A Matrix Multi-Tool for the HP 35s Programmable Calculator
October 17, 2007
This is my second attempt at a large program for the new HP 35s programmable scientific calculator from Hewlett-Packard. My previous program addressed the curve fitting shortcomings of the 35s (compared to the HP-41C/CV/CX with Advantage Pac, or the HP-42S). This program is a start at doing the same for matrix functionality.
The HP-15C was the first calculator to introduce comprehensive support for matrix operations. HP quickly added equivalent functionality to the 41C series with the Advantage Pac, and the 42S came with these operations built in as well. The HP 35s does not have any general purpose matrix operations built in, although it can solve 2×2 and 3×3 linear systems.
The program I present here is a long way from being as powerful as the facilities provided by the 15C, 41C/Advantage, or 42S, but it does provide several useful matrix operations in one simple tool. It is loosely modelled after the Advantage Pac matrix program, which provides an easy-to-use subset of the functionality of the matrix library.
What it Does
Given an N×N matrix A, and an N-element vector b, the matrix multi-tool will do the following:
- Compute A^{-1}, the inverse of A.
- Compute the determinant of A.
- Solve the system of linear equations, Ax=b, giving column vector x.
- Quickly solve additional Ax=b systems for different b vectors.
- Perform matrix-vector multiplication of A and b.
The matrix multi-tool uses Gaussian elimination with partial pivoting to intially compute A^{-1} and simultaneously solve Ax=b. The determinant is also computed during this operation. The inverse is left in A, and x is left in b. When solving additional systems of equations for the same A and different b vectors, the program evaluates A^{-1}b to yield x.
If you’re only interested in A^{-1} and/or A‘s determinant, you can omit entering values for b and ignore the solution of x.
You can also use the built-in matrix-vector multiplication routine directly if you just want to multiply a matrix by one or more vectors.
Thanks to the HP 35s’ support for complex numbers, the matrix multi-tool works with complex matrices too.
Using the Program
Before running the matrix multi-tool, make sure flag zero is clear by pressing FLAGS CF 0
(the program doesn’t clear this flag itself because it is used to maintain information between invocations).
Start the program by pressing XEQ M ENTER
. This will display the main menu, which looks like this:
1A2b 3SoL4Un5×
The menu choices are:
- Go to dimension/edit/view menu for matrix A.
- Go to edit/view menu for column vector b.
- Compute inverse and determinant of A, and solve Ax=b for x.
- Unsolve, restoring original A and most recent b.
- Multiply matrix A by vector b.
Pressing R/S
without selecting a choice exits the matrix program (this is important, to ensure flag 10 is cleared so equations will work in other programs).
Entering Matrix A
Press 1 R/S
from the main menu to display the matrix A menu:
1DIM 2ED 3VIEW
The menu choices are:
- Specify the dimension N for N×N matrix A and N-vector b.
- Edit the entries in matrix A.
- View the entries in A.
Pressing R/S
without selecting a choice will return you to the main menu.
Specifying the Matrix Dimension
Press 1 R/S
to specify the dimensions of N×N matrix A, and consequently N-vector b. The program will display:
N=
3.0000
The currently specified dimension is displayed. Enter the desired dimension and press R/S
(or just press R/S
to keep the existing dimension). The lowest allowed dimension is 2 and the highest depends on the amount of free memory. If the matrix multi-tool program is the only one in memory, you can work with a matrix of dimension up to 19×19.
If you enter a dimension that is too small, the program will display N TO SMALL
. Pressing R/S
will return you to the matrix A menu. If you enter a dimension that is too large, the calculator will display INVALID (I)
and the program will halt. You’ll have to restart it by pressing XEQ M ENTER
.
After you’ve specified the dimension, the program will automatically proceed to the matrix editing mode described below.
