable and that trajectories starting close to 0 0 will depart from it along a spiral Figure 6 14 shows a solution to the van der Pol equation starting from near the equilibrium for y 1 generated from the following MLAB commands XA XA XA XA XA XA XX RESET FCT Y1 T T Y2 FCT Y2 T T Y1 MUx 1 Y172 Y2 INITIAL Y1 0 0 INITIAL Y2 0 01 MU 1 M INTEGRATE Y1 T Y2 T 0 10 101 DRAW M COL 2 4 N M ROW 1 101 20 DRAW N COL 2 4 LT NONE PT XPT LABEL 0 10 2 VIEW Complex numbers may be manipulated in MLAB using the CPROD CDIV and CPOW operators The standard and matrix operators will wo
276. of the two numbers NROWS MA and NROWS MB and NCOLS MA MB and NCOLS MA MB are the larger of the two numbers NCOLS MA and NCOLS MB In the operations below let ME1 be an m by n matrix expression aij and ME2 be an m by n matrix expression b The general forms of the above matrix arithmetic operations are ME1 ME2 The result is an m by n matrix cij with m the larger of m and m and n the larger of n and n Define aij 41 4 i4m 1 mod m 1 n 1 mod n When gt m or j gt n and bij D1 i m 1 mod m 14 j n 1 mod n when i gt m or j gt n Then the result is Cig Gig bije ME1 result is the m by n matrix a j ME1 ME2 equivalent to ME1 ME2 Table 3 24 Matrix expressions with matrix arguments The Euclidean length v of an n dimensional vector v V1 V2 co Un is given by the formula 76 In MLAB the LENGTH operation on a matrix computes Euclidean vector lengths as described in Table 3 25 More precisely the rows of the m by n matrix argument are treated as n dimensional vectors and an m by 1 column vector of Euclidean lengths is computed Observe the following MLAB dialog E E a 3 by 2 matrix UNE 3 4 0 5 0 5 p 2 LENGTH E a 3 by 1 matrix 1 5 2 707106781 3 2 23606798 The general form for vector length computation is LENGTH ME1 If ME1 is an m by n matrix expression
277. oints in the curve matrix get the labels For example the LABEL clause LABEL 2 1 2 amp 2 10 2 119 Figure 4 7 Example of 7 parameter LINETYPE with values 1 1 1 1 0 05 0 will cause the numbers 0 2 0 4 0 6 0 8 1 0 to be printed at the second fourth sixth eighth and tenth points in the curve matrix respectively Each label number is printed in digital form in the picture and near the corresponding curve matrix point For example the digits 1 2 3 and 4 shown near the curve matrix points in Figure 4 2 were generated by the statements M SHAPE 4 2 LIST 9 1200 1 1 350 2 5 650 0 200 DRAW M LABEL 1 4 VIEW Label matrices are frequently used to number the tick marks on the axes of a graph In Figure 4 2 for example label matrices specified as 2 3 and 300 1500 300 were used to number the x axis and the y axis The x and y axes in Figure 4 2 were actually created by XAXIS and YAXIS statements which will be described later The LABELSIZE clause specifies the heights of the label digits in frame fraction units if omitted the heights of the label digits are 015 horizontal frame fraction units The label size can also be expressed in horizontal image fraction units or units of user x coordinates by including the word IFRACT or WORLD respectively The LABELFONT clause specifies the number of the particular font table that the label characters are to be taken from The default clause LABELFONT 5 is
278. ollowed by seven or fewer letters and or digits 0 9 and ext is an optional extension of one to three characters On MSWindows Macintosh OS7 8 9 X and Linux Unix the filename and extension are not limited to eight characters and three characters respectively Also on DOS MSWindows and Macintosh OS7 8 9 systems the filename and extension are not case sensitive on Macintosh OSX NeXTStep Unix and Linux Unix the filename and extension are case sensitive In Chapter 1 for example a save file named TUTOR SAV was created Other examples of acceptable names for disk files are EXPR2A 1 JONES FILE1 LST ST F4 and MLAB LOG Other systems allow various extended forms of filenames but DOS names are a lowest common denominator which we will use herein Certain extensions have special meanings by convention For MLAB users the log filename MLAB LOG and the extensions DAT DO PS and LST have special roles which will be described later Save files print files plot files read files and do files will all be accessed in the directory specified by the MLAB control string FILEDIR For example e on DOS or MSWindows systems if FILEDIR MLAB WORK KINETICS then the MLAB command PRINT mat will create the file AMLAB WORK KINETICS mat 1st e on Linux Unix systems if FILEDIR home George then the MLAB command PRINT mat will create the file home George mat 1st e on Macintosh OSX systems if FILEDIR Users csi then the MLAB command
279. olumn positions as follows M SE1 SE2 If SE1 and SE2 are positive integers i and 7 respective ly then the value returned is the matrix entry in row 1 and column 7 of M if it exists and is undefined other wise In general M SE1 SE2 has the same meaning as MCINT SE1 INT SE2 making scalars into integers An error message appears if a nonexistent matrix element is specified M SE1 This expression is normally used for column vectors matrices with 1 column and is exactly the same as M SE1 1 Table 2 9 Construction of matrix entries 2 3 FUNCTIONS AND THE FUNCTION COMMAND You may employ the FUNCTION command to symbolically define a function of one or more vari ables by specifying a corresponding scalar expression The FUNCTION command is described in Table 2 10 Defined functions may then be used in subsequent commands and scalar expressions This is helpful for several reasons First very complicated expressions need not be given all at once but can be defined part by part using intermediate defined functions Second the combined use of defined functions and matrices gives you powerful techniques for numerical computation and picture generation Third certain mathematical operations are available for direct use only in FUNCTION command scalar expressions although they can be used indirectly in assignment com mands and other MLAB commands by reference to the defined functions Finally the curve fitting an
280. on in the MLAB folder on the desktop select the MLAB application file with a single mouse click and then select Make Alias from the Finder s File menu Then drag the MLAB Alias icon to the desktop Then to run MLAB simply double click on the MLAB Alias icon In a standard Linux Unix with X windows system the MLAB executable directory is specified during the MLAB installation process usually usr local lib mlab Your home directory is the directory you are in when you log in i e home username On these systems MLAB runs in a bash shell terminal window To run MLAB you must first establish a set of X server processes and then create a terminal emulator process such as XTERM with an accompanying window if this is not done already You must then enter that terminal window and cd change 155 directory to the desired working directory Then you can issue the command mlab to run MLAB In NeXTStep Unix systems the MLAB executable directory is specified during the installation process If running MLAB from a NeXTStep Unix terminal window the home directory is the directory you are in when you log in and the current directory is the directory you are in when you run MLAB by typing mlab MLAB can also be run from the NeXTStep Workspace i e desktop by either double clicking on the MLAB icon or by double clicking on the name of an MLAB do file in the file browser window In this latter case ML
281. ons 141 For a curve or axis identifier C the general forms of the CURVEM and LABELM operators are CURVEM C the curve matrix of the curve or axis with identifier C LABELM C the label matrix of the curve or axis with identifier C Table 4 13 CURVEM and LABELM operators A window curve axis or title can be renamed by the use of a special assignment statement The form of window renaming assignment commands is shown in Table 4 14 For example the command W3 UIW would cause the window U1W to be renamed as W3 with U1W becoming an unknown identifier Similarly the command SA2 TEMPC15 would rename the curve TEMPC15 as SA2 with TEMPC15 becoming an unknown identifier The general forms of the renaming assignment commands are U WandU C for arbitrary unknown window and curve identifiers U W and C re spectively Table 4 14 Window renaming assignment commands 4 8 SIMPLIFYING MLAB GRAPHICS There are several techniques available in MLAB for reducing the labor of making pictures in MLAB First there is an alternative form of the WINDOW command as described in Table 4 15 Rather that explicitly defining the user rectangle s limits an ADJUST clause can be given in the WINDOW command Roughly speaking the statement WINDOW ADJUST WNICE IN W1 142 makes the user rectangle the smallest rectangle that nicely contains all the curve matrix points of curves in t
282. ons are not evaluated Assignment of scalars to the resulting matrix entries are made by rows the first scalar is assigned to the 1 1 element of the resulting matrix subsequent scalars fill the first row before filling subsequent rows Table 3 4 KREAD expression simple form For example observe the following MLAB dialog A 3 M KREAD 3 2 Type in numbers for KREAD End lines with lt Enter gt 1 E100 A 4 5 023 7 5 9 8 1 27 833 TYPE M M a 3 by 2 matrix 1E100 1 4 2 5 0 023 Be T5 9 8 In this example the line 50 1 E100 A 4 5 023 7 5 9 8 1 27 833 was typed in response to the KREAD prompt Note that although A was previously defined as a scalar the letter A which was typed in response to the KREAD operator was ignored by MLAB Also of the nine scalar values typed in the response the last three values 1 27 and 833 were ignored by MLAB The first and second scalars 1 E100 and 4 were assigned to the first row of the resultant matrix the third and fourth scalars 5 and 023 were assigned to the second row of the resultant matrix and the fifth and sixth scalars 7 5 and 9 8 were assigned to the third row of the resultant matrix The second argument in the KREAD expression may be omitted if a column vector is to be entered Therefore M LIST 1 3 2 7 8 22 and M KREAD 5 Type in numbers for KREAD End lines with lt Enter gt 1327 8 22 result in the same matrix M The RE
283. ontal frame extent If the PTSIZE clause is provided with only a scalar expression SE1 the value of the scalar is interpreted as the size of the point type in units of the horizontal frame extent Alternate units can be specified by following the scalar expression SE1 with any one of the words FFRACT IFRACT or WORLD FFRACT stands for horizontal frame fraction units IFRACT stands for horizontal image fraction units and WORLD stands for horizontal user coordinates For example the clause 117 PT SQUARE PTSIZE 1 5 WORLD would put a square measuring 1 5 units of the user x coordinates on a side at each point in the curve matrix If the point type is a quoted character the PTFONT clause can be included to specify which character font is to be used when the quoted character is drawn at each point in the curve matrix The scalar expression in the PTFONT clause should evaluate to a value s such that 1 lt s lt 34 or 14 lt s lt 2 On all systems the positive numbers 1 2 34 specify one of the Hershey fonts On MSWindows and Macintosh systems but not on Linux Unix or DOS systems the negative numbers 2 3 14 specify one of the TrueType fonts Tables showing the 34 different Hershey fonts and the 13 different TrueType fonts available in MLAB are shown in Appendix A The clause PT z PTFONT 13 would cause the Greek letter the character corresponding to lower case z ASCIIT character number 122 in font number 13 as
284. operly if the input matrix is not sorted on column one The SORT operator explained in Section 3 4 should be used if necessary It is sometimes necessary to perform LOOKUP s on data in multicolumn matrices The matrix operations explained elsewhere in this lesson can usually be used to transform your data into the proper form 87 The general form of the LOOKUP matrix operator is the following LOOKUP ME1 SE1 where ME1 is a two column matrix in sort by column 1 and SE1 is the value to be searched for in ME1 COL 1 The result is the value from the second column of the row whose first column value is equal to SE1 If SE1 does not exist in column one of ME1 then linear interpolation is used If NCOLS ME1 gt 2 then only the first two columns of ME1 are considered Table 3 31 LOOKUP matrix operator 3 9 THE FOR COMMAND It sometimes happens that a sequence of similar MLAB commands is needed to achieve some objective For example suppose a 5 by 5 matrix M with all elements on the diagonal and below set equal to 1 and all elements above the diagonal set equal to O is desired M could be created with the following MLAB commands M ROW 1 COL 1 1 M ROW 2 COL 1 2 1 M ROW 3 COL 1 3 1 M ROW 4 COL 1 4 1 M ROW 5 COL 1 5 1 Because this is repetitious MLAB provides a capability for repeated execution of an MLAB command The repeated command usually makes reference to an index variable an unknown or sca
285. or NOT CONVERGED the final weighted sum of squared error hopefully the minimum and the number of iterations executed There are three other types of quantities in the final summary error and dependency values for each fitted parameter to be discussed in Section 7 13 and the weighted root mean square error The weighted root mean square error is computed by the formula S m p where S is the final weighted sum of squared error m is the number of data points and p is the number of fitted parameters assuming m gt p If the sum of squared error criterion is used this quantity is a root mean square estimate for the length of the vertical line segment between measured and predicted values for this fitted model and data If the weights are the reciprocal of variances of each data point then the weighted root mean square error is a root mean square estimate of the length of the vertical line segment between measured and predicted values for this fitted model and data where each segment length is measured in units of the standard deviation of the data points It is possible to distinguish between active and inactive constraints during the curve fitting process If b1 b2 bp are curve fitting parameters and the linear constraint ri bi r2 b Tp bp lt or gt Iro is active then r bo1 r2 bo2 Tp bop is nearly equal to ro where bo1 bo2 bop are the current values for the fitted parameters That is
286. ork on a current window W2 Window W1 may be redisplayed while still retaining but not displaying window W2 by the MLAB commands BLANK W2 138 UNBLANK W1 Curves subsequently added to a previously blanked window are blanked as well For example if X were an unknown identifier and U were a defined window then the MLAB commands BLANK U DRAW X M INU would cause the window U to contain a blanked curve X in addition to all of the other curves of U which would be blanked The general forms of the BLANK and UNBLANK commands are BLANK D1 D2 Dn UNBLANK D1 D2 Dn where n gt 1 and D1 D2 Dn is a list of n identifiers each being a curve title or window or a 3D graphical data item All listed curves and all curves belonging to listed windows are put in blanked condition or in unblanked condition as specified by the word BLANK or UNBLANK Table 4 12 BLANK and UNBLANK commands When using a slow graphical device one may want to blank some curves in a picture to avoid delays in redrawing the picture after changes For example a list of curves can be blanked in a window W by an appropriate BLANK command such as BLANK CA CC CD and then either certain of the curves can be selectively unblanked or all of the curves in the window W can be unblanked by executing UNBLANK W The TYPE command can be used to obtain information about windows curves axes and titles as well
