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HP 15c User's Manual

1. Keystrokes Display f MATRIX 1 Sets beginning row and column numbers in Ro and R to 1 Display shows the previous result f USER Activates User mode 1 STO LA A 1 1 Row 1 column 1 of A Displayed momentarily while A key held down 1 0000 Value of ay 2 STO A 2 0000 Value of a12 3 STO LA 3 0000 Value of a43 4 STOI TA 4 0000 Value of ay 5 STO LA 5 0000 Value of ay 6 STO LA 6 0000 Value of az RCLILA A 1 1 Recalls element in row 1 column 1 Ro and R were reset in preceding step 1 0000 Value of a1 RCL A 2 0000 Value of a4 RCL A 3 0000 Value of aj3 RCL A 4 0000 Value of az RCL A 5 0000 Value of dz RCL A 6 0000 Value of a33 f USER 6 0000 Deactivates User mode Checking and Changing Matrix Elements Individually The calculator provides two ways to check recall and change store the value of a particular matrix element The first method uses storage registers Ro and R in the same way as described above except that the row and column numbers aren t automatically changed when User mode is deactivated The second method uses the stack to define the row and column numbers 146 Section 12 Calculating with Matrices Using R and R To access a particular matrix element store its row number in Ro and its column number in R These numbers won t cha
2. Keystrokes Display f CLEAR 0 0000 Clears statistical storage registers R through R and the stack f FIX 2 0 00 Limits display to two decimal places like the data 4 63 ENTER 4 63 O 2 1 00 First data point 4 78 ENTER 4 78 20 2 2 00 Second data point 6 61 ENTER 6 16 40 2 3 00 Third data point 7 21 ENTER 7 21 60 2 4 00 Fourth data point 7 78 ENTER 7 78 80 5 00 Fifth data point RCL 3 200 00 Sum of x values x kg of nitrogen RCL 4 12 000 00 sum of squares of x values Hi RCL 5 31 01 Sum of y values y grain yield RCL 6 200 49 sup of squares of y values RCL 7 1 415 00 Sum of products of x and y values Zxy 52 Section 4 Statistics Functions Correcting Accumulated Statistics If you discover that you have entered data incorrectly the accumulated statistics can be easily corrected Even if only one value of an x y data pair is incorrect you must delete and re enter both values 1 Key the incorrect data pair into the Y and X register 2 Press LY J 2 to delete the incorrect data 3 Key in the correct values for x and y 4 Press 2 Alternatively if the incorrect data point or pair is the most recent one entered and 2 has been pressed you can press LY LSTx 9 J 2 to x remove the incorrect data
3. Indirect Addressing If Ri contains G will address GTO 1 or GSB LL will transfer to 0 Ro f LBL O 9 Ro f LBL 9 10 Ro 0 11 Ri nee 19 Ro f_ LBL 9 20 Roo A For R2 0 only Continued on next page 108 Section 10 The Index Register and Loop Control Indirect Addressing If Ri contains i will address LJ or GSB transfer to For R gt 0 only Index Register Arithmetic Direct STO or RCLJ x L LLJ Storage or recall arithmetic operates with the ndex moker in the same manner as upon other data storage registers page 43 Indirect STO or RCL L L J Lx L 0 carries out storage or recall arithmetic with the contents of the data storage register addressed by the integer portion of the number 0 to 65 in the Index register See the above table Exchanging the X Register Direct f x exchanges contents between the X register and the Index register Works the same as X n does with registers 0 through 9 Indirect f Lx G exchanges contents between the X register and the data storage register addressed by the number 0 to 65 in
4. Iterations Operation 0 1 2 3 4 ISG 0 00602 2 00602 4 00602 6 00602 8 00602 skip next line DSE 6 00002 4 00002 2 00002 0 00002 skip next line Examples Examples Register Operations Storing and Recalling Keystrokes Display f CLEAR REG Clears all storage registers 12 3456 12 3456 STO I 12 3456 Stores in Ry 7 yx 2 6458 STO G 2 6458 Storage in R3 by indirect addressing Ry 12 3456 RCL I 12 3456 Recalls contents of Ry 112 Section 10 The Index Register and Loop Control Keystrokes Display RCL G 2 6458 Indirectly recalls contents of R 2 f x 2 2 6458 Check same contents recalled by directly addressing R 2 Exchanging the X Register Keystrokes Display f lixsl tI 12 3456 Exchanges contents of R and X register RCL I 2 6458 Present contents of Ry f x 0 0000 Exchanges contents of R which is zero with X RCL GD 2 6458 f x lt 2 2 6458 Check directly address R3 Storage Register Arithmetic Keystrokes Display 10 STO LI 10 0000 Adds 10 to Ri RCL I 12 6458 New contents of R old 10 gj STOJ 3 1416 Divides contents of R2 by 7 i RCL G 0 8422 New contents of R 2 f x3 2 0 8422 Check directly address R 3
5. CF ENG LFIX GSB MATRIX SCI STO DIM Lf g HYP ISG RCL SF TEST DSE F GTO HYP LBL RESULT SOLVE xs If you make a mistake while keying in a prefix for a function press f CLEAR PREFIX to cancel the error The CLEAR PREFIX key is also used to show the mantissa of a displayed number so all 10 digits of the number in the display will appear for a moment after the Changing Signs PREFIX key is pressed Pressing CHS change sign will change the sign positive or negative of any displayed number To key in a negative number press digits have been keyed in Keying in Exponents For a negative exponent press CHS after its EEX j enter exponent is used when keying in a number with an exponent First key in the mantissa then press LEEX and key in the exponent CHS after keying in the exponent For example to key in Planck s constant 6 6262x10 Joule seconds and multiply it by 50 CHS may also be pressed after EEX and before the exponent with the same result unlike the mantissa where digit entry must precede CHS 20 Section 1 Getting Started Keystrokes Display 6 6262 6 6262 EEX 6 6262 00 The 00 prompts you to key in the exponent 3 6 6262 03 6 6262x10 4 6 6262 34 6
6. Section 7 Program Editing 89 Make any necessary modifications in the program to also find and display s the length of the circular arc cut by 0 in radians according to the equation s r Complete the following table 6 r e s 45 50 90 100 270 100 Answers 38 2683 and 39 2699 141 4214 and 157 0796 141 4214 and 471 2389 A possible new sequence is BIA ADE AEX STO o 20x rey STO 1 2E SN x OPSE TIPSE RCo RCLJ1 FRAD x JJRIN Section 8 Program Branching and Controls Although the instructions in a program are normally executed sequentially it is often desirable to transfer execution to a part of the program other than the next line Branching in the HP 15C may be simple or it may depend on a certain condition By branching to a previous line it is possible to execute part of a program more than once a process called looping The Mechanics Branching The Go To GTO Instruction Simple branching that is unconditional branching is carried out with the instruction GTO label In a running program GTO will transfer execution to the next appropriately labeled program or routine not to a line number The calculator searches forward in memory wrapping around throug
7. Example After keying in the preceding data Farmer realizes he misread a smeared figure in his lab book The second y value should have been 5 78 instead of 4 78 Correct the data input Keystrokes Display 4 78 4 78 Keys in the data pair we want to replace ENTER and deletes the accompanying statistics 20 9 4 00 The n value drops to four 5 78 5 78 Keys in and accumulates the replacement ENTER data pair 20 2 5 00 The n value is back to five We will use these statistics in the rest of the examples in this section Note that these methods of data deletion will not delete any rounding errors that may have been generated in the statistics registers This difference will not be serious unless the erroneous pair has a magnitude that is enormous compared with the correct pair in such a case it would be wise to start over Section 4 Statistics Functions 53 Mean The x function computes the arithmetic mean average of the x and y values using the formulas shown in appendix A and the statistics accumulated in the relevant registers When you press 9 x the contents of the stack lift two registers if stack lift is enabled one if not the mean of x x is copied into the X register as the mean of y y is copied simultaneously into the Y register Press to view y Example From the corrected statistics data we h
8. T y y y y Z X X xX X Y 4 0000 53 1301 53 1301 53 1301 X 3 5 0000 0 0000 7 Keys gji gt P g JLCLx J 7 Stack Stack No stack enabled disabled lift Imaginary X Register All enabling functions provide for a zero to be placed in the imaginary X register when the next number is keyed or recalled into the display Neutral Operations Stack Lift Some operations like FIX J are neutral that is they do not alter the previous status of the stack lift Thus if you disable the stack lift by pressing ENTER then press f FIX n and key in a new number that number will write over the number in the X register and the stack will not lift Similarly if you have previously enabled the stack lift by executing say Vx then execute a FIX instruction followed by a digit entry sequence the stack will lift The following operations are neutral on the HP 15C FX GRD R S SCI GTO CHS nnn CLEAR PREFIX P R ENG BST CLEAR REG ED DEG SST CLEAR RAD MEM PSE Imaginary X Register The above operations are also neutral with respect to clearing the imaginary X register All digit entry functions are also neutral during digit entry After digit entry termination CHS and EEX are lift enabling is disabling That is th
9. Example Loop Control with DSE Remember the program in section 8 which used a loop to calculate radioactive decay Refer to page 93 This program used a test condition x gt y to exit the loop when the calculated result passed the given limit 50 As we ve seen in this section there s another way to control loop execution through a stored loop counter that is monitored by the ISG or DSE function Section 10 The Index Register and Loop Control 113 Here is a revision of the original radioisotope decay program This time we will limit the program to three executions of the loop rather than setting a specific limit value This example uses DSE with a loop control number in R of 3 00001 n US initial loop counter 4 t 4 decrement value test goal value Make the following changes to the program assuming it is in memory A loop counter will be stored in R and a line number in the Index register Keystrokes Display g LP R 000 Program mode GTOJICHS 013 013 43 30 9 The second of the two loop test condition lines eje 011 42 31 Delete lines 013 and 012 f DSE 2 012 42 5 2 Add your loop counter function counter stored in R3 GTOJ I 013 22 25 Go to given line number 015 Now when the loop counter stored in R2 has reached zero it will skip line 013 and go on to 014 the RTN in
10. gt P converts to polar form and R converts to rectangular form as described later in this section page 133 For the trigonometric functions the calculator considers numbers in the real and imaginary X registers to be expressed in radians regardless of the current trigonometric mode To calculate trigonometric functions for values given in degrees use gt RAD j to convert those values to radians before executing the trigonometric function Two Number Functions The following functions operate on both the real and imaginary parts of the numbers in the X and Y registers and place the real and imaginary parts of the answer into the X registers Both stacks drop just as the ordinary stack drops after a two number function not in Complex mode 0 GRIEG pF Stack Manipulation Functions When the calculator is in Complex mode the following functions simultaneously manipulate both the real and imaginary stacks in the same way as they manipulate the ordinary stack when the calculator is not in Complex mode The x y function for instance will exchange both the real and imaginary parts of the numbers in the X and Y registers X y R4 Rt ENTER LSTx Refer to the HP I5C Advanced Functions Handbook for definitions of complex trigonometric functions and further information about doing calculations in Complex mode 132 Sect
11. Alternatively you can clear program memory which will erase all programs in memory and position you to line 000 To do so press Lf CLEAR PRGM in Program mode Program Begin A label instruction f_ LBL followed by a letter LA through E or number 0 through 9 or 0 through 9 is used to define the beginning of a program or routine The use of labels allows you to quickly select and run one particular program or routine out of several Keystrokes Display f CLEAR 000 Clears program memory and PRGM sets to line 000 start of program memory f BU A 001 42 21 11 Recording a Program Any key pressed operator or constant will be recorded in memory as a programmed instruction Except the nonprogrammable functions which are listed on page 80 68 Section 6 Programming Basics Keystrokes Display 2 002 2 x 003 20 9 004 9 005 48 8 006 8 007 10 VX 008 11 Given h in the X register lines 002 to 008 calculate 2h 98 Program End There are three possible endings for a program e 9 RIN return will end a program return to line 000 and halt e LR S will stop a program without moving to line 000 e The end of program memory contains an automatic LRTN Keystrokes Display g JLRTN 009 43 32
12. Keystrokes Display g P R 4 0000 06 Run mode 10 CHS ENTER Baca Same initial estimates 20 CHS 20 f SOLVE 2 0 4000 Second root STOJ 1 0 4000 Stores root for deflation R R 0 0000 Deflated function value Now modify your subroutine to eliminate the second root Keystrokes Display g J P R 000 Program mode g LBST 9 030 10 Line before RTN BST xy 031 34 Brings x into X register RCL 1 032 45 1 033 30 Deflation for second root 034 10 Appendix D A Detailed Look at SOLVE 237 Again use the same initial estimates to find the next root Keystrokes g JLP R 10 CHS ENTER 20 CHS f SOLVE STOJ 2 R LR Keystrokes g JLP R J 2 g LBST BST X y RCL 2 Find the fourth root Keystrokes g LP R 10 CHS ENTER 20 CHS f SOLVE STO 3 R LR 2 Display 0 0000 Run mode 10 0000 Soaks Same initial estimates 20 8 4999 Third root 8 4999 Stores root for deflation 1 0929 07 Deflated function value near Zero Now change your subroutine to eliminate the third root Display 000 Program mode 034 10 Line before RTN 035 34 Brings x into X register 036 45 2 037
13. Week 1 2 3 Cabbage kg 186 141 215 Broccoli kg 88 92 116 Calculating the Residual The HP 15C enables you to calculate the residual that is the matrix Residual R YX where R is the result matrix and X and Y are the matrices specified in the X and Y registers This capability is useful for example in doing iterative refinement on the solution of a system of equations and for linear regression problems For example if C is a possible solution for AX B then B AC indicates how well this solution satisfies the equation Refer to the HP I5C Advanced Functions Handbook for information about iterative refinement and linear regression The residual function IMATRIX 6 uses the current contents of the result matrix and the matrices specified in the X and Y registers to calculate the residual defined above The residual is stored in the result matrix replacing the original result matrix A matrix specified in the X or Y register can not be the result matrix Using MATRIX 6 rather than X and gives a result with improved accuracy particularly if the residual is small compared to the matrices being subtracted To calculate the residual 1 Enter the descriptor of the Y matrix into the Y register Enter the descriptor of the X matrix into the X register 2 3 Designate the R matrix as the result matrix 4 Press _f MATRIX 6 The residual repl
14. Intermediate Program Stops Optional if this is the last program in memory Use f LPSE pause as a program instruction to momentarily stop a program and display an intermediate result Use more than one PSE for a longer pause Use a LR S run stop instruction to stop the program indefinitely The program will remain positioned at that line You can resume program execution from that line by pressing the keyboard Running a Program R S during Run mode that is from Run Mode Switch back to Run mode when you are done programming J LP R J Program execution must take place in Run mode Section 6 Programming Basics 69 Keystrokes Display g J P R Run mode no PRGM annunciator displayed The display will depend on any previous result The position in program memory does not change when modes are switched Should the calculator be shut off it always wakes up in Run mode Executing a Program In Run mode press f letter label or GSB digit or letter label This addresses a program and starts its execution The display will flash running Keystrokes Display 300 51 300 51 Key a value for h into the X register f A 7 8313 The result of executing program A The number of seconds it takes an object dropped from 300 51 meters high to hit the gr
15. Keystrokes Display g J P R 000 Program mode f CLEAR 000 PRGM f B02 001 42 21 2 6 002 6 0 003 0 x 004 20 9 005 9 4 006 4 4 007 4 Appendix D A Detailed Look at SOLVE 235 Keystrokes Display 008 30 x 009 20 3 010 3 0 011 0 0 012 0 3 013 3 014 40 x 015 20 6 016 6 1 017 1 7 018 7 1 019 1 020 40 x 021 20 2 022 2 8 023 8 9 024 9 0 025 0 026 30 RTN 027 43 32 In Run mode key in two large negative initial estimates such as 10 and 20 and use SOLVE to find the most negative root Keystrokes Display g J P R Run mode 10 CHS ENTER 10 0000 ae ae peer 20 CHS 20 f SOLVE 2 1 6667 First root STOJ 0 1 6667 Stores root for deflation Rt LRY 4 0000 06 Function value near zero 236 Appendix D A Detailed Look at SOLVE Return to Program mode and add instructions to your subroutine to eliminate the root just found Keystrokes Display g ILP R 000 Program mode g BST 9 026 30 Line before RTN BST xsy 027 34 Brings x into X register RCL J O 028 45 0 Rn 029 30 Divides by x a where a is known root A 030 10 Now use the same initial estimates to find the next root
16. Note that these are the only operations in which the blue 9 key precedes a gold letter key Section 12 Calculating with Matrices 147 Example Recall the element in row 2 column 1 of matrix A from the previous example Use the stack registers Keystrokes Display 2 ENTER 1 1 Enters row number into Y register and column number into X register RCL 9 LA 4 0000 Value of a21 Storing a Number in All Elements of a Matrix To store a number in all elements of a matrix simply key that number into the display then press MATRIX followed by the letter key specifying the matrix Matrix Operations In many ways matrix operations are like numeric calculations Numeric calculations require you to specify the numbers to be used often you define a register for storing the result Similarly matrix calculations require you to specify one or two matrices that you want to use A matrix descriptor is used to specify a particular matrix For many calculations you also must specify a matrix for storing the result This is the result matrix Because matrix operations usually require many individual calculations the calculator flashes the running display during most matrix operations Matrix Descriptors Earlier in this section you saw that when you press LRCL MATRIX followed by a letter key specifying a matrix the name of the matrix appears at the left of the display
17. ceeeee Uncertainty and the Display Format Conditions That Could Cause Incorrect Results 0008 Conditions That Prolong Calculation Time Obtaining the Current Approximation to an Integral For Advanced Information eccsceeseeseesnsesneeeteeeneennees 205 209 209 209 210 210 211 212 213 213 213 215 215 215 216 217 217 218 218 220 220 222 226 233 238 239 239 239 240 240 241 245 249 254 257 258 Contents 11 Appendix F Batteries icstuiiaccisexyeavtthie ea veins 259 Low Power Indication cccccccssceesseceeseeceteeecsteeecseeeeeseeenes 259 Installing New Batteries ccccceesscceeeceestseeenteeeeeeenaees 259 Verifying Proper Operation Self Tests ccccccseeeeseeeetees 261 Function Summary and Index ceeeeeeeseeeeeeeeeeeeeeees 262 Complex Functions 2ec cessiesaaeMastecteeveateeaserate seta testers 262 CONVErSIONS sos sestiens tee liet sent el a E debe E EA ages 262 DIgitiENtty tists oean Seating theca nadie abe eae te A 262 Display Control eeens tonaren teas ea Medico cada oes 263 Hyperbolic FUNCHIONS ennir r aas 263 Index Register Control cccccsceccseceesteeeeeeecsteeeeseeeeeeeenes 263 Logarithmic and Exponential Functions 263 Mathematics tcc a tiiets ite Aiea e weaned emaeeas 264 M irix FUnCHONS eisa naa a eee ees a 264 Nimber Alteration vers c2cdees taceeevaee deeestkectsaneisg aes ARN 265 Percentage
18. Keystrokes Display 0 ENTER gerne Initial estimates 10 CHS 10 f SOLVE 0 2 0000 The second root RY 2 0000 A previous estimate of the root RY 0 0000 Value of f x at second root 184 Section 13 Finding the Roots of an Equation You have now found the two roots of f x 0 Note that this quadratic equation could have been solved algebraically and you would have obtained the same roots that you found using SOLVE Graph of f x The convenience and power of the SOLVE key become more apparent when you solve an equation for a root that cannot be determined algebraically Example Champion ridget hurler Chuck Fahr throws a ridget with an upward velocity of 50 meters second If the height of the ridget is expressed as h 5000 1 e 2004 how long does it take for it to reach the ground again In this equation h is the height in meters and f is the time in seconds Solution The desired solution is the positive value of t at which h 0 Use the following subroutine to calculate the height Keystrokes Display LS P R 000 Lf LBLILA 001 42 21 11 Begin with label 2 002 2 Subroutine assumes t is loaded in X and Y registers 0 003 0 004 10 Section 13 Finding the Roots of an Equation 185 Keystrokes Display CHS 005 16 1 20 er 006 12 CHS 007 16 ee 1 008 1 009 40 dee 5 010 5 0 011
19. Section 6 Programming Basics 71 The program to calculate this information uses these formulas and data base area 17 volume base area x height 77h surface area 2 base areas side area 2r 2nrh Radius r Height h Base Area Volume Surface Area 2 5cm 8 0 cm 4 0 10 5 4 5 4 0 TOTALS Method 1 Enter an r value into the calculator and save it for other calculations Calculate the base area nr store it for later use and add the base area to a register which will hold the sum of all base areas 2 Enter h and calculate the volume narh Add it to a register to hold the sum of all volumes 3 Recall r Divide the volume by r and multiply by 2 to yield the side area Recall the base area multiply by 2 and add to the side area to yield the surface areas Sum the surface areas in a register Do not enter the actual data while writing the program just provide for their entry These values will vary and so will be entered before and or during each program run Key in the following program to solve the above problem The display shows line numbers and keycodes the row and column location of a key which will be explained under Further Information Keystrokes Display g LP R 000 Sets calculator to Program mode PRGM displayed f CLEAR PRGM 000 Clears program memory Starts at line 000 72 Section 6 Programming Basics
20. ccccccecsseceeseeeeeeeeesteeeeseeeeteeeesas 62 Continuous Memory ccceceeeeeeeeceeceeeceeeseenteaeeeeeereeeeeaaeas 62 SIGHS 5 EEE T E ahora tev AS 62 6 Contents Resetting Continuous Memory eeeeeseeeeeteeeeeeeeeeeennees Part Il HP 15C Programming 21 0vcsesdsayinagcsezereg dans Section 6 Programming Basics ccceeeeeeeteeeeees The Mechanics phrionsa aE E EAE REE Creating a Programore ea eE site es Loading a Program c ccceesceceseeesteceeseeeseeeeneeeeateees Intermediate Program Stops cccccsseceereeceteeeesteeeeseeeees Running a Program sseseseseseseseserererererererererererererererereree How t Enter Data ccccccccecsessennseescsovsesnsenadnaseoerssersenads Program Memory ssssesesessseseserererererereerrererererererererereree Further Information cecceeseceseeeeesseecteeeneesneecniecniesneenees Program Instructions ccc eeeeeeeeteeeeereeeeeeetenetenenenenenene Instruction Coding izes eeen ma ranean eee Memory Configuration ccccccessceeseeceeseecnteeeenseeeeeeees Program Boundaries ccccccsseceesseceeeeeceneeeeneeesseeeees Unexpected Program Stops cccccccssceestecseeeesteeessseeees Abbreviated Key Sequences ccceeesseceeeecstseeeteeeaees User Mode arroue snenie a ee deste widowed Polynomial Expressions and Horner s Method 00 Nonprogrammable Functions ccceesceceeeceeseeeeeneeeees elo eaa AEE T A E A E T Sect
21. lost 5 0000 row number col number Th x lt N RCL a A Using Matrix Operations in a Program If the calculator is in User mode during program entry when you enter a STOJ or RCLJ LA J through LE J J instruction to store or recall a matrix element a U replaces the dash usually displayed after the line number When this line is executed in a running program it operates as though the calculator were in User mode That is the row and column numbers in Ro and R are automatically incremented according to the dimensions of the specified matrix This allows you to access elements sequentially The USER annunciator has no effect during program execution In addition when the last element is accessed by the User STO or RCL instruction when Ro and R are returned to 1 program execution skips the next line This is useful for programming a loop that stores or recalls each matrix element then continues executing the program For example the following sequence squares all elements of matrix D Th For all matrix elements except last Section 12 Calculating with Matrices 177 T7 For last matrix element MATRIX 7 row norm and MATRIX 8 Frobenius norm functions also operate as conditional branching instructions in a program If the X register contains a matrix descriptor these functions calculate the norm in the usu
22. Keystrokes Display f J LBLIT A 001 42 21 11 Assigns this program the label A STOJ 0 002 44 0 Stores the contents of X register into Ro r must be in the X register before running the program g x 003 43 11 Squares the contents of the X register which will be r g 004 43 26 x 005 20 ar the BASE AREA of a can STO 4 006 44 4 Stores the BASE AREA in R4 STO L 1 007 44 40 1 Keeps a sum of all BASE AREAS in R R S 008 31 Stops to display BASE AREA and allow entry of the h value x 009 20 Multiplies h by the BASE AREA giving VOLUME f LPSE 010 42 31 Pauses briefly to display VOLUME STO 2 011 44 40 2 Keeps a sum of all can VOLUMES in Ro RCL 0 012 45 0 Recalls r 013 10 Divides VOLUME by r 2 014 2 x 015 20 2 trh the SIDE AREA of a can RCL 4 016 45 4 Recalls the BASE AREA of the can 2 017 2 Multiplies base area by two for x 018 20 top and bottom Keystrokes STO 3 g RTN Section 6 Programming Basics 73 Display 019 40 020 44 40 3 021 43 32 Now let s run the program Keystrokes Display g P R f CLEAR REG 25 2 5 fA 19 6350 or GSB LA 8 8 R S 157 0796 164 9336 4 4 R S 50 265
23. Keystrokes Display 300 51 300 51 f JLA 7 8313 1050 f JLA 14 6385 Height of the Eiffel Tower Falling time you calculated earlier The time seconds for a stone to reach the ground after release from a blimp 1050 m high 16 The HP 15C A Problem Solver With this program loaded you can quickly calculate the time of descent of an object from different heights Simply key in the height and press f JLA J Find the time of descent for objects released from heights of 100 m 2 m 275 m and 2 000 m The answers are 4 5175 s 0 6389 s 7 4915 s and 20 2031 s That program was relatively easy You will see many more aspects and details of programming in part II For now turn the page to take an in depth look at some of the calculator s important operating basics Part HP 15C Fundamentals Section 1 Getting Started Power On and Off The ON key turns the HP 15C on and off To conserve power the calculator automatically turns itself off after a few minutes of inactivity Keyboard Operation Primary and Alternate Functions Most keys on your HP 15C perform one primary and two alternate shifted functions The primary function of any key is indicated by the character s on the face of the key The alternate functions are indicated by the gold characters printed above the key and the blue characters printed on the lower face of the key e To select the primary functi
24. RESULT f RESULT A through E STOJA E through STOJLS If A through E G STO MATRIX through E Results result matrix Calculates residual in result matrix Calculates row norm of matrix specified in X register Calculates Frobenius or Euclidean norm of matrix specified in X register Calculates determinant of matrix specified in X register Place LU in result matrix Transforms Z into Z Recalls value from specified matrix using row and column numbers in Roand Ry Recalls value from specified matrix using row and column numbers in Y and X registers Recalls dimensions of specified matrix into X and Y registers Displays descriptor of specified matrix Displays descriptor of result matrix Designates specified matrix as result matrix Stores value from display into element of specified matrix using row and column numbers in Ro and Rj Stores value from Z register into element of specified matrix using row and column numbers in Y and X registers If matrix descriptor is in display copies all elements of that matrix into corresponding elements of specified matrix If number is in display stores that value in all elements of specified matrix Section 12 Calculating with Matrices 179 Keystroke s Results STO RESULT Designates m
25. 1 for different 5z 3 conformal mappings Evaluate for z 1 2i Answer 0 3902 0 0122i One possible keystroke sequence is f LBL A ENTER ENTER 2 x 1 x y 5 x 3 R S Re Im 3 Try your hand at a complex polynomial and rework the example on g RTN page 80 You can use the same program to evaluate P z 5z 2z where z is some complex number Load the stack with z 7 Oi and see if you get the same answer as before Answer 12 691 0000 0 00002 Now run the program for z 1 i Answer 24 0000 4 00007 Section 11 Calculating With Complex Numbers 137 For Further Information The HP 15C Advanced Functions Handbook presents more detailed and technical aspects of using complex numbers in various functions with the HP 15C Applications are included The topics include Accuracy considerations Principal branches of multi valued functions Complex contour integrals Complex potentials Storing and recalling complex numbers using a matrix Calculating the nth roots of a complex number Solving an equation for its complex roots Using SOLVE and in Complex mode Section 12 Calculating With Matrices The HP 15C enables you to perform matrix calculations giving you the capability to handle advanced problems with ease The calculator can work w
26. A routine that combines SOLVE and requires 23 registers of space For Further Information In appendix D Advanced Use of SOLVE additional techniques and explanations for using SOLVE are presented These include e How SOLVE works e Accuracy of the root e Interpreting results e Finding several roots e Limiting estimation time Section 14 Numerical Integration Many problems in mathematics science and engineering require calculating the definite flx integral of a function If the function is denoted by f x and the interval of integration is a to b the integral can be expressed mathematically as g r fide B amp The quantity Z can be interpreted geometrically as the area of a region bounded by the graph of f x the x axis and the limits x a and x b When an integral is difficult or impossible to evaluate by analytical methods it can be calculated using numerical techniques Usually this can be done only with a fairly complicated computer program With your HP 15C however you can easily do numerical integration using the L integrate key Using The basic rules for using L are 1 In Program mode key in a subroutine that evaluates the function f x that you want to integrate This subroutine must begin with a label instruction f J LBL label and end up with a value for f x in the X register
27. Provided that f x is nonnegative throughout the interval of integration t The ZF function does not use the imaginary stack Refer to the HP 15C Advanced Functions Handbook for information about using L in Complex mode 194 Section 14 Numerical Integration 195 In Run mode 2 Key the lower limit of integration a into the X register then press ENTER to lift it into the Y register 3 Key the upper limit of integration b in to the X register 4 Press Lf x Day followed by the label of your subroutine Example Certain problems in physics and engineering require calculating Bessel functions The Bessel function of the first kind of order 0 can be expressed as Jo x 1f cos xsin d0 TAO Find J t cos sin d0 TAO In Program mode key in the following subroutine to evaluate the function f0 cos sin 0 Keystrokes Display g JLP R 000 Program mode f CLEAR PRGM 000 Clear program memory f JLBL O 001 42 21 0 Begin subroutine with a LBL instruction Subroutine assumes a value of 0 is in X register SIN 002 23 Calculate sin 8 COS 003 24 Calculate cos sin 6 g JLRIN 004 43 32 Now in Run mode key the lower limit of integration into the Y register and the upper limit into the X register For this particular problem you also need to specify Radians mode for th
28. calculation The minimum value for n that will affect uncertainty is 6 A number in R less than 6 will be interpreted as 6 248 Appendix E A Detailed Look at A se dx b 0 5x10 dx a This integral is calculated using the samples of 5 x in roughly the same ways that the approximation to the integral of the function is calculated using the samples of f x i Because A is proportional to the factor 10 the uncertainty of an approximation changes by about a factor of 10 for each digit specified in the display format This will generally not be exact in SCI or ENG display format however because changing the number of digits specified may require that the function be evaluated at different sample points so that 5 x 10 would have different values Note that when an integral is approximated in FIX display format m x 0 and so the calculated uncertainty in the approximation turns out to be A 0 5x10 b a Normally you do not have to determine precisely the uncertainty in the function To do so would frequently require a very complicated analysis Generally it s more convenient to use SCI or ENG display format if the uncertainty in the function s values can be more easily estimated as a relative uncertainty On the other hand it s more convenient to use FIX display format if the uncertainty in the function s values can be more easily
29. 1 GTO 0 g RTN nnnnn 9 Therefore xxx 0 and by default yy yy cannot be zero Displays current value of nnnnn Value in R is decremented and tested Skip a line if nnnnn lt test value Continue loop if nnnnn gt test value 0 Tests whether value in display is greater than 0 so loop will continue when nnnnn has reached 0 but display still only shows 1 0 Section 10 The Index Register and Loop Control 115 To display fixed point notation for all possible decimal places on the HP 15C Keystrokes g P R f B Display Run mode 000000000 00000000 0000000 000000 00000 0000 000 00 0 Display at _f_ PSE Jinstruction OCORrN WB UO 1 OO Display when program halts Further Information Index Register Contents Any value stored in the Index register can be referenced in three different ways e Using LI manipu e Using I like any other storage register The value in R can be lated as it is stored recalled exchanged added to etc as a control number The absolute value of the integer portion in R is a separate entity from the fractional portion For indirect branching flag control and display format control with I J only this portion is used For loop control the fractional portion is also used but separately from the integer portion Using Q
30. LSST Instruction To move only one line at a time forward through program memory press SST J single step This function is not programmable In Program mode SST will move the memory position forward one line and display that instruction The instruction is not executed If you hold the key down the calculator will continuously scroll through the lines in program memory In Run mode will display the current program line while the key is held down When the key is released the current instruction is executed the result displayed and the calculator steps forward to the next program line to be executed 82 Section 7 Program Editing 83 The Back Step LBST Instruction To move one line backwards in program memory press back step in Program or Run mode This function is not programmable will scroll with the key held down in Program mode Program instructions are not executed Deleting Program Lines Deletions of program instructions are made with LJ back arrow in Program mode Move to the line you want to delete then press Any remaining following lines will be renumbered to stay in sequence Pressing in Run mode does not affect program memory but is used for display clearing Refer to page 21 Inserting Program Lines Additions to a program are made by moving to the line preceding the point of insertion Any instruction you key
31. Memory Allocation Should you ever get an Error 10 you have run up against limitations of the HP 15C memory If you learn how to reallocate memory you can greatly increase your ability to store information in the HP 15C The HP 15C memory consists of 67 registers R to Res and the Index register divided between data storage and programming advanced function capability The initial configuration is e 46 registers for both programming and the advanced functions SOLVEJ L the imaginary stack and MATRIX functions At seven bytes of memory per register this is worth 322 program bytes if no memory is dedicated to advanced functions e 21 registers for data storage Ro to Ro Ro to Ro and the Index register 76 Section 6 Programming Basics Initial Memory Configuration STORAGE REGISTERS Ry RotoR g COMMON REGISTERS R20 to Res Permanent available for Registers programming R s Ro 322 program bytes R1 available if no memory R2 R2 used for advanced R3 R3 functions R R R Statistics s 4 20 Registers Rs Rs Ro Re Ree Re3 R7 R7 Rea Rg Rg Res Rg Rg Movable Boundary Allocatable Registers shaded R Memory is reallocated by telling the calculator which data storage register shall be the highest data register all other registers are left for programming and advanced functions Keystrokes Display 60 DMJ 60 0000 Reo and below allocated to data storage five Ro to Res rema
32. Program mode Test for x range Branch for x gt 10 3x 3x 45 3x 45 x 3x 45 x 7 350 End subroutine Subroutine for x gt 10 10 1000 End subroutine Execute SOLVE using initial estimates of 7 and 14 to start at the outer end of the beam and search for a point of zero shear stress Appendix D A Detailed Look at 229 Keystrokes Display g P R Run mode K VENTER a One Initial estimates 14 14 f_ SOLVE 2 10 0000 Possible root R LR 1 000 0000 Stress not zero The large stress value at the root points out that the SOLVEJ routine has found a discontinuity This is a place on the beam where the stress quickly changes from negative to positive Start at the other end of the beam estimates of 0 and 7 and use SOLVE again Keystrokes Display g ENTER s0909 Initial estimates f SOLVE 2 3 1358 Possible root R4 ERY 2 0000 07 Negligible stress Beame s beam has zero shear stress at approximately 3 1358 meters and an abrupt change of stress at 10 0000 meters Graph of Q versus x When no root is found and Error 8 is displayed you can press or any one key to clear the display and observe the estimate at which the function was closest to zero By also reviewing the numbers in the Y and Z registers you can often determine the nature of the function near the root estimate and use this in
33. RIN Nested Subroutines If you attempt to call a subroutine that is nested more than seven levels deep GSB the calculator will halt and display Error 5 when it encounters the instruction at the eighth level Note that there is no limitation other than memory size on the number of nonnested subroutines or sets of nested subroutines that you may use Section 10 The Index Register and Loop Control The Index register Rj is a powerful tool in advanced programming of the HP 15C In addition to storage and recall of data the Index register can use an index number to e Count and control loops e Indirectly address storage registers including those beyond R o Rio e Indirectly branch to program line numbers as well as to labels e Indirectly control the display format e Indirectly control flag operations The and Keys Direct Versus Indirect Data Storage With the Index Register The Index register is a data storage register that can be used directly with I I J or indirectly with G The difference is important to note G The I function uses the The i function uses the absolute number itself in the Index value of the integer portion of the register number in the Index register to address another data storage register This is called indirect addressing N
34. Stack Lift in Complex Mode Stack lift operates on the imaginary stack as it does on the real stack the real stack behaves identically in and out of Complex mode The same functions that enable disable or are neutral to lifting of the real stack will enable disable or be neutral to lifting of the imaginary stack These processes are explained in detail in section 3 and appendix B In addition every nonneutral function except 4 and CLx causes the clearing of the imaginary X register when the next number is entered That is these functions cause a zero to be placed in the imaginary X register when the next number is keyed in or recalled Refer to the stack diagrams above for illustrations This feature allows you to execute calculator operations using the same key sequences you use outside of Complex mode Manipulating the Real and Imaginary Stacks Re Im real exchange imaginary Pressing f Re 1m will exchange the contents of the real and imaginary X registers thereby converting the imaginary part of the number into the real part and vice versa The Y Z and T registers are not affected Press Lf_ Re Im twice restore a number to its original form Re Im also activates Complex mode if it is not already activated Temporary Display of the Imaginary X Register Press f G to momentarily display the imaginary part
35. Two Number Functions and ENTER A two number function must have two numbers present in the calculator before executing the function _ J LX and are examples of two number functions Terminating Digit Entry When keying in two numbers to perform an operation the calculator needs a signal that digit entry is terminated for the first number This is done by pressing to separate the two numbers If on the other hand one of the numbers is already in the calculator as the result of a previous operation you do not need to use the ENTER key All functions except the digit entry keys themselves have the effect of terminating digit entry Notice that regardless of the number a decimal point always appears and a set number of decimal places are displayed when you terminate digit entry as by pressing ENTER Chain Calculations In the following calculations notice that The ENTER key is used only for separating the sequential entry of two numbers The operator is keyed in only after both operands are in the calculator The result of any operation may itself become an operand Such intermediate results are stored and retrieved on a last in first out basis New digits keyed in following an operation are treated as a new number The digit keys CHS EEX and le Section 1 Getting Started 23 Example Calculate
36. displays and executes the current program line then steps to next line to be executed page 82 PSE Pause Halts program execution for about 1 second to display contents of X register then resumes execution page 68 R S Run Stop Begins program execution from current line number in program memory Stops execution if program is running page 68 RTN Return Causes Programming Summary and Index calculator to return to line 000 and halt execution if running page 68 If ina subroutine merely returns to line after GSB page 101 SF Set flag true Sets designated flag 0 to 9 Flags 0 through 7 are user flags flag 8 signifies Complex mode and flag 9 signifies an overflow condition page 92 CF Clear flag false Clears designated flag 0 to 9 page 92 F Is flag set Tests for designated flag If set program execution continues If cleared program execution skips one line before continuing page 92 X lt y x 0 TEST 0 through 9 Conditional tests Each test compares value in X register against 0 or value in Y register as indicated If true calculator executes instruction in next line of program memory If false calculator skips one line in program memory before resuming execution page 91 x 0 and TEST 0 5 and 6 are also valid for complex