Editing the Matrix A
To edit the entries of A, press 2 R/S
from the matrix A menu. The program will begin in the upper left corner of the matrix and proceed left to right, top to bottom (the same way you read English text). For each entry, it will first briefly display the indices of that entry (the number above the indices is N):
2.0000
[1.0000,1.0000]
After a brief pause, execution will automatically continue, and the program will display the existing value and prompt for the new value of the entry whose indices were just displayed:
A?
1.2345
To keep the existing value, just press R/S
. Otherwise, enter the new value before pressing R/S
. If you’ve forgotten which entry you are about to change, you can press the x
↔y
key to retrieve the row and column indices to the display. If you’ve already performed calculations that have altered the stack, just press R
↓ a few times. The indices are probably still on the stack (as a bracketed vector).
At any time during editing, you can change which entry is being edited. When the program is stopped awaiting a new value for an entry, you can store new row and column indices into memory registers R
and C
respectively. When you then enter a new value and press R/S
, it will be stored in the newly specified location and editing will resume from there.
After editing the last entry of A, the program will return to the matrix A menu.
Viewing the Matrix A or A^{-1}
To examine A (or result matrix A^{-1}), press 3 R/S
from the matrix A menu. The program will proceed in the same way as during editing, first briefly displaying the indices of each entry. After a brief pause, the program will display the value of the entry whose indices were just displayed:
A=
-3.1416
Press R/S
to proceed to the next entry. If you store new row and column indices into memory registers R
and C
while the program is stopped, the next entry displayed will be the one whose indices you specified, and viewing will continue from there.
When you’ve examined the last entry in A, the program will return to the matrix A menu.
Entering the Vector b
Pressing 2 R/S
from the main menu will display the vector b menu:
2ED 3VIEW
The menu choices are:
- Edit the entries in vector b.
- View the entries in b.
Note that there is no choice number 1. The choices are numbered 2 and 3 for consistency with the menu for matrix A.
Pressing R/S
without selecting a choice will return you to the main menu.
Editing the Vector b
Press 2 R/S
from the vector b menu to edit the entries of b. The program will begin at the top of b. Editing b proceeds the same way as editing A, except that there is only a single index (the row). During editing, you can change which entry is being edited by storing a new row number in memory register R
.
After editing the last entry of b, the program will return to the vector b menu.
Viewing the Vector b or x
To examine b (or solution vector x after solving the linear system), press 3 R/S
from the vector b menu. Viewing b (or x) proceeds the same way as viewing A, except that there is only a single index. During viewing, you can change which entry is to be viewed next by storing a new row number in memory register R
.
Solving the System Ax=b
If you’re at the matrix A or vector b menus, press R/S
to return to the main menu. Then press 3 R/S
to solve the system of equations. First, the program will display a time estimate (in seconds):
T=
4.9680
Press R/S
to continue. After a while (a long while for a really large system), the matrix multi-tool will display the computed determinant:
D=
1278.3225
Press R/S
to return to the main menu. The results can be found as follows:
- A^{-1} will have replaced A. You can examine A^{-1} by pressing
1 R/S
at the main menu, and then3 R/S
at the matrix A menu. - Solution vector x will have replaced b. You can examine x by pressing
2 R/S
at the main menu, and then3 R/S
at the vector b menu. - The determinant is stored in memory register
D
, which you can examine by pressingRCL D
.
Sometimes there is no solution. If this turns out to be the case, the program will display the message SINGULAR
. Pressing R/S
will then return you to the main menu.
Solving for Another Vector b
You can solve additional systems for the same matrix A and different vectors b by entering new values for the entries of b, and then running the solver again. So long as A^{-1} has not been overwritten by editing A, the matrix multi-tool will solve these additional systems by just computing x = A^{-1}b, which is much faster (O(N^{2})) than the full solution process (O(N^{3})).
Restoring A and b
If you have not overwritten A^{-1} or x since the last system you’ve solved, you can undo the solution by pressing 4 R/S
from the main menu. This will restore matrix A and the most recently entered vector b. If you try to perform the undo operation when A^{-1} or x have been modified, the program displays the message CANNOT UNDO
and returns to the main menu.