287. ot zero unless there is perfect agreement between the data and the model Moreover it increases to arbitrarily large values if any data point is moved a sufficient vertical distance away from the function graph Note also that it is the vertical line segment to the function graph that is important not the line segment from the data point to the nearest point on the function graph In Figure 7 1 for example the line segment from the point A on the function graph to the leftmost data point B is shorter than the vertical line segment from B to the function graph but this does not affect the sum of squared error criterion If the experimental measurements can not be made with equal precision for all data points then the sum of squared error criterion may be unsatisfactory for curve fitting For example suppose an experiment involves 1 injecting experimental animals with a drug containing a radioactive label and 2 then doing radioactive counts of blood samples taken at periodic intervals after injection Such an experiment may show several orders of magnitude decreases in radioactive count readings over the time period the animal is monitored Because of the way that radioactive samples are prepared the expected error of the experimental measurement might be roughly proportional to the number of radioactive counts obtained So the large early measurements may have errors which are several orders of magnitude larger than the later measurements Then th
288. ould be obtainable with high confidence However this will not occur for all models There are models where small changes in the pattern of noise in the data can make large changes in the values of the fitted parameters The degree of instability of the fitted parameter values is measured by the dependency values given in the final summary section of the FIT command output Each parameter has an associated dependency value which is a number between 0 and 1 Dependency values close to 1 say 98 or more indicate instability in the fitted value for this parameter That is a high dependency implies that almost as good a curve fit could be obtained even if the dependent parameter were constrained to be somewhat different from its optimum value In effect changes in the dependent parameter can be compensated by appropriate changes in the other fitted parameters without much increase in the weighted sum of squared error The error column included in the FIT command s final summary section may also help to estimate the statistical stability of the fitted parameter values If the model is linear and the data errors are normally and independently distributed with a common variance and if there are no active linear constraints at the final optimal set of parameter values then 1 the fitted parameters are maximum likelihood estimates in the statistical sense and 2 the error values are unbiased estimated standard deviations of these parameter value
289. ows of the curve matrix For line types 6 7 9 10 11 and 12 if ME1 is an n by 2 matrix a then a11 is the marker value x A point symbol is displayed at a point a 1 aja of the curve matrix if aj x for i 2 3 n For line type 6 a solid line is drawn from a 1 aja to ai 1 1 Qi 1 2 if both aj and aj411 are unequal to x for i 2 3 n 1 Line type 7 is similar to line type 6 except that each line connecting a given set of points headed by the marker value a x will be drawn with a dashed line whose dash length is a 2 inches if aj 1 a dash length of infinity i e a solid line will be drawn For a new curve ME1 must be supplied but a curve name Cn and clauses IN W PT 0 and LT 1 will be supplied as needed To modify an existing curve the curve name must be supplied and the curve matrix window pointtype and linetype will remain at their previous values if the modifying clause is omitted Table 4 6 DRAW command short form UNVIEW DRAW Ci IN PIC DELETE W 111 VIEW PIC Points and lines from a DRAW command lying outside the user rectangle are not drawn and lines intersecting the user rectangle show only the part lying within the user rectangle In many cases the most difficult part of making a picture in MLAB is the preparation of curve matrices or matrix expressions for use in DRAW commands This requires competence in the scalar and matrix techniques given in Ch
290. ows unchanged Similarly specified columns of a matrix can be directly replaced For example recall the MLAB dialog in Chapter 1 which created the matrix DAY DAY 6 15 DAY COL 2 LIST 111 078 095 126 243 180 202 387 465 831 The second command above expands the column vector DAY to a 2 column matrix with the first column unchanged and the second column as specified by the LIST expression A matrix can also be created by a ROW or COL matrix assignment command if an unknown identifier instead of 62 In general COL matrix operations have one of the following forms ME1 COL ME2 For an arbitrary m by n matrix expression ME1 and ME2 a k by 1 column vector of integers 71 J2y Jk With all j between 1 and n the result is an m by k matrix with it column equal to column j of ME1 for i 1 2 k All entries of ME2 are made into integer values by applying INT if necessary and an error message appears if a nonexistent column of ME1 is indicated Use with NCOLS ME2 gt 1 is equivalent to ME1 COL LIST ME2 ME1 COL SE1 equivalent to ME1 COL LIST SE1 ME1 COL SE1 SE2 SEn equivalent to ME1 COL LIST SE1 SE2 SEn Table 3 14 COL matrix operators the identifier of an existing matrix is given Note that a combination of the principle of matrix expansion by cyclic extension and the principle of matrix expansion by zero extension is used to supply missing matrix elements Matr
291. p function graph of a curve 198 eL 10 mM Mp Figure 6 7 Histogram of random data 0 0 2 0 Suppose we have a matrix X containing measurements of a random event We know that the values of X are between two values P and Q and we want to see an estimate of their density function over this range Such a graph is called a histogram and it gives the distribution of the values of X in B buckets ranging from P to Q The following commands will graph the histogram of 100 random measurements ranging from zero to one into fifty equally spaced buckets as a bar graph using the HISTO operator RESET P 03 Q 13 B 50 X RAN ON LIST 0 P Q 77100 DRAW BARGRAPH HISTO X B WINDOW O TO 1 O TO 10 VIEW XK XA XA XA XX The bar graph of this histogram matrix is shown in Figure 6 7 It should be remembered that all of these operators use and return matrices which represent points 199 The general forms of the INTERPOLATE SMOOTH BARGRAPH and HISTO matrix operators are INTERPOLATE ME1 ME2 SMOOTH ME1 BARGRAPH ME1 HISTO ME1 The INTERPOLATE operator will return the points of a curve with values taken at ME2 interpolated from ME1 For values in ME2 for which points exist in ME1 the same values are maintained NCOLS ME2 must equal NCOLS ME1 1 for INTERPOLATE to work The SMOOTH operator will take a matrix as input and return points of a smoothed curve with values taken at the same places
292. ped then the USE command will destroy the old column vector Y by replacement MLAB SAVE commands create files in a compact binary form for economical storage Therefore attempting to print or edit SAV files will produce unintelligible output from a disk file that was created by an MLAB SAVE command Save files created by one operating system version of MLAB cannot be USEd by another operating system version of MLAB You can only USE such a file on the same operating system in which it was created The SAVE and USE commands are sometimes useful within a single MLAB session For example one might want a series of graphs which differ only in the data and certain title information Here a window named PIC can be created containing a common framework for the picture series axes and labels common legends and descriptive information and so on Then the command SAVE PIC creates a file named PIC SAV to save this framework Now adding data curves and titles to the framework on the screen creates the first picture The commands PIC1 PIC BLANK PIC1 USE PIC change the window name PIC to PIC1 blank PIC1 and then retrieve PIC for starting the second picture PIC2 A similar technique can be used to create the remaining pictures in the series A saved picture may have some or all of its curve data items blanked For example one could put two blanked TITLES having different titles into PIC and then unblank the string curve wanted fo
293. ped without interrupting DO file execution After all DO file com mands are executed an asterisk prompt appears All contents of the specified device or file are interpreted as MLAB commands separated by semicolons and are executed in sequence Table 5 6 DO command have been saved in RUN DO Note that the word DO appears twice in the command first as part of the syntax of the FOR command and then to specify the DO command At first the MLAB DO command seems to be just a labor saving device of no great importance The experienced MLAB user however finds that DO files are very useful for avoiding large amounts of repetition For example systems of interrelated FUNCTION definition statements can be put into DO files thus avoiding the necessity of retyping long and complex expressions Designing and testing an MLAB DO file is similar to writing a computer program and is usually more difficult than executing MLAB commands interactively This is because one must foresee in advance all the possibilities that may arise There is a special use of the TYPE command for issuing messages from the DO file to the display The message is enclosed in quotation marks and becomes the argument of the TYPE command For example the commands TYPE K IS THE BINDING VALUE TYPE K in a DO file will lead to the computer response 170 K IS THE BINDING VALUE during execution of the DO command for that file This message could be us
294. pressions are more flexible and efficient than the LIST expression and are recommended for MLAB matrix input in most cases The simple LIST expression of n scalar arguments n gt 1 has the form LIST SE1 SE2 SEn It corresponds to a column vector of length n i e an n row by 1 column matrix with entries SE1 SE2 SEn respectively Table 3 3 LIST expression simple form For example observe the following MLAB dialog XA 30 M1 LIST SIND XA XA 90 2E4 TYPE M1 Mi a 3 by 1 matrix 1 0 5 2 60 3 20000 49 Data for a two dimensional matrix can be input from the keyboard by using an appropriate KREAD expression as described in Table 3 4 The KREAD expression of 2 scalar arguments both gt 1 has the form KREAD SE1 SE2 It allows the user to type entries for a matrix with SE1 rows and SE2 columns from the keyboard When MLAB executes this form of the KREAD operator it prints the message Type in numbers for KREAD End lines with lt Enter gt and waits for the user to type SE1 SE2 scalars from the keyboard The numbers may be typed in free format separated by tabs spaces letters or commas Input lines are terminated by pressing the ENTER key If the user hits the Escape key before entering the specified number of scalars then the matrix row containing the final number is completed with zeros and no more rows are created Excess numbers are ignored Scalar expressi
295. produces the nt term in the Fibonacci series 0 1 1 2 3 5 8 which is mathematically defined as 0 for x 0 FIB z lt for x 1 FIB x 1 FIB x 2 forx gt 1 This function can be defined in MLAB with the statement FUNCTION FIB X IF X lt 2 THEN X ELSE FIB X 1 FIB X 2 The TYPEOUT operator summarized in Table 2 12 is often useful in debugging recursively defined functions and in debugging functions in general TYPEOUT X will return the value of X and will type out this value as a side effect The following dialog illustrates this FUNCTION FACT X IF X 1 THEN 1 ELSE TYPEOUT X FACT X 1 A FACT 4 2 6 zi 24 TYPE A A 24 When MLAB prints something due to a TYPEOUT command the line of typed output begins with a double colon When evaluating FACT 4 MLAB expanded the statement to TYPEOUT 4 TYPEOUT 3 TYPEOUT 2x 1 and executed the inner parenthesized parts first 32 The general form of the TYPEOUT operator is TYPEOUT SE1 The value SE1 is returned and is typed out as a side effect This is use ful in debugging function definitions If a matrix expression is used in place of SE1 then erroneous results will occur Table 2 12 TYPEOUT scalar operator 2 6 INDEXED SUMS PRODUCTS AND DEFINITE INTE GRALS Three common mathematical notations the indexed sum the indexed product and the definite integral have corresponding SUM PRODUCT and INTEG
296. qua tions 296 estimated weights 233 Euclidean length of a vector 76 EVAL 36 evaluating functions 85 86 EWT 233 exclamation point 57 EXIT 13 exponential functions 19 exponential model 4 7 225 227 253 exponentiation 18 extracting diagonals of a matrix 72 extracting matrix diagonals 213 factorial 31 fast Fourier transfors 217 FCT 24 281 FFRACT 117 121 129 133 Fibonacci series 32 FILEDIR 157 159 filenames 157 files 154 FILL 200 203 filing polygons 190 200 203 finding roots of functions 36 first order differential equation 280 FIT 7 226 244 245 250 257 298 301 302 FITNORM 250 251 fixed number of members sequences 57 FLATNESS 208 floating point exceptions 18 30 folders 154 FONT 129 133 font for titles 133 font size 133 font tables 134 314 FOR 88 89 94 125 169 FOR command compound form 90 FORMAT 117 122 127 format of axis label 127 format of curve label 117 Fourier transforms 217 FRAME 100 101 147 frame fraction units 99 117 121 frame rectangle 98 99 103 146 FRAMEBOX 105 147 FUNCTION 7 24 112 255 281 function 15 function arguments 24 function definitions 23 255 function identifiers 26 function parameters 23 FUNCTIONS 26 functions 26 functions defined piece wise 26 gamma function 19 GAUSSF 19 GEAR 295 Gear Method 294 Gear Shrager Method 294 GEAR2 295 generalized polynomia
297. r the current picture 162 53 THE READ READON KREAD AND KSREAD FUNC TIONS In Chapter 3 it was shown that the KREAD matrix expression KREAD SE1 SE2 can be used to read typed in numerical values into an MLAB matrix at the keyboard The LIST operator is another convenient way to directly prepare a matrix containing some particular numbers If however a matrix has more than 20 or 30 entries it is usually preferable to prepare a text file containing the data rather than to type the data during the MLAB session You can use any desired text editing program that creates ASCII text files such as EDIT or NOTEPAD on MSWindows Teach Text or SimpleText on Macintosh Emacs or vi on Linux Unix Macintosh OSX and NeXTStep Unix or the builtin MLAB text editor on DOS You can launch most of these editors by typing the MLAB command EDIT FILE lt filename gt Text files are conveniently examined changed and augmented Such a text file can then be read to form a matrix by using the MLAB READ operator If the file name in an MLAB READ operation is a proper MLAB identifier consisting of letter digits and dot characters only then surrounding quotation marks may be omitted Such an unquoted file name will have the extension DAT appended to it For example M14 READ ROD1 150 2 would access the first 300 numbers of the disk file ROD1 DAT to construct the 150 row 2 column MLAB matrix M14 A READ matrix expression may refer to a t