37. from the preceding examples Re Im Re Im Re Im T e f e f e f Z 17 144 17 144 17 144 Yi 4 0 4 0 4 0 Xi 4 0 10 0 0 10 Keys 10 f Re Im Continue with any operation Note that pressing f Re Im simply exchanges the numbers in the real and imaginary X registers and not those in the remaining stack registers 130 Section 11 Calculating With Complex Numbers Storing and Recalling Complex Numbers The STO and RCL functions act on the real X register only therefore the imaginary part of a complex number must be stored or recalled separately The keystrokes to do this can be entered as part of a program and executed automatically To store a ib from the complex X register to R and R3 you can use the sequence STO 1 f J Re Im STO 2 You can follow this by f Re Im to return the stack to its original condition if desired To recall a ib from R and R you can use the sequence RCL 1 RCLJ2 ff I If you wish to avoid disturbing the rest of the stack you can recall the number using the sequence RCL 2 f Re Im Le RCL 1 In Program mode use 9 LCLx instead of J Operations With Complex Numbers Almost all functions performed on real numbers will yield the s
38. limitations 75 77 217 requirements for advanced functions 218 219 requirements for programming 218 stack See Stack status display 215 registers in 213 215 Metal box dimensions example 189 191 Minima finding with SOLVE 230 Modes trigonometric 26 278 Subject Index Multiple roots 234 N Negative numbers 19 in Complex mode 124 125 Nested calculations 38 Neutral operations 211 Nonprogrammable functions 80 Normalizing statistics data 50 null display 144 149 Numerical integration 194 204 O ON and off 18 to reset Continuous Memory 63 to set decimal point display 61 Overflow condition 45 61 100 P P R 66 68 Pause LPSE 68 Percent difference A J 29 Percentage functions 29 30 Permutations function Pyx 47 Phasor notation 133 Pi 24 Polar coordinates 30 in Complex mode 133 135 Power function L 29 Prefix keys 19 PRGM annunciator 66 82 Program control indirect 107 109 111 data entry techniques 69 70 end 68 77 entering 66 68 labels 67 77 loading 66 loop counters 109 112 114 116 mode 66 68 86 Subject Index 279 position changing 82 86 running 68 69 starting 69 stops 68 78 Program execution 69 after GSB 101 after GTO 97 after overflow 100 after test 92 from or through labels 78 79 Program lines instructions 67 74 deleting 83 86 inserting 83
39. page 215 Logarithmic and Exponential Functions LN Computes natural logarithm page 28 e Natural antilogarithm Raises e to power of number in display X register page 28 LOG Computes common logarithm base 10 page 28 10 Common antilogarithm Raises 10 to power of number in display X register page 28 y Raises number in Y register to power of 264 number in display X register enter y then x Causes the stack to drop page 29 Mathematics JWA Arithmetic operators cause the stack to drop page 29 VX Computes square root x page 25 x Computes the square of x page 25 x Calculates the factorial n of x or Gamma function T of 1 x page 25 Vx Computes reciprocal page 25 For matrix use refer to Matrix Functions page 264 7 Places value of x in display page 24 Solves for real root of a function f x with the expression for f x defined by the user in a labeled subroutine page 180 Function Summary and Index S2 Integrate Computes the definite integral of f x with the expression f x defined by the user in a labeled subroutine page 194 Matrix Functions DIM Dimensions a matrix of a given name LA to LE J LI J page 141 RESU
40. reallocated registers in memory and any of the statistics registers no longer exist Error 3 will be displayed when you try to use CLEAR 2 J or 27 Appendix C explains how to reallocate memory In one variable statistical calculations enter each data point x value by keying in x and then press 2 In two variable statistical calculations enter each data pair the x and y values as follows Key y into the display first Press ENTER The displayed y value is copied into the Y register Key x into the display Press 2 The current number of accumulated data points n will be displayed The x value is saved in the LAST X register and y remains in the Y register 2 disable stack lift so the stack will not lift when the next number is keyed in TD ce 50 Section 4 Statistics Functions In some cases involving x or y data values that differ by a relatively small amount the calculator cannot compute s r linear regression or and will display Error 2 This will not happen however if you normalize the data by keying in only the difference between each value and the mean or approximate mean of the values This difference must be added back to the calculations of x y and the y intercept L R For example if your x values were 665999 666000 and 666001 you should enter the data as 1 0 and 1 then add 666000 back to the rel
41. 0 i sin 0 re polar at ib r40 phasor imaginary real gt R and gt P can be used to interconvert the rectangular and polar forms of a complex number They operate in Complex mode as follows f converts the polar or phasor form of a complex number to its gt R rectangular form by replacing the magnitude r in the real X register with a and replacing the angle in the imaginary X register with b g converts the rectangular coordinates of a complex number to the gt P polar or phasor form by replacing the real part a in the real X register with r and replacing the imaginary part b in the imaginary X register with 60 gt F gt R i These are the only functions in Complex mode that are affected by the current trigonometric mode setting That is the angular units for 0 must correspond to the trigonometric mode indicated by the annunciator or absence thereof a qan bo Section 11 Calculating With Complex Numbers 135 Example Find the sum 2 cos 65 i sin 65 3 cos 40 i sin 40 and express the result in polar form In phasor form evaluate 265 3 240 Keystrokes Display g DEG Sets Degrees mode for any polar rectangular conversions 2 2 0000 65 Lf LI 2 0000 C annunciator displayed Complex mode activated f gt R 0 8452
42. 014 015 016 45 017 018 43 019 020 021 022 023 024 21 12 5 0 22 1 21 15 4 0 21 1 44 1 1 40 34 16 14 16 1 40 10 1 20 6 0 43 32 45 1 1 40 20 43 32 Program mode Start at B if payments to be made at the beginning Flag 0 clear false indicates advance payments Go to main routine Start at E if payments to be made at the end Flag 0 set true indicates payment in arrears Routine 1 main routine Stores i from X register 1 Puts n in X 1 i in Y n i 1 i 1 i Recall division with R 7 to get i Vi Multiplies quantity by P Flag 0 set End of calculation if flag 0 set for payments in arrears Recalls i 1 i Multiplies quantity by final term End of calculation if flag 0 clear Section 8 Program Branching and Controls 97 Now run the program to find the total amount needed in an account from which you want to take 250 month for 48 months Enter the periodic interest rate as a decimal fraction that is 0 005 per month First find the sum needed if payments will be made at the beginning of the month payments in advance then calculate the sum needed if payments will be made at the end of the month in arrears Keystrokes Display g P R Set to Run mode 250 ENTER 250 0000 Monthly payment 48 ENTER 48 0000 Payment periods 4 y
43. 3141592654 Round Off Error As mentioned earlier the HP 15C holds every value to 10 digits internally It also rounds the final result of every calculation to the 10th digit Because the calculator can provide only a finite approximation for numbers such as 7 or 2 3 0 666 a small error due to rounding can occur This error can be increased in lengthy calculations but usually is insignificant To accurately assess this effect for a given calculation requires numerical analysis beyond our scope and space here Refer to the HP 15C Advanced Functions Handbook for a more detailed discussion Special Displays Annunciators The HP 15C display contains eight annunciators that indicate the status of the calculator for various operations The meaning and use of these annunciators is discussed on the following pages Low power indication page 62 USER User mode pages 79 and 144 fandg Prefixes for alternate functions pages 18 19 RAD and GRAD Trigonometric modes page 26 Cc Complex mode page 121 PRGM Program mode page 66 Section 5 The Display and Continuous Memory 61 Digit Separators The HP 15C is set at power up so that it separates integral and fractional portions of a number with a period a decimal point and separates groups of three digits in the integer portion with a comma You can reverse this setting to conform to the numerical convention used in many countries To do so turn off the calculator Press and hold
44. 37 in matrix functions 141 156 175 176 with 202 SOLVE 181 183 192 226 Z Z register 32 in matrix functions 174 176 with 202 with SOLVE 181 183 192 226 Product Regulatory amp Environment Information Federal Communications Commission Notice This equipment has been tested and found to comply with the limits for a Class B digital device pursuant to Part 15 of the FCC Rules These limits are designed to provide reasonable protection against harmful interference in a residential installation This equipment generates uses and can radiate radio frequency energy and if not installed and used in accordance with the instructions may cause harmful interference to radio communications However there is no guarantee that interference will not occur in a particular installation If this equipment does cause harmful interference to radio or television reception which can be determined by turning the equipment off and on the user is encouraged to try to correct the interference by one or more of the following measures e Reorient or relocate the receiving antenna e Increase the separation between the equipment and the receiver e Connect the equipment into an outlet on a circuit different from that to which the receiver is connected e Consult the dealer or an experienced radio or television technician for help Modifications The FCC requi
45. Assumes position is at line 000 020 020 44 40 3 Moves position to line 020 or use SST instruction STO 3 Section 7 Program Editing 85 Keystrokes Display ae 019 40 Line 020 deleted g BST hold 016 45 4 The next line to edit is line 016 LRCLJ 4 ae 015 20 Line 016 deleted 2 016 45 2 Line 016 changed to RCL 2 GTO 011 011 44 40 2 Moves to line 011 STOJ or hold BST 2 e 010 42 31 Line 011 deleted g BST hold 008 31 Stop Single stepping backwards to line 008 R S J 007 44 40 1 R S deleted 1 008 45 1 Line 008 changed to RCL 1 g LBST 007 44 40 1 Back step to line 007 e 006 44 4 Line 007 STO L_ 1 deleted e 005 20 Line 006 LSTO 4 deleted STO 2 006 44 2 Changed to STO 2 The replacement of a line proceeds like this 015 x 016 RCL 4 017 2 ees Further Information Single Step Operations Single Step Program Execution If you want to check the contents of a program or the location of an instruction you can single step through the program in Program mode If on the other hand running the program produces an error or you suspect that a portion of the program is faulty 86 Section 7 Program Editing you can check the program by executing it stepwise This is done by p
46. CR2032 Lithium batteries Battery life depends on how the calculator is used If the calculator is being used to perform operations other than running programs it uses much less power Low Power Indication A battery symbol shown in the upper left corner of the display when the calculator is on signifies that the available battery power is running low When the battery symbol begins flashing replace the battery as soon as possible to avoid losing data Use only a fresh battery Do not use rechargeable batteries Warning There is the danger of explosion if the battery is a incorrectly replaced Replace only with the same or equivalent type recommended by the manufacturer Dispose of used batteries according to the manufacturer s instructions Do not mutilate puncture or dispose of batteries in fire The batteries can burst or explode releasing hazardous chemicals Replacement battery is a Lithium 3V Coin Type CR2032 Installing New Batteries To prevent memory loss never remove two old batteries at the same time Be sure to remove and replace the batteries one at a time 259 260 Appendix F Batteries To install new batteries use the following procedure 2 CPi fl 1 With the calculator turned off slide the battery cover off 2 Remove the old battery 3 Insert anew CR2032 lithium battery making sure that the positive sign is facing outward 4 Remove and insert the other
47. Complex Matrix You can calculate the inverse of a complex matrix by using the fact that Z Z To calculate inverse Zi of a complex matrix Z 1 Store the elements of Z in memory in the form either of Z or of Z 2 Recall the descriptor of the matrix representing Z into the display 3 If the elements of Z were entered in the form ZE press f Pyx to transform Z into Z 4 Press f MATRIX 2 to transform Z into 5 Designate a matrix as the result matrix It may be the same as the matrix in which Z is stored 6 Press x This calculates Z which is equal to Z The values of these matrix elements are stored in the result matrix and the descriptor of the result matrix is placed in the X register 7 Press f MATRIX 3 to transform Z into Z7 8 If you want the inverse in the form ZS press 9 Cy x You can derive the complex elements of Z by recalling the elements of Z or Z and then combining them as described earlier Example Calculate the inverse of the complex matrix Z from the previous example 4 7 pete le 2 Keystrokes Display RCL MATRIX LA A 4 2 Recalls descriptor of matrix A f MATRIX 2 A 4 4 Transforms Z into Z and redimensions matrix A 166 Section 12 Calculating with Matrices Keystrokes Display f_ RESUL
48. Converts polar to rectangular form real part a displayed 3 3 0000 40 Lf UI 3 0000 f R 2 2981 Converts polar to rectangular form real part a displayed 3 1434 gJ gt P 4 8863 Converts rectangular to polar form r displayed f G hold 49 9612 0 in degrees release 4 8863 Problems By working through the following problems you will see that calculating with complex numbers on the HP 15C is as easy as calculating with real numbers In fact once your numbers are entered most mathematical operations will use exactly the same keystrokes Try it and see 2i 8 6i 4 W50 2 asi 1 Evaluate 136 Section 11 Calculating With Complex Numbers Keystrokes Display 2 f Re im 0 0000 8 CHS ENTER 8 0000 6 Lf LI 8 0000 3 Ly 352 0000 x 1 872 0000 4 ENTER 4 0000 5 Wx 2 2361 2 CHS LX 4 4721 fj UI 4 0000 295 4551 2 ENTER 5 vx 2 2361 4 CHS x 8 9443 fj Ul 2 0000 E 9 3982 f D 35 1344 9 3982 2 Write a program to evaluate the function values of z represents a linear fractional transformation a class of 2i Display shows real part 8 6i 8 6i 2i 8 6i 2 5 4 A5i 2i 8 6i 4 251 2 4V5i Real part of result Answer 9 3982 35 1344i
49. Data Storage With The Index Register Indirect Program Control With the Index Register Program Loop Control The Mechanics Index Register Storage and Recall ccccccsseeeereeeeeees Index Register Arithmetic 0 cccccceeesceceseceesseeeeseeenteeees Exchanging the X Register ccccccesseseteseeeeeeeteeneees Indirect Branching With TJ eceeeeeeseeeeeeteeeteenteentees Indirect Flag Control With I oo eee eeceeereeeteesteenteeneees Indirect Display Format Control With T eneee Loop Control with Counters ISG and DSE Examples Examples Register Operations ccccccecssseeesseeeeeeees Example Loop Control With DSE eccere Example Display Format Control ccccseeeeesteeeteennees Further Information Index Register Contents cc cccsseceeseecseeeeesteeeeeeeneeeees 92 93 95 97 98 98 99 101 101 101 102 102 105 105 105 106 106 106 107 107 107 107 108 108 108 109 109 109 111 111 112 114 115 115 7 8 Contents ISG and DSE Indirect Display Control cccseeeeeeeeeeeeeeeeeeeeeeeees Part Ill HP 15C Advanced Functions e 000e0 Section 11 Calculating With Complex Numbers The Complex Stack and Complex Mode cccceeeeeeeeees Creating the Complex Stack c ccccceessecssteeesseeeeseeenaees Deact
50. Falling time seconds Programmed Solutions Suppose you wanted to calculate falling times from various heights The easiest way is to write a program to cover all the constant parts of a calculation and provide for entry of variable data Writing the Program The program is similar to the keystroke sequence you used above A label is useful to define the beginning of a program and a return is useful to mark the end of a program Also the program must accommodate the entry of new data Loading the Program You can load a program for the above problem by pressing the following keys in sequence The display shows information which you can ignore for now though it will be useful later The HP 15C A Problem Solver 15 Keystrokes Display g JLP R J 000 f CLEAR PRGM 000 f LBLI A 001 42 21 2 002 x 003 9 004 e 005 8 006 007 VX 008 gJ RTN 009 43 g J P R 7 8313 11 20 48 8 10 11 32 Sets HP 15C to Program mode PRGM annunciator on Clears program memory This step is optional here Label A defines the beginning of the program The same keys you pressed to solve the problem manually Return defines the end of the program Switches to Run mode No PRGM annunciator Running the Program Enter the following information to run the program
51. O f LEX CI L LSCI I and Lf ENG I will format the display in their customary manner refer to pages 58 59 using the number in Rj integer part only for n which must be from 0 to 9 SG and DSE Loop Control With Counters The ISG increment and skip if greater than and DSE decrement and skip if less than or equal to functions control loop execution by referencing and altering a loop control number in a given register Program execution skipping a line or not then depends on that number The key sequence is Lf_ LISG J DSE register number This number is 0 to 9 0 to 9 I or J The Loop Control Number The format of the loop control number is nnnnn is the current counter value nnnnn xxxyy where XXX is the test goal value and yy Is the increment of decrement value Except when using 7 section 14 110 Section 10 The Index Register and Loop Control For example the number 0 05002 in a storage register represents ISG and DSE DSE next line if the loop counter nnnnn is either greater than than or equal to mnnnn x x X y y 0 0 5 0 0 2 Start count at ao ji t count by twos Count up to 50 Operation Each time a program encounters it increments or decrements nnnnn the integer port
52. ON press and hold J release ON then release _ ON Repeating this sequence will set the calculator to the previous display convention Keystrokes Display 12345 67 12 345 67 ON Le 12 345 6700 ON Le 12 345 6700 Error Display If you attempt an improper operation such as division by zero an error message Error followed by a digit will appear in the display For a complete listing of error messages and their causes refer to appendix A To clear the Error display and restore the calculator to its prior condition press any key You can then resume normal operation Overflow and Underflow Overflow When the result of a calculation in any register is a number with a magnitude greater than 9 999999999x10 9 999999999x10 is placed in the affected register and the overflow flag flag 9 is set Flag 9 causes the display to blink When overflow occurs in a running program execution continues until completion of the program and then the display blinks The blinking can be stopped and flag 9 cleared by pressing J QN or 9 LCF 9 Underflow If the result of a calculation in any register is a number with a magnitude less than 1 000000000x10 that number will be replaced by zero Underflow does not have any other effect Recall that display does not include the last three digit
53. Ro and R to 1 Activates User mode Stores a41 Stores a12 Stores a Stores dy Stores a3 Stores a32 Stores a4 Stores a42 Dimensions matrix B to be 4x1 Stores value 0 in all elements of B Specifies value 5 for row 1 column 1 Stores value 5 in b41 Recalls descriptor for matrix B Places descriptor for matrix A into X register moving descriptor for matrix B into Y register Section 12 Calculating with Matrices 171 Keystrokes Display f _JIMATRIX 2 A 4 4 Transforms A into A f_J RESULT LC A 4 4 Designates matrix C as result matrix Calculates X and stores Cc 4 1 in C g Cyx Cc 2 2 Transforms X into X RCL C 0 0372 Recalls c11 RCL JLC 0 1311 Recalls c12 RCL JLC 0 0437 Recalls c31 RCL ILC 0 1543 Recalls c22 f J USER 0 1543 Deactivates User mode f _JIMATRIX 0 0 1543 Redimensions all matrices to 0x0 The currents represented by the complex matrix X can be derived from C L 0 0372 0 131 i Po Solving the matrix equation in the preceding example required 24 registers of matrix memory 16 for the 4x4 matrix A which was originally entered as a 4x2 matrix representing a 2x2 complex matrix and four each for the matrices B and C each representing a 2x1 complex matrix However you would have used four fewer registers if the result matrix were matrix B Not
54. The SOLVE function does not use the imaginary stack Refer to the HP I15C Advanced Functions Handbook for information about complex roots 180 Section 13 Finding the Roots of an Equation 181 The basic rules for using SOLVE are 1 In Program mode key in a subroutine that evaluates the function f x that is to be equated to zero This subroutine must begin with a label instruction f J LBL label and end up with a result for f x in the X register In Run mode Key two initial estimates of the desired root separated by ENTER into the X and Y registers These estimates merely indicate to the calculator the approximate range of x in which it should initially seek a root of f x 0 Press f SOLVEJ followed by the label of your subroutine The calculator then searches for the desired zero of your function and displays the result If the function that you are analyzing equals zero at more than one value of x the routine will stop when it finds any one of those values To find additional values you can key in different initial estimates and use again Immediately before addresses your subroutine it places a value of x in the X Y Z and T registers This value is then used by your subroutine to calculate f x Because the entire stack is filled with the x value this number is continually available to your subroutine The use of this technique is
55. and LCLx disable the stack Clearing the Real X Register Pressing L or 9 J CLx with the calculator in Complex mode clears only the number in the real X register it does not clear the number in the imaginary X register Example Change 6 8i to 7 8i and subtract it from the previous entry Use f Re Im or Lf G to view the imaginary part in X Assume a b c and d represent parts of complex numbers Re Im Re Im Re Im Re Im T a b ajb a b b Z cl d c d c d b Yi 6 0 6 0 6 0 d Xi 6 8 0 8 7 8 1 8 Keys 7 or other operation Since clearing disables the stack as explained above the next number you enter will replace the cleared value If you want to replace the real part with zero after clearing use ENTER or any other function to terminate digit entry otherwise the next number you enter will write over the zero the imaginary part will remain unchanged You can then continue with any calculator function 126 Section 11 Calculating With Complex Numbers Clearing the Imaginary X Register To clear the number in the imaginary X register press f Re Im then press LJ Press f Re Im again to return the zero or any new number keyed in to the imaginary X register Example Replace 1 8i by 1
56. at the right of the real stack is not created until f is pressed Recall also that the shading of the stack indicates that those contents will be written over when the next number is keyed in or recalled Section 11 Calculating With Complex Numbers 123 Re Im Re Im Re Im Re Im Re Im T 9 E le LS i lie an 2S chlo Weill onde Ue el g Zo Y 7 i 6 TE is ze i 6 O X 6 i 2 i 2 i 3 i 2 3 Keys 2 ENTER 3 f LI The execution of f I causes the entire stack to drop the T contents to duplicate and the real X contents to move to the imaginary X register When the second complex number is entered the stacks operate as shown below Note that ENTER lifts both stacks Re Im Re Im Re Im Re Im T 7 0 7 0 6 0 6 0 Z 7 0 6 0 2 3 2 3 Y 6 0 2 3 4 0 4 0 Xi 2 3 4 0 4 0 5 0 Keys 4 ENTER 5 Re Im Re Im Re Im T 6 0 6 0 6 0 Z 2 3 6 0 6 0 Y 4 0 2 3 6 0 xX 5 0 4 5 6 8 Keys fJU A second method of entering complex numbers is to enter the imaginary part first then use Reim and J This method is illustrated under Entering Complex Numbers With _ J page 127 124 Section 11 Calculating With Complex Numbers
57. can then handle the case of not finding a root such as by choosing new initial estimates or changing a function parameter The use of SOLVE as an instruction in a program utilizes one of the seven pending returns in the calculator Since the subroutine called by SOLVE utilizes another return there can be only five other pending returns Executed from the keyboard on the other hand SOLVE itself does not utilize one of the pending returns so that six pending returns are available for subroutines within the subroutine called by SOLVE Remember that if all seven pending returns have been utilized a call to another subroutine will result in a display of Error 5 Refer to page 105 Section 13 Finding the Roots of an Equation 193 Restriction on the Use of SOLVE The one restriction regarding the use of SOLVE is that SOLVE cannot be used recursively That is you cannot use SOLVE in a subroutine that is called during the execution of SOLVE If this situation occurs execution stops and Error 7 is displayed It is possible however to use with S thereby using the advanced capabilities of both of these keys Memory Requirements SOLVE requires five registers to operate Appendix C explains how they are automatically allocated from memory If five unoccupied registers are not available SOLVEJ will not run and Error 10 will be displayed
58. ccsceceeseecsseceeneeeeseecneeeeeneeeeeeeees Trigonometric Operations eseeeeeseeeeeceeeseenteaeeeeeeeenes Time and Angle Conversions ccccccssceeseceteeeetteeeeeeees Degrees Radians Conversions cccccceeseeceeceesteeeeseeeees Logarithmic FUACHONS seninem e aE T anes eaeee ds Hyperbolic Functions sssssssssseisseissersserssersserssersserssers Two Number Functions cccccceesceceseeeeseeceneeecnueeeeneeensees The Power FUNCHON moesson trace Ean aE aE i Percentages ssesiretesietsieatil ae TE E EN Polar and Rectangular Coordinate Conversions 0 Section 3 The Automatic Memory Stack LAST X and Data Storage eena eaen sa een AE NAY Contents 5 The Automatic Memory Stack and Stack Manipulation 32 Stack Manipulation Functions s s s esere 33 The LAST X Register and ESTx ceeeeeeeeceesseeeesteeeeeeees 35 Calculator Functions and the Stack ccccccseesseeeeeees 36 Order of Entry and the ENTER Key ccceeeeeteeeereees 37 Nested Calculations ccccccceccseceeseeseeeeeseeeeesseeeteeeenas 38 Arithmetic Calculations With Constants ccccceereeeeees 39 Storage Register Operations ceeeeesseeteteceeeeeeeeneeaes 42 Storing and Recalling Numbers cccscecsseeeesteeeeeeees 42 Clearing Data Storage Registers ccesssecsseeeesteeeeeeees 43 Storage and Recall Arithmetic ccccccesees
59. condition is true and it skips one instruction if the condition is false A instruction is often placed right after a conditional test making it a conditional branch that is the GTO branch is executed only if the test condition is met Program Execution After Test If True If False ooo 015 f LBt 1 016 L lt Y o er aed 019 020 am Er Flags Another conditional test for programming is a flag test A flag is a status indicator that is either set true or clear false Again execution follows the Do if True Rule it proceeds sequentially if the flag is set and skips one line if the flag is clear The HP 15C has eight user flags numbered 0 to 7 and two system flags numbered 8 Complex mode and 9 overflow condition The system flags are discussed later in this section All flags can be set cleared and tested as follows e LY SEF x will set flag number n 0 to 9 e 9 CF x will clear flag number n e 9j E n will check if flag n is set A flag n that has been set remains set until it is cleared either by a CF n instruction or by clearing resetting Continuous Memory Section 8 Program Branching and Controls 93 Examples Example Branching and Looping A radiobiology lab wants to predict the diminishing radioactivity of a test amount of BIT a radioisotope Write a program to figure the radioactiv
60. ices A eel a nee 266 Prefix Keys henna a oelsastaciena sie Meastaeks 266 Probability ss eroen e eee aE EREE 266 Stack M nipulati oiris on a 266 Stotisties seeur sav ES e E a a REA eae 267 SIEL o EEE E A A 267 Trigonometry iee a AE sven E ee 268 Programming Summary and Index cceeeeeeeeeeee 269 Subject Index aaa te N E 271 The HP 15C A Problem Solver The HP 15C Advanced Programmable Scientific Calculator is a powerful problem solver convenient to carry and easy to hold Its continuous memory retains data and program instructions indefinitely until you choose to reset it Though sophisticated it requires no prior programming experience or knowledge of programming languages to use it The new HP 15C is a modern re release of the original HP 15C introduced in 1982 While the battery life of the new version is now estimated to be 1 year for normal use the calculator is now at least 150 times faster than the original The low power indicator gives you plenty of warning before the calculator stops functioning The HP 15C also conserves power by automatically shutting its display off if it is left inactive for a few minutes But don t worry about losing data any information contained in the HP 15C is saved by Continuous Memory A Quick Look at Your Hewlett Packard calculator uses a unique operating logic represented by the ENTER key that differs from the logic in most other calculators You will find that using E
61. in display X register page 26 SIN COS MAN Compute arc sine arc cosine or arc tangent respectively of number in display X register page 26 Programming Summary and Index P R Program Run mode Sets the calculator to Program mode PRGM annunciator on or Run mode PRGM annunciator cleared page 66 MEM Displays current status of calculator memory number of registers dedicated to data storage the common pool and program memory page 215 MEM Displays current status of calculator memory number of registers dedicated to data storage the common pool and program memory page 215 Back arrow In Program mode deletes displayed instruction from program memory All subsequent instructions are moved up page 83 LBL Label Used with the label designations below to denote the start of a program routine page 67 A JLB JLCJLD ILE J0 123456789 0 1 2 3 4 5 6 7 8 9 Label designations When preceded by LBL define the beginning of a program routine page 67 Also used without LBL to initiate execution of a specific routine page 69 USER Activates and deactivates User mode which exchanges the primary white and gold alternate functions LA through LE J of the top left five functions page 69 User mode also affects the matr
62. in will be added following the line currently in the display To alter an instruction first delete it then add the new version Examples Let s refer back to the can volume program on page 71 in section 6 and make a few changes in the instructions The can program as listed below is assumed to be in memory starting on line 001 Deletions If we don t need the summed base area volume and surface area values we can delete the storage register additions lines 007 011 and 020 Changes To eliminate the need to stop the program to enter the height value A change the R S instruction to a RCL 1 instruction because of the above deletions R is no longer being used and store h in R before running the program To clean things up let s also alter STO 4 line 006 to STO 2 and 4 old line 016 to RCL 2 since we are no longer using R and R3 The editing process is diagrammed on the next page 84 Section 7 Program Editing Original Version Edited Version o 007 sTO 1 change _ 006 007 008 009 010 011 012 j B delete change delete MIM 013 io Let s start at the end of the program and work backwards In this way deletions will not change the line numbers of the preceding lines in the program E change 017 018 g RTN delete Keystrokes Display g 000 Program mode
63. indicated Analyze the original function and its derivatives algebraically It may be possible to determine its behavior for x values near the known root A Taylor series representation for example may indicate the multiplicity of a root Limiting the Estimation Time Occasionally you may desire to limit the time used by SOLVE to find a root You can use two possible techniques to do this counting iterations and specifying a tolerance Appendix D A Detailed Look at SOLVE 239 Counting Iterations While searching for a root SOLVE typically samples your function at least a dozen times Occasionally SOLVE may need to sample it one hundred times or more However SOLVE will always stop by itself Because your function subroutine is executed once for each estimate that is tried it can count and limit the number of iterations An easy way to do this is with an ISG instruction to accumulate the number of iterations in the Index register or other storage register If you store an appropriate number in the register before using SOLVE your subroutine can interrupt the SOLVE algorithm when the limit is exceeded Specifying a Tolerance You can shorten the time required to find a root by specifying a tolerable inaccuracy for your function Your subroutine should return a function value of zero if the calculated function value is less than the specified tolerance This toleran
64. into the display Press f If not already in Complex mode this creates the imaginary stack and displays the C annunciator cana eta 122 Section 11 Calculating With Complex Numbers Example Add 2 3i and 4 5i The operations are illustrated in the stack diagrams following the keystroke listing Keystrokes Display f LFIX 4 2 ENTER 2 0000 Keys real part of first number into real Y register 3 3 Keys imaginary part of first number into real X register fJ UI 2 0000 Creates imaginary stack moves the 3 into the imaginary X register and drops the 2 into the real X register 4 ENTER 4 0000 Keys real part of second number into real Y register 5 5 Keys imaginary part of second number into real X register f I 4 0000 Copies 5 from real X register into imaginary X register copies 4 from real Y register into real X register and drops stack 6 0000 Real part of sum f GJ hold 8 0000 Displays imaginary part release 6 0000 of sum while the G key is held This also terminates digit entry The operation of the real and imaginary stacks during this process is illustrated below Assume that the stack registers have been loaded already with the numbers shown as the result of previous calculations Note that the imaginary stack which is shown below
65. is 29 39 73 C 37 51 95 66 90 168 156 Section 12 Calculating with Matrices pong the Equation AX B The J function is useful for solving matrix equations of the form AX B Y where A is the coefficient matrix B is the constant matrix and X is the x solution matrix The descriptor of the constant matrix B should be entered in the Y register and the descriptor of the coefficient matrix A should be entered in the X register Pressing then calculates the solution X A B Remember that the function replaces the coefficient matrix by its LU decomposition and that this matrix must not be specified as the result matrix Furthermore using rather than LY and X gives a solution faster and with improved accuracy At the beginning of this section you found the solution for a system of linear equations in which the constant matrix and the solution matrix each had one column The following example illustrates that you can use the HP 15C to find solutions for more than one set of constants that is for a constant matrix and solution matrix with more than one column Example Looking at his receipts for his last three deliveries of cabbage and broccoli Silas Farmer sees the following summary If A is a singular matrix that is one that doesn t have an inverse then the HP 15C modifies the LU form of A by an amount that is usually small c
66. mean 53 permutations 47 probability 47 standard deviation 53 Statistics registers 49 50 Status indicators 60 S Storage and recall STO RCL 42 43 44 complex numbers 130 direct with _I J 106 107 indirect 106 107 111 matrices 144 149 176 matrix elements 143 144 147 149 52 282 Subject Index Storage arithmetic 43 Storage registers 42 allocation 42 215 217 arithmetic 43 clearing 43 statistics 42 49 Subroutine levels 102 105 limits 102 105 nesting example 103 returns 101 105 using with SOLVE 180 181 192 System flags 92 99 T T register 32 33 in matrix functions 174 176 with 202 TAN ANJ 26 TEST 91 Tracing 82 Transpose 150 151 154 Trigonometric modes in Complex mode 121 134 Trigonometric operations 26 U U display 176 Uncommitted registers 213 215 217 Underflow in any register 61 storage register arithmetic 45 with SOLVE 223 User flags 92 User mode 69 79 with matrices 143 176 V Vector arithmetic using statistics functions 57 W Wrapping 86 90 Subject Index 283 X X exchange X 42 X exchange Y x y 34 X register 32 35 37 42 60 209 210 imaginary 210 211 in matrix functions 141 156 175 176 with 202 with SOLVE 181 183 102 226 Y y intercept finding 54 Y register 32
67. noninteger out of Complex mode y 0 and x lt 0 or e in Complex mode y 0 and Re x lt 0 vx where out of Complex mode x lt 0 Vx where x 0 LOG where out of Complex mode x lt 0 or e in Complex mode x 0 LN where out of Complex mode x lt 0 or e in Complex mode x 0 SIN where out of Complex mode x gt l COS where out of Complex mode x gt l STO where x 0 RCL where the contents of the addressed register 0 A where the value in the Y register is 0 HYP COS where out of Complex mode x lt 1 HYP TAN where out of Complex mode x S Cyx Pyx where 205 206 Appendix A Error Conditions e xory is noninteger ex lt Oory lt 0 e x gt y e xory gt 10 Error 1 Improper Matrix Operation Applying an operation other than a matrix operation to a matrix that is attempting a nonmatrix operation while a matrix is in the relevant register whether the X or Y register or a storage register Error 2 Improper Statistics Operation X n 0 S n lt l Pr n lt l LR n lt l Error 2 is also displayed if division by zero or the square root of a negative number would be required during computation with any of the following formulas Dex a gt y y n n s M Ss o N r 2 XV n n 1 Y Vantn ly VM N
68. number of decimal places you specify up to nine depending on the size of the integer portion Exponents will be displayed if the number is too small or too large for the display At power up the HP 15C is in _FIX is Lf 4 format The key sequence is FIX n Keystrokes Display 123 4567895 123 4567895 f JLFIX 4 123 4568 f JLFIX 6 123 456790 Display is rounded to six decimal places Ten places are stored internally f CFIX 4 123 4568 Usual FIX 4 display 58 Section 5 The Display and Continuous Memory 59 Scientific Notation Display SCI scientific format displays a number in scientific notation The sequence _f_ SCI n specifies the number of decimal places to be shown Up to six decimal places can be shown since the exponent display takes three spaces The display will be rounded to the specified number of decimal places however if you specify more decimal places than the six places the display can hold that is LSCI 7 8 or 9 rounding will occur in the undisplayed seventh eighth or ninth decimal place With the previous number still in the display Keystrokes Display f SCI 6 1 234568 02 Rounds to and shows six decimal places f SCI 8 1 234567 02 Rounds to eight decimal places but displays only six Engineering Notation Display ENG engineerin
69. numbers and matrix descriptors pages 132 174 0x40 1x gt 0 TEST 2x lt 0 TEST 3 x gt 0 4x lt 0 Sx y TEST 6x y TEST 7 x gt y 8x lt y TEST 9x gt y DSE Decrement and skip if equal to or less than Decrements counter value in given register as stipulated Skips one program line if new counter value is equal to or less than specified test value page 109 LISG Increment and skip if greater than Increments counter value in given register as stipulated Skips one program line if new counter value is greater than specified test value page 109 Subject Index Page numbers in bold type indicate primary references page numbers in regular type indicate secondary references A Abbreviated key sequences 78 Absolute value ABS 24 Allocating memory 42 213 219 Altering program lines 83 Annunciators complex 121 list of 60 PRGM 32 66 trigonometric 26 Antilogarithms common and natural 28 Arithmetic operation 29 37 Asymptotes horizontal 230 Automatic incrementing of row and column numbers 143 B Back stepping _BST 83 Bacterial population example 41 Battery life 259 Battery replacement 260 259 260 Bessel functions 195 197 Branching conditional 91 98 177 192 indirect 108 109 112 114 115 simple 90 C C annunciator 99 121 Can volume and area example 70 74 Chain calculations 22 23 38 Changing signs 19 in Complex mod
70. possible because lt Executing present and Clx disable stack lift Re Im will also create an imaginary stack if one is not already Example Enter 9 8i without moving the stack and then find its square Keystrokes Display Le 0 0000 Prevents stack lift when the next digit 8 is keyed in Omit this step if you d rather save what s in X and lose what s in T 8 8 Enter imaginary part first f Re Im 7 0000 Displays real part Complex mode activated lt lt 0 0000 Disables stack Otherwise it would lift following Re Im 9 9 Enters real part digit entry not terminated g x 17 0000 Real part f G hold 144 0000 Imaginary part release 17 0000 Re Im Re Im Re Im Re Im T a b a b a b a b Zi c d cjd cjd c d Yi e f e f e f e f X 4 7 0 7 8 7 7 8 Keys 8 f Re Im 128 Section 11 Calculating With Complex Numbers lt x lt N A Keys Entering a Real Number Re Im Re Im Re Im Re Im a b a b a b b c d c d d d e f e f e f 7 8 0 8 9 8 17 144 Aa 9 gj Lx You have already seen two ways of entering a complex number There is a shorter way to enter a real number simply key it or recall it into the display just as yo
71. register to zero page 21 J In Run mode removes the last digit keyed in or clears the display if digit entry has been terminated page21 Statistics 2 Accumulates numbers from X and Y registers into storage registers R through R7 page 49 27 Removes numbers in X and Y registers from storage registers R through R for correcting 2 accumulations page 52 X Computes mean of x and y values accumulated by page 53 S s Computes sample standard deviations of x and y values accumulated by page 53 S A yr Linear estimate Function Summary and Index and correlation coefficient Computes estimated value of y p for a given value of x by least squares method and places result in X register Computes the correlation coefficient r of the accumulated data and places result in Y register page 55 L R Linear Regression Computes the y intercept and slope for the linear function best approximating the accumulated data The value of the y intercept is placed in the X register the value of the slope is placed in the Y register page 54 RAN Random number Yields a pseudorandom number as generated from a seed stored using STO RAN page 48 CLEAR Clears contents of the statistics