Performing Matrix-Vector Multiplication
The built-in matrix-vector multiplication routine that the matrix multi-tool uses to solve additional systems when matrix A hasn’t changed is also available directly for your use. After entering A and b as described earlier, pressing 5 R/S
from the main menu will compute the matrix-vector product of A and b.
The result vector will replace b, and can be examined the same way b can, by pressing 2 R/S
at the main menu, and then 3 R/S
at the vector b menu. The matrix A is unaltered. Another multiplication of A by a different b vector can be carried out by entering the new vector and invoking the multiplication.
Note that the undo operation (4 R/S
from the main menu) does not apply to matrix-vector multiplication.
Example
Solve the following system of equations:
3.8x_{1} + 7.2x_{2} = 16.5
1.3x_{1} – 0.9x_{2} = -22.1
Keystrokes | Display | Description |
---|---|---|
XEQ M ENTER | 1A2b 3SoL4Un5× | Start the matrix multi-tool |
1 R/S | 1DIM 2ED 3VIEW | Select matrix A |
1 R/S | N? | Select the DIMension command |
2 R/S | [1,1] A? |
Dimension is 2×2 |
3.8 R/S | [1,2] A? |
A_{11}=3.8 |
7.2 R/S | [2,1] A? |
A_{12}=7.2 |
1.3 R/S | [2,2] A? |
A_{21}=1.3 |
0.9 +/- R/S | 1DIM 2ED 3VIEW | A_{22}=-0.9 Return to A menu |
R/S | 1A2b 3SoL4Un5× | Return to main menu |
2 R/S | 2ED 3VIEW | Select vector b |
2 R/S | [1] B? |
EDit command |
16.5 R/S | [2] B? |
b_{1}=16.5 |
22.1 +/- R/S | 2ED 3VIEW | b_{2}=-22.1 Return to b menu |
R/S | 1A2b 3SoL4Un5× | Return to main menu |
3 R/S | T= 1.4720 | Start solver and display time estimate |
R/S | D= -12.7800 | Continue solver and display determinant |
R/S | 1A2b 3SoL4Un5× | Solution completed |
2 R/S | 2ED 3VIEW | Select vector b (now x) |
3 R/S | [1] B= -11.2887 |
Select the VIEW command x_{1}=-11.2887 |
R/S | [2] B= 8.2496 |
x_{2}=8.2496 |
R/S | 2ED 3VIEW | Return to b menu |
R/S | 1A2b 3SoL4Un5× | Return to main menu |
Performance
Compared to the built-in matrix operations of an HP-15C or HP-42S, the matrix multi-tool is quite slow, but still quite usable for real problems. Furthermore, due to the larger built-in memory of the HP 35s, it can solve larger problems than the 15C or 42S can. The following table gives the approximate times for solving different sizes of systems using the matrix multi-tool:
Size (N) | Solve Ax = b | Compute x = A^{-1}b or Multiply Ab |
---|---|---|
2 | 4 sec | 1 sec |
3 | 8 sec | 2 sec |
4 | 14 sec | 3 sec |
5 | 26 sec | 4 sec |
6 | 43 sec | 5 sec |
7 | 1 min | 7 sec |
8 | 1½ min | 8 sec |
9 | 2 min | 10 sec |
10 | 3 min | 13 sec |
11 | 4 min | 15 sec |
12 | 5½ min | 18 sec |
13 | 7 min | 21 sec |
14 | 8½ min | 25 sec |
15 | 10½ min | 28 sec |
16 | 13 min | 32 sec |
17 | 15½ min | 36 sec |
18 | 18 min | 41 sec |
Program Listing
Line | Instruction | Comments |
---|---|---|
M001♦ | LBL M | Display main menu |
M002 | CLx | |
M003 | SF 10 | |
M004♦ | EQN 1A2b 3SoL4Un5× | Text of main menu (see note 1) |