298. r assignment command is summarized in Table 2 1 For scalars the two forms of the assignment command are unknown identifier scalar expression scalar identifier scalar expression An underscore symbol _ may be substituted for an equal sign to indicate an assignment Table 2 1 Scalar assignment command 16 The commands A 003 and B 4 above are of the first type Suppose that the next command of the tutorial session had been B A 3 This command would have been of the second type changing the value of the scalar B from 4 to 3 003 since the scalar expression A 3 represents the number 003 3 3 003 The second form of the assignment command can be self referring For example the command B B 2 gives B a new value equal to half the current value The examples A 3 and B 2 above suggest the first two rules for constructing and evaluating scalar expressions First scalar constants are scalar expressions which are always evaluated to their indicated numeric value and scalar identifiers are scalar expressions which are evaluated to their current value within the MLAB session Second scalar expressions can be constructed by symbolically indicating the arithmetic operations of addition negation or subtraction multiplication division and raising to a power or The circumflex 7 here means the character SHIFT 6 present on most keyboards not a control character Fo
299. r example assuming that all identifiers below are scalars the following are scalar expressions 1 5 2 XA 3 3 74 MASS WIDTH DEPTH K1 Note that blanks may be inserted between identifiers and operations in MLAB expressions they are ignored Scalar expressions involving two or more arithmetic operations may have several possible evalua tions For example does 2 73 1 have value 7 or 4 As in algebraic notation and most programming languages two methods of removing the ambiguity are available First parentheses may be used to explicitly specify the order of operations For example the scalar expressions 2 73 1 and 2 7 3 1 express the two possible interpretations of 2 73 1 and the operation within the parenthe ses is performed first in each case Complicated scalar expressions may have nested parentheses with expressions inside parentheses evaluated in turn to obtain the final value For example the MLAB scalar expression A0 2 A1 T A2 T T corresponds to the algebraic expression usually written as ag 2a1t agt 17 Note that multiplication which is denoted by juxtaposition in algebraic notation must be explic itly specified with an asterisk for a multiplication sign in MLAB Second where parentheses are not given it may be cumbersome to parenthesize a complicated scalar expression fully MLAB follows the usual algebraic hierarchy of operations That is raising to a power is performed first then multiplica
300. r the window curve and curve matrix should reveal the situation Insert temporary TYPE or other commands into the DO file to monitor logic of the file as well as some of the intermediate results Execute DO file command sequences by typing them interactively and examining the computed results Several samples of input data may be needed to detect errors caused by expres sions that work for some cases but not in full generality 175 Sufficient persistence with the above methods will usually enable the user to prepare DO files On Linux Unix and Macintosh OSX systems the execution of an MLAB command can be inter rupted by typing C CONTROL C Then the currently executing MLAB command is suspended and the message Do you really want to quit MLAB y YES NO is printed If you type y MLAB will terminate and leave you at the bash shell prompt If you type MLAB then prints the prompt Resume calculation or return to top level t TOP LEVEL RESUME If you respond with MLAB will continue with the suspended operation If you type t then MLAB responds Command in progress aborted MLAB state indeterminate Safest Action type EXIT to return to DOS then restart MLAB Riskier but often worthwhile Save your data objects SAVE IN MYFILE then EXIT restart MLAB and restore the objects USE MYFILE On MSWindows and DOS systems the execution of an MLAB command can be interrupted by holding the SHIFT and CONTROL keys
301. rator 6 7 COVARIANCE AND CORRELATION MLAB contains two operators for computing the covariance and correlation matrices of a set of n tuple observations Given an n column matrix of data D whose rows are n tuple observations COV D is the sample covariance matrix of this data and CORR D is the sample correlation matrix The CORR and COV operators are described in Table 6 8 In particular if the rows of D are samples of the vector of random variables 11 22 the i j element of COV D is the unbiased estimate of the covariance between x and zj and the i j element of CORR D is the unbiased estimate of the correlation between x and zj If D is a 1 column matrix then COV D 1 1 is the sample variance of the numbers in D For example given the data matrix D as listed below the associated covariance and correlation matrices are listed TYPE D D a 8 by 2 matrix 1 0 8188943 1 595087 2 0 9785145 0 5698285 3 0 2872429 0 347616 215 4 0 1407133 1 090237 5 1 610475 0 70895 6 2 233432 0 478001 7 0 9479984 0 5481901 8 1 367598 1 999153 TYPE COV D a 2 by 2 matrix 1 1 67393876 645592141 2 645592141 1 24619036 TYPE CORR D a 2 by 2 matrix dos 1 446988405 2 446988405 1 We can compute these values directly for comparison purposes as follows XK XA XA XA XA X X gt gt XX XA XA XX FUNCTION MN X SUM I 1 NROWS X X CI NROWS X FUNCTION COVAR
302. re edition Almost all MLAB users find that making pictures is a considerable part of their interaction with MLAB MLAB graphics constructions are done using special MLAB commands Although it is easy to make simple pictures with these commands complex pictures can be more intricate to construct One advantage of having a command based facility available is that your graphics constructions can be scripted in a do file for easy repetitive use 4 1 GRAPHICAL DISPLAY CONTROL COMMANDS There are two MLAB commands used for overall control of graphical pictures VIEW and UNVIEW The VIEW and UNVIEW commands described in Table 4 1 are like on and off switches for graphical pictures When running MLAB on systems with a graphical window based user interface such as Linux Unix with X Windows Macintosh MSWindows or NextStep Unix the VIEW command causes a sec ond window apart from the window containing MLAB commands to appear on the screen This window can be moved iconized resized or zoomed unzoomed using the usual window controls Call this second window the MLAB picture It is tempting to call the MLAB picture a window 96 but in MLAB window has another meaning The UNVIEW command makes the MLAB picture disappear When running MLAB on a DOS PC the VIEW command causes the MLAB commands on the screen to be replaced with a full screen picture which remains on the screen until the user strikes a key Once a key is struck the pict
303. reasing A summary is provided in Table 3 8 In general the arithmetic sequence matrix expression is given by SE1 SE2 SE3 If a SE1 b SE2 and d SE3 the result is a column vector with components a a d a 2 d a n 1 d such that a n 1 d lt b lt a 4 n difd gt 0 a n 1 d gt b gt a 4 n difd lt 0 If d 0 an error message appears Ifd lt 0 and a lt b then d is used If d gt 0 and a gt b then d is used SE1 SE2 exactly the same as SE1 SE2 1 Table 3 8 Arithmetic sequence matrix expression Note that scalar expressions may be used to specify a b and d For example observe the following MLAB dialog X1 X2 3 Pp 56 R X1 X2 X2 X1 50 TYPE R R a 51 by 1 matrix 1 1 2 1 04 3 1 08 45 terms omitted 49 2 92 50 2 96 51 3 The MLAB colon and exclamation point expression a b c summarized in Table 3 9 generates the column vector of the finite arithmetic sequence which goes from a to b in c steps For example e 1 4 4 equals LIST 1 2 3 4 e 10 1 5 equals LIST 10 7 75 5 5 3 25 1 e 1 2 1 5 4 equals LIST 1 2 1 3 1 4 1 5 e 0 1 100 describes a column vector with the 100 elements 0 010101 020202 989899 1 95 terms omitted Note that the sequence may be increasing or decreasing The fixed number of members sequence matrix expression is given by SE1 SE2 SE3 If a SE1 b SE2 and c SE3 with c gt 0 the result i
304. rential equation systems computing the symbolic partial derivatives may exhaust the available space for holding functions In this event it may still be possible to proceed using a numerical Jacobian The scalar ERRFAC is the tolerance level It specifies the error that will be acceptable in a single step computation of one row of the solution matrix of the solver operation It is a small positive number such as 001 or 0001 For example the tolerance level 0001 specifies that a 01 error or less in computing the next row of the solution matrix is acceptable assuming that the preceding rows are correct Note that this does not prevent the possibility of buildup of substantial errors over many steps due to round off and truncation Decreasing the value of ERRFAC usually increases the accuracy of the computation up to a certain point Decreasing the tolerance level however tends to decrease the step size and so increases the amount of numerical computation and hence numerical error So there are limits to the improvement obtainable by decreasing the tolerance level If the step size becomes too small the procedure may multiply the tolerance level by 10 295 in order to prevent the solution matrix from growing too large and to avoid the use of too much computer time When this occurs a typical error message will be t 199 995597 errfac 0 001 eqn 7 truncation error 2 853873 The scalar control variable DISASTERSW determines how e
305. rix of any size can easily be constructed using matrix editing operations For example if A M and N are MLAB scalars such that M and N are integers then A N 77M describes an M by N matrix with all entries having the scalar value A 69 Numbers can be sorted into an increasing sequence using the SORT matrix expression of MLAB as described in Table 3 21 For example a column vector can be sorted as shown in the following MLAB dialog TYPE V V a 5 by 1 matrix 2 2 4 1 3 po 1 5 0 7 OPPWNEF TYPE SORT V 1 a 5 by 1 matrix OPWNEF An arbitrary matrix M can be sorted so that a specified column k is an increasing sequence That is the sort process permutes the rows of the matrix so that the entries in the k column are an increasing sequence The sort is stable in that rows with the same entry in column k of the sorted matrix will appear in the same relative order as they were in the unsorted matrix Therefore one can sort a list of row vectors into lexicographic order by sorting successively on several columns as shown in the following MLAB dialog TYPE M M a 4 by 3 matrix 1 1 5 6 2 7 4 8 3 7 3 9 4 1 8 6 N SORT M 2 TYPE N N a 4 by 3 matrix 70 1 7 3 9 2 T 4 8 3 1 5 6 4 1 8 6 R SORT N 1 TYPE R R a 4 by 3 matrix 2 1 yel NN 7 BwWN Be Ae w 09 Ol ODO 8 By sorting first on column 2 of the martix M the matrix N is obtained
306. rk with complex numbers because corresponding 211 0 3 0 04 0 22 0 49 0 74 Figure 6 14 A solution to the van der Pol equation 212 The general form of the EIGEN matrix operator is given by EIGEN ME1 Let ME1 be an n by n real matrix The result is an 2n 2 by n matrix The first two rows contain the real parts and imaginary parts re spectively of the eigenvalues of ME1 The remaining 2n rows con tain the real and imaginary parts of the eigenvectors of the correspond ing eigenvalues Table 6 5 EIGEN matrix operator matrix elements will correctly match up Complex number multiplication division and exponen tiation are achieved with the CPROD CDIV and CPOW operators which are defined in Table 6 6 These operators each take two 2 column matrices A and B for each matrix the first column rep resents the real part and the second column represents the imaginary part of a list of complex numbers CPROD A B is a matrix with two columns where each row is the product of the complex numbers in the corresponding row of A and B If A and B do not have the same number of rows the matrix with the fewer number of rows is cyclically extended Similarly CDIV A B and CPOW A B return 2 column matrices that are the complex quotients and powers of the complex numbers in rows of matrices A and B These functions are useful in dealing with the discrete Fourier transform operators and will be discussed there in that con
307. rms y x c for j 1 2 n Caution If the derivatives of unknown functions are involved circular definitions must be avoided Suppose T denotes the MLAB identifier used for the independent variable and Y1 Y2 YN denote the function identifiers for the unknown functions Y1 Y2 Yn the corresponding system of differential equa tions is specified by the MLAB commands FUNCTION Y1 T T FUNCTION Y2 T T SE1 SE2 FUNCTION YN T T SEn INITIAL Y1 SEO SEn 1 INITIAL Y2 SEO SEn 2 INITIAL YN SEO SE2n Table 8 1 Solving first order differential equations systems 310 Here scalar expression SEj denotes the expression h x y y2 2 Yn 2 with T substituted for x YJ for y x YJ T for yj x YJ T C for yj c j 1 2 n In particular note that YJ is treated just like the scalar expression YJ T and YJ T is treated just like the scalar expression YJ T T The expression SEO should be the common argument u for the indepen dent variable in the initial conditions It is usually a scalar constant or a scalar variable and we recommend that exactly the same scalar expres sion be used in all of the initial conditions for a system The scalar expressions SEn 1 SEn 2 SE2n specify the values 21 Za Zn Of the unknown functions at the argument u the expressions oc curring in the initial conditions should not contain any of the identifiers Y1 Y2
308. rner has vertices at 7 5 5 9 and 9 9 d The half circle in the lower left corner has radius 2 and the circle s center is at 7 7 e The spiral in the center is described in polar coordinates r a by the equation r a 20 for 0 lt a lt 20 Hint use the parametric form with x and y expressed as functions of a 6 The data points in Figure 4 18 are 10 1 82 26 1 70 41 1 58 52 1 88 20 1 35 and 37 95 for group A and 48 1 70 45 1 42 82 1 31 65 85 77 97 and 87 56 for group B A square was drawn at the x coordinate and y coordinate mean values for group A and a triangle was drawn for the means of group B Reproduce Figure 4 18 149 Figure 4 17 Five practice drawings 150 Body weight grams A Experimental B Contro o2 25 50 75 100 R Test Score Figure 4 18 Practice drawing using squares and triangles 151 Aa oa 2 Y Ne gt NK J o er ee MTIAB Se te SoS Figure 4 19 Practice drawing using font tables 152 10 Reproduce Figure 4 19 by using the first 28 font tables available in MLAB Suppose MA MB MZ are 26 MLAB matrices for drawing script capital Latin letters If drawn with LINETYPE 6 each will display the corresponding letter script A for MA etc filling the user rectangle O TO 1 O TO 1 The marker value MA 1 1 MB 1 1 etc is 10 for all the matrices Construct MLAB commands by which you could make a monogr