72. register Rj see the diagram of the registers on the inside back cover Six registers R to Rj are also used for statistics calculations The number of available data storage registers can be increased or decreased The DIM function which is used to reallocate registers in calculator memory is discussed in appendix C Memory Allocation The lowest numbered registers are the last to be deallocated from data storage therefore it is wisest to store data in the lowest numbered registers available Storing and Recalling Numbers STO store When followed by a storage register address 0 through 9 or 0 through 9 this function copies a number from the display X register into the specified data storage register It will replace any existing contents of that register RCL recall Similarly you can recall data from a particular register into the display by pressing RCL followed by the register address This brings a copy of the desired data into the display the contents of the storage register remain unaltered X3 X exchange Followed by 0 through 9 this function exchanges the contents of the X register and the addressed data storage register This is useful to view storage registers without disturbing the stack All storage register operations can also be performed with the Index register using I or G which is covered in section 10 an
73. registers R to R7 page 49 267 Storage STO Store Stores a copy of a number into the storage register specified 0 to 9 0 to 9 UI J page 42 Also used for storage register arithmetic new register contents old register contents L J L xj LE J display page 44 RCL Recall Recalls a copy of the number from the storage register specified 0 to 9 0 to 9 LI J page 42 Also used for storage register arithmetic new display old display L J x l register contents page 44 CLEAR REG Clears contents of all storage registers to zero page 43 STx Recalls into the display the number present before the previous operation page 35 268 Trigonometry DEG Sets decimal Degrees mode for trigonometric functions indicated by absence of GRAD or RAD annunciator page 26 Not operative for complex trigonometry Function Summary and Index RAD Sets Radians mode for trigonometric functions indicated by RAD annunciator page 26 GRD Sets Grads mode for trigonometric functions indicated by GRAD annunciator page 26 Not operative for complex trigonometry SIN J COS TAN Compute sine cosine or tangent respectively of number
74. retain the display The display will be four numbers dd uu pp b where dd the number of the highest numbered register in the data storage pool making the total number of data registers dd 2 because of Ro and R uu the number of uncommitted registers in the common pool pp the number of registers containing program instructions and b the number of bytes left before uu is decremented to supply seven more bytes of program memory and pp is incremented The initial status of the HP 15C at power up is 19 46 0 0 The movable boundary between the data storage and common pools is always between Ragand Rag 1 Memory Reallocation There are 67 registers in memory worth seven bytes each Sixty four of these registers R to Res are interconvertible between the data storage and common pools The DIM G Function If you should require more common space as for programming or more data storage space but not both simultaneously you can make the necessary register reallocation using DIM G The procedure is MEM is nonprogrammable t DIM dimension is so called because it is also used with A through CE or 1 to dimension matrices Above however it is used with i to dimension the size of the data storage pool 216 Appendix C Memory Allocation 1 Place dd the number of the highest data storage registe
75. running a program Keeping in mind how the stack moves with subsequent calculations and how the stack can be manipulated as with x y it is possible to write a program to use variables which have been keyed into the X Y Z and T registers 2 Direct entry Enter the data as needed as the program runs Write a run stop instruction into the program where needed so the program will stop execution Enter your data then press R S to restart the program Do not key variable data into the program itself Any values that will vary should be entered anew with each program execution Program Memory At power up Continuous Memory reset the HP 15C offers 322 bytes of program memory and 21 storage registers Most program steps instructions use one byte but some use two The distribution of memory capacity can be altered as explained in appendix C The maximum attainable program memory is 448 bytes with the permanent storage registers R Ro and R remaining maximum number of storage registers is 67 with no program memory Example Mother s Kitchen a canning company wants to package a ready to eat spaghetti mix containing three different cylindrical cans one of spaghetti sauce one of grated cheese and one of meatballs Mother s needs to calculate the base areas total surface areas and volumes of the three different cans It would also like to know per package the total base area surface area and volume
76. the prediction process By permitting more accurate predictions than might otherwise occur properly chosen estimates greatly facilitate the determination of the root you seek The SOLVEJ algorithm will always find a root provided one exists within the overflow bounds if any one of four conditions are met e Any two estimates have function values with opposite signs The function is monotonic meaning that f x either always decreases or else always increases as x is increased 222 Appendix D A Detailed Look at The function s graph is either convex everywhere or concave flx everywhere x e The only local minima and maxima of the function s graph f x occur singly between adjacent zeros of the function x In addition it is assumed that the SOLVE algorithm will not be interrupted by an improper operation Accuracy of the Root When you use the SOLVE key to find a root of an equation the root is found accurately The displayed root either gives a calculated function value f x exactly equal to zero or else is a 10 digit number virtually adjacent to the place where the function s graph crosses the x axis Any such root has an accuracy within two or three units in the 10th significant digit In most situations the calculated root is an accurate estimate of the theoretical infinitely precise root of the equation However certain conditions can cause the finite accuracy of the calculator to give a res
77. too large relative to certain features of the functions being integrated Consider an integral where the interval of integration is wide enough to require excessive calculation time but not so wide that it would be calculated incorrectly Note that because f x xe approaches zero very quickly as x approaches the contribution to the integral of the function at large values of x is negligible Therefore you can evaluate the integral by replacing oo the upper limit of integration by a number not so large as 10 say 10 Appendix E A Detailed Look at 255 Keystrokes Display 0 0 000 00 Keys lower limit into Y register EEX 3 1 03 Keys upper limit into X register f 4 1 1 000 00 Approximation to integral x y 1 824 04 Uncertainty of approximation This is the correct answer but it took almost 60 seconds To understand why compare the graph of the function over the interval of integration which looks about identical to that shown on page 252 to the graph of the function between x 0 and x 10 f x 0 10 By comparing the two graphs you can see that the function is interesting only at small values of x At greater values of x the function is uninteresting since it decreases smoothly and gradually in a very predictable manner lt As discussed earlier the L algorithm will sample the function with higher densities of sample points until the disparity between succ
78. upper limit appears in X register f 4 1 1 3825 00 Integral approximated in SCI j4 X y 1 7091 05 Uncertainty of SCI 4 approximation The uncertainty indicates that this approximation is accurate to at least four decimal places Note that the uncertainty of the SCI 4 approximation is about one hundredth as large as the uncertainty of the SCI 2 approximation In general the uncertainty of any L approximation decreases by about a factor of 10 for each additional digit specified in the display format Provided that f x is still calculated accurately to the number of digits shown in the display Section 14 Numerical Integration 203 In the preceding example the uncertainty indicated that the approximation might be correct to only four decimal places If we temporarily display all 10 digits of the approximation however and compare it to the actual value of the integral actually an approximation known to be accurate to a sufficient number of decimal places we find that the approximation is actually more accurate than its uncertainty indicates Keystrokes Display xy 1 382 00 Return approximation to 5 display f CLEAR PREFIX 1382459676 All 10 digits of approximation The value of this integral correct to eight decimal places is 1 38245969 The calculator s approximation is accurate to seven decimal places rather than only four In f
79. would vary Appendix C Memory Allocation 217 When converting registers note that e You can convert registers from the common pool only if they are uncommitted If for example you try to convert registers which contain program instructions you will get an Error 10 insufficient memory e You can convert occupied registers from the data storage pool causing a loss of stored data An Error 3 results if you try to address a lost that is nonexistent register Therefore it is good practice to store data in the lowest numbered registers first as these are the last to be converted Program Memory As mentioned before each register consists of seven bytes of memory Program instructions use one or two bytes of memory Most program lines use one byte those using two bytes are listed on page 218 The maximum programming capacity of the HP 15C is 448 program bytes 64 convertible registers at seven bytes per register At power up memory can hold up to 322 program bytes 46 allocated registers at seven bytes per register Automatic Program Memory Reallocation Within the common register pool program memory will automatically expand as needed One uncommitted register at a time starting with the highest numbered register available will be allocated to seven bytes of program memory Conversion of Uncommitted Registers to Program Memory Program Bytes a E Ree gt 8to 14 R21 gt 30910315 R20 gt 316t0
80. x x 1 G Keystrokes Display g P R 000 Program mode f J LBL 1 001 42 21 1 g ABS 002 43 16 1 003 1 004 40 g RTN 005 43 32 Section 13 Finding the Roots of an Equation 187 Because the absolute value function is minimum near an argument of zero specify the initial estimates in that region for instance 1 and 1 Then attempt to find a root Keystrokes Display g P R Run mode 1 ENTER SEPARE Initial estimates 1 CHS 1 f SOLVE 1 Error 8 This display indicates that no root was found 0 0000 Clear error display As you can see the HP 15C stopped seeking a root of f x O when it decided that none existed at least not in the general range of x to which it was initially directed The Error 8 display does not indicate that an illegal operation has been attempted it merely states that no root was found where presumed one might exist based on your initial estimates If the HP 15C stops seeking a root and displays an error message one of these three types of conditions has occurred e If repeated iterations all produce a constant nonzero value for the specified function execution stops with the display Error 8 If numerous samples indicate that the magnitude of the function appears to have a nonzero minimum value in the area being searched execution stops with the display Error 8 e If an imprope
81. x 0 TEST 0 TEST 5 TEST 6 Conditional tests for matrix descriptors in the X or X and Y registers x 0 and TEST 0 x 0 test the quantity in the X register for zero Matrix descriptors are considered nonzero TEST 5 x y and TEST 6 x y test if the descriptors in X and Y are the same The result affects program execution skip one line if false page 174 Number Alteration ABS Yields absolute value of number in display page 24 FRAC Leaves only fractional portion of number in display X register by truncating integer portion page 24 INT Leaves only integer portion of number in display X 266 register by truncating fractional portion page 24 RND Rounds mantissa of entire 10 digit number in X register to match display format page 24 Percentage Percent Computes x value in display of number in the Y register page 29 Unlike most two number functions does not drop the stack A Percent difference Computes percent of change between number in Y register and number in display page 30 Does not drop the stack Prefix Keys f Pressed before a function key to select the gold function printed above that key page 18 g Pressed before a function key to select Function Summary and Index
82. y value must be entered first the x value second Upon executing 9 PJ r will appear in the display Press X y X exchange Y to bring 0 out of the Y register and into the display X register 0 will be returned as a value between 180 and 180 between z and 7 radians or between 200 and 200 grads Rectangular Conversion Pressing Section 2 Numeric Functions 3 f RJ rectangular converts a set of polar coordinates magnitude r angle 0 into rectangular coordinates x y 0 must be entered first then r Upon executing Lf _ RJ x will be displayed first press X3 YJ to display y fa pl L gt R Keystrokes Display g DEG 5 ENTER 5 0000 10 10 gJ gt P 11 1803 KEY 26 5651 30 ENTER 30 0000 12 12 f R 10 3923 XZY 6 0000 Set to Degrees mode no annunciator y value x value r 0 rectangular coordinates converted to polar coordinates 0 r x value y value Polar coordinates converted to rectangular coordinates Section 3 The Automatic Memory Stack LAST X and Data Storage The Automatic Memory Stack and Stack Manipulation HP operating logic is based on a mathematical logic known as Polish Notation developed by the noted Polish logician Jan Lukasiewicz Wookashye veech 1878 1956 Conventional algeb
83. 0 0 012 0 0 013 0 x 014 20 5000 1 e X3 015 34 Brings another t value into X register 2 016 2 0 017 0 0 018 0 x 019 20 2001 020 30 500001 e 200r g RTN 021 43 32 Switch to Run mode key in two initial estimates of the time for example 5 and 6 seconds and execute SOLVE Keystrokes Display g J P R Run mode 5 ENTER 5 0000 ah Initial estimates 6 6 f_ SOLVE LA 9 2843 The desired root Verify the root by reviewing the Y and Z registers Keystrokes Display RY 9 2843 A previous estimate of the root RY 0 0000 Value of the function at the root showing that h 0 186 Section 13 Finding the Roots of an Equation Fahr s ridget falls to the ground 9 2843 seconds after he hurls it a remarkable toss Graph of h versus t When No Root Is Found You have seen how the key estimates and displays a root of an equation of the form f x 0 However it is possible that an equation has no real roots that is there is no real value of x for which the equality is true Of course you would not expect the calculator to find a root in this case Instead it displays Error 8 Example Consider the equation Ixl 1 which has no solution since the absolute value function is never negative Express this equation in the required form Ixnl 1 0 and attempt to use SOLVE to find a solution Graph of f
84. 00 Function value g Rt 9 Rt 1 0000 20 Restore the stack f SOLVE 0 Error 8 lt lt 1 1250 20 Another x value RY 1 5626 16 Previous value R 2 0000 Same function value In each of the three cases SOLVEJ initially searched for a root in a direction suggested by the graph around the initial estimate Using 3 third case 10 as the initial estimate SOLVE found the of Teone horizontal asymptote value of 1 0000 Using 1 as the initial estimate a minimum of 0 3788 at x 2 1213 was found Using 10 as the initial estimate the function was essentially constant at a value of 2 0000 for the small range of x that was sampled Finding Several Roots Many equations that you encounter have more than one root For this reason you will find it helpful to understand some techniques for finding several roots of an equation The simplest method for finding several roots is to direct the root search in different ranges of x where roots may exist Your initial estimates specify the range that is initially searched This method was used for all examples in section 13 You can often find the roots of an equation in this manner Another method is known as deflation Deflation is a method by which roots are eliminated from an equation This involves modifying the equation so that the first roots found are no longer roots but the rest of the roots remain roots If a function f x has a value of zero at x a t
85. 06 FRALTARREBRAF X RERARSA SMRELERSASR BRPA PS SBBY SIT 11363 2006 FRATARBER ROGAX SRARFRE SR BRON BARS AK BSFS2003 1F27R RF SFSBLSST RARE SMR 2002 95 EC BES FRRARRHSSHRMATFSLSLEHSRALRSAS 288
86. 1 column 1 Activate User mode by pressing Lf USER With the calculator in User mode after each element is stored or recalled the row number in Ro or the column number in R is automatically incremented by 1 as shown in the example following If you are storing elements key in the value of the element to be stored in row 1 column 1 Press STO or RCL followed by the letter key specifying the matrix Repeat steps 4 and 5 for all elements of the matrix The row and column numbers are incremented according to the dimensions of the matrix you specify While the letter key specifying the matrix is held down after STO or RCL is pressed the calculator displays the name of the matrix followed by the row and column numbers of the element whose value is being stored or recalled If the letter key is held down for longer than about 3 seconds the calculator displays null doesn t store or recall the element value and doesn t increment the row and column numbers Also the stack registers aren t changed After the last element of the matrix has been accessed the row and column numbers both return to 1 Example Store the values shown below in the elements of the matrix A dimensioned above Be sure matrix A is dimensioned to 2x3 A a ap 43 2 3 A Ay ayy 45 6 Section 12 Calculating with Matrices 145
87. 204 Section 14 Numerical Integration Memory Requirements J gt requires 23 registers to operate Appendix C explains how they are automatically allocated from memory If 23 unoccupied registers are not available will not run and Error 10 will be displayed he x A routine that combines L and SOLVE also requires 23 registers of space For Further Information This section has given you the information you need to use L with confidence over a wide range of applications In appendix E more esoteric aspects of are discussed These include e How L works e Accuracy uncertainty and calculation time e Uncertainty and the display format e Conditions that could cause incorrect results e Conditions that prolong calculation time e Obtaining the current approximation to an integral Appendix A Error Conditions If you attempt a calculation containing an improper operation say division by zero the display will show Error and a number To clear an error message press any one key This also restores the display prior to the Error display The HP 15C has the following error messages The description of Error 2 includes a list of statistical formulas used Error O Improper Mathematics Operation Illegal argument to math routine J where x 0 y where e out of Complex mode y lt 0 and x is
88. 3 43 16 CHS 014 16 1 015 1 0 016 232 Appendix D A Detailed Look at Keystrokes e 3 g JLRIN SOLVE Display 017 10 x 10 018 12 019 40 no p eet 020 3 021 40 pp 0 ppe 022 43 32 Use SOLVE with the following single initial estimates 10 1 and 10 Keystrokes g JLP R J 10 ENTER f SOLVE 4 RY RY RtjL9 Rt SOLVE 7ft RY m NTER SOLVE 0 RtJL9 Rt SOLVE tHe a ft R RY EEX CHS ENTER 0 Display 10 0000 Error 8 455 335 48 026 721 85 1 0000 455 4335 Error 8 48 026 721 85 1 0000 1 0000 Error 8 2 1213 2 1471 0 3788 2 1213 Error 8 2 1213 0 3788 1 0000 20 Run mode Single estimate Best x value Previous value Function value Restore the stack Another x value Same function value an asymptote Single estimate Best x value Previous value Function value Restore the stack Same x value Same function value a minimum Single estimate Appendix D A Detailed Look at 233 Keystrokes Display f SOLVE 0 Error 8 p 1 0000 20 Best x value RY 1 1250 20 Previous value RY 2 00
89. 30 Deflation for third root 038 10 Display 1 0929 07 10 0000 ENA 20 Same initial estimates 8 5001 Fourth root 8 5001 Stores root for reference 0 0009 Deflated function value near Zero 238 Appendix D A Detailed Look at Using the same initial estimates each time you have found four roots for this equation involving a fourth degree polynomial However the last two roots are quite close to each other and are actually one root with a multiplicity of 2 That is why the root was not eliminated when you tried deflation once at this root Round off error causes the original function to have small positive and negative values for values of x between 8 4999 and 8 5001 for x 8 5 the function is exactly zero f x in 10 s 30 Graph of f x In general you will not know in advance the multiplicity of the root you are trying to eliminate If after you have attempted to eliminate a root SOLVE finds that same root again you can proceed in a number of ways Use different initial estimates with the deflated function in an attempt to search for a different root Use deflation again in an attempt to eliminate a multiple root If you do not know the multiplicity of the root you may need to repeat this a number of times Examine the behavior of the deflated function at x values near the known root If the function s calculated values cross the x axis smoothly either another root or a greater multiplicity is
90. 35 The LAST X Register and STx The LAST X register a separate memory register preserves the value that was last in the display before execution of a numeric operation Pressing g J LSTx LAST X places a copy of the contents of the LAST X register into the display X register For example The LSTx feature saves you from having to re enter numbers you want to use again as shown under Arithmetic Calculations With Constants page 39 It can also assist you in error recovery such as executing the wrong function or keying in the wrong number For example suppose you mistakenly entered the wrong divisor in a chain calculation Keystrokes Display 287 ENTER 287 0000 12 9 22 2481 Oops The wrong divisor g LSTx 12 9000 Retrieves from LAST X the last entry to the X register the incorrect divisor before was executed Unless that operation was X s or ER which don t use or preserve the value in the display X register but instead calculate from data in the statistics storage registers R2 to R7 For a complete list of operations which save x in LAST X refer to appendix B 36 Section 3 The Memory Stack LAST X and Data Storage Keystrokes Display x 287 0000 Reverses the function that produced the wrong answer 13 9 L 20 6475 The corre
91. 5 10 5 10 5 R S 527 7876 364 4247 4 5 4 5 R S 63 6173 SIDE AREA BASE AREA SURFACE AREA Keeps a sum of all SURFACE AREAS in R3 Ends the program and returns program memory to line 000 Sets calculator to Run mode PRGM cleared Clears all storage registers The display does not change Enter r of the first can Starts program A BASE AREA of first can running flashes during execution Enter h of first can Then restart program VOLUME of first can SURFACE AREA of first can Enter r of the second can BASE AREA of second can Enter h of second can VOLUME of second can SURFACE AREA of second can Enter r of the third can BASE AREA of third can 74 Section 6 Programming Basics Keystrokes Display 4 4 Enter h of third can R S 254 4690 VOLUME of third can 240 3318 SURFACE AREA of third can RCL 1 133 5177 Sum of BASE AREAS RCL 2 939 3362 Sum of VOLUMES RCL 3 769 6902 Sum of SURFACE AREAS The preceding program illustrates the basic techniques of programming It also shows how data can be manipulated in Program and Run modes by entering storing and recalling data input and output using ENTER STO LRCL J storage register arithmetic and programmed stops Further Information Program Instructions Each digit decimal point and function key is considered an instruction and is stored in one line of program memory An instructio
92. 5i Re Im Re Im Re Im Re Im Re Im T b alb a b alb b Z d c d c d c d d Y f e f e f e f f X 1 8 8 1 0 1 5 1 1 5 Keys Rexim 5 Re Im continue with any operation Clearing the Real and Imaginary X Registers If you want to clear or replace both the real and imaginary parts of the number in the X register simply press J which will disable the stack and enter your new number Enter zeros if you want the X register to contain zeros Alternatively if the new number will be purely real including 0 Oi you can quickly clear or replace the old complex number by pressing LR followed by zero or the new real number Example Replace 1 5i with 4 7i Re Im Re Im Re Im Re Im Re Im T b alb c d c d c d Z d c d e f e f c d Y f e f 4 5 4 5 e f X 1 5 0 5 4 5 7 0 4 7 Keys 4 ENTER 7 fj Ul continue with any operation Entering CLx can also be used with Section 11 Calculating With Complex Numbers 127 Complex Numbers with e The clearing functions and Re Im as an alternative method of entering and clearing complex numbers Using this method you can enter a complex number using only the X register without affecting the rest of the stack This is
93. 6262x10 CHS 6 6262 34 6 6262x10 ENTER 6 6262 34 Enters number 50 x 3 3131 32 Joule seconds Note Decimal digits from the mantissa that spill into the exponent field will disappear from the display when you press but will be retained internally To prevent a misleading display pattern EEX will not operate with a number having more than seven digits to the left of the radix mark decimal point nor with a mantissa smaller than 0 000001 To key in such a number use a form having a greater exponent value whether positive or negative For example 123456789 8x10 can be keyed in as 1234567 898x10 0 00000025x10 can be keyed in as 2 5x10 The CLEAR Keys Clearing means to replace a number with zero The clearing operations in the HP 15C are the table is continued on the next page Clearing Sequence Effect g JLCLx J Clears display X register 4 In Run mode Clears last digit or entire display In Program mode Deletes current instruction f CLEAR Lz Clears statistics storage registers display and the memory stack described in section 3 Section 1 Getting Started 21 Clearing Sequence Effect f_ CLEAR PRGM In Run mode Repositions program memory to line 000 In Program mode Deletes all program memory f CLEAR REG Clears all data storage
94. 86 Program memory 67 70 75 217 219 automatic real location 217 218 clearing 67 moving in 67 Q Quadratic equation solving 181 R Ro and Rj using to access matrix elements 143 146 176 RAD 26 Radioisotope example 93 94 Random number generator RAN 48 Random number storage and recall 48 Recall arithmetic 44 Recalling accumulated statistics data 50 Recalling numbers RCL 42 44 with matrices 144 149 176 Reciprocal Lx 25 with matrix 150 Rectangular coordinates 31 in Complex mode 133 135 Registers converting 215 217 Reset Continuous Memory 63 Residual 159 Result matrix 147 148 150 152 Return RTN 68 77 Returns pending 101 105 192 204 Reverse Polish Notation 32 Re Im 124 127 280 Subject Index Rice yield example 50 56 Ridget hurling example 184 186 224 226 Roll down 34 Roll up 34 Roots eliminating 233 234 237 Roots meaningless 188 191 Rounding RND 24 Rounding in the display 59 Round off errors 52 60 with SOLVE 223 237 Row norm 150 177 Run Stop LR S J 68 91 running display 69 147 182 S Scalar operations 151 153 SCI 58 Scientific notation 58 Scrolling 82 Secant line calculation example 102 Self tests 261 Service information 267 270 Shear stress example 227 228 SINJ SIN J 26 Sine integra
95. 9 17 4 4 Keystrokes Display 9 ENTER 9 0000 Digit entry terminated 17 26 0000 9 17 4 22 0000 9 17 4 4 5 5000 9 17 4 4 Even more complicated problems are solved in the same manner using automatic storage and retrieval of intermediate results It is easiest to work from the inside of parentheses outwards just as you would with calculations on paper Example Calculate 6 7 x 9 3 Keystrokes Display 6 ENTER 6 0000 First solve for the intermediate result of 6 7 7 U 13 0000 9 ENTER 9 0000 Then solve for the intermediate result of 9 3 3L 6 0000 x 78 0000 Then multiply the intermediate results together 13 and 6 for the final answer Try your hand at the following problems Each time you press ENTER or a function key in a calculation the preceding number is saved for the next operation 16 x 38 13 x 11 465 0000 4x 17 12 10 5 4 0000 23 13 x 9 1 7 412 1429 V6 4 0 8 12 5 0 72 0 5998 Section 2 Numeric Functions This section discusses the numeric functions of the HP 15C excluding statistics and advanced functions The nonnumeric functions are discussed separately digit entry in section 1 stack manipulation in section 3 and display control in section 5 The numeric functions of the HP 15C are used in the same way whether execu
96. HP 15C Owner s Handbook HP Part Number 00015 90001 Edition 2 4 Sep 2011 Legal Notice This manual and any examples contained herein are provided as is and are subject to change without notice Hewlett Packard Company makes no warranty of any kind with regard to this manual including but not limited to the implied warranties of merchantability non infringement and fitness for a particular purpose In this regard HP shall not be liable for technical or editorial errors or omissions contained in the manual Hewlett Packard Company shall not be liable for any errors or incidental or consequential damages in connection with the furnishing performance or use of this manual or the examples contained herein Copyright 2011 Hewlett Packard Development Company LP Reproduction adaptation or translation of this manual is prohibited without prior written permission of Hewlett Packard Company except as allowed under the copyright laws Hewlett Packard Company Palo Alto CA 94304 USA Introduction Congratulations Whether you are new to HP calculators or an experienced user you will find the HP 15C a powerful and valuable calculating tool The HP 15C provides e 448 bytes of program memory one or two bytes per instruction and sophisticated programming capability including conditional and unconditional branching subroutines flags and editing e Four advanced mathematics capabilities complex number calcula
97. LT Designates the matrix into which the result of certain matrix operations is placed page 148 USER User mode Row and column numbers in Ro and R are automatically incremented each time STO or RCL LA to LE J GDJ is pressed page 144 STO and RCL A to LE J G Stores or recalls matrix elements using the row and column numbers in Rg and R pages 144 146 STO 9 and RCL 9 LA to LE J a Stores or recalls matrix elements using the row and column numbers in the Y and X registers page 146 STO RCL MATRIX LA to LE Stores or recalls matrices for the specified matrix pages 142 147 and STO and RCL RESULT Stores or recalls descriptor of the result matrix page 148 RCL DIM LA through LE J LL J Recalls the dimensions of the given matrix into the Y row and X column registers page 142 x Inverts the matrix whose descriptor is displayed and places the result in the specified result matrix The descriptor of the result matrix is then displayed page 150 Lx Adds subtracts or multiplies the corresponding elements of two matrices or of one matrix and a scalar Stores in r
98. NTER makes nested and complicated calculations easier and faster to work out Let s get acquainted with how this works For example let s look at the arithmetic functions First we have to get the numbers into the machine Is your calculator on If not press ON Is the display cleared To display all zeros you can press _J CLx that is press g then To perform arithmetic key in the first number press ENTER to separate the first number from the second then key in the second number and press L J L j LX J or J The result appears immediately after you press any numerical function key If you have not used an HP calculator before you will notice that most keys have three labels To use the primary function the one printed in white on top of the key just press that key For those printed in gold or blue press the gold f key or the blue 9 key first 12 The HP 15C A Problem Solver 13 The display format used in this handbook is FIX 4 the decimal point is fixed to show four decimal places unless otherwise mentioned If your calculator does not show four decimal places you may want to press f_ LFIX_ 4 to match the displays in the examples Manual Solutions Run through the following two number calculations It is not necessary to clear the calc
99. T A 4 4 Designates B as the result B matrix Vx b 4 4 Calculates Z Z and places the result in matrix B f_ MATRIX 3 b 4 2 Transforms Z into 20y The representation of Z in partitioned form is contained in matrix B 0 0254 0 2420 0 0122 ior Real Part 0 2829 I 0 1691 0 1315 J Pema Part Multiplying Complex Matrices The product of two complex matrices can be calculated by using the fact that YX YX To calculate YX where Y and X are complex matrices 1 Store the elements of Y and X in memory in the form either of Lock 2 Recall the descriptor of the matrix representing Y into the display 3 If the elements of Y were entered in the form of Y press f Pyx to transform Y into Y 4 Press f MATRIX 2 to transform Y into Y 5 Recall the descriptor of the matrix representing X into the display 6 If the elements of X were entered in the form X9 press f Pyx to transform X into X 7 Designate the result matrix it must not be the same matrix as either of the other two Section 12 Calculating with Matrices 167 8 Press X to calculate YX YX The values of these matrix elements are placed in the result matrix and the descriptor of the result matrix is placed in the X register 9 If you want the product in the form YX press 9 Cy x Note that you don t transform X i
100. TEST 6 x y Refer to sections 11 and 12 for more information Flags As a conditional test can be used to pick an option by comparing two numbers in a program a flag can be used to pick an option externally Usually a flag is set or cleared first thing in a program by choosing a different starting point using different labels depending on the condition or mode you want refer to the example on page 95 Section 8 Program Branching and Controls 99 In this way a program can accommodate two different modes of input such as degrees and radians and make the correct calculation for the mode chosen You set a flag if a conversion needs to be made for instance and clear it if no conversion is needed Suppose you had an equation requiring temperature input in degrees Kelvin although sometimes your data might be in degrees Celsius You could use a program with a flag to allow either a Kelvin or Celsius input In part such a program might include f J LBLILC Start program at C for degrees Celsius 9 CF 7 Flag 7 cleared false GTO 1 f LBL D Start program at D for degrees Kelvin g LSF 7 Flag 7 set true f LBL 1 Assuming temperature in X register gJ LF 7 Checks for flag 7 checks for Celsius or Kelvin input GTO 2 If set Kelvin input goes to a later routine skipping the next few instructio
101. The as a reference to the contents of another storage register i key uses the indirect addressing system shown in the tables on pages 107 and 108 In turn the contents of that second register may be used as a loop control number in the fashion described above This is also true for the value in any storage register used for indirect loop control 116 Section 10 The Index Register and Loop Control ISG and DSE For the purpose of loop control the integer portion the counter value of the stored control number can be up to five digits long nnnnn xxxyy The counter value nnnnn is zero if not specified otherwise Xxx in the decimal portion of the control number must be specified as a three digit number For example 5 must be 005 xxx is zero if not specified otherwise Whenever ISG or LDSE is encountered nnnnn is compared internally to xxx which represents the end level for incrementing or decrementing yy must be specified as a two digit number yy cannot be zero so if left or specified as 00 the value for yy defaults to 1 The value nnnnn is altered by the amount of yy each time the loop runs through ISG or DSE Both yy and xxx are reference values which do not change with loop execution Indirect Display Control While you can use the Index register to format the display manually that is from the keyboa
102. Waste Equipment by Users in Private Household in the European Union This symbol on the product or on its packaging indicates that this product must not be disposed of with your other household waste Instead it is your responsibility to dispose of your waste equipment by handing it over to a designated collection point for the recycling of waste electrical and electronic equipment The separate collection and recycling of your waste equipment at the time of disposal will help to conserve natural resources and ensure that it is recycled in a manner that protects human health and the environment For more information about where you can drop off your waste equipment for recycling please contact your local city office your household waste disposal service or the shop where you purchased the product Chemical Substances HP is committed to providing our customers with information about the chemical substances in our products as needed to comply with legal requirements such as REACH Regulation EC No 1907 2006 of the European Parliament and the Council A chemical information report for this product can be found at www hp com go reach Perchlorate Material special handling may apply This calculator s Memory Backup battery may contain perchlorate and may require special handling when recycled or disposed in California BEASBERTE Pb Hg Cd ACHE SRR SRR Crivi PBB PBDE o o o RRRAEASARERSAMASRAACHS 2 BSENT 11363 20
103. Y X Y X Result is stored in result matrix Result matrix may be X or Y Example Calculate C B A where A and B are defined in the previous example 12 3 13 5 A andB p 5 9 p 9 17 Keystrokes Display f J RESULT C Designates C as result matrix RCL MATRIX B b 2 3 Recalls descriptor of matrix B This step can be skipped if descriptor is already in X register RCL MATRIX LA A 2 3 Recalls descriptor of matrix A into X register moving descriptor of matrix B to Y register 154 Section 12 Calculating with Matrices Keystrokes Display Cc 2 3 Calculates B A and stores values in redimensioned result matrix C 1 2 The resultis C 9 3 4 8 Matrix Multiplication With matrix description in both the X and Y registers you can calculate three different matrix products The table below shows the results of the three functions for a matrix X specified in the X register and a matrix Y specified in the Y register The matrix X is the inverse of X and the matrix Y is the transpose of Y Pressing Calculates x YX f MATRIX 5 YX xX y Result is stored in result matrix For J the result matrix can be Y but not X For the others the result matrix must be other than X or Y Note When you use the function to evaluate the expressio
104. a height of 107 meters How long does it take for his remarkable toss described on page 184 in section 13 to reach 107 meters Solution The desired solution is the value of t at which h 107 Enter the subroutine from page 184 that calculates the height of the ridget This subroutine can be used in a new function subroutine to calculate Jt h 107 The following subroutine calculates f t Keystrokes Display g P R 000 Program mode f LBL B 001 42 21 12 Begin with new label GSB A 002 32 11 Calculates h t 1 003 1 0 004 0 7 005 7 Calculates h t 107 006 30 g JLRTN 007 43 32 Appendix D A Detailed Look at SOLVE 225 In order to find the first time at which the height is 107 meters use initial estimates of 0 and 1 second and execute SOLVE using LB Keystrokes Display g P R Run mode 0 ENTER oe Initial estimates 1 1 f SOLVE 4 1718 The desired root B RY 4 1718 A previous estimate of the root RY 0 0000 Value of f t at root It takes 4 1718 seconds for the ridget to reach a height of exactly 107 meters It takes approximately two seconds to find this solution However suppose you assume that the function h t is accurate only to the nearest whole meter You can now change your subroutine to give f t 0 whenever the ca
105. a natural manner As you have seen working through this handbook most calculations do not require you to think about the operation of the automatic memory stack There are occasions however especially as you delve into programming when you need to know the effect of a particular operation upon the stack The following explanation should help you Digit Entry Termination Most operations on the calculator whether executed as instructions in a program or pressed from the keyboard terminate digit entry This means that the calculator knows that any digits you key in after any of these operations are part of a new number The only operations that do not terminate digit entry are the digit entry keys themselves O through 9 CHS EEX Stack Lift There are three types of operations on the calculator based on how they affect stack lift These are stack disabling operations stack enabling operations and neutral operations When the calculator is in Complex mode each operation affects both the real and imaginary stacks The stack lift effects are the same In addition the number keyed into the display real X register after any operation except 4 or CLx is accompanied by the placement of a zero in the imaginary X register 209 210 Appendix B Stack Lift and the LAST X Register Disabling Operations Stack Lift There are four stack disabling opera
106. aces the original result matrix R The descriptor of the result matrix is placed in the X register 160 Section 12 Calculating with Matrices Using Matrices in LU Form As noted earlier two matrix operations calculating a determinant and solving the matrix equation AX B create an LU decomposition of the matrix specified in the X register The descriptor of such a matrix has two dashes following the matrix name A matrix in LU form has elements that differ from the elements of the original matrix However the descriptor for a matrix in LU form can be used in place of the descriptor for the original matrix for operations involving the inverse of the matrix and for the determinant operation That is either the original matrix or its LU decomposition can be used for these operations Vx for the matrix in the X register MATRIX 9 For these three functions using the LU form of the matrix to be inverted gives a result that is identical to that using the original matrix As an example if you solved the matrix equation AX B matrix A would be changed to its LU form If you wanted to change the B matrix and solve the equation again you could do so without changing the A matrix the LU matrix will give the correct solution For all other matrix operations a matrix that is an LU decomposition is not recognized as representing its original matrix Instead the elements of the LU matrix are used just a
107. act since the uncertainty of an approximation is calculated very conservatively the calculator s approximation in most cases will be more accurate than its uncertainty indicates However normally there is no way to determine just how accurate an approximation is For a more detailed look at the accuracy and uncertainty of L approximations refer to appendix E Using in a Program s can appear as an instruction in a program provided that the program is not called as a subroutine by L itself In other words cannot be used recursively Consequently you cannot use L to calculate multiple integrals if you attempt to do so the calculator will halt with Error 7 in the display However s can appear as an instruction in a subroutine called by SOLVE The use of as an instruction in a program utilizes one of the seven pending returns in the calculator Since the subroutine called by LA utilizes another return there can be only five other pending returns Executed from the keyboard on the other hand itself does not utilize one of the pending returns so that six pending returns are available for subroutines within the subroutine called by L 4 Remember that if all seven pending returns have been utilized a call to another subroutine will result in a display of Error 5 Refer to page 105