M005 | CF 10 | |
M006 | x=0? | |
M007 | RTN | |
M008 | 1 | |
M009 | x=y? | |
M010 | GTO M036 | Go to dimension/enter/view A menu |
M011 | R↓ | |
M012 | 2 | |
M013 | x=y? | |
M014 | GTO M057 | Go to enter/view b menu |
M015 | R↓ | |
M016 | 3 | |
M017 | x≠y? | |
M018 | GTO M021 | Not the solve command |
M019 | XEQ M208 | Solve Ax = b |
M020 | GTO M001 | Return to main menu |
M021♦ | R↓ | |
M022 | 4 | |
M023 | x≠y? | |
M024 | GTO M027 | Not the undo command |
M025 | XEQ M183 | Restore A and b |
M026 | GTO M001 | Return to main menu |
M027♦ | R↓ | |
M028 | 5 | |
M029 | x≠y? | |
M030 | GTO M034 | Not the multiply command |
M031 | XEQ M396 | Multiply A and b |
M032 | CF 2 | Vector b has been overwritten |
M033 | GTO M001 | Return to main menu |
M034♦ | XEQ M179 | Invalid command |
M035 | GTO M001 | Return to main menu |
M036♦ | CLx | |
M037 | SF 10 | |
M038 | EQN 1DIM 2ED 3VIEW | |
M039 | CF 10 | |
M040 | x=0? | |
M041 | GTO M001 | Return to main menu |
M042 | SF 1 | |
M043 | 1 | |
M044 | x=y? | |
M045 | GTO M074 | Dimension A |
M046 | R↓ | |
M047 | 2 | |
M048 | x=y? | |
M049 | GTO M095 | Edit A |
M050 | R↓ | |
M051 | CF 1 | |
M052 | 3 | |
M053 | x=y? | |
M054 | GTO M095 | View A |
M055 | XEQ M179 | Invalid command |
M056 | GTO M036 | Return to A menu |
M057♦ | CLx | |
M058 | SF 10 | |
M059 | EQN 2ED 3VIEW | |
M060 | CF 10 | |
M061 | x=0? | |
M062 | GTO M001 | Return to main menu |
M063 | SF 1 | |
M064 | 2 | |
M065 | x=y? | |
M066 | GTO M141 | Edit b |
M067 | R↓ | |
M068 | CF 1 | |
M069 | 3 | |
M070 | x=y? | |
M071 | GTO M141 | View b |
M072 | XEQ M179 | Invalid command |
M073 | GTO M057 | Return to b menu |
M074♦ | INPUT N | Prompt for new size (default is old size) |
M075 | IP | |
M076 | 2 | |
M077 | x>y? | Is dimension at least 2? |
M078 | GTO M081 | Dimension is too small |
M079 | XEQ M085 | Set up memory |
M080 | GTO M095 | Proceed to matrix A editing mode |
M081♦ | SF 10 | Dimension is too small |
M082 | EQN N TOO SMALL | |
M083 | CF 10 | |
M084 | GTO M036 | Return to A menu |
M085♦ | RCL N | Compute fence location 2N^2+N = N(2N+1) |
M086 | ENTER | |
M087 | ENTER | |
M088 | + | 2N |
M089 | 1 | |
M090 | + | 2N+1 |
M091 | x | N(2N+1) |
M092 | STO I | |
M093 | STO (I) | Set up matrix end-of-memory fence |
M094 | RTN | |
M095♦ | 1 | Initialize row and column indices |
M096 | STO R | |
M097 | STO C | |
M098♦ | XEQ M133 | Compute index of A[R,C] |
M099 | STO I | |
M100 | RCL R | |
M101 | RCL N | |
M102 | x<y? | Is it past the bottom of A? |
M103 | GTO M036 | If so, return to A menu |
M104 | EQN [R,C] | Flash [row,col] on display |
M105 | PSE | |
M106 | FS? 1 | Are we editing? |
M107 | GTO M119 | Skip pre-increment of R,C |
M108♦ | 1 | Increment row/column before view |
M109 | STO+ C | |
M110 | RCL C | Is C still within [1..N]? |
M111 | RCL− N | |
M112 | x≤0? | |
M113 | GTO M116 | If yes, don’t increment row |
M114 | STO C | Reset column to 1 |