309. ror criteria associated with all the clauses If Q is typed during execution of a FIT command the operation terminates at the completion of the current iteration Setting the MLAB scalar LSQRPT equal to 8 its default value before giving the FIT command causes MLAB output to be suppressed during execution of the command except for the final summary Setting LSQRPT equal to 0 before giving the FIT command causes all typed output to be suppressed Typing R during execution of a FIT command causes a line or two of the iteration output to be typed Table 7 8 FIT commands for a functional form 278 Consider the general problem of curve fitting a functional form y f z1 2 vey Lp bi ba bp C1 C2 Gy where 1 2 n are the arguments of f bj bo bp are the para meters of f to be fitted and c1 C2 Cy are other parameters of f which are assigned fixed values To set up the functional form in MLAB the FUNCTION command FUNCTION F X1 X2 Xn SE1 is used Here F denotes the MLAB identifier corresponding to the function f and X1 X2 Xn is a list of MLAB identifiers naming the function arguments denoted by x1 2 amp n above The scalar expression SE1 describes the function f and involves the arguments z1 LQ En the fitted parameters b1 b2 bp and the fixed parameters C1 C2 Cq as required The MLAB user chooses either an MLAB scalar or a
310. rror tolerance violations are handled The alternatives for DISASTERSW are e 2 do not report errors and increase tolerance indefinitely as required e 1 do not report errors and increase tolerance as required If the internal value of ERRFAC becomes greater than 1 2 then terminate the integration e 0 report errors and increase tolerance as required If the internal value of ERRFAC becomes greater than 1 2 then terminate the integration e 1 report errors and increase tolerance five times If it would be necessary to increase the tolerance again then terminate the integration e 2 report errors and increase tolerance five times If it would be necessary to increase the tolerance again then continue the solution with fixed step size e 3 report errors and increase tolerance as required If the internal value of ERRFAC becomes greater than 1 2 then continue the solution with fixed step size e 4 force the solution from the beginning with a large fixed step size of 1 80 of the total integration interval e 5 force the solution for one step Do not increase the tolerance Do not report any error e 6 force the solution for one step Do not increase the tolerance Type out an error report The default value of DISASTERSW is 6 Errors of the sort regulated by DISASTERSW are associated with the presence of singularities in the derivative functions or with steep changes in the solution of the differential equation system i e
311. rs R and G are the distributions r and g i above then the commands RT DFT R GT DFT G F IDFT RT COL 1 CDIV RT COL 2 3 GT COL 2 3 compute the desired distribution f i The DFT IDFT and REALDFT operators are described in Table 6 9 6 9 SUMMARY In the following let SE1 SE2 denote arbitrary scalar expressions and ME1 ME2 denote arbitrary matrix expressions 1 Matrix combination operators are a MESH ME1 ME2 b CROSS ME1 ME2 2 Matrix interpolation is done by a INTERPOLATE ME1 ME2 221 The general forms of the discrete Fourier transform operators are DFT ME1 IDFT ME1 REALDFT ME1 DFT ME1 takes the rows of ME1 to specify n NROWS ME1 equally spaced sample values of a complex valued function F The first column of ME1 is taken as the real parts of values of F and the optional second column is taken as the imaginary part of values of F The Fourier trans form of the periodic extension of F as specified by ME1 is returned DFT ME1 has three columns the first is the frequency values and the second and third are the associated complex coefficients If DFT ME1 Q then f t Char Qna iQnz exp 27iQ 1t where b 2ll0827 for n NROWS ME1 A power of 2 fast Fourier transform algorithm is used If n is not a power of 2 then inter polation is used and thus the equalities listed here are only approximate IDFT ME1 takes a matrix identic
312. rted to P and E is kg C t Hence the differential equation C t a ki S t E t ka k3 C t k S t Eo C t k2 k3 C t describes how the concentration of C will change with time Since the rate at which P will be formed from C is kz C t the differential equation P t kz C t 284 describes how the concentration of P will change with time In addition to the three differential equations giving S t C t and P t by expressions involving the unknown functions S t C t and P t and known constants there are initial conditions previously given for time t 0 as follows S 0 So C 0 0 and PO 0 In MLAB the FUNCTION and INITIAL commands are used to set up the system of differential equations as shown in the command sequence FUNCTION S T T K1 S E0 C K2 C FUNCTION C T T K1x S E0 C K2 K3 C FUNCTION P T T K3 C INITIAL S 0 SO INITIAL C 0 0 INITIAL P O XK XA XX X Il o Again differential equations are treated as MLAB function definitions for the derivatives S T C T and P T of the unknown functions S C and P The INITIAL command of MLAB sets up initial conditions for S C and P as required to complete the differential equation system n first order differential equations will require n initial conditions To numerically evaluate S C and P appropriate values are first assigned to all scalar constants and th
313. rts while solving differential equations 296 RIGHT 122 133 134 143 rightward tick mark 106 ROOT 36 roots of a function 36 ROSE 125 ROTATE 72 192 rotating columns of a matrix 72 rotating rows of a matrix 72 row concatenation 64 ROW matrix assignment command 64 ROW matrix operator 62 row number of matrix containing maximum element 72 row number of matrix containing minimum element 72 row replication operator 67 RTICK 106 128 running DOS MLAB 156 running Linux Unix MLAB 156 running Macintosh MLAB 155 344 running MSWindows MLAB 155 running NeXTStep MLAB 156 running Unix MLAB 156 SAVE 12 160 168 save file 12 154 158 160 scalar 15 scalar arithmetic 18 scalar assignment 16 scalar expressions 16 scalar identifiers 22 scalar matrix arithmetic 73 SCALARS 22 scalars 22 scientific notation 16 screen fraction units 99 scrolling 12 searching through a matrix 86 setting diagonal elements of a matrix 89 shaded circle 201 shading polygons 190 200 SHAPE 60 61 SHIFT 129 133 shifting text 134 shifting titles 129 133 shooting method 300 short dash line type 106 simplifying graphics 142 sine 19 singular value decomposition of a matrix 213 215 SIZE 129 133 SMARKER 106 SMOOTH 194 199 204 smooth curves 112 smoothing functions 194 SOLID 106 solid line extending to the bottom image bor der 106 solid line extending
314. s corresponding to all initial estimates for fitted parameters and known values for all fixed parameters of the system Initial estimates should be generated as carefully as possible If close estimates of the fitted parameters can not be obtained then it may be useful to search for good initial estimates by evaluating the weighted sum ofsquared error for appropriate trial selections 5 Execute an MLAB FIT command directly or via a DO command which specifies the list of fitted parameters unknown function or derivative data matrix and weight vector The list of fitted parameters may contain matrix names in which case all elements of listed matrices are fitted parameters Multiple clauses each specifying an MLAB defined functional form or an unknown function or a derivative of an unknown function of the system with its weighted data may be used in the differential equation FIT command if all the data matrices have the same first column 6 After examining the FIT command output modify the model continue curve fitting or prepare graphical displays as needed In order to prepare graphical displays solve the differential equation system by using the INTEGRATE operator In the following let F denote a function identifier in MLAB defined functions Z and Y1 Y2 Yn denote unknown functions of the differential equations system csi denote a constraint set name N1 and N2 denote positive scalar constants S denote a scalar identifier S
315. s assuming that circular definitions are avoided For example consider the commands 25 FUNCTION GA X AO A1 X A2 Xx X FUNCTION GB T1 T2 C GA T1 T2 ATAN2 Dx T1 T2 Here GB makes reference to the defined function GA The parameters of a defined function consist of the scalar identifiers in the defining scalar expression which are not contained in the argument list as well as any parameters of defined functions referred to in the defining scalar expression For example the parameters of GB are C D AO A1 and A2 Here C and D are identifiers which appear in the defining expression for GB but are not arguments and AO A1 and A2 are parameters for GA which are used in defining GB A defined function may have any number of parameters including zero Note that matrix element references look exactly like defined function references for functions of one or two variables In effect a matrix can be regarded as a function defined by table lookup There is no real problem with this ambiguity since F 2 3 can only refer to one data item F which must be either a matrix or a defined function It is recommended that the MLAB user always specify the same number of arguments in any references to a defined function as was specified in the FUNCTION command defining it How ever there are conventions for handling defined function references with too many arguments extra arguments are ignored or too few arguments arguments not suppl
316. s data d draws a graph of the exponential curve and the computed data The m data points are to be represented by horizontal bars of POINTTYPE STAR intersecting with vertical bars which extend one standard deviation above and one standard deviation below the horizontal bar The unbiased sample standard deviation of each data point is computed from the n 1 repeated measurements for the dependent variable of the point n gt 2 is assumed 4 Suppose that one has two data matrices MA and MB to curve fit to the functional form h t sin c t k with respect to parameters c and k Furthermore suppose that c represents a physical constant which is the same for both experiments but k represents a calibration constant that is different for the two different data matrices How could MLAB be used to curve fit for the best overall fit to c 5 Which of the following define linear models a The functional form h s t a et c sin t with respect to a b and c b The functional form FE 2 3 t 5 b sin k t with respect to b 276 c The functional form in part b with respect to b and k d The functional form g x y sin u x cos u x y eloslu with respect to u Caution the expression for g x y can be simplified 6 Log into MLAB and test your answers to exercises 1 4 by first assigning numerical values to scalars and matrices that were not specified 277 The basic
317. s a column vector with components a a d a 2 d b a c 1 d where d b a c 1 Table 3 9 Fixed number of members arithmetic sequence matrix expression Scalar expressions may be used to specify a b and c as the following MLAB dialog shows 57 N 99 100 X1 3 X2 4 R X1 X2 100x X2 X1 TYPE R a 100 by 1 matrix 2 3 3 01010101 3 02020202 95 terms omitted 3 98989899 4 The arithmetic sequence matrix operations a b d a b and a b c are some of the most frequently used MLAB operations 3 4 MATRIX MANIPULATION OPERATIONS In using MLAB one often needs to rearrange elements in matrices select submatrices of existing matrices and join matrices together to form larger matrices There are a number of MLAB matrix expressions and matrix assignment commands which are useful for such matrix editing For example given an m by n matrix A the n by m matrix obtained by interchanging the rows and columns of A called the transpose of A is frequently denoted by A in mathematics texts The apostrophe is also used to denote the matrix transpose operation in MLAB as shown in Table 3 10 and the MLAB dialog below TYPE 1 3 a 1 by 3 matrix 1 2 3 Here the transpose of a column vector is seen to be a row vector The matrix DAY from the Tutorial of Chapter 1 can be transposed as follows USE TUTOR1 USING W CS1 DAY B A M Y 58 In genera
318. s in MLAB will be considered in this chapter 2 1 DATA TYPES In the Chapter 1 introductory tutorial it was seen that MLAB users can create numerical data either as individual numbers called scalars e g A B or as rectangular arrays of numbers called matrices e g M DAY Two forms of symbolic data were also shown namely user defined functions e g Y and constraint sets e g CS1 There are twelve data types in MLAB scalar matrix string string array function constraint set initial condition window curve title 3D window and surface These MLAB data types will be discussed in the appropriate places The individual data item names chosen by the user are called identifiers Like most computer languages MLAB considers arbitrary sequences of letters and digits to be identifiers provided that the first character is a letter In Macintosh and Linux Unix identifiers may be up to 35 characters long in MSWindows identifiers may be up to 27 characters long and in DOS identifiers may be up to 15 characters long For example X AREA T23A8 or PICTURE11 are valid identifiers but 6AU4A is not At each moment during an MLAB session each identifier corresponds to at most one data item scalar matrix etc It is called a scalar identifier if it corresponds to a scalar data item a matrix identifier if it corresponds to a matrix data item and an unknown identifier if it does not correspond to any data item One can use appropriate
319. s of B such that Aij Bli t m j if k lt m n the elements of the resulting matrix A are the elements of B such that Ai j Bir i 1 m j 1 mod k If ME1 is not a column vector then LIST ME1 is used in its place Table 3 12 SHAPE expression general form The operation 1 1 1 m j 1 mod k means compute the value of 1 m gt j 1 divide by k take the remainder and add 1 This effectively extends column vector B with copies of itself There are two matrix operations ROW and COL that are used to form submatrices of existing matrices by selecting certain rows or columns The ROW operator is described in Table 3 13 and the COL operator is described in Table 3 14 For example if DM is a 6 by 20 matrix in MLAB then DM ROW 3 6 COL 6 10 is a matrix expression denoting the 4 by 5 submatrix of DM formed from the elements in rows 3 4 5 and 6 and columns 6 7 8 9 and 10 of the matrix DM Note that the colon notation for sequences of consecutive integers is used to indicate the selected rows and columns this is a commonly used form of the ROW and 61 COL matrix expressions However ROW and COL operations can also be used to rearrange the order of the rows or columns and to repeat or skip rows or columns For example DM ROW 2 5 1 5 denotes a 4 by 20 matrix with the first row the same as the second row of DM the second and fourth rows the same as the fifth row of DM and the third row the same