108. ae Appendix C Memory Allocation The Memory Space cccceescecesscecsseceeceeeeeeeecneeesseeeneeeeenas REGISIELS Menten wct cc tepsekcensotameasseiadenes EEEE cesar EE cane Memory Status IMEM ceeseseeceeceeeceeeseenenaeeeeeeeeeeees Memory Reallocation cccccscccsscceeseeeeseeccneeeeneeeteeeenaees The DIM G2 Function oo ceecccccceeeesseeeteceenteeeeseeeeneeees Restrictions on Reallocation o sessies Program Memory iis cestieds cccbecn et catiece cde conn pateaeeeeceeeas hasta Automatic Program Memory Reallocation Two Byte Program Instructions eceeesesseceeeeeeeenneaeees Memory Requirements for the Advanced Functions Appendix D A Detailed Look at SOLVE How SOLVE Works ten vie sees ri dodgy dered and deeds Accuracy of the ROOM 2 sc dersstacezesssae de nt E O EERE Interpreting Results ccccceccssceesseceeeeeseseecsteeeeseeenteeens Finding Several Roots cc ccccsscessseceeseeeeeeeecsteeesseeeeteeees Limiting the Estimation Time Counting Iterations cccceeeeeseeneeneeeeeeeeeeneneeeeeeeeneees Specifying a Tolerance cccccceeeseeeeseecenteeenteeeeeeenteeees For Advanced Information ceccecceesesseesnsestseenesneennees Appendix E A Detailed Look at How ZF Works ooo eecececeeescnseseeeeecesecneeeeeeneeseeseeeaeenes Accuracy Uncertainty and Calculation Time
109. ainty indicates that the displayed digits of the approximation might not include any digits that could be considered accurate Actually this approximation is more accurate than its uncertainty indicates Keystrokes Display xy 7 79 03 Return approximation to display f CLEAR PREFIX hold 7785820888 All 10 digits of SCI 2 approximation The actual value of this integral correct to five significant digits is 7 7805x10 Therefore the error in this approximation is about 7 7858 7 7805 x10 5 3x10 This error is considerably less than the uncertainty 1 45x10 The uncertainty is only an upper bound on the error in the approximation the actual error will generally be smaller Now calculate the integral in SCI 3 and compare the accuracy of the resulting approximation to that of the SCI 2 approximation Keystrokes Display f SCI 3 7 786 03 Changes display format to LSCI J 3 R LRY 3 142 00 Rolls down stack until upper limit appears in X register f o0 7 786 03 Integral approximated in SCI 3 Xsy 1 448 04 Uncertainty of SCI 3 approximation X3y 7 786 03 Returns approximation to display f CLEAR hold 7785820888 All 10 digits of SCI 3 approximation 244 Appendix E A Detailed Look at All 10 digits of the app
110. al manner and program execution continues with the next program line If the X register contains a number program execution skips the next line In both cases the original contents of the X register are stored in the LAST X register This is useful for testing whether a matrix descriptor is in the X register during a program Summary of Matrix Functions Keystroke s g Cy x through LE J DIM A MATRIX 0 MATRIX 1 MATRIX 2 MATRIX 3 MATRIX 4 l Lenco Unc Ol MATRIX 5 Results Transforms Z into Z Changes sign of all elements in matrix specified in X register Dimensions specified matrix Dimensions all matrices to 0x0 Sets row and column numbers in Ro and R to 1 Transform Z into Z Transforms Z into Z Calculate transpose of matrix specified in X register Multiplies transpose of matrix specified in Y register with matrix specified in X register Stores in 178 Section 12 Calculating with Matrices Keystroke s f_ MATRIX 6 f_ MATRIX 7 f_ MATRIX 8 f_ MATRIX 9 f Pyx RCL A EJ through RCLILY A throug YD h E RCL DIM LA through E D A through E RCL MATRIX h RCL
111. alf is the imaginary part Therefore by inspection of matrix C gz 10000 2 8500x10 4 0000x10 1 0000 11 0000x107 3 8000x107 FI 11 10 1 0000x107 1 0500x10 As expected a 1 Of JO O ZZ i 0 1 0 0 Solving the Complex Equation AX B You can solve the complex matrix equation AX B by finding X A B Do this by calculating X A B To solve the equation AX B where A X and B are complex matrices 1 Store the elements of A and B in memory in the form either of Z or of ZS 2 Recall the descriptor of the matrix representing B into the display 3 If the elements of B were entered in the form BS press Lf Pyx to transform B into B Section 12 Calculating with Matrices 169 4 Recall the descriptor of the matrix representing A into the display 5 If the elements of A were entered in the form of A press Lf Pyx to transform A into A 6 Press MATRIX 2 to transform A into A 7 Designate the result matrix it must not be the same as the matrix representing A 8 Press this calculates X The values of these matrix elements are placed in the result matrix and the descriptor of the result matrix is placed in the X register 9 If you want the solution in the form X press 9 Cy x Note that you don t transform B into B You can derive the com
112. already familiar with HP calculators The use of SOLVE and L requires a knowledge of HP 15C programming Contents The HP 15C A Problem Solver cccccccceessssseeeeeeeeees A Quick Look at ENTER cccccceccesssscceeeeeeeesssteseeeeeesenes Manual Solutions 0 c cccccccecsssssseecececeesesseseeeeeesenenessaas Programmed Solutions c ccccccceesseceeeeesneeessteeeesseeeteeees Part HP 15C Fundamentals 0 cseeccceeeeeeeeeee Section 1 Getting Started Adis nacsanemamariarn Power On and Oth ne enn Sail ee E EE Keyboard Operation cccccccccsceesseceeeeeeseeeecsseeessseeetetens Primary and Alternate Functions ccccccssseeesteeeereeees Prefix KEYS n a nesses setter sekuetees Serve EH eer ES Changing Signs cccceessecesscecsseeeeseeeeeeeecnteeeeseeenteeess Keying in Exponents 00 0 0 ccc ceeeeeeeeeeeeeeeeeeseneneneneneneneaeaes The CLEAR KEY Snr Seema teeth E AS Display Clearing Clx and 49 a a Calc lations s aeien era e aE ETN One Number Functions ccccccseseeeseeceeseecseeeeeneeeeeeees Two Number Functions and ENTER ccccceeseeeeereees Section 2 Numeric Functions seeeeeeeeeeeeneeeeees Pi raa en odaltuc tan Grated duce reit iae ana a E ea Number Alteration Functions cccccsseeeseecseeeeesteeeseeees One Number Functions ccccceecceceeeceeeeeeeneeeesueeeenseeneees General Functions c
113. ame answer whether executed in or out of Complex mode assuming the result is also real In other words Complex mode does not restrict your ability to calculate with real numbers Any functions not mentioned below or in the rest of this section Calculating With Complex Numbers ignore the imaginary stack You can use the HP 15C matrix function described in section 12 to make storing and recalling complex numbers more convenient By dimensioning a matrix to be nx2 n complex numbers can be stored as rows of the matrix This technique is demonstrated in the HP 15C Advanced Functions Handbook section 3 under Applications The exceptions are P and R which operate differently in Complex mode in order to facilitate converting complex numbers to polar form page 133 Section 11 Calculating With Complex Numbers 131 One Number Functions The following functions operate on both the real and imaginary parts of the number in the X register and place the real and imaginary parts of the answer back into those registers vx Lx LN LOG LY 10 e LABS P R All trigonometric and hyperbolic functions and their inverses also belong to this group The ABS function gives the magnitude of the number in the X registers the square root of the sum of the squares of the real and imaginary parts the imaginary part of the magnitude is zero
114. and the number of rows followed by the number of columns appears at the right The matrix name is called the descriptor of the matrix Matrix descriptors can be moved among the stack and data storage registers just like a number that is using STO RCL ENTER etc Whenever a matrix descriptor is displayed in the X register the current dimensions of that matrix are shown with it You use matrix descriptors to indicate which matrices are used in each matrix operation The matrix operations discussed in the rest of this section 148 Section 12 Calculating with Matrices operate on the matrices whose descriptors are placed in the X register and for some operations the Y register Two matrix operations calculating a determinant and solving the matrix equation AX B involve calculating an LU decomposition also known as an LU factorization of the matrix specified in the X register A matrix that is an LU decomposition is signified by two dashes following the matrix name in the display of its descriptor Refer to page 160 for using a matrix in LU form The Result Matrix For many operations discussed in this section you need to define the matrix in which the result of the operation should be stored This matrix is called the result matrix Other matrix operations do not use or affect the result matrix This is noted in the descriptions of these operations Such an operation either replaces the original
115. any numbers which may be in the stack Stack Lift No Stack Lift or Drop TERR Notice the number in the T register remains there when the stack drops allowing this number to be used repetitively as an arithmetic constant Stack Manipulation Functions ENTER Pressing ENTER separates two numbers keyed in one after the other It does so by lifting the stack and copying the number in the display X register into the Y register The next number entered then writes over the value in the X register there is no stack lift The example below shows what happens as the stack is filled with the numbers 1 2 3 4 The 34 Section 3 The Memory Stack LAST X and Data Storage shading indicates that the contents of that register will be written over when the next number is keyed in or recalled R roll down Rt roll up and X Y X exchange Y R4 and Rt roll the contents of the stack registers up or down one register one value moves between the X and the T register No values are lost Xx y exchanges the numbers in the X and Y registers If the stack were loaded with the sequence 1 2 3 4 the following shifts would result from pressing LR R J and x3 y Keys RY Section 3 The Memory Stack LAST X and Data Storage
116. ary X register while the key is held page 124 SF 8 Sets flag 8 which activates Complex mode page 121 CF 8 Clears flag 8 deactivating Complex mode page 121 Conversions R Converts polar magnitude r and angle 0 in X and Y registers respectively to rectangular x and y coordinates page 31 For operation in Complex mode refer to page 134 gt P Converts x y rectangular coordinates placed in X and Y registers respectively to polar magnitude r and angle 0 page 30 For operation in Complex mode refer to page 134 262 gt H MS Converts decimal hours or degrees to hours minutes seconds or degrees minutes seconds page 27 gt H Converts hours minutes seconds or degrees minutes seconds to decimal hours or degrees page 27 RAD Converts degrees to radians page 27 gt DEG Converts radians to degrees page 27 Digit Entry ENTER Enters a copy of number in X register display into Y register used to separate multiple number entries pages 22 37 CHS Change sign of number or exponent of 10 in display pages 19 124 263 EEX Enter exponent next digits keyed in are exponents of 10 page 19 O through 9 keys page 22 Decimal point page 22 digit Disp
117. atrix Operations in a Program sssr Summary of Matrix Functions ccccccceesecsseeeesseeeeeeenaees For Further Information Section 13 Finding the Roots of an Equation Using SOLVE When No Root Is Found ccccccsseceeseeeeeeeeseeeessseeneeeeenas Choosing Initial Estimates c ccccecscsceeseeeeeeeeesseeeeeeenaees Using SOLVE in a Program cecceeesceeseeeeeseeesteeesseeeeteees Restriction on the Use of SOLVE n Memory Requirements For Further Information Section 14 Numerical Using AX eee Accuracy of F Integration esseere Using L in a Program oo eeeececeeececeeeceeeeeeeeeeeeeeeeeees Memory Requirements For Further Information 149 149 151 153 154 156 159 160 160 161 164 165 166 168 173 173 173 174 174 176 177 179 180 180 186 188 192 193 193 193 194 194 200 203 204 204 9 10 Contents Appendix A Error Conditions ccescececeeeesreeeeeenneeees Appendix B Stack Lift and the LAST X Register 5 Digit Entry Termination oo cccccecceceeceeeeeeeeeeeeeeeeeeeeeees Stack lita jected atin ita nea eed adele Disabling Operations ccccccccceeseeeeececsteeeesseeeteeeeaes Enabling Operations ccccecccceeeeeeeeeeenseeesseeeneeeeeses Neutral Operations c cccccccecsseceeseeeeeeeeenteeeeseeeeteeeesas LAST X Register asio a lating waded iett
118. atrix specified in X register as result matrix f_ USER Row and column numbers in Ro and R are automatically incremented each time STO J or RCL LA through LE J J is pressed Vx Inverts matrix specified in X register Stores in result matrix Use f x if User mode is on j If matrix descriptors specified in both X and Y registers adds or subtracts corresponding elements of matrices specified If matrix descriptor specified in only one of these registers performs addition or subtraction with all elements in specified matrix and scalar in other register Stores in result matrix x If matrix descriptors specified in both X and Y registers calculates product of specified matrices as YX If matrix specified in only one of these registers multiplies all elements in specified matrix by scalar in other register Stores in result matrix If matrix descriptors specified in both X and Y registers multiplies inverse of matrix specified in X register with matrix specified in Y register If matrix specified in only Y register divides all elements of specified matrix by scalar in other register If matrix specified in only X register multiplies each element of inverse of specified matrix by scalar in other register Stores in result matrix For Further Information The HP I5C Advanced Func
119. ave already entered and accumulated calculate the average fertilizer application x and average grain yield y for the entire range Keystrokes Display SIX 40 00 Average kg of nitrogen x for all cases Xsy 6 40 Average tons of rice y for all cases Standard Deviation Pressing JJ LS_ computes the standard deviation of the accumulated statistics data The formulas used to compute s the standard deviation of the accumulated x values and s the standard deviation of the accumulated y values are given in appendix A This function gives an estimate of the population standard deviation from the sample data and is therefore termed the sample standard deviation When you press 9 Ls J the contents of the stack registers are lifted twice if stack lift is enabled once if not s is placed into the X register and s is placed into the Y register Press X Y to view s When your data constitutes not just a sample of a population but all of the population the standard deviation of the data is the true population standard deviation denoted o The formula for the true population standard deviation differs by a factor of 4 n 1 n from the formula used for the s function The difference between the values is small for large n and for most applications can be ignored But if you want to calculate the exact value of the population standard deviation for an ent
120. ave keystrokes when addressing calling up programs for execution Pressing f USER will exchange the primary functions and f shifted functions of the LA through LE keys only In User mode USER annunciator displayed f shift A B C D E a4 JX e TO ye Vx Primary J shift gt x LN LOG A Press _9 USER again to deactivate User mode Polynomial Expressions and Horner s Method Some expressions such as polynomials use the same variable several times for their solution For example the expression fx Ax B C Dx E uses the variable x four different times A program to solve such an equation could repeatedly recall a stored copy of x from a storage register A shorter programming method however would be to use a stack which has been filled with the constant refer to Loading the Stack with a Constant page 41 Horner s Method is a useful means of rearranging polynomial expressions to cut calculation steps and calculation time It is especially expedient in SOLVE and L two rather long running functions that use subroutines This method involves rewriting a polynomial expression in a nested fashion to el
121. ay as the result of a calculation a recall or keying in Press STO 3 Press L J LX or L Key in the register address 0 to 9 0 to 9 The Index register discussed in section 10 can also be used 44 Section 3 The Memory Stack LAST X and Data Storage The number in the register is determined as follows For storage arithmetic number in display new contents _ old contents of register of register x TH m x lt N ao JLO Keys Recall Arithmetic Recall arithmetic allows you to perform arithmetic with the displayed value and a stored value without lifting the stack that is without losing any values from the Y Z and T registers The keystroke sequence is the same as for storage arithmetic using RCL in place of STO For recall arithmetic contents of new display old display x registert r e x lt N a Keys RCL JLO Section 3 The Memory Stack LAST X and Data Storage 45 Example Keep a running count of your newly blooming crocuses for two more days Keystrokes Display 8 STO 0 8 0000 Places the total number of blooms as of day 2 in Ro 4 STO 0 4 0000 Day 3 adds four new blooms to those already blooming 3 STO 0 3 0000 Day 4 adds three new blooms 24 RCL 0 9 0000 Subtracts total number of blooms summed in Ro 15 f
122. battery as in steps 2 through 3 Make sure that the positive sign on each battery is facing outward 5 Replace the battery cover Note Be careful not to press any keys while the battery is out of the calculator If you do so the contents of Continuous Memory may be lost and keyboard control may be lost that is the calculator may not respond to keystrokes 6 Press ON to turn on the power If for any reason Continuous Memory has been reset that is if its contents have been lost the display will show Pr Error Pressing any key will clear this message Appendix F Batteries 261 Verifying Proper Operation Self Tests If it appears that the calculator will not turn on or otherwise is not operating properly use the following procedures to access the test system 1 Turn the calculator off 2 Press and HOLD the 9 and ENTER keys keep both keys held down for the next step 3 Press the ON key while both 9 and ENTER keys are held down from Step 2 above 4 Release the ON key 5 Release the _J_ and ENTER keys You will be presented with a main test screen that displays the following 1 L 2 C 3 H e Press 1 to perform the LCD test all LCD segments will be turned on Press any key to exit e Press 2 to perform the checksum test and see the copyright messages Press any key to go from one screen to the next until you ret
123. before performing certain matrix operations Appendix C describes how memory is organized how to determine the number of registers currently available for storing matrix elements and how to increase or decrease that number Dimensioning a Matrix To dimension a matrix to have y rows and x columns place those numbers in the Y and X registers respectively and then execute f DIM followed by the letter key specifying the matrix 1 Key the number of rows y into the display then press ENTER to lift it into the Y register Y number of rows 2 Key the number of columns x into the X register 3 Press f DIM followed by a X n mber ot columns letter key A through E that specifies the name of the matrix The matrix functions described in this section operate on real matrices only In Complex mode the imaginary stack is ignored during matrix operation However the HP 15C has four matrix functions that enable you to calculate using real representations of complex matrices as described on pages 160 173 t You don t need to press f before the letter key Refer to Abbreviated Key Sequences on page 78 142 Section 12 Calculating with Matrices Example Dimension matrix A to be a 2x3 matrix Keystrokes Display 2 2 0000 Keys number of rows into Y register 3 3 Keys number of columns into X register f DIMJLA 3 0000 Dimensions ma
124. ber or pair of parentheses and work outward as you would for a manual calculation Otherwise you may need to place an intermediate result into a storage register For example consider the calculation of 3 4 5 6 7 Keystrokes Display 6 ENTER 7 13 0000 Intermediate result of 6 7 5 x 65 0000 Intermediate result of 5 6 7 4 69 0000 Intermediate result of 4 5 6 7 3 x 207 0000 Final result 3 4 5 6 7 The following sequence illustrates the stack manipulation in this example The stack automatically drops after each two number calculation and then lifts when a new number is keyed in For simplicity throughout the rest of this handbook we will not show arrows between the stacks Section 3 The Memory Stack LAST X and Data Storage 39 lt x lt N aA Keys lt x lt N a Keys Arithmetic Calculations With Constants There are three ways without using a storage register to manipulate the memory stack to perform repeated calculations with a constant 1 Use the LAST X register 2 Load the stack with a constant and operate upon different numbers Clear the X register every time you want to change the number operated upon 3 Load the stack with a constant and operate upon an accumulating number Do not change the number in the X register LAST X Use your constant in the X register that is en
125. ble in the final approximation the algorithm terminates placing the current approximation in the X register and its uncertainty in the Y register It is extremely unlikely that the errors in each of three successive approximations that is the differences between the actual integral and the approximations would all be larger than the disparity among the approximations themselves Consequently the error in the final approximation will be less than its uncertainty Although we can t know the error in the final approximation the error is extremely unlikely to exceed the displayed uncertainty of the approximation In other words the uncertainty estimate in the Y register is an almost certain upper bound on the difference between the approximation and the actual integral Accuracy Uncertainty and Calculation Time The accuracy of an approximation does not always change when you increase by just one the number of digits specified in the display format though the uncertainty will decrease Similarly the time required to calculate an integral sometimes changes when you change the display format but sometimes does not Example The Bessel function of the first kind of order four can be expressed as J4 x cos 40 xsin0 d0 m o The relationship between the display format the uncertainly in the function and the uncertainty in the approximation to its integral are discussed later in this appendi
126. ccess of the routine in locating a root depends primarily upon the nature of the function it is analyzing and the initial estimates at which it begins searching The mere existence of a root does not ensure that the casual use of the SOLVE key will find it If the function f x has a nonzero horizontal asymptote or a local minimum of its magnitude the routine can be expected to find a root of f x 0 only if the initial estimates do not concentrate the search in one of these unproductive regions and of course if a root actually exists Choosing Initial Estimates When you use to find the root of an equation the two initial estimates that you provide determine the values of the variable x at which the routine begins its search In general the likelihood that you will find the particular root you are seeking increases with the level of understanding that you have about the function you are analyzing Realistic intelligent estimates greatly facilitate the determination of a root The initial estimates that you use may be chosen in a number of ways If the variable x has a limited range in which it is conceptually meaningful as a solution it is reasonable to choose initial estimates within this range Frequently an equation that is applicable to a real problem has in addition to the desired solution other roots that are physically meaningless These usually occur because the equation being analyzed is appropriate only between certain l
127. ce that you specify should correspond to a value that is negligible for practical purposes or should correspond to the accuracy of the computation This technique eliminates the time required to define the estimate more accurately than is justify by the problem The example on page 224 uses this method For Advanced Information In the HP I15C Advanced Functions Handbook additional advanced techniques and applications for using SOLVE are presented These topics include e Using with polynomials e Solving a system of equations e Finding local extremes of a function e Using SOLVE for financial problems e Using SOLVE in Complex mode e Solving an equation for its complex roots Appendix E A Detailed Look at Section 14 Numerical Integration presented the basic information you need to use L This appendix discusses more intricate aspects of L that are of interest if you use L often How Works The algorithm calculates the integral of a function f x by computing a weighted average of the function s values at many values of x known as sample points within the interval of integration The accuracy of the result of any such sampling process depends on the number of sample points considered generally the more sample points the greater the accuracy If f x could be evaluated at an infinite number of sample points the algorithm could neglecting the li
128. ct answer Calculator Functions and the Stack When you want to key in two numbers one after the other you press ENTER between entries of the numbers However when you want to key in a number immediately following any function including manipulations like LR you do not need to use ENTER Why Executing most HP 15C functions has this additional effect e The automatic memory stack is lift enabled that is the stack will lift automatically when the next number is keyed or recalled into the display e Digit entry is terminated so the next number starts a new entry Keys There are four functions ENTER CLx 2 and 2 that disable stack lift They do not provide for the lifting of the stack when the next number is keyed in or recalled Following the execution of one of these functions a new number will simple write over the currently displayed number instead of causing the stack to lift Although the stack lifts when ENTER is pressed it will not lift when the next number is keyed in or recalled The operation of ENTER illustrated on page 34 shows how ENTER thus disables the stack In most cases the above effects will come so naturally that you won t even think about them Ce will also disable the stack lift if digit entry is terminated making clear the
129. d on this difference which is the uncertainty of the approximation For example if the integral Si 2 is 1 6054 0 0001 the approximation to the integral is 1 6054 and its uncertainty is 0 0001 This means that while we don t know the exact difference between the actual integral and its approximation we do know that it is highly unlikely that the difference is bigger than 0 0001 Note the first footnote on page 200 202 Section 14 Numerical Integration If the uncertainty of an approximation is larger than what you choose to tolerate you can decrease it by specifying a greater number of digits in the display format and repeating the approximation Whenever you want to repeat an approximation you don t need to key the limits of integration back into the X and Y registers After an integral is calculated not only are the approximation and its uncertainty placed in the X and Y registers but in addition the upper limit of integration is placed in the Z register and the lower limit is placed in the T register To return the limits to the X and Y registers for calculating an integral again simply press R R Example For the integral in the expression for J 1 you want an answer accurate to four decimal places instead of only two Keystrokes Display f SCI 4 1 8826 03 Set display format to SCI 4 R4 LRY 3 1416 00 Roll down stack until
130. d with matrices section 12 Section 3 The Memory Stack LAST X and Data Storage 43 The above are stack lift enabling operations so the number remaining in the X register can be used for subsequent calculations If you address a nonexistent register the display will show Error 3 Example Springtime is coming and you want to keep track of 24 crocuses planted in your garden Store the number of crocuses blooming the first day and add to this the number of new blooms the second day Keystrokes Display 3 STO 0 3 0000 Stores the number of first day blooms in Ro Turn the calculator off Next day turn it back on again RCL 0 3 0000 Recalls the number of crocuses that bloomed yesterday 5 8 0000 Adds today s new blooms to get the total blooming crocuses Clearing Data Storage Registers Pressing f CLEAR clear registers clears the contents of all data storage registers to zero It does not affect the stack or the LAST X register To clear a single data storage register store zero in that register Resetting Continuous Memory clears all registers and the stack Storage and Recall Arithmetic Storage Arithmetic Suppose you not only wanted to store a number but perform arithmetic with it and store the result in the same register You can do this directly without using by using the following procedure 1 Have your second operand besides the one in storage in the displ
131. described on page 41 Example Use SOLVE to find the values of x for which fx x 3x 10 0 Using Horner s method refer to page 79 you can rewrite f x so that it is programmed more efficiently Six a 3 x 10 In Program mode key in the following subroutine to evaluate f x Keystrokes Display g JLP R 000 Program mode f CLEAR PRGM 000 Clear program memory 182 Section 13 Finding the Roots of an Equation Keystrokes Display f LBL O 001 42 21 0 Begin with LBL instruction Subroutine assumes stack loaded with x 3 002 3 003 30 Calculate x 3 x 004 20 Calculate x 3 x 1 005 1 0 006 0 007 30 Calculate x 3 x 10 g RTN 008 43 32 In Run mode key two initial estimates into the X and Y registers Try estimates of 0 and 10 to look for a positive root Keystrokes Display g P R Run mode O ENTER 0 0000 te 10 10 Initial estimates You can now find the desired root by pressing _f_ SOLVE 0 When you do this the calculator will not display the answer right away The HP 15C uses an iterative algorithm to estimate the root The algorithm analyzes your function by sampling it many times perhaps a dozen times or more It does this by repeatedly executing your subroutine Finding a root will usually require about 2 to 10 seconds but sometimes the p
132. display of its descriptor after the preceding step Store the elements of B in memory in the form either of B or of BS Recall the descriptor of the matrix representing A into the display Recall the descriptor of the matrix representing B into the display If the elements of B were entered in the form BS press f _ PyxJ to transform B into B Press f_ MATRIX 2 to transform B into Designate the result matrix it must not be the same matrix as either of the other two Press X J Press f IMATRIX 4 to transpose the result matrix Redimension the result matrix to have half the number of rows as indicated in the display of its descriptor after the preceding step Press RCL RESULT to recall the descriptor of the result matrix Press f_ MATRIX 4 to calculate X If you want the solution in the form X press 9 Cyx Section 12 Calculating with Matrices 173 A problem using this procedure is given in the HP 15C Advanced Functions Handbook under Solving a Large System of Complex Equations Miscellaneous Operations Involving Matrices Using a Matrix Element With Register Operations If a letter key specifying a matrix is pressed after any of the following function keys the operation is performed using the matrix element specified by the row and column numbers in Ro and Rj just as though it we
133. ds to be used at more than one point in a program memory space can be conserved by storing those instructions as a single subroutine The Mechanics Go To Subroutine and Return The GSB go to subroutine instruction is executed in the same way as the condition GSB label like GTO branch with one major difference it establishes a pending return GTO line with the corresponding label label transfers program execution to the A J to LE J 0 to 9 or 0 to 9 However execution then continues until the first subsequent RINJ instruction is encountered at which point execution transfers back to the instruction immediately following the last there GSB instruction and continues on from Subroutine Execution f J LBLILA GSB L 1 g JLRIN END Execution transfers to line 000 and halts gt Lf JILBULs JI LLSIULRTN RETURN Execution transfers back to original routine after GSBI _ 1 A or instruction followed by a letter label is an abbreviated key sequence no f necessary Abbreviated key sequences are explained on page 78 101 102 Section 9 Subroutines Subroutine Limits A subroutine can call up another subroutine and that sub
134. e GD sequence used to view the imaginary X register 212 Appendix B Stack Lift and the LAST X Register LAST X Register The following operations save x in the LAST X register Except when used as a matrix function t Li uses the LAST X register in a special way as described in appendix E x HYP COS HYP TAN A x gt H MS gt P TAN gt H gt R ABS SIN 7 gt DEG Pyx CoS gt RAD Cy x CINT MAN LN 2 RND HYP SIN e Vx HYP COS LOG Dr x HYP TAN TO MATRIX VX HYP CSIN ye rail 5 through 9 Appendix C Memory Allocation The Memory Space Storage registers program lines and advanced function execution all draw on a common memory space in the HP 15C The availability of memory for a specific purpose depends on the current allocation of memory as well as on the total memory capacity of the calculator Registers Memory space in the HP 15C is allocated on the basis of registers This space is partitioned into two pools which strictly define how a register may be used There is always a combined total of 67 registers in these two pools e The data storage pool contains registers which may be used only for data storage At power
135. e 124 125 271 272 Subject Index in matrices 177 CHS J 19 Clearing blinking in display 100 complex numbers 125 127 display 21 memory 63 operations 20 21 overflow condition 45 61 prefix keys 19 statistics registers 49 Coefficient matrix 156 Combinations function Cy x 47 Common pool 213 Complex arithmetic example 132 Complex conjugate forming 125 Complex matrix inverting 162 164 165 multiplying 162 164 166 storing elements 161 transforming 162 164 Complex mode 120 121 activating 99 120 121 133 deactivating 121 mathematics functions in 131 stack lift in 124 Complex numbers clearing 125 127 converting polar and rectangular forms 133 135 entering 121 127 128 129 storing and recalling 130 Conditionals indirect 109 111 112 116 Conditional tests 91 98 192 in Complex mode 132 with matrix descriptors 174 Constant matrix 156 Constants calculations with 39 42 using in arithmetic calculations 35 39 42 Subject Index 273 Continuous Memory duration of 62 resetting clearing 63 what it retains 43 48 58 61 62 Conventions handbook 18 Conversions degrees and radians 27 polar and rectangular coordinates 30 31 time and angle 26 27 Correcting accumulated statistics data 52 Correlation coefficient find the r 55 56 COS COS 26 Counters in program loops 98 112 114 Crocus example 43 Cumulative calculations 41 D Da
136. e following formula to find the new balance after each payment New Balance Old Balance x 1 i Payment The first part of the key sequence would be 1 01 ENTER ENTER ENTER 1000 For each payment execute x 100L Balance after six payments 446 32 Store 100 in Rs Then 1 Divide the contents of Rs by 25 2 Subtract 2 from the contents of Rs 3 Multiply the contents of R by 0 75 4 Add 1 75 to the contents of Rs 5 Recall the contents of Rs Answer 3 2500 Section 4 Statistics Functions A word about the statistics functions their use is based on an understanding of memory stack operation Section 3 You will find that order of entry is important for most statistics calculations Probability Calculations The input for permutation and combination calculations is restricted to nonnegative integers Enter the y value before the x value These functions like the arithmetic operators cause the stack to drop as the result is placed in the X register Permutations Pressing f Pyx calculates the number of possible different arrangements of y different items taken in quantities of x items at a time No item occurs more than once in an arrangement and different orders of the same x items in an arrangement are counted separately The formula is P y Combinations Pressing 9 Cy x calculates the number of possibl
137. e interval of integration By calculating a weighted average of the function s values at the sample points the algorithm approximates the integral of f x Unfortunately since all that the algorithm knows about f x are its values at the sample points it cannot distinguish between f x and any other function that agrees with f x at all the sample points This situation is depicted in the illustration on the next page which shows over a portion of the interval of integration three of the infinitely many functions whose graphs include the finitely many sample points 250 Appendix E A Detailed Look at f x x With this number of sample points the algorithm will calculate the same approximation for the integral of any of the functions shown The actual integrals of the functions shown with solid lines are about the same so the approximation will be fairly accurate if f x is one of these functions However the actual integral of the function shown with a dashed line is quite different from those of the others so the current approximation will be rather inaccurate if f x is this function The L 4 algorithm comes to know the general behavior of the function by sampling the function at more and more points If a fluctuation of the function in one region is not unlike the behavior over the rest of the interval of integration at some iteration the algorithm will likely detect the fluctuation When this happens the nu
138. e it s more convenient to use SCI display format when calculating most integrals we ll use when calculating integrals in subsequent examples Note Remember that once you have set the display format you can change the number of digits appearing in the display by storing a number in the Index register and then pressing Lf LFIX J LL f LSCI LI J or L ENG I J as described in section 10 This capability is especially useful when L is executed as part of a program It is possible that integrals of functions with certain characteristics such as spikes or very rapid oscillations might be calculated inaccurately However this possibility is very small The general characteristics of functions that could cause problems as well as techniques for dealing with them are discussed in appendix E The accuracy of a calculated function depends on such considerations as the accuracy of empirical constants in the function as well as round off error in the calculations These considerations are discussed in more detail in the HP 15C Advanced Functions Handbook t The reason for this is discussed in appendix E Section 14 Numerical Integration 201 Because the accuracy of any integral is limited by the accuracy of the function as indicated in the display format the calculator cannot compute the value of an inte
139. e particular function and generally can be determined only by trying it Furthermore if you do get a more accurate answer it will come at the cost of about double the calculation time This unavoidable trade off between accuracy and calculation time is important to keep in mind if you are considering decreasing the uncertainty in hopes of obtaining a more accurate answer The time required to calculate the integral of a given function depends not only on the number of digits specified in the display format but also to a certain extent on the limits of integration When the calculation of an integral requires an excessive amount of time the width of the interval of integration that is the difference of the limits may be too large compared with certain features of the function being integrated For most problems however you need not be concerned about the effects of the limits of integration on the calculation time These conditions as well as techniques for dealing with such situations will be discussed later in this appendix Uncertainty and the Display Format Because of round off error the subroutine you write for evaluating f x cannot calculate f x exactly but rather calculates F X f EG 0 where 6 x is the uncertainty of f x caused by round off error If f x relates to a physical situation then the function you would like to integrate is not f x but rather 246 Appendix E A Detailed Look at F x f x 6
140. e sample points must be the same in the region where the function is interesting To achieve the same density of sample points the total number of sample points required over the larger interval is much greater than the number required over the smaller interval Consequently several more iterations are required over the larger interval to achieve an approximation with the same accuracy and therefore calculating the integral requires considerably more time Because the calculation time depends on how soon a certain density of sample points is achieved in the region where the function is interesting the calculation of the integral of any function will be prolonged if the interval of integration includes mostly regions where the function is not interesting Fortunately if you must calculate such an integral you can modify the problem so that the calculation time is considerably reduced Two such techniques are subdividing the interval of integration and transformation of variables These methods enable you to change the function or the limits of integration so that the integrand is better behaved over the interval s of integration These techniques are described in the HP 15C Advanced Functions Handbook Appendix E A Detailed Look at 257 Obtaining the Current Approximation to an Integral When the calculation of an integral is requiring more time than you care to wait you may want to stop and display the current approximation You can ob
141. e sets of y different items taken in quantities of x items at a time No item occurs more than once in a set and different orders of the same x items in a set are not counted separately The formula is y C My x Examples How many different arrangements are possible of five pictures which can be hung on the wall three at a time Keystrokes Display 5 ENTER 3 3 Five y pictures put up three x at a time f Pyx 60 0000 Sixty different arrangement possible 47 48 Section 4 Statistics Functions How many different four card hands can be dealt from a deck of 52 cards Keystrokes Display 52 ENTER 4 4 Fifty two y cards dealt four x at a time g Cy x 270 725 0000 Number of different hands possible The maximum size of x or y is 9 999 999 999 Random Number Generator Pressing f RAN random number will generate a random number part of a uniformly distributed pseudo random number sequence in the range 0 lt r lt l At initial power up including reset of Continuous Memory the HP 15C random number generator will use zero as a seed to initiate a random number sequence Any time you generate a random number that number becomes the seed for the next random number You can initiate a different random number sequence by storing a new seed for the random number generator Repetition of a random number seed will produce repetiti
142. e that since X and B are not restricted to be vectors that is single column matrices X and B could have required more memory The HP 15C contains sufficient memory to solve using the method described above the complex matrix equation AX B with X and B having up to six columns if A is 2x2 or up to two columns if A is 3x3 The allowable number of columns doubles if the constant matrix B is used as the result matrix If X and B have more columns or if A is 4x4 you can solve the equation using the alternate method below This method differs from the preceding one in that it involves separate inversion and multiplication operations and fewer registers If all available memory space is dimensioned to the common pool MEM 1 64 0 0 Refer to appendix C Memory Allocation 172 10 11 12 13 14 15 Press f_ MATRIX 2 17 18 19 20 Section 12 Calculating with Matrices Store the elements of A in memory in the form either of A or of AS Recall the descriptor of the matrix representing A into the display If the elements of A were entered in the form A press f Pyx to transform A into A Press f_ MATRIX 2 to transform A into A Press STO RESULT to designate the matrix representing A as the result matrix Press x to calculate A Redimension A to have half the number of rows as indicated in the