M115 | STO+ R | Compute next row index |
M116♦ | FS? 1 | Were we called by edit’s post-increment? |
M117 | RTN | If so, return |
M118 | REGZ | Bring [R,C] back into x register (see note 2) |
M119♦ | RCL (I) | Fetch matrix element |
M120 | STO A | Make it the default |
M121 | FS? 1 | |
M122 | GTO M125 | Edit entry |
M123 | VIEW A | |
M124 | GTO M098 | Skip editing code |
M125♦ | INPUT A | Prompt for A[R,C] |
M126 | CF 0 | Previous solve is no longer undoable |
M127 | XEQ M133 | Recompute index in case user changed it |
M128 | STO I | |
M129 | R↓ | Retrieve value to store in A[R,C] |
M130 | STO (I) | |
M131 | XEQ M108 | Increment row/column after edit |
M132 | GTO M098 | Process next entry of A |
M133♦ | RCL C | Compute index of A[R,C] |
M134 | 1 | |
M135 | − | |
M136 | RCL× N | N(C-1) |
M137 | RCL+ R | |
M138 | 1 | |
M139 | − | N(C-1) + R-1 |
M140 | RTN | |
M141♦ | 1 | Initialize row index |
M142 | STO R | |
M143♦ | XEQ M171 | Compute index of b[R] |
M144 | STO I | |
M145 | RCL R | |
M146 | RCL N | |
M147 | x<y? | Is R past the end of b? |
M148 | GTO M057 | If so, return to b menu |
M149 | EQN [R] | Flash [row] on display |
M150 | PSE | |
M151 | FS? 1 | Are we editing? |
M152 | GTO M156 | Skip pre-increment of R,C |
M153 | 1 | Increment row before view |
M154 | STO+ R | |
M155 | R↓ | Bring [R] back into x register |
M156♦ | RCL (I) | Fetch vector element |
M157 | STO B | Make it the default |
M158 | FS? 1 | |
M159 | GTO M162 | Edit entry |
M160 | VIEW B | |
M161 | GTO M143 | Skip editing code |
M162♦ | INPUT B | Prompt for b[R] |
M163 | CF 2 | Previous solve no longer undoable |
M164 | XEQ M171 | Recompute index in case user changed it |
M165 | STO I | |
M166 | R↓ | Retrieve value to store in b[R] |
M167 | STO (I) | |
M168 | 1 | |
M169 | STO+ R | Increment row after edit |
M170 | GTO M143 | Process next entry of b |
M171♦ | RCL N | Compute index of b[R] |
M172 | x^{2} | |
M173 | ENTER | |
M174 | + | 2N^2 – beginning of b |
M175 | RCL+ R | |
M176 | 1 | |
M177 | − | 2N^2 + R-1 |
M178 | RTN | |
M179♦ | SF 10 | |
M180 | EQN INVALID CMD | |
M181 | CF 10 | |
M182 | RTN | |
M183♦ | FS? 0 | Flag 0 has to be set to be able to undo |
M184 | GTO M186 | If it is, go check flag 2 |
M185 | GTO M188 | Can’t undo if flag 0 not set |
M186♦ | FS? 2 | Flag 2 also has to be set for undo |
M187 | GTO M192 | If it is, use Gaussian elimination to undo |
M188♦ | SF 10 | Display error message |
M189 | EQN CANNOT UNDO | |
M190 | CF 10 | |
M191 | RTN | Return to main menu |
M192♦ | XEQ M215 | Call Gaussian elimination routine |
M193 | RCL D | Restore determinant to pre-undo value |
M194 | 1/x | |
M195 | STO D | |
M196 | CF 0 | Clear flag 0 since we’ve unsolved back to A |
M197 | CF 2 | Clear flag 2 since we’re back to b |
M198 | RTN | Return to main menu |
M199♦ | RCL N | Routine to set up I, J, and E for solve or multiply |
M200 | x^{2} | N^2 |
M201 | STO I | Set I to N^2 for counting |