320. s of permitted matrix expressions involving arithmetic operations with scalar and matrix arguments excluding raising to a power are given below SE1 ME1 result is an m by n matrix b with bij aij for alli lt mandj lt n ME1 SE1 equivalent to SE1 ME1 SE1 ME1 result is an m by n matrix bij with bj aij for alli lt mandj lt n ME1 SE1 equivalent to SE1 ME1 SE1 ME1 result is an m by n matrix bij with bij x aij for alli lt mandj lt n ME1 SE1 equivalent to SE1 ME1 ME1 SE1 equivalent to 1 SE1 ME1 Table 3 23 Matrix expressions with scalar matrix arguments TYPE M1 gt a 2 by 3 matrix ie l 5 2 2 3 7 0 TYPE M1 M2 a 2 by 3 matrix 9 10 Le 0 2 6 13 2 TYPE M1 M2 a 2 by 3 matrix 1 14 i 2 2 0 1 2 If MA and MB are matrices of different sizes then rows and columns are cyclically replicated in MA and MB as needed in order to compute MA MB or MA MB This is similar to the matrix expansion by cyclic extension principle for matrix editing operations and ROW or COL matrix assignment commands For example observe the MLAB dialog 75 TYPE U V U a 2 by 1 matrix 1 6 2 4 V a 1 by 2 matrix Lee el TYPE U V a 2 by 2 matrix 1 13 5 2 11 3 The dimensions of a matrix sum or difference are the smallest possible In other words NROWS MA MB and NROWS MA MB are the larger
321. scribed the MLAB DO command causes execution of one or more MLAB com mands separated by semicolons which are read in character form from some computer input device In practice DO commands almost always refer to disk files The general form of the DO command is presented in Table 5 6 Suppose TEST DO consists of MLAB commands ending with DO TESTA followed by a semicolon and TESTA DO consists of MLAB commands ending with DO TESTB followed by a semicolon 168 The general form of the MLAB PLOT command is PLOT W1 W2 IN filename Wi W2 are one or more window identifiers If no window identifiers are specified all defined windows will be included filename is the name of a file If the specified filename corresponds to a file that already exists in the MLAB working directory the user will be prompted to specify if the existing file should be kept or over written If no file extension is provided in filename and the plotting device specified by a prior PLOTDEV command is a PostScript device then ps is used as the extension If no file extension is provided in filename and the plotting device specified by a prior PLOTDEV command is an HP Graphics Language device then 1j is used as the extension If no filename is provided and the plot device is a PostScript device then the first unused name in the sequence mlabpO ps mlabp1 ps etc is created containing the current picture If no filename is provided and th
322. sen as the current location for the next iteration At least one trial point in the direction of steepest descent will be examined If there is no such point which depends upon the convergence factor chosen then the search process converges at the current location Note that any constraints that have been imposed will determine boundaries for the region to be searched In fact such linear constraints determine convex polygonal regions possibly infinite in the latitude longitude plane When curve fitting p parameters the linear constraints determine a convex polygonal region in the Euclidean p space of the fitted parameters like a gem with facets The MLAB strategy for generating trial points will prevent choosing parameter values outside the region determined by the constraints If the current location is near the boundary of the region determined by the constraints then the direction of steepest descent is projected onto the region boundary if necessary The initial estimates of the parameters can have a critical effect upon the success of the search Clearly the bottom of the valley is more likely to be found if the search process is started from a side of the valley rather than from a distant point Obtaining initial estimates of the unknown parameters based upon scientific understanding of the experiment is very desirable for curve fitting nonlinear models This is less important for linear models because of the linear model analysis built
323. should be used The input to KREAD ignores Z in distinction to READ CON which terminates with either Z or the ESCape key The MLAB operator KSREAD returns a string which is read from the keyboard The string to be entered is terminated by the first white space character i e space tab or LF CR RE TURN ENTER key which appears in the input Entry must be terminated by pressing the RETURN ENTER key Thus the KSREAD function provides a convenient means of entering a file name for use in a do file 5 4 THE PRINT COMMAND In previous chapters it was shown that the MLAB TYPE command can be used to display infor mation about current data items The MLAB PRINT command is similar to the TYPE command except that MLAB data information is directed to a file by the PRINT command The general form of the PRINT command is shown in Table 5 4 For example the same descriptive information is generated by the two MLAB commands below TYPE NROWS M NCOLS M M PRINT NROWS M NCOLS M M IN MDATA The TYPE command causes the two scalars NROWS M and NCOLS M and the matrix M given in the command to be typed to the MLAB command window The PRINT command causes a computer disk file named MDATA LST to be created containing the same information in character form This file is an ASCII text file it is suitable for printing unlike the MLAB SAVE command or for reading by another program After exiting from MLAB the user can either use the MLAB comma
324. shown in Appendix A Figure A 8 to be drawn at each point in the curve matrix If the PTFONT clause is not included in a DRAW statement the default font is font number 5 In the short form of the DRAW command the LINETYPE clause included either a scalar expression or a name to specify the type of line segment to be used in connecting x y coordinates of the curve matrix The first 5 linetypes are variations of the pattern a thin first line segment followed by a first gap followed by a second thin line segment followed by a second gap For example LT SOLID fits this pattern with the gaps having zero extent Instead of using a name or number in the LINETYPE clause it is possible to use seven numbers separated by commas and enclosed by parentheses to specify the line type as follows LINETYPE SE1 SE2 SE3 SE4 SE5 SE6 SE7 SE1 specifies the length of the first line segment in the pattern SE2 specifies the length of the gap SE3 specifies the length of the second line segment and SE6 specifies the thickness of the two line segments SE7 specifies how the line segments are filled e if SE7 evaluates to 1 then the line segments are not filled e if SE7 evaluates to 0 then the line segments are solid filled 118 e if SE7 evaluates to k gt 0 then the line segments are filled with k horizontal thin line segments The scalar expression SE4 specifies the length of a tic mark that is placed in the center of the first gap If SE4 is po
325. sions to be explained below The first command changes M from an unknown identifier to a matrix identifier corresponding to the matrix 6 15 05 The second command causes the current matrix associated with M to be changed to the matrix symbolically described by POINTS Y M Note that the second type of matrix assignment command can be self referring For example the matrix expression POINTS Y M which describes the new matrix assigned to M makes reference to the current value 6 15 05 of the matrix M when the second matrix assignment command above is executed Matrix expressions are similar to scalar expressions in several respects If a matrix with a certain MLAB identifier is created during an MLAB session then one can refer to that matrix by its identifier in matrix expressions that are part of subsequent MLAB commands There are various matrix operations that can be indicated in matrix expressions in order to input and edit matrices do matrix arithmetic evaluate functions at many points and so on Parentheses indicate the order of operations in matrix expressions involving several operations The use of parentheses is recommended in matrix expressions having several operations To give the forms of matrix expressions let ME1 ME2 MEn denote arbitrary matrix expressions that the MLAB user chooses as arguments for the operations in question ME1 ME2 etc will also indicate the matrices rectangular arrays of numbers obtained by evaluat
326. sions can be compared by any of the six comparison scalar operations lt gt lt gt and NOT For example the MLAB function RCP defined by FUNCTION RCP T IF T NOT O THEN 1 T ELSE O will compute reciprocals except that RCP 0 O The error condition DIVISION BY ZERO will not occur when RCP 0 is evaluated because only the correct clause THEN or ELSE is numerically evaluated for a conditional scalar expression occurring in an MLAB function definition As de scribed in Table 2 11 a comparison expression is regarded as a scalar expression which evaluates to either 0 or 1 The Boolean operators NOT AND and OR also form scalar expressions which evaluate to 0 or 1 Note that comparison or logical operations may appear in any scalar expression just like other scalar operations For example the command FUNCTION F X NOT X gt 0 X gt 0 EXP Ax X is an alternative method for correctly defining the function shown in Figure 2 1 However be careful with parentheses when using such expressions For example NOT X gt 0 1 is interpreted as NOT X gt 0 1 rather than NOT X gt 0 1 by MLAB 30 In general arithmetic comparison logical and conditional scalar expressions work normally if 1 equals true and 0 equals false Arithmetic comparisons values SE1 lt SE2 1 if SE1 lt SE2 0 if SE1 gt SE2 SE1 gt SE2 1 if SE1 gt SE2 0 if SE1 lt SE
327. sions will be arranged into a linear constraint by collecting terms The following dialog shows an example of this rearrangement process CONSTRAINTS PRC A 2 B 1 gt 2xB A 6 TYPE PRC constraints PRC A 2 B gt 0 4 Coefficients of A and B and the constant term were computed by collecting the appropriate terms Note that it is permitted to use a scalar variable other than a parameter in a linear constraint but not as a coefficient of a parameter For example in curve fitting the parameters A and B A 2 B gt K is a valid constraint and K may be assigned any value before the FIT command using this constraint is executed However the forms K A lt B and A K B 240 The general form of the MLAB CONSTRAINTS command is CONSTRAINTS csi Here csi is either an unknown MLAB identifier or the identifier of an existing MLAB constraint set which is to be replaced If csi is an exist ing constraint set then all its previous contents are deleted The quantity in braces represents a set of linear constraints entered as algebraic inequalities involving sums of terms of the form sc si sc si or si sc for any scalar constant sc and any scalar identifier si If there is more than one constraint constraints must be separated by commas MLAB will rearrange each constraint expression into a standard linear constraint form by collecting on the left side all terms which involve scalar identifiers and b
328. sitive the tick mark extends equally to both sides of the gap if SE4 is negative the tick mark extends to one side of the gap Similarly SE5 specifies the length of a tic mark that is placed in the center of the second gap For example a line consisting of e solid dashes of width 05 horizontal frame fraction units and length 1 horizontal frame fraction units e gaps of width 1 horizontal frame fraction units e tick marks in every other gap that are 05 horizontal frame fraction units wide and 1 hori zontal frame fraction units long is displayed in an MLAB picture shown in Figure 4 7 by the following commands UNVIEW DELETE W WINDOW 1 TO 10 1 TO 10 IN W DRAW C 1 10 LINETYPE 1 1 1 1 0 05 0 IN W VIEW The LABEL clause of the extended DRAW command is used to place numbers near some or all of the x y coordinates in the curve matrix It includes a label matrix which has m rows and either 1 or 2 columns If the label matrix has 1 column the successive numbers in the column vector label successive points in the curve For example LABEL 1 1 1 will cause the first point in the curve matrix to be labelled with the number 1 the second point in the curve matrix to be labelled with the number 2 and the tenth point in the curve matrix to be labelled with the number 1 If the label matrix has 2 columns the first column is interpreted as a list of numerical labels and the second column specifies which p
329. st of text strings specifying a set of k choices to be made For Linux with X Windows Macintosh and MSWindows systems a radio button appears to the left of each text string and a CONTINUE button appears beneath the menu Generally MENUCHOICE will be used within a DO file By use of the up and down arrow keys and the TAB key the user may select one member of this list On Linux Unix with X Windows Macintosh and MSWindows systems the radio button corresponding to the currently selected string is filled in On DOS the currently selected string is displayed in reverse video When the RETURN ENTER key is pressed or the CONTINUE button is clicked the dialog window is removed from the screen and an integer value is returned that is the sequence number in 1 2 k of the selected string For DOS systems the optional scalar or array C specifies the color choice as an integer for each of the elements of the menu in this order the title text the title box border and the title background the menu text the menu box border and the menu background If C is a 2 column matrix column 1 contains the background color while column 2 contains the foreground color If C is a 1 column matrix it contains the background color and the text color defaults to white Row 1 of the array C pertains to the title row 2 pertains to the menu items row 3 pertains to the title box and row 4 pertains to the menu box If rows 2 4 are omitted the information from row 1
330. straints Observe that the above FIT command specifies 1 a list of parameters A B C for curve fitting 2 a functional form given by the MLAB function F 3 the data matrix D 4 the associated weight vector V and 5 the constraint set Z1 The MLAB FIT command for curve fitting a functional form requires these operands except that the phrase WITH WEIGHT V can be omitted if all weights are equal to one and the phrase CONSTRAINTS Z1 can be omitted if there are no constraints The following MLAB dialog shows the changes that occur as a result of the previous FIT command TYPE A B C A 3 89352415 B 1 40348841 C 0 15838198 TYPE FUNCTIONS currently defined FUNCTIONS F A F C F F B TYPE SOSQ 245 SOSQ 260577588 TYPE STDEST STDEST a 3 by 1 matrix 1 931970633 2 866608321 3 171116012 TYPE DEPVALS DEPVALS a 3 by 1 matrix 1 0 96551634 2 994265251 3 987432487 TYPE COVP COVP a 3 by 3 matrix 1 868569261 772576329 0 14386197 2 172576329 751009982 145976277 3 0 14386197 145976277 2 92806894E 2 As a result of the FIT command 1 the values of the parameters A B and C changed 2 three new functions were defined which correspond to the derivatives of F with respect to each of the curve fitting parameters 3 the scalar SOSQ was assigned the value of the best weighted sum of squared error 4 the 3 element column vector STDEST was assigned the errors