143. e trigonometric functions 196 Section 14 Numerical Integration Keystrokes Display g P R Run mode O ENTER 0 0000 Key lower limit 0 into Y register g T 3 1416 Key upper limit n into X register 9g J RAD 3 1416 Specify Radians mode for trigonometric functions x Now you are ready to press _f_ L 0 to calculate the integral When you do so you ll find that just as with the calculator will not display the result right away as it does with other operations The HP 15C calculates integrals using a sophisticated iterative algorithm Briefly this algorithm evaluates f x the function to be integrated at many values of x between the limits of integration At each of these values the calculator evaluates the function by executing the subroutine you write for that purpose When the calculator must execute the subroutine many times as it does when you press s you can t expect any answer right away Most integrals will require on the order of 2 to 10 seconds but some integrals will require even more Later on we ll discuss how you can decrease the time somewhat but for now press _f_ L 4 0 and take a break or read ahead while the HP 15C takes care of the drudgery for you e Keystrokes Display f ILO 2 4040 f cos sin d In general don t forget to multiply the value of the i
144. ears x 12 months 005 0 005 Monthly interest rate as a decimal fraction fl B 10 698 3049 Deposit necessary for payments to be made in advance Repeat stack entries f E 10 645 0795 Deposit necessary for payment to be made in arrears The difference between this deposit and the tuition cost 12 000 represents interest earned on the deposit Further Information Go to In contrast to the nonprogrammable sequence GTO CHS nnn the programmable sequence label cannot be used to branch to a line number but only to program label a line containing f LB label Execution continues from the point of the new label and does not return to the original routine unless given another GTO instruction GTO label can also be used in Run mode that is from the keyboard to move to a labeled position in program memory No execution occurs It is possible to branch under program control to a particular line number by using indirect addressing discussed in section 10 98 Section 8 Program Branching and Controls Looping Looping is an application of branching which uses a GTO instruction to repeat a portion of the program A loop can continue indefinitely or may be conditional A loop is frequently used to repeat a calculation with different variables At the same time a counter which increments with each loop may be included to keep t
145. ection 13 Finding the Roots of an Equation Keystrokes Display 003 30 x 004 20 x 6 x 8 005 8 005 40 x 007 20 x 6 x 8 x 4 008 4 x 009 20 4 x 6 x 8 x 7 010 7 011 48 5 012 5 013 30 g JLRTN 014 43 32 It seems reasonable that either a tall narrow box or a short flat box could be formed having the desired volume Because the taller box is preferred larger initial estimates of the height are reasonable However heights greater than 2 decimeters are not physically possible because the metal is only 4 decimeters wide Initial estimates of 1 and 2 decimeters are therefore appropriate Find the desired height Keystrokes Display g P R Run mode 1 ENTER 1 0000 KY 2 2 Initial estimates f SOLVE 3 1 5000 The desired height RY 1 5000 Previous estimate RY 0 0000 f x at root Section 13 Finding the Roots of an Equation 191 By making the height 1 5 decimeters a 5 0x1 0x1 5 decimeter box is specified If you ignore the upper limit on the height and use initial estimates of 3 and 4 decimeters still less than the width you will obtain a height of 4 2026 decimeters a root that is physically meaningless If you use small initial estimates such as 0 and 1 decimeter you will obtain a height of 0 2974 decimeter producing an undesirably Graph of f x short flat bo
146. ed by the letter of the matrix specified in Rj This is not actually a matrix operation only the letter in the matrix descriptor is used Conditional Tests on Matrix Descriptors Four conditional tests x 0 TEST 0 x 0 TEST 5 x y and TEST 6 x y can be performed with matrix descriptors in the X and Y registers Conditional tests can be used to control program execution as described in section 8 If a matrix descriptor is in the X register the result of x 0 will be false and the result of TEST 0 will be true regardless of the element values in the matrix If matrix descriptors are in the X and Y registers when TEST 5 or TEST 6 conditional test is performed x and y are equal if the same descriptor is in the X and Y registers and not equal otherwise The comparison is made between the descriptors themselves not between the elements of the specified matrices Other conditional tests can t be used with matrix descriptors Stack Operation for Matrix Calculations During matrix calculations the contents of the stack registers shift much like they do during numeric calculations For some matrix calculations the result is stored in the result matrix The arguments one or two descriptors or numbers in the X register or the X and Y registers are combined by the operation and the descriptor of the result matrix is placed in
147. eletions After an insertion the display will show the instruction you just added After a deletion the display will show the line prior to the deleted now nonexistent one If all space available in memory is occupied the calculator will not accept any program instruction insertions and Error 4 will be displayed Initializing Calculator Status The contents of storage registers and the status of calculator settings will affect a program if the program uses those registers or depends on a certain status setting If the current status is incorrect for the program being run you will get incorrect results Therefore it is wise to clear registers and set relevant modes either just prior to running a program or within the program itself A self initializing program is more mistake proof but it also uses more program lines Calculator initializing functions are f CLEAR L2 L CLEAR PRGM f CLEAR REG 9 DEG 9 RAD 9 GRD L9 SF and 9 CF Problems It is good programming technique to avoid using identical program labels This shouldn t be hard since the HP 15C provides 25 different labels To ensure against duplication of labels you can clear program memory first 1 The following program is used by the manager of a savings and loan company to compute the future values of savings acco
148. entire display like CLx Otherwise it is neutral For a further discussion of the stack refer to appendix B Section 3 The Memory Stack LAST X and Data Storage 37 Keys g Clx Order of Entry and the ENTER Key An important aspect of two number functions is the positioning of the numbers in the stack To execute an arithmetic function the numbers should be positioned in the stack in the same way that you would vertically position them on paper For example 98 98 98 98 215 15 x15 15 As you can see the first or top number would be in the Y register while the second or bottom number would be in the X register When the mathematics operation is performed the stack drops leaving the result in the X register Here is how a subtraction operation is executed in the calculator The same number positioning would be used to add 15 to 98 multiply 98 by 15 or divide 98 by 15 38 Section 3 The Memory Stack LAST X and Data Storage Nested Calculations The automatic stack lift and stack drop make it possible to do nested calculations without using parentheses or storing intermediate results A nested calculation is solved simply as a series of one and two number operations Almost every nested calculation you are likely to encounter can be done using just the four stack registers It is usually wisest to begin the calculation at the innermost num
149. ents Individually Storing a Number in All Elements of a Matrix ecee Matrix Operations Matrix Descriptors 0 0 0 eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeees The Result Matrix 116 116 119 120 120 120 121 121 121 124 124 124 125 128 129 130 130 131 131 131 132 133 133 135 137 138 140 141 142 142 143 143 145 147 147 147 148 Copying a Matrix Contents One Matrix Operations cceeeeseesnetececeeeeeenenteneeeeeens Scalar Operations Arithmetic Operations cccceseesseesteseteenecnteeeaeeeneens Matrix Multiplication Solving the Equation AX Buu cecccccccceesteeeteeeesseeeeseeees Calculating the Residual c ccccccesseceeseeesseeeesteeeneees Using Matrices in LU Form cc eeeeeeeeeeeseeeeenenentnenenenenes Calculations With Complex Matrices cccccessseeesteeeees Storing the Elements of a Complex Matrix ceceeeee The Complex Transformations Between Z and Z Inverting a Complex Matrix ceceeseeteceteceneeeeeeeeneeees Multiplying Complex Matrices ossee Solving the Complex Equation AX B esce Miscellaneous Operations Involving Matrices 1ceeeee Using a Matrix Element With Register Operations Using Matrix Descriptors in the Index Register Conditional Tests on Matrix Descriptors ccceeeereees Stack Operation for Matrix Calculations cecceeceereeenee Using M
150. ept X y intercept slope Keys f L R xsy The slope and y intercept of the least squares line of the accumulated data are calculated using the equations shown in appendix A Section 4 Statistics Functions 55 Example Find the y intercept and slope of the linear approximation of the data and compare to the plotted data on the graph below Grain Yield metric tons hectare 7 50 6 50 0 20 40 60 80 Nitrogen Application kg hectare Keystrokes Display f JIL R 4 86 y intercept of the line 0 04 Slope of the line Linear Estimation and Correlation Coefficient When you press f f r the linear estimate y is placed in the X register and the correlation coefficient r is placed in the Y register To display r press X y 56 Section 4 Statistics Functions Linear Estimation With the statistics accumulated an estimated value for y denoted y can be calculated by keying in a proposed value for x and pressing _JL r An Estimated value for x denoted x can be calculated as follows 1 Press f JL RJ 2 Key in the known y value 3 Press xy x y Correlation Coefficient Both linear regression and linear estimation presume that the relationship between the x and y data values can be approximated by a linear function The correlation coefficient r is a deter
151. er limit of integration so the calculator can calculate the integral of a function that is undefined there Only when the endpoints of the interval of integration are extremely close together or the number of sample points is extremely large does the algorithm evaluate the function at the limits of integration 200 Section 14 Numerical Integration Accuracy of The accuracy of the integral of any function depends on the accuracy of the function itself Therefore the accuracy of an integral calculated using is limited by the accuracy of the function calculated by your subroutine To specify the accuracy of the function set the display format so that the display shows no more than the number of digits that you consider accurate in the function s values If you specify fewer digits the calculator will compute the integral more quickly but it will presume that the function is accurate to only the number of digits specified in the display format We ll show you how you can determine the accuracy of the calculated integral after we say another word about the display format You ll recall that the HP 15C provides three types of display formatting FIX J LSCIJ and ENG Which display format should be used is largely a matter of convenience since for many integrals you ll get about the same results using any of them provided that the number of digits is specified correctly considering the magnitude of the function Becaus
152. er the function has any quick wiggles relative to the interval of integration If you re not familiar with the function and you have reason to suspect that it may cause problems you can quickly plot a few points by evaluating the function using the subroutine you wrote for that purpose If for any reason after obtaining an approximation to an integral you have reason to suspect its validity there s a very simple procedure you can use to verify it subdivide the interval of integration into two or more adjacent subintervals integrate the function over each subinterval then add the resulting approximations This causes the function to be sampled at a brand new set of sample points thereby more likely revealing any previously hidden spikes If the initial approximation was valid it will equal the sum of the approximations over the subintervals Conditions That Prolong Calculation Time In the preceding example page 251 you saw that the algorithm gave an incorrect answer because it never detected the spike in the function This happened because the variation in the function was too quick relative to the width of the interval of integration If the width of the interval were smaller you would get the correct answer but it would take a very long time if the interval were still too wide For certain integrals such as the one in that example calculating the integral may be unduly prolonged because the width of the interval of integration is
153. es are placed in the X and Z registers A function value of zero is the expected result However a nonzero function value is also acceptable because it indicates that the function s graph apparently crosses the x axis within an infinitesimal distance from the calculated root In most such cases the function value will be relatively close to Zero The number in the T register is the same number that was left in the Y register by the final execution of your function subroutine Generally this number is not of interest Appendix D A Detailed Look at 227 Special consideration is required for a different type of situation in which finds a root with a nonzero function value If your function s graph has a discontinuity that crosses the x axis specifies as a root an x value adjacent to the discontinuity This is reasonable because a large change in the function value between two adjacent values of x might be the result of a very rapid continuous transition Because this cannot be resolved by the algorithm the root is displayed for you to interpret A function may have a pole where its magnitude approaches infinity If the function value changes sign at a pole the corresponding value of x looks like a possible root of your equation just as it would for any other discontinuity crossing the x axis However for such functions the function value placed into the Z register when that root is found will be relatively large I
154. essive approximations becomes sufficiently small In other words the algorithm samples the function at increasing numbers of sample points until it has sufficient information about the function to provide an approximation that changes insignificantly when further samples are considered 256 Appendix E A Detailed Look at If the interval of integration were 0 10 so that the algorithm needed to sample the function only at values where it was interesting but relatively smooth the sample points after the first few iterations would contribute no new information about the behavior of the function Therefore only a few iterations would be necessary before the disparity between successive approximations became sufficiently small that the algorithm could terminate with an approximation of a given accuracy On the other hand if the interval of integration were more like the one shown in the graph on page 252 most of the sample points would capture the function in the region where its slope is not varying much The few sample points at small values of x would find that values of the function changed appreciably from one iteration to the next Consequently the function would have to be evaluated at additional sample points before the disparity between successive approximations would become sufficiently small In order for the integral to be approximated with the same accuracy over the larger interval as over the smaller interval the density of th
155. estimated as an absolute uncertainly FIX display format may be inappropriate to use leading to peculiar results when you are integrating a function whose magnitude and uncertainty have extremely small values within the interval of integration Likewise SCI display format may be inappropriate to use also leading to peculiar results if the magnitude of the function becomes much smaller than its uncertainty If the results of calculating an integral seem strange It may be more appropriate to calculate the integral in the alternate display format Appendix E A Detailed Look at 249 Conditions That Could Cause Incorrect Results Although the L4 algorithm in the HP 15C is one of the best available in certain situations it like nearly all algorithms for numerical integration might give you an incorrect answer The possibility of this occurring is extremely remote The s algorithm has been designed to give accurate results with almost any smooth function Only for functions that exhibit extremely erratic behavior is there any substantial risk of obtaining an inaccurate answer Such functions rarely occur in problems related to actual physical situations when they do they usually can be recognized and dealt with in a straightforward manner he As discussed on page 240 the L algorithm samples the function f x at various values of x within th
156. esult matrix page 152 155 For two matrices multiplies inverse of matrix in X by matrix in Y For only one matrix if matrix in Y divides all elements of matrix by scalar in X if matrix in X multiplies each element of inverse of matrix by the scalar in Y Stores in result matrix pages 152 155 CHS changes sign of all elements in matrix specified in X register page 150 MATRIX 0 through 9 Matrix operations MATRIX 0 Dimensions all matrices to 0x0 page 143 MATRIX 1 Sets row and column numbers in Ro and R to 1 page 143 MATRIX 2 Complex transform Z to Z page 164 MATRIX 3 inverse Function Summary and Index complex transform Z to Z page164 MATRIX 4 Transpose X to X page 150 MATRIX 5 Transpose multiply Y and X to Y X page 154 MATRIX 6 Calculates residuals in result matrix page 159 MATRIX 7 Calculates row norm of matrix specified in X register page 150 MATRIX 8 Calculates Frobenius norm of matrix specified in X register page 150 MATRIX 9 Calculates determinant of matrix specified in X register also does LU decomposition of the matrix page 150 Cy x Transforms matrix stored in partitioned form Z to complex form Z9 page 162 Pyx Transforms matrix stored in complex form Z 9 to 265 partitioned form Z 3 page 162
157. evant results The statistics of the data are compiled as follows Register Contents R n Number of data points accumulated n also appears in the X register R3 X Summation of x values R4 Ex Summation of squares of x values Rs Xy Summation of y values R6 Ly Summation of squares of y values R7 xxy Summation of products of x and y values You can recall any of the accumulated statistics to the display X register by pressing and the number of the data storage register containing the desired statistic If you press RCL 2 Xy and Xx will be copied simultaneously from R and R respectively into the X register and the Y register respectively The sequence RCL 2 lifts the stack twice if stack lift is enabled once if not and then enables stack lift Example Agronomist Silas Farmer has developed a new variety of high yield rice and has measured the plant s yield as a function of fertilization Use the 2 function to accumulate the data below to find the values for Xx Xx Ly Ly and Lxy for nitrogen fertilizer application x versus grain yield y Section 4 Statistics Functions 51 X MIEROGENA PLUED 0 00 20 00 40 00 60 00 80 00 kg per hectare x GRAIN YIELD Y metric tons per 4 63 4 78 6 61 7 21 7 78 hectare y A hectare equals 2 47 acres
158. ex mode There will be times however when you will need Complex mode to perform certain operations on real numbers such as 5 Without Complex mode such as operation would result in an Error 0 improper math function To activate Complex mode at any time and without disturbing the stack contents set flag 8 before executing the function in question Example The arc sine sin of 2 404 normally would result in an Error 0 Assuming 2 404 in the X register the complex value arc sin 2 404 can be calculated as follows Keystrokes Display g LSF 8 Activates Complex Mode 9 J SINJ 1 5708 Real part of arc sin 2 404 f D hold 1 5239 Imaginary part of arc sin 2 404 release 1 5708 Display shows real part again when i is released Polar and Rectangular Coordinate Conversions In many applications complex numbers are represented in polar form sometimes using phasor notation However the HP 15C assumes that any complex numbers are in rectangular form Therefore any numbers in polar or phasor form must be converted to rectangular form before performing a function in Complex mode i Pressing Lf Re Im twice will accomplish the same thing The sequence C CI is not used because it would combine any numbers in the real X and Y registers into a single complex number 134 Section 11 Calculating With Complex Numbers mo _ i0 r cos
159. f CLEAR PRGM 000 001 f LBL 9 002 9 RAD 003 STO 0 004 x57 005 STO 0 006 GSB 3 007 CHS 008 x57 009 GSB 3 010 L 011 RCL 0 012 9 RTN SUBROUTINE 013 f LBL 3 014 9 x 015 LG LSTx 016 LSIN 017 L 018 9 RIN Section 9 Subroutines 103 Not programmable Start main program Radians mode Stores x2 in Ro Brings x into X x into Y Qo x in Ro Transfer to subroutine 3 with x Return from subroutine 3 yL Brings x into X register Transfer to subroutine with x2 Return from subroutine 3 y2 y Recalls x2 x from Ro and calculates y2 y X2 x1 Program end return to line 000 Start subroutine 3 2 x Recall x sin x x sin x which equals y Return to origin in main program Calculate the slope for the following values of x and xz 0 52 1 25 1 1 0 81 0 98 Remember to use routine with a digit label GSB 9 rather than Lf Answers 1 1507 0 8415 1 1652 9 when addressing a 104 Section 9 Subroutines Example Nesting The following subroutine labeled 4 calculates the value of the expression yx y z f as part of a larger calculation in a larger program The subroutine calls upon ano
160. f the pole occurs at a value of x that is exactly represented with 10 digits the subroutine may try that value and halt prematurely with an error indication In this case the operation will not be completed Of course this may be avoided by the prudent use of a conditional statement in your subroutine Example In her analysis of the stresses in a structural component design consultant Lucy I Beame has determined that the shear stress can be expressed as g 3x 45x 350 for0 lt x lt 10 1000 for 10 lt x lt 14 where Q is the shear stress in newtons per square meter and x is the distance from one end in meters Write a subroutine to compute the shear stress for any value of x Use to find the location of zero shear stress 228 Appendix D A Detailed Look at SOLVE Solution The equation for the shear stress for x between 0 and 10 is more efficiently programmed after rewriting it using Horner s method Keystrokes g P R f LBL 2 1 0 g ilx lt y GTO 9 9 LCLx 3 x 4 5 x x 3 5 0 g RIN f LBL 9 EEX 3 g RTN Display 000 001 42 21 2 002 1 003 0 004 43 10 005 22 9 006 43 35 007 3 008 20 009 4 010 5 011 30 012 20 013 20 014 3 015 5 016 0 017 40 013 43 32 019 42 21 9 020 26 021 3 022 43 32 Q 3x 45 x 350 for0 lt x lt 10
161. formation constructively 230 Appendix D A Detailed Look at If the algorithm terminates its search near a local minimum of the function s magnitude clear the Error 8 display and observe the ffx numbers in the X Y and Z registers by rolling down the stack If the value of the x function saved in the Z register is relatively close to zero it is possible that a root of your equation has been found the number returned in the X register may be a 10 digit number very close to a theoretical root You can explore this potential minimum further by rolling the stack until the returned estimates are back in the X and Y registers and then executing SOLVE again using these numbers as initial estimates If an actual minimum has been found Error 8 will again be displayed and the number in the X register will be approximately the same as before but possibly closer to the actual location of the minimum Of course you may deliberately use to find the location of a local minimum of the function s magnitude However in this case you must be careful to confine the search in the region of the minimum Remember tries hard to find a zero of the function If the algorithm stops searching and displays Error 8 because it is working on a horizontal asymptote when the value of the function is essentially constant for a large range of x the estimates in X and Y registers usually are significantly different from each other The number i
162. g format displays numbers in an engineering notation format in a manner similar to SCI except In engineering notation the first significant digit is always present in the display The number you key in after f ENG specifies the number of additional digits to which you want to round the display e Engineering notation shows all exponents in multiples of three Keystrokes Display 012345 0 012345 f ENG 12 03 Rounds to the first digit after 1 the leading digit f J ENG 3 12 35 03 10 Lx 123 5 03 Decimal shifts to maintain multiple of three in exponent f FIX 4 0 1235 Usual FIX 4 format Therefore the display shows no distinction among SCI 7 8 and 9 unless the number rounded up is a 9 which carries a 1 over into the next higher decimal place 60 Section 5 The Display and Continuous Memory Mantissa Display Regardless of the display format the HP 15C always internally holds each number as a 10 digit mantissa and a two digit exponent of 10 For example z is always represented internally as 3 141592654x10 regardless of what is in the display When you want to view the full 10 digit mantissa of a number in the X register press f CLEAR PREFIX To keep the mantissa in the display hold the PREFIX key down Keystrokes Display 9 La 3 1416 f CLEAR PREFIX hold
163. gh conditional and unconditional branching As we step through the fundamentals of programming we ll rework the falling object program illustrated in the Problem Solver page 14 Loading a Program Program Mode Press 9 LP R program run to set the calculator to Program mode PRGM annunciator on Functions are stored and not executed when keys are pressed in Program mode Keystrokes Display g LP R 000 Switches to Program mode PRGM annunciator and line number 000 displayed 66 Section 6 Programming Basics 67 Location in Program Memory Program memory and therefore the calculator s position in program memory is demarcated by line numbers Line 000 marks the beginning of program memory and cannot be used to store an instruction The first line that contains an instruction is line 001 Program lines other than 000 do not exist until instructions are written for them You can start a program at any existent line designated nnn but it is simplest and safest to start an independent program as opposed to a subroutine at the beginning of program memory As you write any existing program lines will be preserved and bumped down in program memory Press GTO CHS 000 Gn Program or Run mode to move to line 000 without recording the GTO statement In Run mode f CLEAR PRGM will also reset the calculator to line 000 without clearing program memory
164. gral exactly but rather only approximates it The 3 HP 15C places the uncertainty of an integral s approximation in the Y register at the same time it places the approximation in the X register To determine the accuracy of an approximation check its uncertainty by pressing XS YJ Example With the display format set to SCI 2 calculate the integral in the expression for J from the example on page 197 Keystrokes Display 0 0 0000 Key lower limit into Y register gJ a 3 1416 Key upper limit into X register g RAD 3 1416 If not already in Radians mode f LSCI 2 3 14 00 Set display format to SCI 2 f wji 1 3 00 Integral approximated in SCI 2 8 X 1 8 Uncertainty of SCI 2 8 03 approximation The integral is 1 38 0 00188 Since the uncertainty would not affect the approximation until its third decimal place you can consider all the displayed digits in this approximation to be accurate In general though it is difficult to anticipate how many digits in an approximation will be unaffected by its uncertainty This depends on the particular function being integrated the limits of integration and the display format No algorithm for numerical integration can compute the exact difference between its approximation and the actual integral But the algorithm in the HP 15C estimates an upper boun
165. h line 000 if necessary and resumes execution at the first line containing the proper label Looping If a GTO instruction specifies a label at a lower numbered line that is a prior line the series of instructions between the GTO and the label will be executed repeatedly possibly indefinitely The continuation 90 Section 8 Program Branching and Controls 91 of this loop can be controlled by a conditional branch an R S instruction written into the loop or simply by pressing any key during execution which stops the program 015 f LBL 7 T bass Conditional Tests Another way to alter the sequence of program execution is by a conditional test a true false test which compares the number in the X register either to zero or to the number in the Y register The HP 15C provides 12 different tests two explicit on the keyboard and 10 others accessible using 9 TEST n 1 Direct 9 x lt Y and 9 x 0 2 Indirect 9 TEST n n Test n Test 0 x 0 5 X y 1 x gt 0 6 x y 2 x lt 0 7 X gt y 3 x20 8 X lt y 4 xs0 9 x2y Four of the conditional tests can also be used for complex values as explained in section 11 on page 132 92 Section 8 Program Branching and Controls Following a conditional test program execution follows the Do if True Rule it proceeds sequentially if the
166. he interval of integration Basically the more rapid the variation in the function or its derivatives and the lower the order of such rapidly varying derivatives the less quickly will the _ algorithm terminate and the less reliable will the resulting approximation be lt Appendix E A Detailed Look at 253 Note that the rapidity of variation in the function or its low order derivatives must be determined with respect to the width of the interval of integration With a given number of sample points a function f x that has three fluctuations can be better characterized by its samples when these variations are spread out over most of the interval of integration than if they are confined to only a small fraction of the interval These two situations are shown in the next two illustrations Considering the variations or fluctuations as a type of oscillation in the function the criterion of interest is the ratio of the period of the oscillations to the width of the interval of integration the larger this ratio the more quickly the algorithm will terminate and the more reliable will be the resulting approximation f x l Calculated integral of this function will be accurate l I l tix l Calculated integral of this function may be inaccurate 254 Appendix E A Detailed Look at In many cases you will be familiar enough with the function you want to integrate that you ll know wheth
167. he display If you are keying in both numbers remember that they must be separated by ENTER or any other function like 9 INT or x that terminates digit entry For a two number function the first value entered is considered the y value because it is placed into the Y register for memory storage The second value entered is considered the x value because it remains in the display which is the X register The arithmetic operators L J LX J and are the four basic two number functions Others are given below The Power Function Pressing Y calculates the value of y raised to the x power The base number y is keyed in before the exponent x To Calculate Keystrokes Display 214 2 ENTER 1 4 y 2 6390 os 2 ENTER 1 4 CHS 0 3789 2 2 CHS ENTER 3 DF 8 0000 37 or 243 2 ENTER 3 x y 1 2599 Percentages The percentage functions and A J preserve the value of the original base number along with the result of the percentage calculation As shown in the example below this allows you to carry out subsequent calculations using the base number and the result without re entering the base number Percent The _ function calculates the specified percentage of a base numbe
168. hen the new function fx x a will not approach zero in this region if a is a simple root of j x 0 You can use this information to eliminate a known root Simply 234 Appendix D A Detailed Look at SOLVE add a few program lines at the end of your function subroutine These lines should subtract the known root to 10 significant digits from the x value and divide this difference into the function value In many cases the root will be a simple one and the new function will direct SOLVE away from the known root On the other hand the root may be a multiple root A multiple root is one that appears to be present repeatedly in the following sense at such a root not only does the graph of f x cross the x axis but its slope and perhaps the next few higher order derivatives also equals zero If the known root of your equation is a multiple root the root is not eliminated by merely dividing by the factor described above For example the equation fx x x a 0 has a multiple root at x a with a multiplicity of 3 This root is not eliminated by dividing f x by x a But it can be eliminated by dividing by x a Example Use deflation to help find the roots of 60x 944x 3003x 6171x 2890 0 Using Horner s method this equation can be rewritten in the form 60x 944 x 3003 x 6171 x 2890 0 Program a subroutine that evaluates the polynomial
169. histle in the firehouse steeple The sound level at the firehouse door 3 2 meters from the whistle is 138 decibels Write a program to find the sound level at various distances from the whistle Use the equation L Ly 20 log 7 rg where Lois the known sound level 138 db at a point near the source ro is the distance of that point from the source 3 2 m L is the unknown sound level at a second point and r is the distance of the second point from the source in meters What is the sound level at 3 km from the source r 3 km A possible keystroke sequence is g ILP R Lf J LBLLC 3 2 L9G J LOG 20 Lx CHS 138 LO JLRTIN L9 JLP R J taking 15 program lines and 15 bytes of memory This problem can be solved in a more general way by removing the specific values 3 2 and 138 from the program and instead recalling the Lo and ro values from storage registers or by removing 3 2 and 138 and loading Lo r and r into the stack before execution Lo ENTER r ENTER ro Answer for r 3 km L 78 5606 db A typical large tomato weighs about 200 grams of which about 188 g 94 are water A tomato grower is trying to produce tomatoes of lower percentage water Write a program to calculate the percent change in water content of a given tomato compared to the typical tomato Use a programmed stop to enter the water weigh
170. imes called the sine integral of the form sio as Section 14 Numerical Integration Find Si 2 199 Key in the following subroutine to evaluate the function f x sin x x Keystrokes Display g JLP R 000 f_ LBL 2 SIN 002 003 004 9g RTN 005 001 42 21 2 23 34 10 43 32 Program mode LBL Begin subroutine with a instruction Calculate sin x Since a value of x will be placed in the Y register by the algorithm before it executes this subroutine the X y operation at this point will return x to the X register and move sin x to the Y register Divide sin x by x Now key the limits of integration into the X and Y registers In Radians mode press f Keystrokes Display g JLP R 0 4401 O ENTER 0 0000 2 2 g RAD 2 0000 f LA 2 1 6054 2 to calculate the integral Run mode Key lower limit into Y register Key upper limit into X register If not already in Radians mode Si 2 If the calculator attempted to evaluate f x sin x x at x 0 the lower limit of integration it would terminate with Error 0 in the display signifying an attempt to divide by zero and the integral could not be calculated However the algorithm normally does not evaluate functions at eith
171. iminate exponents greater than 1 Ax Be Cre Dx E Ax Bx Cx D x E Ax Bx C x D x E Ax B x C x D x E 80 Section 6 Programming Basics Example Write a program for 5x 2x as Sx 2 x x x then evaluate forx 7 Keystrokes g P R f LBL B 5 x 2 x x x g LRTN g P R 7 ENTER ENTER ENTER f LB Display 000 001 42 21 12 002 003 004 005 006 007 008 009 7 0000 43 5 20 2 40 20 20 20 32 12 691 0000 Nonprogrammable Functions Assumes position in memory is line 000 If it is not clear program memory Sx 5x 2 5x 2 x 5x 2 x 5x 2 x Returns to Run mode Prior result remains in display Loads the stack X Y Z and T registers with 7 When the calculator is in Program mode almost every function on the keyboard can be recorded as an instruction in program memory The following functions cannot be stored as instructions in program memory CLEAR CLEAR G USER mli lax A lac PREFIX g BST SST PRGM 9g MEM lt g P R ONJ _ GTO CHS nnn ON Section 6 Programming Basics 81 Problems 1 The village of Sonance has installed a 12 o clock w
172. imits of the variable You should recognize this restriction and interpret the results accordingly Section 13 Finding the Roots of an Equation 189 If you have some knowledge of the behavior of the function f x as it varies with different values of x you are in a position to specify initial estimates in the general vicinity of a zero of the function You can also avoid the more troublesome ranges of x such as those producing a relatively constant function value or a minimum of the function s magnitude Example Using a rectangular piece of sheet metal 4 decimeters by 8 decimeters an open top box having a volume of 7 5 cubic decimeters is to be formed How should the metal be folded A taller box is preferred to a shorter one Solution You need to find the height of the box that is the amount to be folded up along each of the four sides that gives the specified volume If x is the height or amount folded up the length of the box is 8 2x and the width is 4 2x The volume V is given by V 8 2x 4 2x x By expanding the expression and then using Horner s method page 79 this equation can be rewritten as V 4 x 6 x4 8 x To get V 7 5 find the values of x for which Six 4 x 6 x 4 8 x 75 0 The following subroutine calculates f x Keystrokes Display g P R 000 Program mode f LBL 3 001 42 21 3 Label 6 002 6 Assumes stack loaded with x 190 S
173. in for programming The optional omission of the f keystroke after another prefix key is explained on page 78 Abbreviated Key Sequences Section 6 Programming Basics 77 Keystrokes Display 1 Lf DIM IG 1 0000 R and Ro allocated for data storage R to Res available for programming and advanced functions 19 f DIM 19 0000 Original allocation Rj9 R and below for data storage R29 to Res for programming and advanced functions RCL DIM G 19 0000 Displays the current highest data register The DIM and MEM memory status functions are described in detail in appendix C Keep in mind that an error message will result given the above memory configuration if 1 You try to address a register higher than Rj R which initially is the highest register allocated to data storage Error 3 2 You have 322 occupied program bytes and try to load more program lines Error 4 3 You try to run an advanced function with insufficient available memory Error 10 Program Boundaries End Not every program needs to end with a RTN or LR S instruction If you are at the end of occupied program memory there is an automatic RIN J instruction so you do not need to enter one This can save you one line of memory On the other hand a program can end by simply transferring execution to another ro
174. into the Y register and a new copy of the constant will be generated in the T register If the variables change as in the preceding example be sure and clear the display before entering the new variable This disables the stack so that the arithmetic result will be written over and only the constant will occupy the rest of the stack If you do not have different arguments that is the operation will be performed upon a cumulative number then do not clear the display simply repeat the arithmetic operation Example A bacteriologist tests a certain strain of microorganisms whose population typically increases by 15 each day a growth factor of 1 15 If she starts with a sample culture of 1000 what will be the bacteria population at the end of each day for four consecutive days Keystrokes Display 1 15 1 15 Growth factor ENTER ENTER 1 1500 Filling the stack ENTER 1000 1 000 Initial culture size 42 Section 3 The Memory Stack LAST X and Data Storage Keystrokes Display x 1 150 0000 Population at the end of day 1 x 1 322 5000 Day2 x 1 520 8750 Day3 x 1 749 0063 Day4 Storage Register Operations When numbers are stored or recalled they are copied between the display X register and the data storage registers At power up initial turn on or Continuous Memory reset the HP 15C has 21 directly accessible storage registers Ro through Ro R o through R o and the Index
175. ion 11 Calculating With Complex Numbers Conditional Tests For programming the four conditional tests below will work in the complex sense x 0 and TEST 0 compare the complex number in the real and imaginary X registers to 0 Oi while TEST 5 and TEST 6 compare the complex numbers in the real and imaginary X and Y registers All other conditional tests besides those listed below ignore the imaginary stack x 0 TEST 0 x 0 TEST 5 x y TEST 6 x y Example Complex Arithmetic The characteristic impedance of a ladder network is given by an equation of the form A Z e where A and B are complex numbers Find Zo for the hypothetical values A 1 2 4 7iand B 2 7 3 2i Keystrokes Display 1 2 ENTER 4 7 LI 1 2000 Enters A into real and imaginary X registers 2 7 ENTER 3 2 Lf LI 2 7000 Enters B into real and imaginary X registers moving A into real and imaginary Y registers 1 0428 Calculates A B VX 1 0491 Calculates Zo and displays real part f amp hold 0 2406 Displays imaginary part of Zp while G is held down release 1 0491 Again displays real part of Zo Section 11 Calculating With Complex Numbers 133 Complex Results from Real Numbers In the preceding examples the entry of complex numbers had ensured the automatic activation of Compl
176. ion 7 Program Editing ivesitwatomsiisrosndiarvormnsyanris The Mechanics ccccecsceceeseeceeececseeeeeeeceaeeecaeeesaeeenaees Moving to a Line in Program Memory nsss Deleting Program Lines c ccccccceesseceesceesteceeseeeneeeeeas Inserting Program Lines cceeeeeeeeeeeeeeeeeneeeneneeeneeenes Examples tierce cai h poh detent eee ante te es Further Information ccccccecsseceeeseceeseeeseeeecseeeseeenseeees Single Step Operations cccccecceceeeeeesseeeesseeeeseeenteeess hime Position eriein niina bee cathe dened cette ecu entne nee Insertions and Deletions cccceesseceeececsteeeeseeeeeeeeenes Initializing Calculator Status cccceccseeeseeesseeeeseeeteeeeees Problems si 228i Phat iies Sevihetie aR aac Poise sees eels Section 8 Program Branching and Controls 06 The Mechanics 3 28 oan ieee Pa ae a ees Branching eeraa en e hte eee eS Conditional T ste e aS Flags Examples Contents Example Branching and Looping n se Example Flags Further Information GoTo Looping Conditional Branching Flags The System Flags Flags 8 and 9 ecceeesseesseeesteeeeeeees Section 9 Subroutines The Mechanics GoTo Subroutine and Return o Subroutine Limits Examples Further Information The Subroutine Return Nested Subroutines Section 10 The Index Register and Loop Control The I and Keys Direct Versus Indirect