M202 | ENTER | |
M203 | + | |
M204 | STO J | Set J to 2N^2 for indexing |
M205 | RCL+ N | 2N^2+N |
M206 | STO E | Set end of memory pointer to 2N^2 + N |
M207 | RTN | |
M208♦ | FS? 0 | Do we still have A’ intact? |
M209 | GTO M392 | Use matrix-vector multiplication to solve |
M210 | XEQ M215 | Call Gaussian elimination |
M211 | SF 0 | Set flag 0 indicating we have A’ |
M212 | SF 2 | Set flag 2 indicating we have x |
M213 | VIEW D | Display determinant |
M214 | RTN | |
M215♦ | RCL N | Estimate time for Gaussian elimination (see note 3) |
M216 | 3 | |
M217 | y^{x} | |
M218 | 0.184 | |
M219 | x | |
M220 | STO T | |
M221 | VIEW T | Display estimate and make program stoppable |
M222♦ | XEQ M199 | Set up I, J, and E for solver |
M223 | CLx | Fill temporary matrix with 0 |
M224♦ | DSE J | Decrement J (will never skip) |
M225 | STO (J) | |
M226 | DSE I | Decrement count |
M227 | GTO M224 | Fill in next 0 entry |
M228♦ | 1 | Fill diagonal of temporary matrix with 1 |
M229 | STO (J) | |
M230 | RCL+ N | N+1 |
M231 | STO+ J | Next diagonal entry of temporary matrix |
M232 | RCL J | |
M233 | RCL E | |
M234 | x>y? | |
M235 | GTO M228 | Fill in next 1 entry |
M236 | 1 | |
M237 | STO D | Initialize determinant to 1 |
M238 | CLx | |
M239 | STO K | Index of first diagonal entry is 0 |
M240 | RCL N | N >= 2 |
M241 | STO R | Number of rows remaining to process |
M242♦ | RCL R | Start of Gaussian elimination |
M243 | 1 | |
M244 | x≥y? | |
M245 | GTO M325 | There’s nothing to do for the last row |
M246 | − | |
M247 | STO R | |
M248 | STO P | Find row with largest value in current column |
M249 | RCL K | Start at current diagonal entry |
M250 | STO I | |
M251 | STO J | |
M252 | RCL (I) | Save current entry as largest seen so far |
M253 | ABS | |
M254 | STO T | |
M255♦ | 1 | Point to next row |
M256 | STO+ I | |
M257 | RCL T | |
M258 | RCL (I) | |
M259 | ABS | |
M260 | x≤y? | Is new entry larger than largest so far? |
M261 | GTO M265 | No, don’t save new entry |
M262 | STO T | Save new largest seen so far |
M263 | RCL I | Save index of largest seen so far |
M264 | STO J | |
M265♦ | DSE P | |
M266 | GTO M255 | Check next row |
M267 | RCL J | Swap rows if necessary |
M268 | RCL K | |
M269 | x=y? | Is the largest in a row other than the current row? |
M270 | GTO M293 | No, so don’t swap rows |
M271 | STO I | Point to diagonal entry of current row |
M272 | RCL T | |
M273 | x≠0? | |
M274 | GTO M279 | Found a non-zero row; okay to swap |
M275 | SF 10 | |
M276 | EQN SINGULAR | |
M277 | CF 10 | |
M278 | RTN | |
M279♦ | RCL D | Swapping rows flips sign of determinant |
M280 | +/− | |
M281 | STO D | |
M282♦ | RCL E | |
M283 | RCL J | |
M284 | x≥y? | Is column index past end of memory? |
M285 | GTO M293 | Yes, done swapping rows |
M286 | x↔ (I) | Swap entries between (I) and (J) |
M287 | x↔ (J) | |
M288 | x↔ (I) | |
M289 | RCL N | Increment pointers to next column |
M290 | STO+ I | |