331. string HEIGHT 30 in the position shown in Figure 4 1 Here JX is an unknown identifier After execution of the TITLE command it becomes the identifier of a title object that has the text string HEIGHT 30 The text string in the command is enclosed in quotation marks to indicate its beginning and end As in the DRAW command the clause IN W puts the new title object JX into the list of items in the window W The SIZE clause specifies character height So SIZE 015 FFRACT indicates that the text string will be drawn 015 horizontal frame fraction units high Upper and lower case letters in a text string appear in the picture as typed Lower case letters and punctuation are printed in correct proportion to upper case characters The clause ANGLE 90 specifies that the text is to be written vertically upward i e at an angle of 90 degrees measured counterclockwise from normal horizontal text The clause FONT 5 specifies that the character set in font table number 5 will be used The clause AT 7 20 WORLD specifies that the first character of the text string is to be drawn at the display point corresponding to the user rectangle point 7 20 That is the starting point of the text string is given in the user s coordinate system for the window The clause SHIFT 0 specifies that the string is not to be shifted from the starting point 7 20 The clause COLOR GREEN specifies that the color of the string is to be green for graphic displays Green appears blac
332. such that Ark lt Qik for i 1 2 m That is the result is the row index r for which a is the minimum for all values in column k In case of ties the result is the least row index Table 3 22 Some useful matrix editing operators In this Matrix Editing Operations section it was shown that the TYPE command can be used for displaying lists of matrix identifiers or more generally matrix expressions That is the form TYPE ME1 ME2 MEn 72 will cause n numerical matrices to be displayed If the matrix expression is an MLAB identifier a heading containing the identifier and the dimensions of the matrix will precede the listed values of the matrix Otherwise only a heading containing the dimensions of the matrix precedes the display of numbers One useful form of the above TYPE command employs the COL operation for handling large matrix displays Where the rows of a given matrix are too long to be accommodated in one line this command form will divide the matrix into convenient parts For example if C is a 10 column matrix to be examined the command TYPE C COL 1 5 C COL 6 10 will cause C to be typed in two parts columns 1 to 5 and columns 6 to 10 This keeps the rows short enough to be shown on one line Thus vertical alignment of the columns is maintained and the wrap around problem for long lines is avoided 3 5 MATRIX ARITHMETIC OPERATIONS There are a number of matrix ar
333. t so this is highly efficient In general creating a hierarchical system of functions is more efficient than most other methods of defining a model Matrix elements can be used for MLAB function parameters This can be convenient when indexed sums appear in the functional forms as for power series Fourier series and so on For example a generalized polynomial model approximating a power series can be set up by the MLAB command 255 FUNCTION Q X C 1 SUM J 1 N C J 1 x X J to represent the polynomial g x co 61 C2 2 H Cni r of degree n Here C is a column vector of length n 1 To curve fit a polynomial to a known data matrix D and weight vector V an integer value is first assigned to the MLAB scalar N to determine the polynomial degree The FUNCTION command above then defines the model Numerical values are next assigned to the parameters by creating an n 1 by 1 MLAB column vector C The MLAB command FIT C Q TO D WITH WEIGHT V now initiates the curve fitting process Each element of the matrix C is treated as an individual parameter for curve fitting just as if each were a separate MLAB scalar Both MLAB scalar and matrix names can be used in the list of parameters of the same model For example h t c11 sin a t c12 cos a t c21 sin b t c22 cos b t could be set up by the command FUNCTION H T C 1 1 SIN A T C 1 2 COS A T C 2 1 SIN B T C 2 2
334. t is the same 10 relative error for the original data yields a constant error log 1 1 after the logarithmic transformation Now the simple unweighted sum of squared error criterion is appropriate only when the measurement errors are expected to be about the same for all data points The logarithmic transformation will usually give more satisfactory results for curve fitting data with experimental error proportional to the magnitude of the measurement Mathematical transformations of data should be used with care If the errors are uniform or come from a Poisson process for example then taking logarithms will usually give worse results than would be obtained from curve fitting the original data 7 2 WEIGHTED SUM OF SQUARED ERROR CRITERION A second more flexible method is available in MLAB for analysis of data obtained with measure ments of unequal precision In this method the agreement between the model and the experimental 229 data is measured by a weighted sum of squared error criterion Each data point has an associated weight that is a non negative number assigned by the data analyst These assignments are made so that data points measured with high precision are given greater weights than those measured with less precision For example reconsider the data of Figure 7 1 Suppose that the measurements outside the interval 1 4 on the x axis are known to be less precise It is decided to assign weights w t 1 2 3 4 as shown below
335. t named W This will cause the new curve Ci where i is some integer to be drawn in the default window W overlapping the window currently in use Suppose the user is drawing in a window named PIC but neglects to add IN PIC to a DRAW command To correct this error use the command sequence 110 The short form of the DRAW command is DRAW C ME1 POINTTYPE ptyp LINETYPE ltyp IN wW The identifier C is either unknown to create a new curve or is an exist ing curve to be modified The n by 2 matrix ME1 becomes the curve matrix for C and curve C is put into window W The user may use DRAW commands such that NCOLS ME1 1 or 2 for 2 dimensional drawing If NCOLS ME1 1 the MLAB adds another column to the curve matrix so that it becomes 1 NROWS ME1 ME1 The curve s point type ptyp can be a scalar expression such that INT ptyp is an integer between 0 and 30 a quoted keyboard character i e c or a predefined point type name The curve s line type 1typ can be a scalar expression such that INT 1typ is an integer between 0 and 19 or a predefined line type name The word POINTTYPE may be abbreviated PT and the word LINETYPE may be abbreviated LT For line types other than 6 7 9 10 11 and 12 the curve is drawn on the dis play by drawing the specified point symbol at each of the n points in the curve matrix and drawing the specified type of line at each of the n 1 lines between points in adjacent r
336. t unit of the cosine oscillations which sum to zx It is of interest to compute the functions M and from x In particular M shows the relative strengths of the various members of the set of oscillations which sum to form x The function M s is called the power spectrum of zx When zx is given numerically by means of n equally spaced samples of x over one period we usually assume x is sampled often enough so that x is smooth between samples and M s is 0 for s gt n 2 p We are given the matrix of input to z to to T x to T X t4 2 T x to 2 T kaye abia 217 where T p n and we wish to compute the values M 0 M 1 p M n 2 p and 0 9 1 p p n 2 p These two sequences of values are called the real discrete Fourier transform of the samples of x The MLAB operator REALDFT can be used to compute the real discrete Fourier transform of x If X is a matrix with n rows and n is a power of 2 REALDFT X uses a fast discrete Fourier transform algorithm and returns a matrix with 3 columns and n 2 1 rows The j row is j 1 p M G D p 9 5 1 p containing the frequency amplitude and phase shift values for the j 1 cosine term in the Fourier series of z If n the number of rows in the input matrix is not a power of 2 REALDFT X first performs a linear interpolation of matrix X at the values X 1 1 X m 1 nn where nn is the greatest power of 2 less than n REALDFT X then computes the re
337. t up the model with linear constraints e evaluate a weighted sum of squared error 24 times for all pairs such that a is among 7 8 9 1 0 1 1 1 2 and k is among 01 02 03 04 and use the pair producing the minimum weighted sum of squared error as initial estimates for a and k Q curve fit the model to the data using these initial estimates 275 2 Design a DO file that inputs a two column matrix D and curve fits polynomials of orders 0 constant 1 straight line 2 quadratic 3 cubic and 4 quartic to D successively using weights all equal to 1 Use a 5 by 1 column vector as the list of polynomial coefficients which are the fitted parameters Also a 5 by 1 column vector is to be created containing the root mean square error for each curve fit 3 Suppose D is an arbitrary m by n matrix where the first column contains measurements of an independent variable and the last n 1 columns contain corresponding repeated measurements of a dependent variable Construct a DO file which a computes an m by 2 matrix M where the first column contains the values of the indepen dent variable and the second column contains the means of the repeated measurements of the dependent variable b computes an m by 1 column vector W of weights by the reciprocal of variances method c curve fits an exponential curve y kK en with fitted parameters K and a using initial estimates K 1 and a 1 to thi
338. tain the numerical values of scalar ex pressions As will be seen in subsequent chapters the TYPE command responds with appropriate information about any MLAB data item or outputs the result of evaluating any of a variety of MLAB expressions The TYPE command is summarized in Table 2 17 Uses of the TYPE command for other kinds of MLAB data items and expressions will be described as such terms are introduced 40 In general derivatives in MLAB may be defined by expressions of the form F X1 X2 Xn F DIFF Xi DIFF X2 DIFF Xn Here F is the identifier of an MLAB function and X1 X2 Xnisa list of n MLAB identifiers usually arguments and parameters of F for n gt l Let DF denote the general DIFF expression shown above Then DF de fines an MLAB function which has the same list of arguments as F The scalar expression defining function DF is obtained by n fold symbolic differentiation of the expression defining F treated as an ordinary or partial derivative with respect to the list of variables X1 X2 Xn If F has m arguments then the scalar expression DF SE1 SE2 SEm can be used for numerical calculation of derivatives Table 2 16 Defining MLAB derivatives 2 10 SUMMARY In the following let SE1 SE2 SEn denote scalar expressions let U denote an unknown identifier let X denote a scalar identifier let F denote a function identifier defined by a FUNCTION command or builtin function
339. tes M 1 1 M 1 2 in the default window The final VIEW command then displays the default window with the curve and title 5 9 SUMMARY In the following let file1 file2 filen denote disk files dev denote an input device name filename and filenamel filename2 filenamen denote user selected disk filenames of one to eight characters ext denote an optional filename extension of one to three characters D1 D2 Dn denote MLAB identifiers SE1 and SE2 denote arbitrary scalar expressions Wname denotes a window name M denotes a matrix identifier and ME1 denotes a column vector 1 File Specification filename ext 2 MLAB Input Matrix and String Expressions READ filename SE1 SE2 READ filename ext SE1 SE2 ext is optional READ dev SE1 SE2 m by n matrices m lt SE1 and n SE2 are read if SE1 and SE2 are omitted column vectors are read until CTRL Z Z or end of file is detected READ CON SE1 SE2 same as READ CON SE1 SE2 READON filename ext SE1 SE2 KREAD prompt SE1 SE2 string prompt is printed and m by n matrix is read row by row KSREAD reads a string MENUCHOICE T S X C displays the title string or string array T and menu options specified by string array S with option number X initially selected and waits for user to select an option the number of the option selected by the user is returned as a scalar value GETSTRINGS T S H X Y C L111
340. text or with the EIGEN operator discussed above 6 6 SINGULAR VALUE DECOMPOSITION OF A MATRIX An a by b matrix M can be written as a product of matrices U P and V UPV M where U is an a by b matrix such that UU I V is a b by b matrix such that VV I and Pisa b by b real diagonal matrix whose diagonal elements 01 gt 02 gt gt On gt 0 are called the singular values of M The number of non zero singular values is the rank of M but since round off may produce small non zero values in place of zero caution is required The MLAB SVD operator described in Table 6 7 will compute U V and the diagonal elements of P from M The matrix returned is equal to GETDIAG P amp U amp V The SVD procedure is also employed in computing the Moore Penrose generalized inverse of a matrix used for matrix inverses in MLAB Singular value decomposition has an important application in analyzing data derived from exper iments in which a number of processes are occurring simultaneously and for which the measured data are linear combinations of some property of each process This set of conditions is not es pecially peculiar it applies for example to optical spectra An optical spectrum is a measured functional relation between the color i e wavelength of light shining upon a transparent sample 213 The general forms of the CPROD CDIV and CPOW operators are CPROD ME1 ME2 CDIV ME1 ME2 CPOW ME1 M
341. th a title specified by string argument T a list of of prompt fields specified by string array argument S a list of initial strings for the form specified by the string array argument H an initially active field specified by the integer scalar X a string field width specified by the integer scalar or vector Y and form layout specified by the integer scalar L GETSTRINGS returns a string array containing the contents of the editable text fields of the form at the time the user strikes the RETURN ENTER key or clicks the CONTONUE button with the mouse The elements of the returned string array can then be converted to scalar values function definitions matrices titles etc Generally GETSTRINGS will be used within a DO file The user selects each text entry field to be edited using the up arrow and down arrow keys the TAB key or the mouse The initial string appearing in a given text entry field may be recovered by pressing the ESCape key The user terminates the form fill in process by pressing the ENTER RETURN key On Linux Unix with X Windows Macintosh and MSWindows systems the form fill in process may also be termi nated by clicking the mouse button while the mouse is on the CONTINUE button Upon quitting 178 The general form of the MENUCHOICE function is MENUCHOICE T S X C where T is either a string or string array defining the title and or instructions for the menu the optional argument S is a string array def