177. ion of the loop control number thereby keeping count of the loop iterations It compares nnnnn to xxx the prescribed test value and exits the loop by skipping the ISG or ISG or less DSE the test value xxx The amount that nnnnn is incremented or decremented is specified by yy With these functions as opposed to the other conditional tests the rule is Skip if True False nnnnn lt xxx For ISG True nnnnn gt xxx instruction f ISG I GTO 1 zii l Y I I instruction r 7 7 exit loop v given nnnnn xxxyy increment nnnnn to nnnnn yy compare it to xxx and skip the next program line if the new value satisfies nnnnn gt xxx This allows you to exit a loop at this point when nnnnn becomes greater than xxx Section 10 The Index Register and Loop Control 111 False nnnnn gt xxx True nnnnn lt xxx instruction HDO loop i GTOJ 1 Y Instruction H dg exit loop For DSE given nnnnn xxxyy decrement nnnnn to nnnnn yy compare it to xxx and skip the next program line if the new value satisfies nnnnn lt xxx This allows you to exit a loop at this point when nnnnn becomes less than or equal to xxx For example loop iterations will alter these control numbers as follows
178. ire population you can easily do so simply add using the mean x of the data to the data before pressing _9 Ls The result will be the population standard deviation If you subsequently correct any of your accu mulated data values remember to delete the first mean value and add the corrected one 54 Section 4 Statistics Functions Example Calculate the standard deviation about the mean calculated above Keystrokes Display g jLs 31 62 Standard deviation about the mean nitrogen application X X 1 24 Standard deviation about the mean grain yield y Linear Regression Linear regression is a statistical method for finding a straight line that best fits a set of two or more data pairs thus providing a relationship between two or more data pairs thus providing a relationship between two variables By the method of least squares f J L RJ will calculate the slope A and y intercept B of the linear equation y Ax B 1 Accumulate the statistics of your data using the 2 key 2 Press f L RJ The y intercept B appears in the display X register The slope A is copied simultaneously into the Y register 3 Press xy to view A As is the case with the functions x and Ls L R causes the stack to lift two registers if it s enabled one if not T Z Y slope y interc
179. irective 2009 125 EC where applicable CE compliance of this product is valid if powered with the correct CE marked AC adapter provided by HP Compliance with these directives implies conformity to applicable harmonized European standards European Norms that are listed in the EU Declaration of Conformity issued by HP for this product or product family and available in English only either within the product documentation or at the following web site www hp eu certificates type the product number in the search field The compliance is indicated by one of the following conformity markings placed on the product For non telecommunications products and for EU harmonized telecommunications products such as Bluetooth within power class below 10mW For EU non harmonized telecommunications products If applicable a 4 digit notified XXXx body number is inserted between CE and Please refer to the regulatory label provided on the product The point of contact for regulatory matters is Hewlett Packard GmbH Dept MS HQ TRE Herrenberger Strasse 140 71034 Boeblingen GERMANY 286 Japanese Notice CORB TIABDAARHRE CT CORE RARE CER FOCEEBWELTETM COREP SIA PT LEY a gt RER TIELCEASNSL SME ES SHCTCEMHVET FRSA CH TIE LU RW BUELT RA VCCI B Korean Notice B2 J121 ol IE IARD AMVASSSS St IIS F 7 a Rsk ALS BHNOe F qo Gas asao eee Bet ASUC Disposal of
180. ith up to five matrices which are named A through E since they are accessed using the corresponding LA through E keys The HP 15C lets you specify the size of each matrix store and recall the values of matrix elements and perform matrix operations for matrices with real or complex elements A summary of matrix functions is listed at the end of this section A common application of matrix calculations is solving a system of linear equations For example consider the equations 3 8x 7 2x2 16 5 1 3x 0 9x2 22 1 for which you must determine the values of x and x2 These equations can be expressed in matrix form as AX B where 3 8 7 2 i 16 5 A gt e and B 13 09 x 22 1 The following keystrokes show how easily you can solve this matrix problem using your HP 15C The matrix operations used in this example are explained in detail later in this section First dimension the two known matrices A and B and enter the values of their elements from left to right along each row from the first row to the last Also designate matrix C as the matrix that you will use to store the result of your matrix calculation C X 138 Section 12 Calculating with Matrices 139 Keystrokes Display g CF 8 Deact
181. ithmic Functions Natural Logarithm Pressing g LN calculates the natural logarithm of the number in the display that is the logarithm to the base e Natural Antilogarithm Pressing of the number in the display that is raises e to the power of that number Common Logarithm Pressing logarithm of the number in the display that is the logarithm to the base 10 e calculates the natural antilogarithm g LOG calculates the common Common Antilogarithm Pressing 10 calculates the common antilogarithm of the number in the display that is raises 10 to the power of that number Keystrokes Display 45 9 LN 3 8067 Natural log of 45 3 4012 Le 30 0001 Natural antilog of 3 4012 12 4578 9 LOG 1 0954 Common log of 12 4578 3 1354 10 1 365 8405 Common antilog of 3 1354 Hyperbolic Functions Given x in the display X register Pressing Calculates f I HYP LSIN hyperbolic sine of x 9 J HYP JLSIN inverse hyperbolic sine of x f HYP COS hyperbolic cosine of x 9 J HYP COS inverse hyperbolic cosine of x f LHYP TAN hyperbolic tangent of x 9 HYP TAN inverse hyperbolic tangent of x Section 2 Numeric Functions 29 Two Number Functions The HP 15C performs two number math functions using two values entered sequentially into t
182. ity at 3 day intervals until a given limit is reached The formula for N the amount of radioisotope remaining after t days is N N 2 where k 8 days the half life of n and No is the initial amount The following program uses a loop to calculate the number of millicuries mci of isotope theoretically remaining at 3 day intervals of decay Included is a conditional test to check the result and end the program when radioactivity has fallen to a given value a limit The program assumes t the first day of measurement is stored in Ro No the initial amount of isotope is stored in R and the limit value for radioactivity is stored in Ro Keystrokes Display g P R 000 Program mode f CLEAR PRGM 000 Optional f LBL LA 001 42 21 11 ma loop returns to this ine RCL O 002 45 0 Recalls current t which changes with each loop f LPSE 003 42 31 Pauses to display t 8 004 8 k 005 10 006 16 tk 2 007 2 008 34 y 009 14 2 94 Section 8 Program Branching and Controls Keystrokes Display RCL x 1 010 45 20 1 Recall multiplication with the contents of R No yielding N the mci of I remaining after t days f LPSE 011 42 31 Pauses to display N RCL 2 012 45 2 Recalls limit value to X register g TEST 9 013 43 30 9 x gt y Tests whether limit value i
183. ivates Complex mode 2 ENTER DIM LA 2 0000 Dimensions matrix A to be 2x2 f MATRIX 1 2 0000 Prepares for automatic entry of matrix elements in User mode f USER 2 0000 Turns on the USER annunciator 3 8 STO LA A 1 1 Denotes matrix A row 1 column 1 A display like this appears momentarily as you enter each element and remains as long as you hold the letter key 3 8000 Stores a11 7 2 STO LA 7 2000 Stores a 1 3 STO A 1 3000 Stores a 9 CHS STO LA 0 9000 Stores dy 2 ENTER 1 f DIM 1 0000 Dimensions matrix B to B be 2xl 16 5 STO B 16 5000 Stores b41 22 1 CHS STO B 22 1000 Stores bo f_ RESULT C 22 1000 Designates matrix C for storing the result Using matrix notation the solution of the matrix equation AX B is where A X A B is the inverse of matrix A You can perform this operation by entering the descriptors for matrices B and A into the Y and X registers and then pressing A descriptor shows the name and dimensions of a matrix Note that if A and B were numbers you could calculate the answer in a similar manner 140 Section 12 Calculating with Matrices Keystrokes RCL MATRIX B RCL MATRIX LA Display running Enters descriptor for B the 2x1 constant matrix Enters descriptor for A the 2x2 coefficient matrix into the X register moving
184. ivating Complex Mode ccceeccecetseeesteeeeseeeeeeeees Complex Numbers and the Stack n e Entering Complex Numbers csccccseceeesteeeeseeeteeees Stack Lift in Complex Mode ccccesseeseseeesteeeeseeeeeeees Manipulating the Changing Signs Real and Imaginary Stacks 0 Clearing a Complex Number c ccceesseesseeeeseeeeees Entering a Real Number cc cccssceeeseeeteeeesteeeeeeeens Entering a Pure Imaginary Number 0 ccscceeeeeee Storing and Recalling Complex Numbers cccee Operations With Complex Numbers cccccceeeeseeeeees One Number Functions 0 ccccccceesseceeeeeceteeeeseeeseeenaes Two Number Functions c ccccccceesseceeseeesneeecsseeeeeeenaes Stack Manipulation Functions s s s reese Conditional Tests Complex Results from Real Numbers nsss Polar and Rectangular Coordinate Conversions 00 Problems For Further Information cccccccccccceseesssseceeeeeeeenesssaeeeeeees Section 12 Calculating With Matrices cceeeeeees Matrix Dimensions Dimensioning a Matrix ceceeeeesee cece eeeeenteeeeeeereneees Displaying Matrix Dimensions cseeeeeeeeeeeeeeeeeeeees Changing Matrix Dimensions 2330 n A wales Storing and Recalling Matrix Elements c cecceeseereeees Storing and Recalling All Elements in Order 00 Checking and Changing Matrix Elem
185. ix use of STO or RCL LA through E User mode automatically increments Ry row number or R column number for storage or recall of matrix elements page 144 GTO Go to Used with GD 269 a label designator listed above or _I to transfer the position of the calculator to the designated label If it is a program instruction program execution continues If itis not a program instruction only the position change occurs page 90 If a negative number is stored in Rj GTO will effect a transfer to a line number page 109 GTO CHS nnn Go to line number Positions calculator to the existing line number specified by nnn Not programmable page 82 GSB Go to subroutine Used with a label designator listed above or start the execution of a given labeled routine Can be used both in a program and from the keyboard in Run mode A LRTN instruction transfers execution back to the first line 270 following the page 101 BST Back step Moves calculator back one or more lines in program memory Also scrolls in Program mode Displays line number and contents of previous program line page 83 SST Single step In Program mode moves calculator forward one or more lines in program memory In Run mode
186. l example 198 199 Single stepping LSST 82 85 Skip if True rule 110 Slope finding the 54 SOLVE 180 181 accuracy 222 226 specifying 238 algorithm 182 187 188 220 222 230 231 conditions necessary for 221 222 constant function value with 187 189 execution time 238 illegal math routine with 187 188 initial estimates with 181 188 192 221 233 237 memory usage 193 nonzero minimum of function with 187 programmed 192 recursive use of 193 restrictions on 193 using as a conditional test 192 using functions with discontinuities 227 using functions with poles 227 using functions with several roots 233 238 with no root 186 188 192 229 Square root vx J 25 Squaring Lx 25 Stack contents with 197 202 drop 33 38 lift 33 36 38 44 209 211 p Subject Index 281 manipulation functions 33 34 in Complex mode 131 imaginary 120 125 used to access matrix elements 146 147 Stack disabling operations 210 Stack enabling operations 210 211 Stack movement 32 33 37 in matrix functions 174 176 with SOLVEJ 181 Standard deviation Ls _ 53 sample vs population 53 Star example 40 Statistics accumulation of data 2 49 Statistics correction of accumulated data Statistics functions combinations 47 correlation coefficient 55 linear estimation 55 linear regression 54
187. lay Control FIX Selects fixed point display mode page 58 SCI J Selects scientific notation display mode page 59 ENG Selects engineering notation display mode page 59 Mantissa Pressing f CLEAR PREFIX displays all 10 digits of the number in the X register as long as the PREFIX key is held down page 60 It also clears any partial key sequences page 19 Function Summary and Index Hyperbolic Functions HYP J SIN HYP COS HYP TAN Compute hyperbolic sine hyperbolic cosine or hyperbolic tangent respectively page 28 HYP LSIN J HYP COs HYP TAN Compute inverse hyperbolic sine inverse hyperbolic cosine or inverse hyperbolic tangent respectively page 28 Index Register Control Index register Rj Storage register for indirect program execution branching with GTO and GSB looping with ISG and indirect flag control and indirect display format control page 107 Also used to enter complex numbers and activate Complex mode page 121 G Indirect operations Used to address another storage register through R for purposes of storage recall storage arithmetic and program loop control page 107 Also used with DIM to allocate storage registers
188. lays descriptor of A matrix to be copied STO JIMATRIX A 2 3 Redimensions matrix B and copies A into B RCL MATRIX b 2 3 Displays descriptor of new B matrix B One Matrix Operations The following table shows functions that operate on only the matrix specified in the X register Operations involving a single matrix plus a number in another stack register are described under Scalar Operations page 151 150 Section 12 Calculating with Matrices One Matrix Operations Sign Change Inverse Transpose Norms Determinant Result in Eoee on Nias Effect on Result Keystroke s A Specified in 7 X register Matrix X register CHS No change Changes signof None all elements Vx Descriptor of None Inverse of Jx in result matrix specified matrix User Mode f IMATRIX 4 Descriptor of Replaced by None transpose transpose f_ MATRIX 7 Row norm of None None specified matrix f IMATRIX 8 Frobenius or None None Euclidean norm of specified matrix f MATRIX 9 Determinant of None t LU decomposi specified tion of specified matrix matrix The row norm is the largest sum of the absolute values of the elements in each row of the specified matrix The Frobenius of Euclidean norm is the square root of the sum of the squares of all elements in the specified matrix t Unless the result matrix is the same matrix s
189. lculated magnitude of f t is less than 0 5 meter Change your subroutine as follows Keystrokes Display g P R 000 Program mode GTO 006 006 30 Line before RTN instruction g LABS 007 43 16 Magnitude of f t 008 48 Accuracy 5 009 5 J TEST 7 DRO aaa tae ot Test for x gt y and return zero if accuracy gt magnitude 0 5 gt I f t 9 LCL J 011 43 35 9g TEST O 012 43 30 0 Test for x 0 and restore 9 J LSTx 013 43 36 f t if value is nonzero 226 Appendix D A Detailed Look at Execute SOLVE again Keystrokes Display g P R O ENTER 0 0000 1 1 f ENTER B 4 0681 RY 4 0681 RY 0 0000 Run mode Initial estimates The desired root A previous estimate of the root Value of modified f t at root After 4 0681 seconds the ridget is at a height of 107 0 5 meters This solution although different from the previous answer is correct considering the uncertainty of the height equation And this solution is found in just under half the time of the earlier solution Interpreting Results The numbers that SOLVE places in the X Y and Z registers help you evaluate the results of the search for a root of your equation Even when no root is found the results are still significant When finds a root of the specified equation the root and function valu
190. lculator at a moment when the display is blank In summary to obtain the current approximation to an integral follow the steps below 1 Press R S to halt the calculator preferably while the display is blank 2 When the calculator halts switch to Program mode to check the current program line e If that line contains the subroutine label return to Run mode and view the LAST X register step 3 258 Appendix E A Detailed Look at e If any other program line is displayed return to Run mode and single step LSST through the program until you reach a RTN instruction keycode 43 32 or line 000 if there is no LRTN Be sure to hold the SST key down long enough to view the program line numbers and keycodes 3 Press J LSTx to view the current approximation If you want to continue calculating the final approximation press the calculator For Advanced Information ej R S This refills the stack with the current x value and restarts The HP 15C Advanced Functions Handbook explores more esoteric aspects of he and its applications These topics include e Accuracy of the function to be integrated e Shortening calculation time e Calculating difficult integrals e Using 4 in Complex mode Appendix F Batteries Batteries The HP 15C is shipped with two 3 Volt
191. m any known zeros 192 Section 13 Finding the Roots of an Equation e Many functions exhibit special behavior when their arguments approach zero You can check your function to determine values of x for which any argument within your function becomes zero and then specify estimates at or near those values Although two different initial estimates are usually supplied when using SOLVE you can also use with the same estimate in both the X and Y registers If the two estimates are identical a second estimate is generated internally If your single estimate is nonzero the second estimate differs from your estimate by one count in the seventh significant digit If your estimate is zero 1x107 is used as the second estimate Then the root finding procedure continues as it normally would with two estimates Using SOLVE in a Program You can use the SOLVE operation as part of a program Be sure that the program provides initial estimates in the X and Y registers just prior to the operation The SOLVE routine stops with a value of x in the X register and the corresponding function value in the Z register If the x value is a root the program proceeds to the next line If the x value is not a root the next line is skipped Refer also to Interpreting Results on page 226 for a further explanation of roots Essentially the instruction tests whether the x value is a root and then proceeds according to the Do if True rule The program
192. matrix with the result of the operation if the result is a matrix such as a transpose or returns a number to the X register if the result is a number such as a row norm Before you perform an operation that uses the result matrix you must designate the result matrix Do this by pressing f RESULT followed by the letter key specifying the matrix If the descriptor of the intended result matrix is already in the X register you can press STO RESULT instead The designated matrix remains the result matrix until another is designated To display the descriptor of the result matrix press RCL RESULT When you perform an operation that affects the result matrix the matrix is automatically redimensioned to the proper size If this redimensioning would require more additional elements than there are available in matrix memory a maximum of 64 for all five matrices then the operation can t be performed This restriction can often be overcome by designating the result matrix to be one of the matrices being operated on However there are certain operations for which the result matrix can not be the same one as either of the matrices being operated on this is noted in the description of these operations The LU decomposition of a matrix A is another matrix in which is encoded a lower triangular matrix L and an upper triangular matrix U whose product LU equals matrix A po
193. mber of sample points is increased until successive iterations yield approximations that take into account the presence of the most rapid but characteristic fluctuations For example consider the approximation of e xe dx 0 Appendix E A Detailed Look at 251 Since you re evaluating this integral numerically you might think naively in this case as youll see that you should represent the upper limit of integration by 10 9 _ which is virtually the largest number you can key into the calculator Try it and see what happens Key in a subroutine that evaluates the function f x xe Keystrokes Display g LPR 000 Program mode f CBU 1 001 42 21 1 CHS 002 1 6 e 003 12 004 20 g LRTN 005 43 32 Set the calculator to Run mode Then set the display format to SCI 3 and key the limits of integration into the X and Y registers Keystrokes Display g JLP R J Run mode f JLSCI 3 Sets display format to SCI 3 O ENTER 0 000 00 Keys lower limit into Y register EEX 99 1 99 Keys upper limit into X register f 7 1 0 000 00 Approximation of integral The answer returned by the calculator is clearly incorrect since the actual integral of f x xe from 0 to is exactly 1 But the problem is not that you represented o by 10 since the actual integral of this functio
194. metric functions operate in the trigonometric mode you select Specifying a trigonometric mode does not convert any number already in the calculator to that mode it merely tells the calculator what unit of measure degrees radians or grads to assign a number for a trigonometric function Pressing g display Degrees are in decimal not minutes seconds form Pressing the display In Pressing g DEG sets Degrees mode No annunciator appears in the RAD sets Radians mode The RAD annunciator appears in Complex mode all functions except P values are in radians regardless of the trigonometric annunciator displayed g and gt R assume GRD sets Grads mode The GRAD annunciator appears in the display Continuous Memory will maintain the last trigonometric mode selected At power up initial condition or when Continuous Memory is reset the calculator is in Degrees mode Trigonometric Functions Given x in the display X register Pressing Calculates SIN sine of x g J SIN arc sine of x COS cosine of x g J COSs arc cosine of x TAN tangent of x 9 TAN arc tangent of x Before executing a trigonometric function be sure that the calculator is set to the desired trigonometric mode Degrees Radians or Grads Time and Angle Conversions Numbers representing time h
195. mination of how closely your data fit a straight line The range is 1 lt r lt 1 with l1 representing a perfectly negative correlation and 1 representing a perfectly positive correlation Note that if you do not key in a value for x before executing _f_ L r the number previously in the X register will be used usually yielding a meaningless value for 7 Example What if 70 kg of nitrogen fertilizer were applied to the rice field Predict the grain yield based on Farmer s accumulated statistics Because the correlation coefficient is automatically included in the calculation you can view how closely the data fit a straight line by pressing X after the y prediction appears in the display y Section 4 Statistics Functions 57 Keystrokes Display 70 L r 7 56 Predicted grain yield in tons hectare X y 0 99 The original data closely approximates a straight line Other Applications Interpolation Linear interpolation of tabular values such as in thermodynamics and statistics tables can be carried out very simply on the HP 15C by using the r function This is because linear interpolation is linear estimation two consecutive tabular values are assumed to form two points on a line and the unknown intermediate value is assumed to fall on that same line Vector Arithmetic The statistical accumulation functions can be used to perform vect
196. mitation imposed by the inaccuracy in the calculated function f x provide an exact answer Evaluating the function at an infinite number of sample points would take a very long time namely forever However this is not necessary since the maximum accuracy of the calculated integral is limited by the accuracy of the calculated function values Using only a finite number of sample points the algorithm can calculate an integral that is as accurate as is justified considering the inherent uncertainty in f x The algorithm at first considers only a few sample points yielding relatively inaccurate approximations If these approximations are not yet as accurate as the accuracy of f x would permit the algorithm is iterated that is repeated with a larger number of sample points These iterations continue using about twice as many sample points each time until the resulting approximation is as accurate as is justified considering the inherent uncertainty in f x 240 Appendix E A Detailed Look at 241 The uncertainty of the final approximation is a number derived from the display format which specifies the uncertainty for the function At the end of each iteration the algorithm compares the approximation calculated during that iteration with the approximations calculated during two previous iterations If the difference between any of these three approximations and the other two is less than the uncertainty tolera
197. month If you wanted to withdraw the monthly payments from a bank account yielding 6 per year compounded monthly which equals 0 5 per month how much must you deposit in the account at the start of the college years to fund monthly payments for the next 4 years The formula is 1 4 if payments are to be made V P IOo i each month in advance and the formula is i each month in arrears v p ea if payments are to be made V is the total value of the deposit you must make in the account P is the size of the periodic payment you will draw from the account i is the periodic interest rate here periodic means monthly since interest is compounded monthly and n is the number of compounding periods months The following program allows for either payment mode It assumes that before the program is run P is in the Z register n is in the Y register and i is in the X register 96 Section 8 Program Branching and Controls Keystrokes g P R f LBL B J CF O GTO 1 f LBL LE g SF O f LBL 1 STOJ1 1 X y CHS y CHS 1 RCL 1 g LF 0 g RINJ RCL 1 1 x J RTN Display 000 001 42 002 43 003 004 42 005 43 006 42 007 008 009 010 011 012 013
198. n A B you must enter the matrix descriptors in the order B A rather than in the order that they appear in the expression The value stored in each element of the result matrix is determined according to the usual rules of matrix multiplication For MATRIX 5 the matrix specified in the Y register isn t changed by this operation even though its transpose is used The result is identical to that obtained using MATRIX 4 transpose and X J This is the same order you would use if you were entering b and a for evaluating a b b a Section 12 Calculating with Matrices 155 For J the matrix specified in the X register is replaced by its LU decomposition The function calculates X Y using a more direct method than does Lx and X J giving the result faster and with improved accuracy Example Using matrices A and B from the previous example calculate C A B 1 2 3 13 5 A and B 4 5 9 7 9 17 Keystrokes Display RCL IMATRIX A 2 3 Recalls descriptor for matrix A A RCL MATRIX b 2 3 Recalls descriptor for matrix B B into X register moving matrix A descriptor into Y register f_ RESULT b 2 3 Designates matrix C as result matrix f_ MATRIX 5 Cc 3 3 Calculates A B and stores result in matrix C which is redimensioned to 3x3 The result matrix C
199. n the Z register is the value of the potential asymptote If you execute SOLVE again using as initial estimates the numbers that were returned in the X and Y registers a horizontal asymptote may again cause Error 8 but with numbers in the X and Y registers that will differ from the previous numbers The value of the function in the Z register would then be about the same as that obtained previously Appendix D A Detailed Look at 231 If Error 8 is displayed as a result of a search that is concentrated in a local flat region of the fi function the estimates in the X and Y registers 4 will be relatively close together or extremely small Execute SOLVE again using for initial estimates the numbers from the X and Y s registers or perhaps two numbers somewhat further apart If the magnitude of the function is neither a minimum nor constant the algorithm will eventually expand its search and find a more significant result Example Investigate the behavior of the function f 3 ge aero as evaluated in the following subroutine Keystrokes Display g LP R 000 Program mode f LBL 0 001 42 21 0 g LABS 002 43 16 CHS 003 16 H e 004 12 xsy 005 34 Bring x value into X register J xe 006 43 11 xe HI x 007 20 e 008 12 2 009 2 x 010 20 2 CHS 011 16 X3y 012 34 Bring x value into X register g LABS 01
200. n X meets or exceeds N in Y 9 RTN 014 43 32 Ifso program ends 3 015 3 If not program continues STO 0 016 44 40 0 Adds 3 days to rin Ro GTOJLA 017 22 11 Goto A and repeat execution to find a new N from a new t Notice that without lines 012 to 014 the loop would run indefinitely until stopped from the keyboard Let s run the program using ft 2 days N 100 mci and a limit value of half of No 50 mci Keystrokes Display g P R Run mode display will vary 2 STO 0 2 0000 th 100 STO 1 100 0000 No 50 STO 2 50 0000 Limit value for N PICA 2 0000 ti 84 0896 Ni 5 0000 to 64 8420 N gt 8 0000 h 50 0000 N3 50 0000 N limit program ends Section 8 Program Branching and Controls 95 Example Flags Calculations on debts or investments can be calculated in two ways for payments made in advance at the beginning of a given period and for payments made in arrears at the end of a given period If you write a program to calculate the value or present value of a debt or investment with periodic interest and periodic payments you can use a flag as a status indicator to tell the program whether to assume payments are made in advance or payments are made in arrears Suppose you are planning the payment of your child s future college tuition You expect the cost to be about 3 000 year or about 250
201. n Y Register Scalar in X Register Matrix in X Register Adds scalar value to each matrix element x Multiplies each matrix element by scalar value Subtracts scalar value from each matrix element Divides each matrix element by scalar value Subtracts each matrix element from scalar value Calculates inverse of matrix and multiplies each element by scalar value Result matrix may be the specified matrix Example Calculate the matrix B 2A then subtract 1 from every element in B From before use Keystrokes f RESULT B T 23 4 59 Designates matrix B as result matrix RCL IMATRIX 2 x A 2 3 Displays descriptor of matrix A b 2 3 Redimensions matrix B to the same dimensions as A multiplies the elements of A by 2 stores those values in the corresponding elements of B and displays the descriptor of the result matrix Section 12 Calculating with Matrices 153 Keystrokes Display iL b 2 3 Subtracts 1 from the elements of matrix B and stores those values in the same elements of B The result which you can view using RCL B in User mode is 13 5 B 7 9 17 With matrix descriptors in both the X and Y registers pressing J or _ calculates the sum or difference of the matrices Arithmetic Operations Pressing Calculates
202. n from 0 to 10 is very close to 1 The reason you got an incorrect answer becomes apparent if you look at the graph of f x over the interval of integration 252 Appendix E A Detailed Look at fix The graph is a spike very close to the origin Actually to illustrate f x the width of the spike has been considerably exaggerated Shown in actual scale over the interval of integration the spike would be indistinguishable from the vertical axis of the graph Because no sample point happened to discover the spike the algorithm assumed that f x was identically equal to zero throughout the interval of integration Even if you increased the number of sample points by calculating the integral in LSCI 9 none of the additional sample points would discover the spike when this particular function is integrated over this particular interval Better approaches to problems such as this are mentioned at the end of the next topic Conditions That Prolong Calculation Time lt You ve seen how the algorithm can give you an incorrect answer when f x has a fluctuation somewhere that is very uncharacteristic of the behavior of the function elsewhere Fortunately functions exhibiting such aberrations are unusual enough that you are unlikely to have to integrate one unknowingly Functions that could lead to incorrect results can be identified in simple terms by how rapidly it and its low order derivatives vary across t
203. n may include prefixes such as Lf _ STO GTO and LBL and still occupy only one line Most instructions require one byte of program memory however some require two For a complete list of two byte instructions refer to Appendix C Instruction Coding Each key on the HP 15C keyboard except for the digit keys 0 through 9 is identified in Program mode by a two digit keycode that corresponds to the key s position on the keyboard Instruction Code STO 1 006 44 40 1 Sixth program line f DSE LT XXX 42 5 25 DSE is just 5 The first digit of a keycode refers to the row 1 to 4 from top to bottom and the second digit refers to the column 1 2 9 O from left to right Exception the keycode for a digit key is simply that digit Section 6 Programming Basics 75 Ge ENG cS ma af DIM DSE TITEI U Gi S Be i CF EAR a ese mon REG PREFIX i EE MS jaan pay PA a FRING nim RND 4 FRAC USER eet Ltt Ean t I INT MEM sea ae meee d HEWLETT PACKARD Keycode 25 second row fifth key Memory Configuration Understanding memory configuration is not essential to your use of the HP 15C It is essential however for obtaining maximum efficiency in memory and programming use The more you program the more useful this knowledge will be Memory configuration and allocation is thoroughly explained in appendix C
204. ng with matrices you ll probably want to redimension all five matrices to 0x0 so that the registers used for storing their elements will be available for program lines or for other advanced functions You can redimension all five matrices to 0x0 at one time by pressing f MATRIX 0 You can dimension a single matrix to 0x0 by pressing 0 f DIM LA through LE Storing and Recalling Matrix Elements The HP 15C provides two ways of storing and recalling values of matrix elements The first method allows you to progress through all of the elements in order The second method allows you to access elements individually Storing and Recalling All Elements in Order The HP 15C normally uses storage registers R and R to indicate the row and column numbers of a hel ie matrix element If the calculator is in User mode mimbar the row and column numbers are automatically oimn incremented as you store or recall each matrix element from left to right along each row from the first row to the last To set the row and column numbers in Ro and R to row 1 column 1 press f MATRIX 1 144 Section 12 Calculating with Matrices To store or recall sequential elements of a matrix 1 2 Be sure the matrix is properly dimensioned Press f MATRIX 1 This stores 1 in both storage registers Ro and R so that elements will be accessed starting at row
205. nge automatically unless the calculator is in User mode To recall the element value after storing the row and column numbers press RCL followed by the letter key specifying the matrix To store a value in that element after storing the row and column numbers place the value in the X register and press STO followed by the letter key specifying the matrix Example Store the value 9 as the element in row 2 column 3 of matrix A from the previous example Keystrokes Display 2 STO 0 2 0000 Stores row number in Ro 3 STO 1 3 0000 Stores column number in Rj 9 9 Keys the new element value into the X register STOJ LA A 2 3 Row 2 column 3 of A 9 0000 Value of a33 Using the Stack You can use the stack registers to specify a particular matrix element This eliminates the need to change the numbers in Ro and Rj To recall an element value enter the row number and column number into the stack in that order Then press RCL 9 followed by the letter key specifying the matrix The element value is placed in the X register The row and column numbers are lost from the stack To store an element value first enter the value into the stack followed by the row number and column number Then press STO L9 followed by the letter key specifying the matrix The row and column numbers are lost from the stack the element value is returned to the X register
206. ns 2 If cleared Celsius input adds 273 to the 7 value in the X register since K C 273 3 f LBL 2 Calculation continues for both modes The System Flags Flags 8 and 9 Flag 8 Setting flag 8 will activate Complex mode described in section 11 turning on the C annunciator If another method is used to activate Complex mode flag 8 will automatically be set Complex mode is deactivated only by clearing flag 8 flag 8 is cleared in the same manner as the other flags 100 Section 8 Program Branching and Controls Flag 9 An overflow condition described on page 61 automatically sets flag 9 Flag 9 causes the display to blink or if a program is running waits until execution is complete and then starts blinking the display Flag 9 may be cleared in three ways e Press LY CF 9 the common procedure for clearing flags e Press LJ This will only clear flag 9 and stop the blinking it will not clear the display e Turn the calculator off Flag 9 is not cleared if the calculator turns itself off If you set flag 9 manually SF 9 it causes the display to blink irrespective of the overflow status of the calculator As usual a program will run to completion before the display starts blinking Therefore flag 9 can be used as a programming tool to provide a visual signal for a selected condition Section 9 Subroutines When the same set of instructions nee
207. ntegral by whatever constants if any are outside the integral In this particular problem we need to multiply the integral by 1 x to get Jy 1 Keystrokes Display 9 ILA 3 1416 0 7652 Jo 1 Section 14 Numerical Integration 197 Before calling the subroutine you provide to evaluate f x the L algorithm just like the algorithm places the value of x in the X Y Z and T registers Because every stack register contains the x value your subroutine can calculate with this number without having to recall it from a storage register The subroutines in the next two examples take advantage of this feature A polynomial evaluation technique that assumes the stack is filled with the value of x is discussed on page 79 Note Since the calculator puts the value of x into all stack registers any numbers previously there will be replaced by x Therefore if the stack contains intermediate results that you ll need after you calculate an integral store those numbers in storage registers and recall them later Occasionally you may want to use the subroutine that you wrote for the L operation to merely evaluate the function at some value of x If you do so with a function that gets x from the stack more than once be sure to fill the stack manually with the value of x by pressing ENTER ENTER ENTER before you execute the subroutine Example The Bessel function
208. nto X You can derive the complex elements of the matrix product YX by recalling the elements of XY or Yx and combining them according to the conventions described earlier Example Calculate the product ZZ where Z is the complex matrix given in the preceding example Since elements representing both matrices are already stored Z in A and Z in B skip steps 1 3 4 and 6 Keystrokes RCL J MATRIX RCL J MATRIX f_J RESULT _C x f USER RCLILC RCLILC RCLILC RCLILC RCLILC RCLILC RCLILC RCLILC f USER Display A b qgqagqas 0000 8500 0000 0000 0000 8000 0000 0500 0500 10 11 11 10 11 10 10 N Displays descriptor of matrix A Displays descriptor of matrix B Designates C as result matrix Calculates Z5 ZZ Activates User mode Matrix C row 1 column 1 Displayed momentarily while last key held down Value of c11 Value of cj Value of c31 Value of c22 Value of c31 Value of c32 Value of c41 Value of c42 Deactivates User mode 168 Section 12 Calculating with Matrices Writing down the elements of C 1 0000 2 8500x 107 ca 24 0000107 1 0000 _ gf 1 0000x107 3 8000x 107 2 1 0000x107 1 0500x 107 where the upper half of matrix C is the real part of ZZ and the lower h
209. o322 Movable Boundary 218 Appendix C Memory Allocation Your very first program instruction will commit R s all seven bytes from an uncommitted register to a program register Your eighth program instruction commits R 4 and so on until the boundary of the common pool is encountered Registers from the data storage pool at power up this is Ryo and below are not available for program memory without reallocating registers using DIM Gi Two Byte Program Instructions The following instructions are the only ones which require two bytes of calculator memory All others require only one byte f BU label T MATRIX 0 to 9 GTO label f Les 2 to 9 0 to 9 g CF nor LL J f DSE 2 to 9 0 to 9 9 JSF nor LL J f USG 2 to 9 0 to 9 Q LF nor LI J STO L J L J LX LE f FIX n or LJ RCL C4 L J GJ CE f LSC nm or LL STO MATRIX LA to E f ENG nor LJ STO LA to LE J in User mode f SOLVE RCL LA to LE J G J in User mode f L STO 9J a RCL LS J G Memory Requirements for the Advanced Functions The four advanced functions require temporary
210. of the first kind of order 1 can be expressed as T n gt f cos 0 xsin d TJO Find 1 T n cos 6 sin d TAO Key in the following subroutine that evaluates the function f0 cos 0 sin 0 Keystrokes Display g P R 000 Program mode f LBL 1 001 42 21 1 Begin subroutine with a label 198 Section 14 Numerical Integration Keystrokes Display 002 23 Calculate sin 0 003 30 Since a value of 8 will be placed into the Y register by the L algorithm before it executes this subroutine the _ operation at this point will calculate 9 sin 8 COS 004 24 Calculate cos 0 sin 8 g JLRTN 005 43 32 In Run mode key the limits of integration into the X and Y registers Be sure that the trigonometric mode is set to Radians then press f L 1 to calculate the integral Finally multiply the integral by 1 m to calculate J 1 Keystrokes Display g J P R Run mode O ENTER 0 0000 Key lower limit into Y register gJr 3 1416 Key upper limit into X register 9 RAD 3 1416 If not already in Radians mode FIU 1 1 3825 cos 0 sin 6 d gla LE 0 4401 J A Example Certain problems in communications theory for example pulse transmission through idealized networks require calculating an integral somet
211. of the number in the X register without actually switching the real and imaginary parts Hold the key down to maintain the display Changing Signs In Complex mode the CHS function affects only the number in the real X register the imaginary X register does not change This enables you to change the sign of the real or imaginary part without affecting the other To key in a negative real or imaginary part change the sign of that part as you enter it If you want to find the additive inverse of a complex number already in the X register however you cannot simply press CHS as you would outside Except for the P and gt R functions as explained in this section page 133 Section 11 Calculating With Complex Numbers 125 of Complex mode Instead you can do either of the following e Multiply by 1 e If you don t want to disturb the rest of the stack press CHS f Rex Im CHS f _ Re Im To find the negative of only one part of a complex number in the X register e Press CHS to negate the real part only e Press f Re im CHS Lf Re Im to negate the imaginary part only forming the complex conjugate Clearing a Complex Number Inevitably you will need to clear a complex number You can clear only one part at a time but you can then write over both parts since
212. ompared to round off error The calculated solution corresponds to that for a nonsingular coefficient matrix close to the original singular matrix Section 12 Calculating with Matrices 157 Week 1 2 3 Total Weight kg 274 233 331 Total Value 120 32 112 96 151 36 Silas knows that he received 0 24 per kilogram for his cabbage and 0 86 per kilogram for his broccoli Use matrix operations to determine the weights of cabbage and broccoli he delivered each week Solution Each week s delivery represents two linear equations one for weight and one for value with two unknown variables the weights of cabbage and broccoli All three weeks can be handled simultaneously using the matrix equation 1 1l diy diy a3 _ 274 233 331 0 24 0 86 d3 dbs tog 120 32 112 96 151 36 or AD B where the first row of matrix D is the weights of cabbage for the three weeks and the second row is the weights of broccoli Keystrokes Display 2 2 0000 Dimensions A as 2x2 matrix ENTER f DIM LA f_ MATRIX 1 2 0000 Sets row and column numbers in Ro and R to 1 f USER 2 0000 Activates User mode 1 STOJLA 1 0000 Stores a STO A 1 0000 Stores aj 24 STOJLA 0 2400 Stores ay 86 STOJLA 0 8600 Stores ay 2 ENTER 3 3 0000 Dimensions B as 2x3 matrix f_ DIM _B 158 Section 12 Calc
213. on of the random number sequence STOJ _f_ RAN will store the X register number 0 lt r lt 1 as a new seed for the random number generator A value for r outside this range will be converted to fit within the range RCL JL f RAN will recall to the display the current random number seed Keystrokes Display 5764 0 5764 Stores 0 5764 as random number seed STO f 0 5764 The f keystroke may be omitted RAN f_ RAN 0 3422 Random number sequence initiated by the f RAN 0 2809 above seed m 0 0000 Passes the spectral test D Knuth The Art of Computer Programming Vol 2 Seminumerical Algorithms Third Edition 1998 Section 4 Statistics Functions 49 Keystrokes Display RCL L 0 2809 Recall last random number generated RAN which is the new seed The f may be omitted Accumulating Statistics The HP 15C performs one and two variable statistical calculations The data is first entered into the Y and X registers Then the 2 function automatically calculates and stores statistics of the data in storage registers R through R These registers are therefore referred to as the statistics registers Before beginning to accumulate statistics for a new set of data press f CLEAR gt to clear the statistics registers and stack If you have
214. on printed on the face of a key press only that key For SOLVE example J e To select the alternate function printed in gold or blue press the like colored prefix key f or L9 J followed by the function key For example f SOLVE 9 x lt yl Throughout this handbook we will observe certain conventions in referring to alternate functions References to the function itself will appear as just the key name in a box such as the MEM function References to the use of the key will include the prefix key such as press 9 MEM References to the four gold functions printed under the bracket labeled CLEAR will be preceded by the word CLEAR such as the CLEAR REG function or press Lf_ CLEAR PRGM Note that the ON key is lower than the other keys to help prevent its being pressed inadvertently 18 Notice that when you press the prefix key an f or g annunciator appears and remains in the display until a function Section 1 Getting Started 19 f or g key is pressed to complete the sequence Prefix Keys 0 0000 f A prefix key is any key which must precede another key to complete the key sequence for a function Certain functions require two parts a prefix key and a digit or other key For your reference the prefix keys are
215. onent of the function s value at x that would appear if the value were displayed in SCI J display format The uncertainty is proportional to the factor 10 which represents the magnitude of the function s value at x Therefore SCI and ENG display formats imply an uncertainty in the function that is relative to the function s magnitude Similarly if a function value is display in FIX n the rounding of the display implies that the uncertainty in the function s values is 8 x 0 5x10 Since this uncertainty is independent of the function s magnitude FIX display format implies an uncertainty that is absolute Each time the L 4 algorithm samples the function at a value of x it also derives a sample of 6 x the uncertainty of the function s value at x This is calculated using the number of digits n currently specified in the display format and if the display format is set to SCI or ENG the magnitude m x of the function s value at x The number A the uncertainty of the approximation to the desired integral is the integral 6 x Although SCI 8 or 9 generally results in the same display as SCI 7 it will result in a smaller uncertainty of a calculated integral The same is true for the ENG format A negative value for n which can be set by using the Index register will also affect the uncertainty of an
216. or f Re Im or 2 by setting flag 8 the Complex mode flag 9 J SF 8 When the calculator is in Complex mode the C annunciator in the display is lit This tells you that flag 8 is set and the complex stack exists In or out of Complex mode the number appearing in the display is the number in the real X register Note In Complex mode signified by the C annunciator the HP 15C performs all trigonometric functions using radians The trigonometric mode annunciator in the display RAD GRAD or blank for Degrees applies to two functions only R and P as explained later in this section Deactivating Complex Mode Since Complex mode requires the allocation of five registers from memory you will have more memory available for programming and other advanced functions if you deactivate Complex mode when you are working solely with real numbers To deactivate Complex mode clear flag 8 keystroke sequence 9 CF 8 The C annunciator will disappear Complex mode is also deactivated when Continuous Memory is reset as described on page 63 In any case deactivating Complex mode dissolves the imaginary stack and all imaginary numbers there are lost Complex Numbers and the Stack Entering Complex Numbers To enter a number with real and imaginary parts Key the real part of the number into the display Press ENTER Key the imaginary part of the number
217. or addition and subtraction Polar vector coordinates must be converted to rectangular coordinates upon entry 0 ENTER r R 2 The results are recalled from R x and Rs y using RCL 2 and converted back to polar coordinates if necessary Remember that for polar coordinates the angle is between 180 and 180 or n and a radians or 200 and 200 grads To convert to a positive angle add 360 or 27 or 400 to the angle For the second vector entered the final keystroke will be either 2 or 27 depending on whether the two vectors should be added or subtracted Section 5 The Display and Continuous Memory Display Control The HP 15C has three display formats FIX use a given number 0 through 9 to specify display format The illustration below shows how the number 123 456 would be displayed specified to four places in each possible mode LSCI J and ENG that f SC4 1 2346 f ENG 4 123 46 f LFIX 4 123 456 0000 05 03 Owing to Continuous Memory any change you make in the display format will be preserved until Continuous Memory is reset The current display format takes effect when digit entry is terminated until then all digits you key in up to 10 are displayed Fixed Decimal Display FIX fixed decimal format displays a figure with the
218. or any initial estimates because the function never equals zero nor changes sign On the other hand the equation xl 1 10 10 has no roots because the left side of the equation is always greater than the right side However because of round off in the calculation of fx del 1 10 7 10 224 Appendix D A Detailed Look at SOLVE the root 1 0000 is found for initial estimates of 1 and 2 By recognizing situations in which round off error may influence the operation of SOLVE you can evaluate the results accordingly and perhaps rewrite the function to reduce the effects of round off In a variety of practical applications the parameters in an equation or perhaps the equation itself are merely approximations Physical parameters have an inherent accuracy or inaccuracy Mathematical representations of physical processes are only models of those processes accurate only to the extent that the underlying assumptions are true An awareness of these and other inaccuracies can be used to your advantage By structuring your subroutine to return a function value of zero when the calculated value is negligible for practical purposes you can usually save considerable time in finding a root with SOLVEJ particularly for cases that would normally take a long time Example Ridget hurlers such as Chuck Fahr can throw a ridget to heights of 105 meters and more In fact Fahr s hurls usually reach
219. ote that the matrix functions and complex functions use the 1 and G keys also but for different purposes Refer to sections 11 and 12 for their usage 106 Section 10 The Index Register and Loop Control 107 Indirect Program Control With the Index Register The I key is used for all forms of indirect program control other than indirect register addressing Hence I not G is used for indirect program branching indirect display format control and indirect flag control Program Loop Control Program loop counting and control can be carried out in the HP 15C by any storage register Ro through Ro R o through Ro or the Index register 1 J Loop control can also be carried out indirectly with i The Mechanics Both _I_ and Gi can be used in abbreviated key sequences omitting the preceding f prefix as explained on page 78 Index Register Storage and Recall Direct STO LI and RCL I Storage and recall between the X register and the Index register operate in the same manner as with other data storage registers page 42 Indirect STO or RCL G stores into or recalls from the data storage register whose number is addressed by the integer portion of the value 0 to 65 in the Index register See the table below and on the next page
220. ound Restarting a Program Press LR S to continue execution of a program that was stopped with a instruction User Mode User mode is an optional condition to save keystrokes when executing letter named programs Pressing f USER will interchange the f_ shifted and primary functions of the A through LE keys You can then execute a program using just one keystroke skipping the f or GSB How to Enter Data Every program must take into account how and when data will be supplied This can be done in Run mode before running the program or during an interruption in the program 1 Prior entry If a variable value will be used in the first line of the program enter it into the X register before starting the program If it will be used later you can store it with STOJ into a storage register and recall it with a programmed RCL within the program 70 Section 6 Programming Basics This is the method used above where h was placed in the X register before running the program No ENTER instruction is necessary because program execution here f JLA both terminates digit entry and enables the stack lift The above program then multiplied the contents of the X register h by 2 The presence of the stack even makes it possible to load more than one variable prior to
221. ours or angles degrees can be converted by the HP 15C between a decimal fraction and a minutes seconds format Section 2 Numeric Functions 27 Hours Decimal Hours lt _ Hours Minutes Seconds Decimal Seconds H h H MMSSs Degrees Decimal Hours lt _ Degrees Minutes Seconds Decimal Seconds D d D MMSSs Hours Degrees Minutes Seconds Conversion Pressing f gt H MS converts the number in the display from a decimal hours degrees format to an hours degree minutes seconds decimal seconds format For example press f gt H MS to convert 1 2345 1 1404 aie Sa Lo seconds to s minutes hours hours Press Lf PREFIX to display the value to all possible decimal places 1140420000 to the hundred thousandth of a second Decimal Hours or Degrees Conversion Pressing 9 H converts the number in the display from an hours degrees minutes seconds decimal seconds format to a decimal hours degrees format Degrees Radians Conversions The DEG and RAD functions are used to convert angles to degrees or radians D d lt oR r The degrees must be expressed as decimal numbers and not in a minutes seconds format Keystrokes Display 40 5 f gt RAD 0 7069 Radians g gt DEG 40 5000 40 5 degrees decimal fraction 28 Section 2 Numeric Functions Logar
222. pecified in the X register If the specified matrix is a singular matrix that is one that doesn t have an inverse then the HP 15C modifies the LU form by an amount that is usually small compared to round off error For 1x the calculated inverse is the inverse of a matrix close to the original singular matrix Refer to the HP 15C Advanced Functions Handbook for further information Section 12 Calculating with Matrices 151 Example Calculate the transpose of matrix B Matrix B was set in preceding examples to Lal 2S3 4 5 9f Keystrokes Display RCL MATRIX LB b 2 3 Displays descriptor of 2x3 matrix B f MATRIX 4 b 3 2 Descriptor of 3x2 transpose Matrix B which you can view using RCL B in User mode is now e ll WN ee Ons Scalar Operations Scalar operations perform arithmetic operations between a scalar that is a number and each element of a matrix The scalar and the descriptor of the matrix must be placed in the X and Y registers in either order Note that the register position will affect the outcome of the _ _ and functions The resulting values are stored in the corresponding elements of the result matrix The possible operations are shown in the following table 152 Section 12 Calculating with Matrices Elements of Result Matrix Operation Matrix in Y Register Scalar i
223. plex elements of the solution X by recalling the elements of X or X and combining them according to the conventions described earlier Example Engineering student A C Dimmer wants to analyze the electrical circuit shown below The impedances of the components are indicated in complex form Determine the complex representation of the currents J and J This system can be represented by the complex matrix equation 10 200 200 1 _ 5 200 200 30 7 0 or AX B 170 Section 12 Calculating with Matrices In partitioned form 10 0 5 0 0 0 Batak anette andB 4 200 200 0 200 170 0 where the zero elements correspond to real and imaginary parts with zero value Keystrokes 4 ENTERJ2 f DIM A f_ MATRIX 1 f J USER 10 STOJLA O STO LA STOJLA STOJLA 200 STOJLA CHS STO LA STOJLA 170 STOJLA 4 ENTER i DIM B O STO IMATRIX _ B 5 ENTER 1 ENTER STO LS JLB RCL J MATRIX LB RCL MATRIX LA Display 2 0000 2 0000 2 0000 10 0000 0 0000 0 0000 0 0000 200 0000 200 0000 200 0000 170 0000 1 0000 0 0000 1 0000 5 0000 b 4 1 A 4 2 Dimensions matrix A to be 4x2 Set beginning row and column numbers in
224. process of searching for a zero of the specified function the algorithm uses the value of the function at two or three previous estimates to approximate the shape of the function s graph The algorithm uses this shape to intelligently predict a new estimate where the graph might cross the x axis The function subroutine is then executed computing the value of the function at the new estimate This procedure is performed repeatedly by the SOLVE algorithm f x If any two estimates yield function values with opposite signs the algorithm presumes that the function s graph must cross the x axis in at least one place in the interval between these estimates The interval is systematically narrowed until a root of the equation is found A root is successfully found either if the computed function value is equal to zero or if two estimates differing by one unit in their last significant digit give function values having opposite signs In this case execution stops and the estimate is displayed 220 Appendix D A Detailed Look at 221 As discussed in section 13 page 186 the occurrence of other situations in the iteration process indicates the apparent absence of a function zero The reason is that there is no way to logically predict a new estimate that is likely to have a function value closer to zero In such cases Error 8 is displayed You should note that the initial estimates you provide are used to begin
225. r 30 Section 2 Numeric Functions For example to find the sales tax at 3 and total cost of a 15 76 item Keystrokes Display 15 76 ENTER 15 7600 Enters the base number the price 3 L9 JL 0 4728 Calculates 3 of 15 76 the tax 16 2328 Total cost of item 15 76 0 47 Percent Difference The A function calculates the percent difference between two numbers The result expresses the relative increase a positive result or decrease a negative result of the second number entered compared to the first number entered For example suppose the 15 76 item only cost 14 12 last year What is the percent difference in last year s price relative to this year s Keystrokes Display 15 76 ENTER 15 7600 This year s price our base number 14 12 LJ A 10 4061 Last year s price was 10 41 less than this year s price Polar and Rectangular Coordinate Conversions The gt P and R functions are provided in the HP 15C for conversions between polar coordinates and rectangular coordinates The angle 0 is assumed to be in the mode whether degrees in a decimal format not a minutes seconds format radians or grads 0 is measured as shown in the illustration at right Polar Conversion Pressing J P polar converts a set of rectangular coordinates x y to polar coordinates magnitude r angle 0 The
226. r argument is used in a mathematical operation as part of your subroutine execution stops with the display Error 0 In the case of a constant function value the routine can see no indication of a tendency for the value to move toward zero This can occur for a function whose first 10 significant digits are constant such as when its graph levels off at a nonzero horizontal asymptote or for a function with a relatively broad local flat region in comparison to the range of x values being tried In the case where the function s magnitude reaches a nonzero minimum the routine has logically pursued a sequence of samples for which the magnitude has been getting smaller However it has not found a value of x at which the function s graph touches or crosses the x axis 188 Section 13 Finding the Roots of an Equation The final case points out a potential deficiency in the subroutine rather than a limitation of the root finding routine Improper operations may sometimes be avoided by specifying initial estimates that focus the search in a region where such an outcome will not occur However the SOLVE routine is very aggressive and may sample the function over a wide range It is a good practice to have your subroutine test or adjust potentially improper arguments prior to performing an operation for instance use ABS prior to yx Rescaling variables to avoid large numbers can also be helpful The su
227. r you want allocated into the display 1 lt dd lt 65 The number of registers in the uncommitted pool and therefore potentially available for programming will be 65 ad 2 Press f DIM G There are two ways to review your allocation e Press RCL DIM G to recall into the stack the number of the highest allocated data storage register dd Programmable e Press 9 MEM as explained above to view a more complete memory status dd uu pp b Keystrokes Display assuming a cleared program memory 1 Lf DIM GD 1 0000 Ry Ro and Ry g MEM hold 1 64 0 0 allocated for data storage Sixty four registers are uncommitted none contain program instructions 19 DIM 19 0000 Rio Ro is the highest numbered G data storage register Forty six RCL DIM 19 0000 registers left in the common pool Restrictions on Reallocation Continuous Memory will maintain the configuration you allocate until a new DIM G is executed or Continuous Memory is reset If you try to allocate a number less than 1 dd 1 If you try to allocate a number greater than 65 Error 10 results If program memory is not cleared the number of uncommitted registers UU is less owing to allocation of registers to program memory pp Therefore pp would be gt 0 and b
228. rack of loop iterations This counter can then be checked with a conditional test to determine when to exit the loop This is shown in the example on page 112 Conditional Branching There are two general applications for conditional branching One is to control loops as explained above A conditional test can check for either a certain calculated value or a certain loop count The other major use is to test for options and pursue one For example if a salesperson made a variable commission depending on the amount of sale you could write a program which takes the amount of sale compares it to a test value and then calculates a specific commission depending on whether the sale is less than or greater than the test value 66 Tests A conditional test takes what is in the X register x and compares it either to zero such as x 0 J or to y that is what is in the Y register such as X lt y For an x y comparison therefore you must have the x and y values juxtaposed in the X and Y registers This might require that you store a test value and then recall it bringing it into the X register Or the value might be in the stack and be moved as necessary using Xs YJ LR or Rt Tests With Complex Numbers and Matrix Descriptors Four of the conditional tests also work with complex numbers and matrix descriptors x 0 J TEST 0 x 0 TEST 5 x y and
229. racy with this function 118 Part Ill HP 15C Advanced Functions Section 11 Calculating With Complex Numbers The HP 15C enables you to calculate with complex numbers that is numbers of the form a ib where ais the real part of the complex number b is the imaginary part of the complex number and i v 1 As you will see the beauty of calculating with the HP 15C in Complex mode is that once the complex numbers are keyed in most operations are executed in the same manner as with real numbers The Complex Stack and Complex Mode Calculations with complex numbers are Pas Real Imaginary performed using a complex stack composed Stack Stack of two parallel four register stacks and two p LAST X registers One of these parallel stacks referred to as the real stack contains the real parts of complex numbers used in calculations This is the same stack used in ordinary calculations The other stack referred to as the imaginary stack contains the imaginary parts of complex numbers used in calculations LAST X I x lt N 4 Creating the Complex Stack The imaginary stack is created by converting five storage registers as described in appendix C when you activate Complex mode it does not exist when the calculator is not in Complex mode 120 Section 11 Calculating With Complex Numbers 121 Complex mode is activated 1 automatically when executing f I
230. raic notation places the algebraic operators between the relevant numbers or variables when evaluating algebraic expressions Lukasiewicz s notation specifies the operators before the variables For optimal efficiency of calculator use HP applied the convention of specifying entering the operators after specifying entering the variable s Hence the term Reverse Polish Notation RPN The HP 15C uses RPN to solve complicated calculations in a straightforward manner without parentheses or punctuation It does so by automatically retaining and returning intermediate results This system is implemented through the automatic memory stack and the ENTER key minimizing total keystrokes The Automatic Memory Stack Registers T Z Y X 0 0000 0 0000 0 0000 Always displayed When the HP 15C is in Run mode no PRGM annunciator displayed the number that appears in the display is the number in the X register 32 Section 3 The Memory Stack LAST X and Data Storage 33 Any number that is keyed in or results from the execution of a numeric function is placed into the display X register This action will cause numbers already in the stack to lift remain in the same register or drop depending upon both the immediately preceding and the current operation Numbers in the stack are stored on a last in first out basis The three stacks drawn below illustrate the three types of stack movement Assume x y z and represent
231. rd this function is most commonly used in programming This capability is especially valuable for the function for which accuracy can be stipulated by specifying the number of digits to be displayed as described in section 14 There are as usual certain display limitations to keep in mind Recall that any display format function merely alters the number of decimal places to which the display is rounded In its memory the calculator always retains a number in scientific notation as a 10 digit mantissa with a two digit exponent The integer portion of the number in the Index register specifies the number of decimal places to which the display is rounded A number less than zero defaults to zero zero decimal places displayed in FIX format while a number greater than 9 defaults to 9 9 decimal places displayed in _FIX Note that in SCI and ENG format modes the maximum display is a seven digit mantissa with a two digit exponent However a format number greater than six and less than or equal to nine will alter the decimal place at which rounding occurs Refer to page 58 59 Section 10 The Index Register and Loop Control 117 An exception is in the case of Ly where the display format number in R may range from 6 to 9 This is discussed in appendix E on page 247 A number less than zero will not affect the display format but will affect accu
232. re a data storage register STO RCL BTO 21 0 3 ROG 1 Gd J ISG xs Using Matrix Descriptors in the Index Register In certain applications you may want to perform a programmed sequence of matrix operations using any of the matrices A through E In this situation the matrix operations can refer to whatever matrix descriptor is stored in the index register Rj If the Index register contains a matrix descriptor e Pressing i after any of the functions listed above performs the operations using the element specified by Ro and R and the matrix specified in Ry e Pressing i after STOJ _9 or RCL J _9 performs the operation using the element specified by the row and column numbers in the Y and X registers and the matrix specified in Ry Also in User mode the row and column numbers in Ro and R are incremented according to the dimensions of the specified matrix 174 Section 12 Calculating with Matrices e Pressing Lf DIM JLI dimensions the matrix specified in Ry according to the dimensions in the X and Y registers e Pressing RCL DIM recalls to the X and Y registers the dimensions of the matrix specified in Ry e Pressing GSB I or GTO _I has the same result as pressing GSB or GTO follow
233. register space from the common register pool Function Registers Needed SOLVE 23 if executed Sy together Complex Stack 5 Matrices 1 per matrix element Appendix C Memory Allocation 219 For SOLVE and L allocation and deallocation of the required register space takes place automatically Memory is thereby allocated only for the duration of these operations Space for the imaginary stack is allocated whenever f Lj if Re Im or L9 J SF 8 is pressed The imaginary stack is deallocated when CF 8 is executed Space for matrix registers is not allocated until you dimension it using DIM Reallocation takes place when you redimension a matrix MATRIX 0 dimensions all matrices to 0 x 0 If you should interrupt a SOLVE or ZF routine in progress by pressing a key you could deallocate its registers by pressing 9 RTN or f CLEAR PRGM in Run mode Appendix D A Detailed Look at SOLVE Section 13 Finding the Roots of an Equation includes the basic information needed for the effective use of the SOLVE algorithm This appendix presents more advanced supplemental considerations regarding SOLVE How SOLVE Works You will be able to use SOLVE most effectively by having a basic understanding of how the algorithm works In the
234. registers f CLEAR PREFIX Clears any prefix from a partially entered key sequence Also temporarily displays the mantissa Display Clearing and The HP 15C has two types of display clearing operations LCLx clear X and back arrow In Run mode e Clx clears the display to zero deletes only the last digit in the display if digit entry has not been terminated by ENTER or most other functions You can then key in a new digit or digits to replace the one s deleted If digit entry has been terminated then acts like CLx Keystrokes Display 12345 12 345 Digit entry not terminated Ka 1 234 Clears only the last digit 9 12 349 Vx 111 1261 Terminates digit entry 0 0000 Clears all digits to zero In Program mode e Clx is programmable it is stored as a programmed instruction and will not delete the currently displayed instruction e lt is not programmable so it can be used for program correction Pressing will delete the entire instruction currently displayed 22 Section 1 Getting Started Calculations One Number Functions A one number function performs an operation using only the number in the display To use any one number function press the function key after the number has been placed in the display Keystrokes Display 45 45 g J LOG 1 6532
235. res the user to be notified that any changes or modifications made to this device that are not expressly approved by Hewlett Packard Company may void the user s authority to operate the equipment 284 Declaration of Conformity for Products Marked with FCC Logo United States Only This device complies with Part 15 of the FCC Rules Operation is subject to the following two conditions 1 this device may not cause harmful interference and 2 this device must accept any interference received including interference that may cause undesired operation If you have questions about the product that are not related to this declaration write to Hewlett Packard Company P O Box 692000 Mail Stop 530113 Houston TX 77269 2000 For questions regarding this FCC declaration write to Hewlett Packard Company P O Box 692000 Mail Stop 510101 Houston TX 77269 2000 or call HP at 281 514 3333 To identify your product refer to the part series or model number located on the product Canadian Notice This Class B digital apparatus meets all requirements of the Canadian Interference Causing Equipment Regulations Avis Canadien Cet appareil num rique de la classe B respecte toutes les exigences du R glement sur le mat riel brouilleur du Canada European Union Regulatory Notice Products bearing the CE marking comply with the following EU Directives e Low Voltage Directive 2006 95 EC e EMC Directive 2004 108 EC e Ecodesign D
236. ressing SST J in Run mode Keystrokes Display g LP R Run mode f CLEAR REG Clear storage registers GTO LA oe to first line of program 8 1 8 0000 Store a can height 2 5 2 5 Enter a can radius SST hold 001 42 21 11 Keycode for line 001 label release 2 5000 Result of executing line 001 SST 002 44 o GIO 0 2 5000 Result SST 003 4311 9 x 6 2500 Result SST 004 43 26 I LT 3 1416 Result SST 005 20 x 19 6350 Result the base area of the can Wrapping SST will not move program position into unoccupied program territory Instead the calculator will wrap around to line 000 In Run mode SST will perform any instructions at the end of program memory such as RTN GTO or GSB Line Position Recall that the calculator s position in program memory does not change when it is shut off or Program Run modes are changed Upon returning to Program mode the calculator line position will be where you left it If you executed a program ending with RTN the position will be at line 000 Therefore if the calculator is left on and shuts itself off you need only turn it on and switch to Program mode the calculator always wakes up in Run mode to be back where you were Section 7 Program Editing 87 Insertions and D
237. rocess will require even more time Press _f_ SOLVE O and sit back while your HP 15C exhibits one of its powerful capabilities The display flashes running while SOLVE is operating Press f FIX 4 to obtain the displays shown here The display setting does not influence the operation of SOLVE t An algorithm is a step by step procedure for solving a mathematical problem An iterative algorithm is one containing a portion that is executed a number of times in the process of solving the problem Section 13 Finding the Roots of an Equation 183 Keystrokes Display f_ SOLVE 0 5 0000 The desired root After the routine finds and displays the root you can ensure that the displayed number is indeed a root of f x 0 by checking the stack You have seen that the display X register contains the desired root The Y register contains a previous estimate of the root which should be very close to the displayed root The Z register contains the value of your function evaluated at the displayed root Keystrokes Display RY 5 0000 A previous estimate of the root RY 0 0000 Value of the function at the root showing that f x 0 Quadratic equations such as the one you are solving can have two roots If you specify two new initial estimates you can check for a second root Try estimates of 0 and 10 to look for a negative root
238. rom the total number of plants 24 9 crocuses have not bloomed RCL 0 15 0000 The number in Ro does not change Overflow and Underflow If an attempted storage or recall arithmetic operation would result in overflow in a data storage register the value in the affected register will be replaced with 9 999999999x10 and the display will blink To stop the blinking clear the overflow condition press J or ON or L9 CF 9 In case of underflow the value in the register will be replaced with zero no display blinking Overflow and underflow are discussed further on page 61 Problems 1 Calculate the value of x in the following equation 8 33 4 5 2 8 33 7 46 0 32 Ao 4 3 3 15 2 75 1 71 2 01 Answer 4 5728 A possible keystroke solution is 4 ENTER 5 2 8 33 x L9 J LSTx 7 46 0 32 LX 3 15 ENTER 2 75 L 4 3 LX 1 71 ENTER 2 01 x L Lx 46 Section 3 The Memory Stack LAST X and Data Storage Use arithmetic with constants to calculate the remaining balance of a 1000 loan after six payments of 100 each and an interest rate of 1 0 01 per payment period Procedure Load the stack with 1 i where i interest rate and key in the initial loan balance Use th
239. routine can call up yet another subroutine This subroutine nesting the execution of a subroutine within a subroutine is limited to stack of subroutines seven levels deep this does not count the main program level The operation of nested subroutines is as shown below Main Program LBL A 4 LBL 4 LBUL2 a4 LBULS 4 LBL L4 1 1 1 1 1 1 1 1 1 GSB 1 d GSB 3 i 0 i gt GsB 2 gt O38 4 i gt L gt v Y v RTN RTN RTN RTN RTN End Examples Example Write a program to calculate the slope of the secant line joining points x y and X2 y2 on the graph shown where y x sin x given x in radians The secant slope is Zo ok 9 o y2 y or x3 sinx 47 sinx 27 27 The solution requires that the equation for y be evaluated twice once for y and once for y2 given the data input for x and x2 Since the same calculation must be made for different values it will save program space to call a subroutine to calculate y The following program assumes that x has been entered into the Y register and xz into the X register MAIN PROGRAM g JLP R J
240. roximations in SCI 2 and SCI 3 are identical the accuracy of the approximation in SCI 3 is no better than the accuracy in SCI 2 despite the fact that the uncertainty in SCI 3 is less than the uncertainty in SCI 2 Why is this Remember that the accuracy of any approximation depends primarily on the number of sample points at which the function f x has been evaluated The algorithm is iterated with increasing numbers of sample points until the disparity among three successive approximations is less than the uncertainty derived from the display format After a particular iteration the disparity among the approximations may already be so much less than the uncertainty that it would still be less if the uncertainty were decreased by a factor of 10 In such cases if you decreased the uncertainty by specifying one more digit in the display format the algorithm would not have to consider additional sample points and the resulting approximation would be identical to the approximation calculated with the larger uncertainty If you calculated the two preceding approximations on your calculator you may have noticed that it did not take any longer to calculate the integral in SCI 3 than in SCI 2 This is because the time to calculate the integral of a given function depends on the number of sample points at which the function must be evaluated to achieve an approximation of acceptable accurac
241. ruction end of memory Unexpected Program Stops Pressing Any Key Pressing any key will halt program execution It will not halt in the middle of an operation This instruction will be completed before the program stops Error Stops Program execution is immediately halted when the calculator attempts an improper operation that results in an Error display To see the line number and keycode of the error causing instruction the line at which the program stopped press any one key to remove the Error message then switch to Program mode If the display is flashing when a program stops an overflow condition exists page 61 Press ON or L9 LCF 9 to stop the blinking Abbreviated Key Sequences In certain cases an f prefix you might expect to include in a key sequence is not needed The rule for using an abbreviated key sequence is the f prefix key is unnecessary after any other prefix key Page 19 contains a list of prefix keys h Section 6 Programming Basics 79 For example f_ LBL f JLA becomes f JILBL A J Lf JLDIMJ_f JGD becomes f DIM G and STO f RAN becomes STO RAN The removal of the f is not ambiguous because the f _ shifted function is the only logical one in these cases The keycodes for such instructions do not include the extraneous f even if you do key it in User Mode User mode is a convenience to s
242. s for a given matrix operation 208 Appendix A Error Conditions or J where the dimensions are incompatible x where e the dimensions are incompatible or e the result is one of the arguments Vx where the matrix is not square scalar matrix where the matrix is not square where e the matrix in the X register is not square e the dimensions are incompatible or the result is the matrix in the X register MATRIX 2 where the input is a scalar or the number of rows is odd MATRIX 3 where the input is a scalar or the number of columns is odd MATRIX 4 where the input is scalar MATRIX 5 where e the input is a scalar e o the dimensions are incompatible or e o the result is one of the arguments MATRIX 6 where e the input is scalar e the dimensions are incompatible including the result or e the result is one of the arguments MATRIX 9 where the matrix is not square RCL DIM I J where contents of R are scalar DIM LI J where contents of R are scalar STO RESULT where the input is scalar Py x where the number of columns is odd Cy x where the number of rows is odd Pr Error Power Error Continuous Memory interrupted and reset because of power failure Appendix B Stack Litt and the LAST X Register The HP 15C calculator has been designed to operate in
243. s of the mantissa 62 Section 5 The Display and Continuous Memory Low Power Indication When a flashing asterisk which indicates low battery power appears in the lower left hand side of the display there is no reason to panic You still have plenty of calculator time remaining at least 10 minutes if you continuously run programs and at least an hour if you do calculations manually Refer to appendix F page 259 for information on replacing the batteries 0 0000 Continuous Memory Status The Continuous Memory feature of the HP 15C retains the following in the calculator even when the display is turned off e All numeric data stored in the calculator e All programs stored in the calculator e Position of the calculator in program memory e Display mode and setting e Trigonometric mode Degrees Radians or Grads e Any pending subroutine returns e Flag settings except flag 9 which clears when the display is manually turned off e User mode setting e Complex mode setting When the HP 15C is turned on it always wakes up in Run mode If the calculator is turned off Continuous Memory will be preserved for a short period while the batteries are removed Data and programs are preserved longer than other aspects of calculator status Refer to appendix F for instructions on changing batteries Section 5 The Display and Continuous Memory 63 Resetting Continuous Memory If at any time you want to rese
244. s they appear in matrix memory and the result is not the result you would obtain using the original matrix Calculations With Complex Matrices The HP 15C enables you to perform matrix multiplication and matrix inversion with complex matrices that is matrices whose elements are complex numbers and to solve systems of complex equations that is equations whose coefficients and variables are complex However the HP 15C stores and operates on only real matrices The capability of doing calculations with complex matrices is completely independent of the capability of doing calculations with complex numbers described in the preceding section You don t need to activate Complex mode for calculations with complex matrices Section 12 Calculating with Matrices 161 Instead calculations with complex matrices are performed by using real matrices derived from the original complex matrices in a manner to be described below and performing certain transformations in addition to the regular matrix operations These transformations are performed by four calculator functions This section will describe how to do these calculations There are more examples of calculations with complex matrices in the HP 15C Advanced Functions Handbook Storing the Elements of a Complex Matrix Consider an mxn complex matrix Z X iY where X and Y are real mxn matrices This matrix can be represented in the calculator as a 2mxn partitioned matrix
245. seeeesteeeeeeees 43 Overflow and Underflow o ececccccceesseeeeeeesteeesseeneeeees 45 Problemi aiara inea rT wledtn eee 45 Section 4 Statistics Functions eeeeeeeeeeeeeeeeeeeeees 47 Probability Calculations c ccccccccseeeeeeeeeeeeeesseeesneeenteeees 47 Random Number Generator ccccccceesseeseseeesseeeeseeenteeees 48 Accumulating Statistics sessist eie ate TE R nS 49 Correcting Accumulated Statistics ccccccssseeesteeeeeees 52 MOGI tics Jae sh cick E rece tee iat aati ditmsrecuta te 53 Standard Deviation ccccccccscceecteeeeeeecseeeesseeeeeeeneeeess 53 Linear Regression rsss aise asa a ko aE a 54 Linear Estimation and Correlation Coefficient 04 55 Other Applications 2 3 22 c deh cee A a 57 Section 5 The Display and Continuous Memory 58 Display Gontroll Gretas ee elo Ae tates 58 Fixed Decimal Display ccccccceesseeeeceesteeeeseeeeeeeenes 58 Scientific Notation Display cccceecccetseeesseeeeseeeeeeeees 59 Engineering Notation Display ccccccccssceesteeeereeees 59 Mantissa Display ccccecceceeeeeeeeeeeeeeeeecneeeeseeeeeeeees 60 Round OM Error cces cess tect net AAE E REES 60 Special Displaysixixse a Nahe eae 60 ANNUNCIGIOIS onroro i a a a AE aE 60 Digit Separators miseen pasee e ao EE a 6l Error Display Carer ira aa a A Ne 6l Overflow and Underflow cccccccceecseeeteeesteeesseeneeeees 61 Low Power Indication
246. ssibly with same rows interchanged The HP 15C Advanced Functions Handbook discusses LU decomposition in detail t Matrix A is automatically designated as the result matrix whenever Continuous Memory is reset Section 12 Calculating with Matrices 149 While the key used for any matrix operation that stores a result in the result matrix is held down the descriptor of the result matrix is displayed If the key is released within about 3 seconds the operation is performed and the descriptor of the result matrix is placed in the X register If the key is held down longer the operation is not performed and the calculator displays null Copying a Matrix To copy the elements of a matrix into the corresponding elements of another matrix use the STO MATRIX sequence 1 Press MATRIX followed by the letter key specifying the matrix to be copied This enters the descriptor of the matrix into the display 2 Press L STO MATRIX followed by the letter key specifying the matrix to be copied into If the matrix specified after does not have the same dimensions as the matrix specified after STO the second matrix is redimensioned to agree with the first The matrix specified after STO dimensioned need not already be Example Copy matrix A from the previous example into matrix B Keystrokes Display RCL MATRIX A 2 3 Disp