342. the corresponding curve matrix point The first word may be either LEFT CENTER or RIGHT and the second word may be either TOP CENTER or BOTTOM Without a PLACE clause the lower left corner of the label box is placed at the offset from the corresponding curve matrix point The clause PLACE LEFT TOP would place the top left corner of the label box at the offset from the curve matrix point The clause FORMAT SE6 SE7 SE8 SE9 SE10 SE11 specifies the format in which the point labels are to be drawn The meanings of the scalar values SE6 SE7 SE8 SE9 SE10 and SE11 are beyond the scope of this guide however a complete description of the format codes appears in the NFORMAT section of The MLAB Reference Manual The clause FORMAT 3 9 0 0 2 0 is supplied if the FORMAT clause is omitted Suppose M is the 5 row by 2 column matrix and W is the window created by the following MLAB commands WINDOW 8 TO 3 4 5 TO 1 25 IN W NO IMAGEBOX IN W M SHAPE 5 2 LIST 0 1 5 4 1 2 1 5 5 2 0 TYPE M M a 5 by 2 matrix 1 0 1 2 0 5 0 4 3 1 0 2 4 1 5 0 5 5 2 0 The LABEL clause with a single column matrix DRAW CX M LABEL 1 5 IN W VIEW produces the curve with lables as shown in Figure 4 8 a In order to label the points with numbers in a different font and size and shift the labels to the right and below the corresponding curve matrix points as shown in Figure 4 8 b the following command could be used
343. the information in a computer disk file named TUTOR1 SAV by typing the following MLAB command SAVE Y M A B DAY CS1 W IN TUTOR1 to which MLAB responds 12 creating save file TUTOR1 SAV The data in TUTORI SAV can be retrieved during this or a future MLAB session using the command USE TUTOR1 You can now terminate the MLAB program and return to the operating system by typing EXIT On computer systems with a graphical user interface you can also terminate the MLAB program by the usual method of terminating an application i e select QUIT from the FILE menu or click the close box in the title bar of the MLAB command window When MLAB begins it opens a file called MLAB LOG and writes all subsequently typed input and printed output to that file Any previously existing file named MLAB LOG will be renamed to MLABO LOG any file named MLABO LOG will first be shifted to be MLAB1 LOG any file named MLAB1 LOG will first be shifted to be MLAB2 LOG any file named MLAB2 LOG that is replaced by MLAB1 LOG will be deleted The computer disk file MLAB LOG now contains the session log which you can print using a print utility program In DOS this is done by typing PRINT MLAB LOG For Linux Unix Macintosh or MSWindows you can open and print the MLAB LOG file with any text editor or word processing program you have available You can also rerun MLAB and type PRINT FILE MLABO LOG Do you see why we print MLABO LOG instead of MLAB LOG
344. tions and divisions are performed from left to right and finally additions and subtractions are performed from left to right For example AO 2 A1 T A2 T T is also an MLAB scalar expression which is equivalent to the algebraic expression shown above As in most programming languages the elementary functions of the real numbers are available in MLAB Some examples of mathematical expressions and corresponding MLAB expressions are given below sin ax cos bx becomes SIN A X 2 COS B X 2 log y a c 1 3 becomes LOG Y A X C 1 3 vVBz 1 becomes SQRT BETA Z 1 exp x arcsin x becomes ABS EXP X ASIN X Of course some scalar expressions are undefined for certain real numbers when such operations as division by zero logarithm of a negative number etc are called for Even if mathematically defined the computed result may cause arithmetic overflow EXP 500 gt 1 7 x 10308 for exam ple or may be numerically meaningless for example SIN 10 750 will not compute accurately since the argument has only sixteen digit precision During scalar expression evaluation warn ing messages will be issued if overflow occurs or undefined functions are found In such cases computationally plausible results are taken as the results and the expression evaluation process continues This can be an important aid in some situations Warning may also be given for the underflow condition where a number of magnitude less than 2 x
345. to 47 in Hershey fonts 1 to 34 317 ascir4 8 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 Font O 2 34536 789 1 SS A Ad DEEP DRETS E POR NV ASIS INS RS IO AA SR A A A IS LV VY VV VV OVE VV VV ewe es A ya O nal ne ee Oe ae ee ue eas TCOCDIDODODADONADONONN O y MMO 0 00 00 09 00 00 00 MM 09 00 00 00 0 0 SY 00 ta wo KRARRAREREARARERARRARREED OHODWODOWOW MW OWOWOOWWOHO tO th th CO LAT WY LO LO LO LO LO IO LO LO LO LO LO LO LO 10 LO LF a LF LO 1 D AS RAR ARS RD RAEE SS Y Y Y anan0ADaamamamnamn anna m man NU CUCU NCQ NON ON CN ANNANN UUU a o HAN OR e ee 9 1 2 3 5 amp 8 9 callao eiejalola loas seo O ANUMTILDIORDODODO ANN AD eee nnn AX AAAAAA lt X monmotooM v __VVVVVV m AS E E o gt a 10 Y mM aY o LO OU Ne Oy do rah cae AS os on mm a nO 00D coCO 00 00 00 0 amp Y nn F RMMNNN oO COWBH LS CO amp 10 LO LO Ld La ly VE Set SS SSS nM OD A MM Mm Mm vv YNNNNNN ad eerenene oD 2006998 Figure A 4 ASCII characters 48 to 63 in Hershey fonts 1 to 34 318 asci 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 Font AB CODE 0050RaK00000000000009098999000 e ex VdOOODO0 MN O ZEZ2ZZZZZZZZZZZAZAALIRRAOT gt KOzzZzZ2e2e SESTIOSSSS3 ASIS SSINSNERRBRAS q no SSSss IMS 10 A lt lt SK SNR SOT wv dl ll el K WSK MN MY My MY My Mi Rt he et SRST SMG
346. to the left of the maximum for S versus c the curve fitting process tries to improve the sum of squared error by trying negative values for c such that c is large The values of f x are negative between 1 and 1 at c 30 and so taking a negative value for c with c large makes the computed graph for f x identically zero between 1 and 1 again as shown in Figure 7 9 The next initial estimate does give a correct answer C 25 FIT C F TOM matherr overflow error in cosh arg 6 495189e 03 return value 1 797693e 308 matherr overflow error in cosh arg 2 598076e 03 return value 1 797693e 308 matherr overflow error in cosh arg 6 495289e 03 return value 1 797693e 308 matherr overflow error in cosh arg 2 598116e 03 return value 1 797693e 308 matherr overflow error in cosh 267 arg 6 495389e 03 return value 1 797693e 308 matherr overflow error in cosh arg 2 598156e 03 return value 1 797693e 308 final parameter values value error dependency parameter 0 7094633599 0 0224051432 o Cc 7 iterations CONVERGED best weighted sum of squares 2 979701e 02 weighted root mean square error 3 859858e 02 weighted deviation fraction 2 585658e 02 R squared 9 731768e 01 Note however that 7 iterations were required for convergence for this case The next initial estimate leads to a common difficulty C 2 FIT C F TOM final parameter values value error dependency parameter 3
347. trix to be drawn with linetype ALTERNATE For example the following commands will draw the polar graph r cos 7 6 using the MESH operator to easily draw and label the polar coordinate axes RESET WINDOW 1 5 TO 1 5 1 5 TO 1 5 IN W1 IMAGE 25 TO 2 25 25 TO 2 25 INCHES IN W1 U 0 150 30 DRAW MESH COSD ON U amp SIND ON U XK XA XA XX 190 Figure 6 2 Graphing in polar coordinates COSD ON U 180 SIND ON U 180 IN W1 LINETYPE ALTERNATE LABEL MESH U U 180 FUNCTION R T COS 7 T DRAW TPOLAR R O PI 05 IN W1 LINETYPE DASHED VIEW These commands draw the polar graph shown in Figure 6 2 The MESH operator is used to draw and label the coordinate axes Drawing the curve is achieved using the standard polar transformations x r cos 0 y r sin 0 These transformations are applied automatically in the built in function TPOLAR which is used instead of POINTS The MESH operator can also be used to draw information in a certain way Consider for example the CDF operator given in Section 3 6 While this operator returns a matrix of values corresponding 191 to the values of the empirical distribution function at a variety of points distribution functions are commonly drawn as step functions as in Figure 6 3 The following commands use the MESH and ROTATE operators to accomplish this RESET FUNCTION GROUP X INT X 30 30 M GROUP ON RAN ON 07730 G CDF M R ROTAT
348. ts 74 matrix expressions with matrix arguments 76 matrix inverse 79 94 matrix joining operators 67 69 matrix operators 189 matrix power operation 79 matrix product 77 79 matrix repeating expression 67 matrix repeating operators 69 matrix row replication operator 67 matrix transpose 58 matrix scalar arithmetic 73 MAXITER 251 MAXPOS 164 201 MAXROW 72 MENUCHOICE 177 178 MESH 189 192 200 METHOD 280 295 301 MINROW 72 MIXED 295 mixed Adams Gear Combination Method 294 MLAB directory 154 155 MLAB picture 103 MLAB scripts 169 MLABEX DO 2 models 253 modifying titles 134 Moore Penrose generalized inverse 79 213 multiple line titles 133 multiplication 18 NAMESW 166 NBAR 106 NCOLS 49 nested FOR commands 89 NO FRAMEBOX 105 NO IMAGEBOX 105 noise 196 non linear models 258 NONE 106 norm of weighted sum of errors 250 normal bar point type 106 normal distribution 19 normal random numbers 21 NORMRAN 21 NOT 29 30 NOT CONVERGED 252 253 NOT 30 NROWS 49 number of columns in a matrix 49 number of rows in a matrix 49 numerical precision 16 OCTAGON 106 octagon point type 106 ODERPT 280 296 301 ODESTR 286 292 OFFSET 117 121 127 133 ON 84 86 on line help 12 optical spectra 214 OR 29 30 ORANGE 125 overflow 18 parabola 143 144 parameter 23 parametric curves 114 parentheses 22 48 parenthesis rul
349. ts can sometimes prevent this overshoot phenomenon Observe the dialog CONSTRAINTS Z C gt 0 C 10 FIT C F TO M CONSTRAINTS Z final parameter values value error dependency parameter 0 7094748974 0 022404762 o Cc 5 iterations CONVERGED best weighted sum of squares 2 979701e 02 weighted root mean square error 3 859858e 02 weighted deviation fraction 2 585643e 02 R squared 9 731768e 01 no active constraints Because the search process was prevented from using negative values for c by the linear constraint c gt 0 it eventually converged to the correct value For nonlinear curve fitting it is usually desirable to use MLAB to make pictures showing the agreement between the data and the fitted model Also the residuals Yi f ti T e Lin by be sag les for i 1 2 3 M can be computed for the m data points These residuals should be positive or negative randomly each about half the time If the residuals are almost all of the same sign in certain regions a poor curve fit or an inappropriate functional form should be suspected For example the residuals are positive and negative for the curve fit of Figure 7 8 apparently randomly Note that the data points fall above and below the function graph with no obvious pattern However the residuals are all positive for the curve fits of Figures 7 9 and 7 10 agreeing with the observation that they are poor curve fits The magnitude of the root mean square
350. ture editions Daniel Kerner Silver Spring MD January 2014 Contents 1 A TUTORIAL MLAB SESSION El THE TUTORIAL oora taana aa a re ne de L2 SUMMARY socorro p aaa ER eR A aaa eRe a 2 NUMERICAL CALCULATION IN MLAB 21 DATA TYPES oo ad RAOR A p aA a A a 2 2 SCALAR ASSIGNMENT STATEMENTS AND SCALAR EXPRESSIONS 2 3 FUNCTIONS AND THE FUNCTION COMMAND 2 4 RELATIONS AND DEFINITION OF FUNCTIONS BY CASES 2 5 RECURSIVE FUNCTIONS AND THE TYPEOUT OPERATOR 2 6 INDEXED SUMS PRODUCTS AND DEFINITE INTEGRALS 2 7 EVALUATING AND FINDING ROOTS OF EXPRESSIONS 2 8 DERIVATIVES OF FUNCTIONS lt lt 300044 ee wk a 29 THE TYPE COMMAND caon ena dp eee ia taa de i 210 SUMMARY o occisa 6 OS 64 a k aaa AAA 2 11 EXERCISES sre neca tos aaora ani m A G ee we oe ee 3 COMPUTATION WITH MATRICES IN MLAB 3 1 MATRIX ASSIGNMENT STATEMENTS AND MATRIX DIMENSIONS 11 13 15 15 16 23 26 31 33 36 37 40 41 45 47 3 2 ENTERING NUMBERS INTO A MATRIX coccoccosorrors rss 3 3 GENERATING ARITHMETIC SEQUENCE VECTORS 3 4 MATRIX MANIPULATION OPERATIONS 4 2426 65 sea eh ee ava ce da 3 5 MATRIX ARITHMETIC OPERATIONS o o 3 6 RANKORDER AND DISTRIBUTION OF VALUES IN A MATRIX TABULATION OF A FUNCTION ec 266 2h a ROS See Ca De eS 38 LINEAR INTERPOLATION is aaie ae ae et Ee ae wk oo THE FOR COMMAND ooo ras dd A AA 310 SUMMARY ie ra ada pedos da da
351. uation specification here S T C T and P T as its first three arguments and a column vector of values for the independent variable here the times 0 10 as its fourth argument The resulting matrix U again has the specified vector as its first column Also columns 2 3 4 5 6 and 7 give the corresponding values for S U I 1 S T UCI 1 C U I 1 C T U I 1 P UC I 1 and P T U I 1 respectively for I 1 2 NROWS U That is each unknown function and its derivative are given successively in the columns of U following the first The order of the columns is the same as the order of the function derivatives specified in the INTEGRATE operator Although it has not been illustrated the column vector of independent variable values given as the last argument to the INTEGRATE operator need not be a sequence of equally spaced or even unidirectional values Any desired sequence of values may be specified Differential equations often arise in situations in which the rate of change at any time is a function of a value at a previous time Such a differential equation is said to have a delay term An example of this is a model of the population of an isolated colony of animals This model will take into account the diminishing of the growth rate as the population increases due to overcrowding and shortage of food This diminishing effect will take place after a certain amount of time r called the lag time The population will
352. uished from other FIT commands in MLAB by the function identifiers F ap pearing in the command Hence the general form of the FIT command for this system is the same as for the FIT command for a defined function in MLAB That is the basic form is FIT Pi P2 Pn F TO ME1 WITH WEIGHT ME2 The function identifier F refers to an identifier among the list of Yi Y2 Yn Y1 T Y2 T YO T Identifiers P1 P2 Pn to be fit may be either scalars or matrices in MLAB The parameters then include the scalars listed and the elements of any matrix listed these parameters may appear in any differential equation expression SEj or initial condition value SEn j for j 1 2 n However no fitted parameter is permitted in SEO the common value of the independent variable T specified in the INITIAL commands The multiple clause form below is also permitted FIT P1 P2 Pn clausel clause2 clauset Table 8 3 FIT command differential equations systems 312 Each clause is of the form F TO ME1 WITH WEIGHT ME2 where F is an MLAB defined functional form or one of the unknown functions or derivatives for the differential equations system listed in the previous paragraph For FIT commands involving systems of differential equations there should be 2 column matrices ME1 specified in each clause F TO ME1 WITH WEIGHT ME2 since F must be a function of one variable T As before each weight matrix
353. umber of iterations has been executed The default value of MAXITER is 20 Zero iterations may be specified in which case a summary for the user defined initial parameter values is typed with no change in the values of the fitted parameters TOLSOS is another control variable that determines convergence of the fitting process It should always be a small positive number If unspecified its value is 001 This value indicates that convergence occurs at the i iteration if the parameters obtained in the i iteration give less than a 1 improvement in the weighted sum of squared errors over the best previous value More precisely suppose the curve fitting process did not converge in the first 1 iterations Then the curve fitting process will converge at the i iteration if and only if Si 1 Si lt convergence factor Si 1 where S denotes the smallest weighted sum of squared error found in the i iteration and S _1 is the weighted sum of squared error at the end of iteration 1 Here So is the initial weighted sum of squared error In all cases Si lt Sii The MLAB response to the FIT command when LSQRPT has its default value of 8 is a final summary of the curve fitting process after completion of the last iteration The final summary contains the final values assigned to the fitted parameters the final Lagrange multipliers an 251 indication whether the curve fitting process converged or not CONVERGED