247. struction thereby ending the program If the loop counter has not yet decreased to zero execution continues with line 013 This branches to line 015 and continues the program and the looping To run the program put t day 1 in Ro No initial isotope batch in R the loop counter in Ro and the line number for branching in the Index register Keystrokes Display g P R Run mode 2 0 2 00000 t 100 STO 1 100 0000 No 3 000001 STO 2 3 0000 Loop counter This instruction could also be programmed 114 Section 10 The Index Register and Loop Control Keystrokes 15 Display CHS STO I f A 15 0000 2 0000 84 0896 5 0000 64 8420 8 0000 50 0000 50 0000 Branch line number Running program loop counter Loop counter 2 Loop counter 1 Loop counter 0 program ends Example Display Format Control The following program pauses and displays an example of FIX display format for each possible decimal place It utilizes a loop containing a DSE instruction to automatically change the number of decimal places Keystrokes g P R f CLEAR PRGM f LBL B 9 STOJ LI f LBL 0 f FIX RC L LI f PSE f DSE GTOJ 0 g TEST
248. t entirely clear the HP 15C Continuous Memory 1 Turn the calculator off 2 Press and hold the QN key then press and hold the key 3 Release the ON key then the key This convention is represented as ON J When Continuous Memory is reset Pr Error power error will be displayed Press any key to clear the display Note Continuous Memory can inadvertently be interrupted and reset if the calculator is dropped or otherwise traumatized Part Il HP 15C Programming Section 6 Programming Basics The next five sections are dedicated to explaining aspects of programming the HP 15C Each of these programming sections will first discuss basic techniques The Mechanics then give examples for the implementation of these techniques Examples and lastly discuss finer points of operation in greater detail Further Information Read only as far as you need to support your use of the HP 15C The Mechanics Creating a Program Programming the HP 15C is an easy matter based simply on recording the keystroke sequence used when calculating manually This is called keystroke programming To create a program out of a series of calculation steps requires two extra manipulations deciding where and how to enter your data and loading and storing the program In addition programs can be instructed to make decisions and perform iterations throu
249. t of the new tomato What is the percent change in water content for a 230 g tomato of which 205 g are water A possible keystroke sequence is f_ LBLJLD 94 ENTER R S enter water weight oft new tomato ENTER R S enter total weight of new tomato 9 JIA 9 JLRTN taking 11 program lines and 11 bytes of memory Answer for the 230 g tomato above the percent change in percent water weight is 5 1804 Section 7 Program Editing There are many reasons to modify a program after you ve already stored it you might want to add or delete an instruction like STOJ LPSE J or R S or you might even find some errors The HP 15C is equipped with several editing features to make this process as easy as possible The Mechanics Making a program modification of any kind involves two steps moving to the proper line the location of the needed change and making the deletion s and or insertion s Moving to a Line in Program Memory The Go To GTO Instruction The sequence GTO CHS nnn will move program memory to line number nnn whether pressed in Run mode or Program mode PRGM displayed This is not a programmable sequence it is for manually finding a specific position in program memory The line number must be a three digit number satisfying 000 lt nnn lt 448 The Single Step
250. ta storage 42 Data storage pool 213 214 Debt payment example 95 Decimal point 22 Decimal point display 61 Deflation 233 234 237 DEG 26 Determinant 150 Digit entry 22 in Complex mode 121 125 127 128 129 termination 22 36 209 Digit separator display 61 DIM J 76 77 215 217 Disabling stack lift 36 Display See also X register blinking 100 clearing 21 error messages 61 full mantissa 60 in Complex mode 121 Display format 58 59 61 effect on Lf 200 241 244 245 249 Do if True rule 92 192 274 Subject Index DSE 109 111 112 116 E EEX J 19 Electrical circuit example 169 171 Enabling stack lift 36 ENG 59 Engineering notation 59 ENTER 12 33 34 36 effect on digit entry 22 29 effect on stack movement 37 41 Entering data for statistical analysis 49 Error conditions 205 208 display 61 stops 78 Errors with L 203 204 with SOLVE 187 192 193 Euclidean norm See Frobenius norm Exchanging the real and imaginary stacks 124 Exponential function See Power function Exponents 19 20 F fj 18 Factorial function Lx J 25 Falling stone example 14 FIX 58 Fixed decimal notation 58 Flag tests 92 98 Flag 8 99 Flag 9 100 Format handbook 2 18 Fractional portion FRAC 24 Frobenius norm 150 177 Functions nonprogrammable 80 Func
251. tain the current approximation but not its uncertainty Pressing while the HP 15C is calculating an integral halts the calculation just as it halts the execution of a running program When you do so the calculator stops at the current program line in the subroutine you wrote for evaluating the function and displays the result of executing the preceding program line Note that after you halt the calculation the current approximation to the integral is not the number in the X register nor the number in any other stack register Just as with any program pressing R S again starts the calculation from the program line at which it was stopped The algorithm updates the current approximation and stores it in the LAST X register after evaluating the function at each new sample point To obtain the current approximation therefore simply halt the calculator single step if necessary through your function subroutine until the calculator has finished evaluating the function and updating the current approximation Then recall the contents of the LAST X register which are updated when the RTN instruction in the function subroutine is executed While the calculator is updating the current approximation the display is blank and does not show running While the calculator is executing your function subroutine running is displayed Therefore you might avoid having to single step through your subroutine by halting the ca
252. ted from the keyboard or in a program Some of the functions such as ABS J are in fact primarily of interest for programming Remember that the numeric functions like all functions except digit entry functions automatically terminate digit entry This means a numeric function does not need to be preceded or followed by ENTER Pi Pressing 9 Z places the first 10 digits of m into the calculator 7 does not need to be separated from other numbers by ENTER Number Alteration Functions The number alteration functions act upon the number in the display X register Integer Portion Pressing 9 LINT replaces the number in the display with the nearest integer of lesser or equal magnitude Fractional Portion Pressing f FRAC replaces the number in the display with its fractional part that is the difference between the number and its integer part Rounding Pressing 9_ RND rounds all 10 internally held digits of the mantissa of the displayed value to the number of digits specified by the current FIX J SCI or ENG display format Absolute Value Pressing 9 LABS yields the absolute value of the number in the display 24 Section 2 Numeric Functions 25 Keystrokes Displa
253. ter it second so that it always will be saved in the LAST X register Pressing J _J LSTx will retrieve the constant and place it into the X register the display This can be done repeatedly 40 Section 3 The Memory Stack LAST X and Data Storage Example Two close stellar neighbors of Earth are Rigel Centaurus 4 3 light years away and Sirius 8 7 light years away Use the speed of light c 3 0108 meters second or 9 5x10 meters year to figure the distances to these stars in meters The stack diagrams show only one decimal place T Z Y 43 x 43 a3 Keys 4 3 ENTER 9 5 EEX 15 T Z y 43 x s aie Keys x 8 7 g LAST X T z 41 16 Y 41 16 Rigel Centaurus is 4 1x10 meters away i 9 5 15 8 3 16 Jo Sirius is 8 3x10 Keys x meters away Section 3 The Memory Stack LAST X and Data Storage 41 Loading the Stack with a Constant Because the number in the T register is replicated when the stack drops this number can be used as a constant in arithmetic operations T lt New constant generation Z Y lt Drops to interact with X register X Keys x Fill the stack with a constant by keying it into the display and pressing ENTER three times Key in your initial argument and perform the arithmetic operation The stack will drop a copy of the constant will fall
254. the Index register See the above table Indirect Branching With 1 The I key but not the G key can be used for indirect branching GTO and subroutine calls GSB _I Only the integer portion of the number in Ry is used GJJ is only used for indirect addressing of storage registers Section 10 The Index Register and Loop Control 109 To Labels If the R value is positive GTOJ I and GSB LL will transfer execution to the label which corresponds to the number in the Index register see the above table For instance if the Index register contains 20 00500 then a GIO I instruction will transfer program execution to f_ LBL LA J See the chart on page 107 To Line numbers If the R value is negative GTO causes branching to that line number using the absolute value of the integer portion of the value in R For instance if R contains 20 00500 then a GTO transfer program execution to program line 020 _ instruction will Indirect Flag Control With 1 SF LI J CF I J or LF will set clear or test the flag 0 to 9 specified in R by the magnitude of the integer portion Indirect Display Format Control With
255. the X register The argument from the X register is placed in the LAST X register Section 12 Calculating with Matrices 175 6 0000 5 0000 4 0000 Keys anit 5 0000 4 0000 5 0000 5 0000 matrix B Keys x vox LS 4 0000 Li x lt N Several matrix functions operate on the matrix specified in the X register only and store the result in the same matrix For these operations the contents of the stack including the LAST X register are not moved although the display changes to show the new dimensions if necessary For the MATRIX 7 MATRIX 8 and 9 functions the matrix descriptor specified in the X register is placed in the LAST X register and the norm or for MATRIX 9 the determinant is placed in the X register The Y Z and T registers aren t changed When you recall descriptors or matrix elements into the X register with the stack enabled other descriptors and numbers already in the stack move up in the stack and the contents of the T register are lost The LAST X register is not changed When you store descriptors or matrix elements the stack and the LAST X register isn t changed In contrast to the operation described above the STOJ _Y and RCL J _9 functions do not affect the LAST X register and operate as shown on the next page 176 Section 12 Calculating with Matrices lost T z vee Y fewnameer SS eee al STO g A
256. the blue function printed below that key page 18 For other prefix keys refer to Display Control keys page 263 Storage keys page 267 and the Programming Summary and Index page 269 CLEAR PREFIX Cancels any prefix keystrokes and partially entered instructions such as f SCI page 19 Also displays the complete 10 digit mantissa of the number in the display page 60 Probability Cy x Combination Computes the number of possible sets of y different items taken x at a time and causes the stack to drop page 47 For matrix use refer to Matrix Functions keys page 264 Py x Permutation Computes the number of possible different arrangements of y different items taken x at a time and causes the stack to drop page 47 For matrix use refer to Matrix Functions keys page 264 Stack Manipulation xy Exchanges contents of X and Y stack registers page 34 X X register exchange Exchanges contents of X register with those of any other named storage register Used with LI G digit or digit address page 42 Re Im Real exchange imaginary Exchanges the contents of the real and imaginary X registers and activates Complex mode page 124 R Rolls down contents of stack page 34 Rt Rolls up contents of stack page 34 CLx Clears contents of display X
257. the descriptor for B into the Y register Temporary display while AB is being calculated and stored in matrix C Descriptor for the result matrix C a 2x1 matrix Now recall the elements of matrix C the solution to the matrix equation Also remove the calculator from User mode and clear all matrices Keystrokes RCL J LC RCL LC f USER f MATRIXJO Display c 1 1 11 2887 8 2496 8 2496 8 2496 Denotes matrix C row 1 column 1 Value of cy x1 Value of c21 x2 Deactivates User mode Clears all matrices The solution to the system of equations is x 11 2887 and x 8 2496 Note The description of matrix calculations in this section presumes that you are already familiar with matrix theory and matrix algebra Matrix Dimensions Up to 64 matrix elements can be stored in memory You can use all 64 elements in one matrix or distribute them among up to five matrices Section 12 Calculating with Matrices 141 Matrix inversion for example can be performed on an 8x8 matrix with real elements or on a 4x4 matrix with complex elements as described later To conserve memory all matrices are initially dimensioned as 0x0 When a matrix is dimensioned or redimensioned the proper number of registers is automatically allocated in memory You may have to increase the number of registers allocated to matrix memory before dimensioning a matrix or
258. ther subroutine a nested subroutine labeled 5 to do the repetitive squaring The program is executed after placing the variables 7 z y and x into the T Z Y and X registers Keystrokes f LBL 4 Start of main subroutine g j Lx x GSB 5 Calculates y and rey GSB 5 Calculates z and vt y 22 GSB 5 Calculates f and Ly VX E R x y z t g RTN End of main subroutine returns to main program f LBL 5 Start of nested subroutine xy g J Lx Calculates a square and adds it to current sum of squares g RTN End of nested sub routine returns to main subroutine If you run the subroutine with its nested subroutine alone using x 4 3 y 7 9 z 1 3 and t 8 0 the answer you get upon pressing GSB 4 is 12 1074 Section 9 Subroutines 105 Further Information The Subroutine Return The pending return condition means that the RIN instruction occurring subsequent to a GSB instruction causes a return to the line following the GSB rather than a return to line 000 This is what makes a subroutine useful and reuseable in different parts of a program it will always return execution to where it branched from even as that point changes The only difference between using a GSB branch and a GTO branch is the transfer of execution after a
259. tions matrix calculations solving for roots and numerical integration e Direct and indirect storage in up to 67 registers This handbook is written for you regardless of your level of expertise The beginning part covers all the basic functions of the HP 15C and how to use them The second part covers programming and is broken down into three subsections The Mechanics Examples and Further Information in order to make it easy for users with varying backgrounds to find the information they need The last part describes the four advanced mathematics capabilities Before starting these sections you may want to gain some operating and programming experience on the HP 15C by working through the introductory material The HP 15C A Problem Solver on page 12 The various appendices describe additional details of calculator operation as well as warranty and service information The Function Summary and Index and the Programming Summary and Index at the back of this manual can be used for quick reference to each function key and as a handy page reference to more comprehensive information inside the manual Also available from Hewlett Packard is the HP 15C Advanced Functions Handbook which provides applications and technical descriptions for the root solving integration complex number and matrix functions Note You certainly do not need to read every part of the manual before delving into the HP 15C Advanced Functions if you are
260. tions one number 22 25 Functions primary and alternate 18 Functions two number 22 29 Subject Index 275 G gj 18 Gamma function x 25 GRD 26 GSB 101 LGTO 90 97 98 GTO CHS J 82 H Horner s Method 79 181 Hyperbolic functions 28 Imaginary stack clearing the 124 creation of 121 123 133 display of 124 stack lift of 124 Index register arithmetic 108 112 display format control 109 114 115 116 exchange with X register 108 112 flag control 109 115 loop control 107 109 111 storage and recall 107 111 115 Indirect addressing 106 108 115 Initialization 87 Instructions 74 Integer portion UNT J 24 Integrate function _ 194 204 accuracy of 200 203 240 241 245 algorithm for 196 240 241 249 251 255 256 display format with 245 249 execution time for 196 200 244 245 254 256 memory usage 204 obtaining an approximation for 257 258 problems with erratic functions 249 254 programmed 203 204 recursive use of 203 subroutines for 194 195 276 Subject Index uncertainty in 202 203 240 244 245 249 Interchanging functions See User mode Interference radio and television 271 Intermediate results 22 38 Interpolation using r 57 ISG 109 111 116 Iterations using ISG and DSE 111 K Keycodes 74 75 Keying in chain calculations 22 e
261. tions Handbook presents more detailed and technical aspects of the matrix functions in the HP 15C including applications The topics include least squares calculations solving nonlinear equations ill conditioned and singular matrices accuracy considerations iterative refinement and creating the identity matrix Section 13 Finding the Roots of an Equation In many applications you need to solve equations of the form fix 0 This means finding the values of x that satisfy the equation Each such value of x is called a root of the equation f x f x 0 and a zero of the function f x ROOT These roots or zeros that are real numbers are called real roots or real zeros For many problems the roots of an equation can be determined x analytically through algebraic manipulation in many other instances this is not possible Numerical techniques can be used to estimate the roots when analytical methods are not suitable When you use the SOLVE key on your HP 15C you utilize an advanced numerical technique that lets you effectively and conveniently find real roots for a wide range of equations Using SOLVE In calculating roots the SOLVE operation repeatedly calls up and executes a subroutine that you write for evaluating f x Actually any equation with one variable can be expressed in this form For example f x a is equivalent to f x a 0 and f x g x is equivalent to f x g x 0
262. tions on the calculator These operations disable the stack lift so that a number keyed in after one of these disabling operations writes over the current number in the displayed X register and the stack does not lift These special disabling operations are ENTER Clx 2 27 Imaginary X Register A zero is placed in the imaginary X register when the next number following ENTER 2 or 27 is keyed or recalled into the display real X register However the next number keyed in or recalled after J or Clx does not change the contents of the imaginary X register Enabling Operations Stack Lift Most of the operations on the keyboard including one and two number mathematical functions like Lx and X are stack enabling operations This means that a number keyed in after one of these operations will lift the stack because the stack has been enabled to lift Both the real and imaginary stacks are affected Recall that a shaded X register means that its contents will be written over when the next number is keyed in or recalled T t Z y y Z Z y X X Y y x 4 0000 4 0000 X x 4 4 0000 3 Keys 4 ENTER 3 Assumes Stack Stack No stack stack lifts disabled lift enabled Refer to footnote page 36 Appendix B Stack Lift and the LAST X Register 211
263. tores a1 5 STOJLA 5 0000 Stores ay 3 LSTOJLA 3 0000 Stores a23 8 STOJLA 8 0000 Stores a24 f USER 8 0000 Deactivates User mode RCL MATRIX LA A 2 4 Display descriptor of matrix A f Pyx A 4 2 Transforms Z into Z and redimensions matrix A 164 Section 12 Calculating with Matrices Matrix A now represents the complex matrix Z in Z form Veal Part AZE se Y imaginary Pan The Complex Transformations Between Z and An additional transformation must be done when you want to calculate the product of two complex matrices and still another when you want to calculate the inverse of a complex matrix These transformations convert between the Z representation of an mxn complex matrix and a 2mx2n partitioned matrix of the following form The matrix created by the MATRIX 2 transformation has twice as many elements as Z For example the matrices below show how is related to Z The transformations that convert the representation of a complex matrix between Z and are shown in the following table Pressing Transforms Into f MATRIX 2 Zz Z f MATRIX 3 Z Zz To do either of these transformations recall the descriptor of Z or into the display then press the keys shown above The transformation is done to the specified matrix the result matrix is not affected Section 12 Calculating with Matrices 165 Inverting a
264. trix A to be 2x3 Displaying Matrix Dimensions There are two ways you can display the dimensions of a matrix e Press RCL MATRIX followed by the letter key specifying the matrix The calculator displays the name of the matrix at the left and the number of rows followed by the number of columns at the right e Press RCL DIM followed by the letter key specifying the matrix The calculator places the number of rows in the Y register and the number of columns in the X register Keystrokes Display RCL MATRIX B b O O Matrix B has 0 rows and 0 columns since it has not been dimensioned otherwise RCL DIM LA 3 0000 Number of columns in A x y 2 0000 Number of rows in A Changing Matrix Dimensions Values of matrix elements are stored in memory in order from left to right along each row from the first row to the last If you redimension a matrix to a smaller size the required values are reassigned according to the new dimensions and the extra values are lost For example if the 2x3 matrix shown at the left below is redimensioned to 2x2 then t 2 3 m2 lost s z 5 6 Section 12 Calculating with Matrices 143 If you redimension a matrix to a larger size elements with the value 0 are added at the end as required by the new dimensions For example if the same 2x3 matrix is re dimensioned to 2x4 then When you have finished calculati
265. u would if the calculator were not in Complex mode As you do so a zero will be placed in the imaginary X register as long as the or Clx as explained on page 124 previous operation was not P The operation of the real and imaginary stacks during this process is illustrated below Assume the last key pressed was not or LCLx and the contents remain from the previous example lt x lt N A Keys Re Im Re Im Re Im a b d e f d f 17 144 f 17 144 4 0 17 144 4 0 4 0 4 ENTER Followed by another number Section 11 Calculating With Complex Numbers 129 Entering a Pure Imaginary Number There is a shortcut for entering a pure imaginary number into the X register when you are already in Complex mode key in the imaginary number and press Lf_ Re Im Example Enter 0 10i assuming the last function executed was not or CLx Keystrokes Display 10 10 Keys 10 into the displayed real X register and zero into the imaginary X register f Re Im 0 0000 Exchanges numbers in real and imaginary X registers Display again shows that the number in the real X register is zero as it should be for a pure imaginary number The operation of the real and imaginary stacks during this process is illustrated below Assume the stack registers contain the numbers resulting
266. ulating with Matrices Keystrokes 274 STOJLB 233 STOJLB 331 STO LB 120 32 STO LB 112 96 STO B 151 36 STOJLB f_ RESULT _D RCL MATRIX LB RCL MATRIX LA RCL LD RCL LD RCL LD RCL LD RCL LD RCL LD f J USER 274 331 112 151 151 2 186 141 215 Display 0000 233 0000 120 9600 3600 0000 3200 3600 3 0000 0000 0000 88 0000 92 0000 116 116 0000 0000 Note that you did not need to press Stores bu Stores by Stores b43 Stores b21 Stores by Stores b 3 Designates matrix D as result matrix Recalls descriptor of constant matrix Recalls descriptor of coefficient matrix A into X register moving descriptor of constant matrix B into Y register Calculates AB and stores result in matrix D Recalls d the weight of cabbage for the first week Recalls d the weight of cabbage for the second week Recalls d3 Recalls d gt Recalls dy Recalls dy3 Deactivates User mode f MATRIX 1 before beginning to store the elements of matrix B This is because after you stored the last element of matrix A the row and column numbers in Ro and R were automatically reset to 1 Section 12 Calculating with Matrices 159 Silas deliveries were
267. ulator between problems If you enter a digit incorrectly press to undo the mistake then key in the correct number To Compute Keystrokes Display 9 6 3 9 ENTER 6 3 0000 9x6 54 9 ENTER 6 x 54 0000 9 6 1 5 9 ENTER 6 1 5000 9 531 441 9 ENTER 6 D7 531 441 0000 Notice that in the four examples Both numbers are in the calculator before you press the function key e ENTER is used only to separate two numbers that are keyed in one after the other Pressing a numeric function key in this case LX L J or L J executes the function immediately and displays the result To see the close relationship between manual and programmed problem solving let s first calculate the solution to a problem manually that is from the keyboard Then we ll use a program to calculate the solution to the same problem with different data 14 The HP 15C A Problem Solver The time an object takes to fall to the ground ignoring air friction is given by the formula where tf time in seconds h height in meters g the acceleration due to gravity 9 8 m s Example Compute the time taken by a stone falling from the top of the Eiffel Tower 300 51 meters high to the earth Keystrokes Display 300 51 ENTER 300 5100 Enter h 2 x 601 0200 Calculates 2h 9 8 61 3286 2h g VX 7 8313
268. ult that appears to be inconsistent with your theoretical expectation Appendix D A Detailed Look at SOLVE 223 If a calculation has a result whose magnitude is smaller than 1 000000000x10 the result is set equal to zero This effect is referred to as underflow If the subroutine that calculates your function encounters underflow for a range of x and if this affects the value of the function then a root in this range may be expected to have some inaccuracy For example the equation x 0 has a root at x 0 Because of underflow SOLVE produces a root of 1 5060 25 for initial estimates of 1 and 2 As another example consider the equation 1 x 0 whose root is infinite in value Because of underflow gives a root of 3 1707 49 for initial estimates of 10 and 20 In each of these examples the algorithm has found a value of x for which the calculated function value equals zero By understanding the effect of underflow you can readily interpret results such as these The accuracy of a computed value sometimes can be adversely affected by round off error by which an infinitely precise number is rounded to 10 significant digits If your subroutine requires extra precision to properly calculate the function for a range of x the result obtained by may be inaccurate For example the equation Ix 5l 0 has a root at x J5 Because no 10 digit number exactly equals V5 the result of using SOLVE is Error 8 f
269. unts according to the formula FV PV 1 D where FV is future value PV is present value i is the periodic interest rate and n is the number of periods Enter PV first into the Y register and n second into the X register before executing the program Given is an annual interest rate of 7 5 so i 0 075 88 Section 7 Program Editing Keystrokes Display f LBL L 1 001 42 21 1 f LFIX J2 002 42 7 2 1 003 1 004 48 0 005 0 Interest 7 006 7 5 007 5 X 008 34 y 009 14 1 3 x 010 20 PV 1 3 g RTN 011 43 32 Load the program and find the future value of 1 000 invested for 5 years of 2 300 invested for 4 years Remember to use GSB to run a program with a digit label Answers 1 435 63 3 071 58 Alter the program to make the annual interest rate 8 0 Using the edited program find the future value of 500 invested for 4 years of 2 000 invested for 10 years Answers 680 24 4 317 85 Create a program to calculate the length of a chord subtended by an angle 0 in degrees on a circle of radius r according to the equation e 2r sin E 2 s 2 No Find 2 when 6 30 and r 25 Answer 12 9410 A possible p program is Lf JILBL LA L9 DEG f LFIX 4 2 x xsy 2 SIN Lx L9 J RTN Assumes Oin Y register and r in X Ti when program is run
270. up Continuous Memory reset this equals 21 registers This pool contains at least three registers at all times Ry Ro and R e The common pool contains uncommitted registers available for allocation to programming matrices the imaginary stack and SOLVE and operation At power up there are 46 uncommitted registers in the common pool The use of SOLVE 4 Complex mode or matrices temporarily requires extra memory space as explained later in this appendix 213 214 Appendix C Memory Allocation MEMORY Permanent Index Register Index Registe Allocatable R2 ERE Sr Rg Hees DATA STORAGE POOL R3 to R gg allocated Ro ae here Initial config uration dd 19 ies Highest numbered Rg data register dd gt R g MOVABLE BOUNDARY after R gq Initially Ria gt oe COMMON POOL Matrix Elements Imaginary Stack SOLVE and Program Lines Number of uncommitted registers uu Number of registers g aa by program ines pp Total allocatable memory 64 registers numbered R through Res dd 1 uu pp matrix elements imaginary stack SOLVE and 64 For memory allocation and indirect addressing data registers R o through R 9 are referred to as Rj through Ro Appendix C Memory Allocation 215 Memory Status MEM To view the current memory configuration of the calculator press 9 MEM memory holding to
271. urn to the main test screen e Press 3 to perform the keyboard test You then need to press EVERY key on the keyboard until all the keys have been pressed at least once the screen will progressively turn off You can press the keys in any order and any number of times Once all the keys have been pressed and the screen is clear press on any key to return to the test screen Press ON to exit the test system This will also turn the calculator off If the calculator detects an error at any point it will display an error message If you still experience difficulty write or telephone Hewlett Packard at an address or phone number listed on the web at www hp com support Function Summary and Index ON Turns the calculator s display on and off page 18 It is also used in resetting Continuous Memory page 63 changing the digit separator page 61 and in various tests of the calculator s operation pages 261 Complex Functions Real exchange imaginary Activates Complex mode establishing an imaginary stack and exchanges the real and imaginary X registers page 124 I Used to enter complex numbers Activates Complex mode establishing an imaginary stack page 121 Also used with DIM to indirectly dimension matrices page 174 For Index register functions refer to Index Register Control keys page 263 i Displays the contents of the imagin
272. utine using GTO section 7 Labels Labels in a program or subroutine are markers telling the calculator where to begin execution Following an _f_ label or GSB label instruction the calculator will search downward in program memory for the For memory allocation and indirect addressing registers R o through R v are referred to as Rio through Rjpo 78 Section 6 Programming Basics corresponding label If need be the search will wrap around at the end of program memory and continue at line 000 When it encounters an appropriate label the search stops and execution begins If a label is encountered as part of a running program it has no effect that is execution simply continues Therefore you can label a subordinate routine within a program more on subroutines in section 9 Since the calculator searches in only one direction from its present position it is possible though not advisable to use duplicate program labels Execution will begin at the first appropriately labeled line encountered 000 ig If an f LA entry starts the search f f LBLITA for A here it then proceeds downward through f JILBL 3 memory wraps around to line 000 and stops at label A Execution then R S to starts and continues ignoring any tA other labels until a halt inst
273. x As an aid for examining the behavior of a function you can easily evaluate the function at one or more values of x using your subroutine in program memory To do this fill the stack with x Execute the subroutine to calculate the value of the function press f letter label or GSB label The values you calculate can be plotted to give you a graph of the function This procedure is particularly useful for a function whose behavior you do not know A simple looking function may have a graph with relatively extreme variations that you might not anticipate A root that occurs near a localized variation may be hard to find unless you specify initial estimates that are close to the root If you have no informed or intuitive concept of the nature of the function or the location of the zero you are seeking you can search for a solution using trial and error The success of finding a solution depends partially upon the function itself Trial and error is often but not always successful Ifyou specify two moderately large positive or negative estimates and the function s graph does not have a horizontal asymptote the routine will seek a zero which might be the most positive or negative unless the function oscillates many times as the trigonometric functions do e If you have already found a zero of the function you can check for another solution by specifying estimates that are relatively distant fro
274. x Provided that f x does not vary rapidly a consideration that will be discussed in more detail later in this appendix 242 Appendix E A Detailed Look at Calculate the integral in the expression for J 1 mT cosa sin dO 0 First switch to Program mode and key in a subroutine that evaluates the function f cos 40 sin 0 Keystrokes g P R f CLEAR f LBL A x COS xs SIN 0 g RTN Display 000 PRGM 000 Program mode 001 42 21 0 002 003 004 005 006 007 008 4 20 34 23 30 24 43 32 Now switch to Run mode and key the limits of integration into the X and Y registers Be sure the trigonometric mode is set to Radians and set the display format to SCI 2 Finally press f L4 J0 to calculate the integral Keystrokes g P R E NTER g T g RAD SCI 2 i xy 0 Display 0 0000 3 1416 3 1416 3 14 00 7 79 03 1 45 03 Run mode Keys lower limit into Y register Keys upper limit into X register Sets the trigonometric mode to Radians Sets display format to SCI J 2 Integral approximated in SCI 2 Uncertainty of SCI J 2 approximation Appendix E A Detailed Look at 243 The uncert
275. x where 62 x is the uncertainty associated with f x that is caused by the approximation to the actual physical situation Since f x f x 6 x the function you want to integrate is F x f x 8 x 85 x or F f a 8 where 6 x is the net uncertainty associated with f x Therefore the integral you want is b Bes P x dx Lf x 6 4x dx ba b fdr dedx J A b where 7 is the approximation to F x dx and A is the uncertainty a associated with the approximation The L algorithm places the number 7 in the X register and the number A in the Y register The uncertainty 5 x of f x the function calculated by your subroutine is determined as follows Suppose you consider three significant digits of the function s values to be accurate so you set the display format to 2 The display would then show only the accurate digits in the mantissa of a function s values for example 1 23 04 Since the display format rounds the number in the X register to the number displayed this implies that the uncertainty in the function s values is 0 005x10 0 5x10 x10 0 5x10 Thus setting the display Appendix E A Detailed Look at 247 format to SCI n or n where n is an integer implies that the uncertainty in the function s values is 8 x 0 5x10 x10 0 5x107 In this formula n is the number of digits specified in the display format and m x is the exp
276. xponents 19 20 one number functions 22 two number functions 22 29 L Labels 67 77 90 97 LAST X register 35 in matrix functions 174 176 operations saved by 212 putting constants in 39 40 to correct statistics data 52 Linear equations solving with matrices 138 156 Linear estimation r 55 56 Linear regression L R 54 Loading the stack with constants 39 41 Logarithmic functions common and natural 28 Loop control number 109 116 Looping 90 98 Low power indication 62 259 LU decomposition 148 155 156 160 Lukasiewicz Jan 32 M Mantissa displaying full 10 digits 60 Matrix complex 160 163 copying 149 descriptors 139 147 160 in Rj 173 174 Subject Index 277 dimensioning 140 142 142 174 dimensions displaying 142 147 equation complex 168 memory 140 171 name See Matrix descriptors partitioned 161 164 Matrix elements accessing individually 145 147 displaying 144 storing and recalling 143 144 147 149 176 Matrix functions using Ry 173 174 using registers 173 arithmetic 153 conditional 177 inverse 150 154 multiplication 154 one matrix 149 151 programmed 176 177 reciprocal 150 residual 159 row norm 150 177 summary 177 179 transpose 150 151 154 Mean Lx 53 MEM 215 Memory allocation 76 215 217 availability 75 77 213 215 configuration initial 75 76 distribution 75 213 214
277. y 123 4567 9 JUNT 123 0000 9 JLSTx CHS L9 JUNT 123 0000 Reversing the sign does not alter digits 9 JLSTx J FRAC 0 4567 1 23456789 CHS g JIRND 1 2346 f CLEAR PREFIX 1234600000 Temporarily displays all release 1 2346 digits in the mantissa g LABS 1 2346 One Number Functions One number math functions in the HP 15C operate only upon the number in the display X register General Functions Reciprocal Pressing L x calculates the reciprocal of the number in the display Factorial and Gamma Pressing f Lx calculates the factorial of the displayed value where x is an integer 0 lt x lt 69 You can also use x to calculate the Gamma function T x used in advanced mathematics and statistics Pressing f Lx calculates r x 1 so you must subtract 1 from your initial operand to get I x For the Gamma function x is not restricted to nonnegative integers Square Root Pressing Vx calculates the positive square root of the number in the display Squaring Pressing _9 Lx2 calculates the square of the number in the display Keystrokes Display 25 Vx 0 0400 8 Lf Lx 40 320 0000 Calculates 8 or T 9 3 9 yx 1 9748 12 3 9 x 151 2900 26 Section 2 Numeric Functions Trigonometric Operations Trigonometric Modes The trigono
278. y For the SCI 3 approximation the algorithm did not have to consider more sample points than it did in SCI 2 so it did not take any longer to calculate the integral Often however increasing the number of digits in the display format will require evaluating the function at additional sample points so that calculating the integral will take more time Now calculate the same integral in SCI J 4 Keystrokes Display f LSCI 4 7 7858 03 SCI 4 display R LRY 3 1416 00 Rolls down stack until upper limit appears in X register f 4 0 7 7807 03 Integral approximated in SCI 4 Appendix E A Detailed Look at 245 This approximation took about twice as long as the approximation in SCI 3 or LSCI J 2 In this case the algorithm had to evaluate the function at about twice as many sample points as before in order to achieve an approximation of acceptable accuracy Note however that you received a reward for your patience the accuracy of this approximation is better by almost two digits than the accuracy of the approximation calculated using half the number of sample points The preceding examples show that repeating the approximation of an integral in a different display format sometimes will give you a more accurate answer but sometimes it will not Whether or not the accuracy is changed depends on th
279. y a matrix of the form shown for Z the transformations used for multiplying and inverting a complex matrix presume that the matrix is represented by a matrix of the form shown for Z The HP 15C provides two transformations that convert the representation of a complex matrix between Z and Z Pressing Transforms Into f J Pyx Zz Zz J Cyx z ze To do either of these transformations recall the descriptor of Z or Z into the display then press the keys shown above The transformation is done to the specified matrix the result matrix is not affected Section 12 Calculating with Matrices 163 Example Store the complex matrix 443i 7 2i Z 145i 348i in the form Te since it is written in a form that shows Z Then transform Z into the form Z You can do this by storing the elements of Z in matrix A and then using the function where poe AS 2 1153 8f Keystrokes Display f_ MATRIX 0 Clears all matrices 2 ENTER 4 4 0000 Dimensions matrix A to be f DIMJLA 2x4 f_ MATRIX 1 4 0000 Sets beginning row and column numbers in Rg and R to l f USER 4 0000 Activates User mode 4 STOJLA 4 0000 Stores a11 3 STOJLA 3 0000 Stores ay 7 STOJLA 7 0000 Stores a43 2 CHS STOJLA 2 0000 Stores ay4 1 LSTOJLA 1 0000 S
280. yap poMdy Prx M n M MY y P n x Xx n M where M n amp x 2x N ny Zy P nXxy Xxdy A and B are the values returned by the operation L RJ where y Ax B Appendix A Error Conditions 207 Error 3 Improper Register Number or Matrix Element Storage register named is nonexistent or matrix element indicated is nonexistent Error 4 Improper Line Number or Label Call Line number called for is currently unoccupied or nonexistent gt 448 or you have attempted to load a program line without available space or the label called does not exist or User mode is on and you did not press Lf before vx Le 10 L or x Error 5 Subroutine Level Too Deep Subroutine nested more than seven deep Error 6 Improper Flag Number Attempted a flag number gt 9 Error 7 Recursive SOLVE or 7 A subroutine which is called by SOLVE also contains a SOLVE instruction a subroutine which is called by J also contains an instruction Error 8 No Root SOLVE unable to find a root using given estimates Error 9 Service Self test discovered circuitry problem or wrong key pressed during key test Error 10 Insufficient Memory There is not enough memory available to perform a given operation Error 11 Improper Matrix Argument Inconsistent or improper matrix argument
281. zP X Real Part Y ImaginaryPart The superscript P signifies that the complex matrix is represented by a partitioned matrix All of the elements of Z are real numbers those in the upper half represent the elements of the real part matrix X those in the lower half represent the elements of the imaginary part matrix Y The elements of Z are stored in one of the five matrices A for example in the usual manner as described earlier in this section For example if Z X 7Y where x x x 11 a ad e al X21 22 y21 22 then Z can be represented in the calculator by x1 42 xoz Ne m 122 Y zi 2 162 Section 12 Calculating with Matrices Suppose you need to do a calculation with a complex matrix that is not written as the sum of a real matrix and an imaginary matrix as was the matrix Z in the example above but rather written with an entire complex number in each element such as Z iti 12 ee X21 iy21 X22 129 This matrix can be represented in the calculator by a real matrix that looks very similar one that is derived simply by ignoring the i and the sign The 2 x 2 matrix Z shown above for example can be represented in the calculator in complex form by the 2 x 4 matrix c 1 Mia X2 Viz Xa Yn Xn Yan The superscript C signifies that the complex matrix is represented in a complex like form Although a complex matrix can be initially represented in the calculator b

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