354. ure disappears and the screen containing the recent history of MLAB commands returns The UNVIEW command has no effect on DOS PCs A second use of the display control commands is to avoid unnecessary delays Redrawing a complex picture may require a few seconds One may want to make several changes to the current picture without waiting for the picture to be redrawn after each change To do this execute the UNVIEW command after the initial VIEW command then give graphics commands modifying the picture and then execute the VIEW command to display the new picture Since it is sometimes impossible to interrupt the drawing of a picture without destroying the MLAB session one must anticipate this problem in order to avoid it with display control commands The general forms of the MLAB picture display commands are VIEW W1 UNVIEW where W1 is an optional list of one or more MLAB window objects VIEW displays the MLAB picture with the named MLAB window objects If no window objects are named VIEW displays all window objects UNVIEW removes the MLAB picture from the display screen Table 4 1 The VIEW and UNVIEW commands MLAB window objects are explained in the next section 4 2 WINDOW CURVE AND TITLE DATA OBJECTS The MLAB data types called curve axis title and window are involved in making two dimensional pictures Curves are data objects which contain certain related elements of a picture A curve might descri
355. variable x unknown functions derivatives of unknown functions delay terms fitted parameters b1 b2 bp and fixed parameters C1 C2 Cg The value u must be a common value of x e Initial estimates for fitted parameters bo1 bo2 bop e Fixed parameter values Co1 Co2 Cog e Constraint set each constraint being one of the forms r by bre b2 1 bp lt ro r bi tro b2 Tp bp ro r bi r2 b2 Tp bp gt ro e Fitted functions y x or y x for a given k k lt n e Weighted sum of squared error S b1 ba dp ul Yrl xi b1 ba bp S b1 b2 bp YE yj zi b1 da bp The standard MLAB curve fitting procedure for differential equations systems follows with op tional steps indicated by an asterisk 1 Create MLAB matrices via USE DO or matrix assignment commands as follows a An m by 2 data matrix b An m vector of data point weights 2 Create an MLAB system of differential equations via a USE DO or FUNCTION and INITIAL commands corresponding to the given system Parameters may be represented by scalars or matrix identifiers 3 Create an MLAB constraint set via a USE DO or CONSTRAINTS command containing all the linear constraints for the model Matrix element parameters can not appear in MLAB linear constraints 303 4 Create MLAB scalars or matrices via USE DO or scalar or matrix assignment command
356. velocity such that dx k t k dt j at all times and passes through the point 4 4 at time t 1 with positive y component Suppose the constant k is to be curve fit using measurements for x t and y t that were obtained at certain times during an experiment and were entered in data matrices DX and DY respectively Then the following MLAB dialog shows the process of fitting to estimate the shared parameter k DX COL 1 1 10 DX COL 2 LIST 4 56 13 14 20 23 29 07 35 53 52 32 63 43 82 37 99 78 118 23 DY COL 1 1 10 DY COL 2 LIST 5 36 10 89 9 33 11 68 12 28 18 63 18 89 15 23 18 5 FUNCTION X T T K T K INITIAL X 1 4 FUNCTION Y T 2 SQRT K T T 2 K T 4 1 5 K K 4 FIT K X TO DX Y TO DY final parameter values value error dependency parameter 1 99700586 0 0304029348 o K 2 iterations CONVERGED best weighted sum of squares 1 603790e 02 weighted root mean square error 2 905340e 00 weighted deviation fraction 2 844207e 02 R squared 9 925548e 01 Table 8 3 and Table 8 4 show the form of the FIT command when differential equations are involved 8 8 SUMMARY Assume the following e Data points coordinates x vi with weights w for i 1 2 m 302 e Differential equation system y 1 h x yi 2 ya 2 Ynl 2 D1 Da bp C1 C2 Cq With initial conditions y u z for j 1 2 n The functions hy hz hn may involve an independent
357. weights of the embryos Check the entries of matrix DAY If any are incorrect type the MLAB command DELETE DAY and then retype the above MLAB commands in order to define the matrix DAY correctly Later you will learn easier ways to make corrections to matrices Our next objective is to display the data on the computer screen Type the MLAB commands DRAW CA DAY POINTTYPE TRIANGLE LINETYPE NONE VIEW The first command causes the 10 by 2 data matrix DAY to be plotted as a list of 10 points in the x y coordinate plane with a small triangle drawn at each data point The second command causes the picture of Figure 1 1 to show in a separate window on Linux Unix with X Windows Macintosh or MSWindows systems or on the entire display screen of DOS systems On Linux Unix with X Windows Macintosh and MSWindows systems type the command UNVIEW 0 9 0 722 0 544 0 266 0 188 Figure 1 1 Plot of dry weights vs age to make the graphics window go away or click the close box in the graphics window On DOS systems the picture will go away if you strike any key The next set of commands will define the model that is describe the family of exponential curves by its functional form Recall that the equation of an exponential curve has the form y x a exp b x where a and b are parameters of the curve To fit an exponential curve to our data we must first define the general equation of the exponential curve Since w
358. with m arguments A1 A2 Am and let DF denote an ordi nary or partial derivative F A1 A2 Ak 1 MLAB Data Types are a b matrix string string array function constraint set g initial condition window 41 The TYPE command has the general form TYPE TL DL 0310 This command causes information describing or evaluating the given terms T1 T2 Tn to be typed on the terminal Each term must be either an MLAB identifier or a permissible MLAB expression The re sponse to any term includes the data type of the identifier or expression plus information about the particular data item corresponding to that term If the term is an MLAB identifier the response lists that identifier along with its value If Tj is an MLAB expression the response lists only the value of the expression TYPE SE1 SE2 SEn types SE1 SE2 SEn TYPE M types the current matrix entries by row starting each row with the row number of the matrix some rows require more than one line of output TYPE F types the FUNCTION command currently defining F TYPE F X1 X2 Xm types the indicated ordinary or partial deri vative if necessary creates and types sym bolic derivative expressions Table 2 17 TYPE command i curve j title k 3D window 1 surface 2 Scalars and matrix entry values have sixteen digit precision and non zero magnitudes from 2 2 x 107908 to 1 7 x
359. x assignment commands given above only the first form that containing ME1 needs separate discussion Let ME1 denote a column vector if NCOLS ME1 gt 1 then LIST ME1 is used of positive integers j1 J2 Jx which represent row numbers of M If necessary the INT function is applied to the entries of ME1 to produce ji j2 jg If any ji is zero or negative an error message appears Suppose M is an m by n matrix and ME2 is an m by n matrix expres sion Then the command M ROW ME1 ME2 causes M to become an m by n matrix where m is the maximum of m and j j2 je First if m gt k or n gt n a warning message is printed that the dimensions of ME2 exceed the dimensions of the left hand side matrix Then if m lt m m m rows of zeros are added to M Next for each i lt k the i row of ME2 replaces row j of M with cyclic extension or truncation of the it row of ME2 as necessary If i gt m so there is no it row of ME2 m is subtracted successively from until the remainder is between 1 and m i e the rows of ME2 are used cyclically to replace rows of M as required Table 3 15 ROW matrix assignment command MA MB a 2 by 3 matrix 1 2 1 3 65 The general forms of the COL matrix assignment commands are as follows M COL ME1 ME2 M COL SE1 ME2 M COL SE1 SE2 SEn ME2 As in ROW matrix assignment commands M denotes an MLAB matr
360. y computing one scalar constant possibly zero to follow the relational symbol Linear constraints cannot contain matrix element parameters However a linear constraint can contain any MLAB scalar with a fixed value which need not be a parameter of the functional form Table 7 5 CONSTRAINTS command for example are not permitted An MLAB linear constraint can not contain a product or a quotient of two scalar variables 7 7 GENERAL PROBLEM OF CURVE FITTING A FUNCTIONAL FORM The problem of curve fitting a functional form model to data may be summarized as shown in Table 7 6 Although a global minimum for S was used in this discussion there are some situations in which an isolated local minimum for S is a satisfactory curve fitting solution 7 8 PREPARING DATA FOR CURVE FITTING OPERATIONS To curve fit with MLAB the user must specify a mathematical model Also the data and weights must be put into tw MLAB matrices a matrix of data points and an optional column vector of weights Any convenient matrix technique can be used to create them For example the data 241 Assume m models of the functional forms fili tay abu Daya bp Y2 Jr L2 En b1 b2 sg b bp Ym fm Zi Dos eg OHO DSi nag 0p involving m dependent variables y Ya Ym N independent variables T1 2 Ln and p parameters b1 b2 bp having unknown values There may also be other parameters in f which are assi
361. y to increase the tolerance again then terminate the integration 306 the value 2 means report errors and increase tolerance five times If it would be necessary to increase the tolerance again then continue the solution with fixed step size the value 3 means report errors and increase tolerance as required If the internal value of ERRFAC becomes greater than 1 2 then continue the solution with fixed step size the value 4 means force the solution from the beginning with a large fixed step size of 1 80 of the total integration interval the value 5 means force the solution for one step Do not increase the tolerance Do not report any error the value 6 means force the solution for one step Do not increase the tolerance Type out an error report The default value is 6 8 9 EXERCISES 1 Numerically solve the following differential equations and graph the solutions a dy 2 2 y with y 0 1 Compare the results with the analytic solution y aa b y 2 y with y 1 1 y 1 1 for zx 1 2 1 using a tolerance level of 002 Compare the result with the analytic solution y 1 zx 2 Numerically solve the ee systems of differential equations and graph the solutions a 4 1000 and dy 0 5 E 16 t with x 0 500 y 0 0 for t 0 2 1 Compare ihe results wih the nal le solutions x t 1000 t 500 and y t 500 t 8 b Y dy 2 y 1 y2 and dya Ya y 1
362. ying each row of the other matrix argument by the corresponding vector component Examples are shown in the following MLAB dialog 79 In general matrix powers are given as follows ME1 SE1 Usually SE1 is an integer n if not an error message is printed Assume ME1 is a square m by m matrix A If n gt 0 the result is A A A n times If n 0 the result is an m by m identity matrix d where dj 1 and dj 0 for i j If n 1 and ME1 is a nonsingular matrix A then the result is the matrix inverse A7 If n lt 1 the result is 471 x A7 x A7 n times As usual the circumflex may be replaced by double asterisks Table 3 27 Matrix power operation TYPE UX UY V UX a 2 by 3 matrix V a 2 by 1 matrix TYPE UX UY a 2 by 3 matrix Ii 2 4 20 2 18 4 3 TYPE V UX a 2 by 3 matrix 8 16 40 9 t 2 12 3 80 Element by element matrix products are computed by component by component multiplication of corresponding columns of the two matrix arguments If two column vectors have unequal lengths then the elements of the shorter column vector are recycled and paired with the remaining components of the longer vector If one matrix has fewer columns than the other then the columns of the narrower matrix are used cyclically first column used again after the last then second column again etc For example if A has twice as many columns
363. ype linetype d title text string position 2 MLAB Commands for making pictures a Display control commands e VIEW W1 e UNVIEW b Window creation and modification e WINDOW SE1 TO SE2 SE3 TO SE4 ADJUST WTYPE IN W e FRAME SE1 TO SE2 SE3 TO SE4 COLOR SES IN W e IMAGE SE1 TO SE2 SE3 TO SE4 COLOR SE5 IN W e NO FRAMEBOX COLOR SE1 IN W e NO IMAGEBOX COLOR SE1 IN W c Curve creation and modification e DRAW C ME1 IN W POINTTYPE lt SE1 c gt PTSIZE SE2 UNITS PTFONT SE3 LINETYPE lt SE4 SE5 SE6 SE7 SE8 SE9 SE10 gt LABEL ME2 LABELSIZE SE11 UNITS LABELFONT SE12 OFFSET SE13 SE14 UNITS PLACE A B FORMAT SE15 SE16 SE17 SE18 SE19 ANGLE SE20 COLOR SE21 New curve defaults name C CO C1 etc W default window POINTTYPE 0 PTSIZE 02 FFRACT PTFONT 5 LINETYPE 1 LABELSIZE 02 FFRACT LABELFONT 5 OFFSET 0 0 FFRACT PLACE LEFT BOTTOM FORMAT 3 5 0 0 2 0 ANGLE 0 COLOR 1 d Axis creation and modification e lt X Y gt AXIS A ME1 POINTTYPE SE1 PTSIZE SE2 UNITS PTFONT SE3 LINETYPE lt SE4 SE5 SE6 SE7 SE8 SE9 SE10 gt LABEL ME2 LABELSIZE SE11 UNITS LABELFONT SE12 OFFSET SE13 SE14 UNITS PLACE A B FORMAT SE15 SE16 SE17 SE18 SE19 SE20 ANGLE SE21 147 COLOR SE22 IN W New axis defaults name W XAXIS W YAXIS or AO A1 etc default window W POINTTYPE 0 PTSIZE 02 FFRACT PTFONT 5 LINETYPE 1 LABELSIZE 02 FFRACT LABELFONT 5 OFFSET 0 0 FFRA
364. ysically printed if a suitable printing device is attached to the computer with the command PRINT FILE lt filename gt where filename is an existing filename in the current working directory Table 5 4 PRINT command or Hewlett Packard LaserJet Graphics Language commands which are binary graphics language commands developed by Hewlett Packard Inc that can be sent to a suitable HP printer MLAB can generate any of six types of output plot files specified with one of the following PLOTDEV commands e PLOTDEV e PLOTDEV e PLOTDEV e PLOTDEV e PLOTDEV e PLOTDEV PSL for black and white PostScript language in landscape mode PSP for black and white PostScript language in portrait mode PSCL for color PostScript language in landscape mode PSCP for color PostScript language in portrait mode LJ2L for black and white HP Graphics Language in landscape mode or LJ2P for black and white HP Graphics Language in portrait mode 167 Here landscape mode refers to the larger dimension of an 8x11 5 inch page oriented horizontally portrait mode refers to the larger dimension of an 8x11 5 inch page oriented vertically If no PLOTDEV command is given PLOTDEV PSL output files are generated If PLOTDEV is set to generate PostScript output i e PLOTDEV PSL PSP PSCL or PSCP then the name of the disk file created by a PLOT command is mlabp ps where is 0 1 2 The PLOT command never modifies an existin

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