M291 | STO+ J | |
M292 | GTO M282 | Swap next column of rows |
M293♦ | RCL R | Subtract multiple of current row from remaining rows |
M294 | STO P | |
M295♦ | RCL K | |
M296 | STO I | Point to diagonal entry of current row |
M297 | RCL+ P | |
M298 | STO J | Point to same column in Pth row |
M299 | RCL (J) | Compute multiplier so entry becomes zero |
M300 | RCL÷ (I) | |
M301 | STO T | |
M302 | CLx | Clear entry in starting column |
M303 | STO (J) | |
M304♦ | RCL N | Increment to next column in both rows |
M305 | STO+ I | |
M306 | STO+ J | |
M307 | RCL E | |
M308 | RCL J | |
M309 | x≥y? | Is column index past end of memory? |
M310 | GTO M315 | Yes, done subtraction |
M311 | RCL (I) | Compute multiple of current row |
M312 | RCL× T | |
M313 | STO− (J) | Subtract from same column of Pth row |
M314 | GTO M304 | Proceed to next column |
M315♦ | DSE P | |
M316 | GTO M295 | Proceed to next row |
M317 | RCL K | Update determinant for current row |
M318 | STO I | |
M319 | RCL (I) | |
M320 | STO× D | |
M321 | 1 | Point to next diagonal entry |
M322 | RCL+ N | |
M323 | STO+ K | |
M324 | GTO M242 | Process next row |
M325♦ | RCL K | Update determinant for last row |
M326 | STO I | |
M327 | RCL (I) | |
M328 | STO× D | |
M329 | RCL N | Start of back substitution |
M330 | STO R | Begin with last (Nth) row |
M331♦ | RCL K | Divide row R by its diagonal entry |
M332 | STO J | |
M333 | RCL (J) | |
M334 | STO T | |
M335 | 1 | Diagonal entry becomes 1 |
M336 | STO (J) | |
M337 | STO P | Initialize counter to be used later |
M338♦ | RCL N | Divide rest of row R by diagonal entry in T |
M339 | STO+ J | Increment pointer to next column |
M340 | RCL E | |
M341 | RCL J | |
M342 | x≥y? | Are we past the end of memory? |
M343 | GTO M347 | Yes, done dividing this row |
M344 | RCL T | Divide entry by T |
M345 | STO÷ (J) | |
M346 | GTO M338 | Proceed to next column of row R |
M347♦ | DSE R | Compute address of previous row |
M348 | GTO M363 | Process previous row |
M349 | RCL N | Done back substitution; swap result into A |
M350 | x^{2} | |
M351 | STO K | Have to move N^2 entries |
M352 | STO J | Point to beginning of temporary matrix |
M353 | CLx | |
M354 | STO I | Point to beginning of A |
M355♦ | RCL (J) | Copy entry from temporary to A |
M356 | STO (I) | |
M357 | 1 | Point to next entry in each |
M358 | STO+ I | |
M359 | STO+ J | |
M360 | DSE K | |
M361 | GTO M355 | Copy next entry |
M362 | RTN | |
M363♦ | RCL K | Get index of diagonal entry of row R |
M364 | STO I | |
M365 | RCL− P | |
M366 | STO J | Index of corresponding entry in row R-P |
M367 | RCL (J) | Fetch and save to use as a multiplier |
M368 | STO T | |
M369 | CLx | Element in this column will become 0 |
M370 | STO (J) | |
M371♦ | RCL N | Update indices to next column in both rows |
M372 | STO+ I | |
M373 | STO+ J | |
M374 | RCL E | |
M375 | RCL J | |
M376 | x≥y? | Are we past the end of memory? |
M377 | GTO M382 | Done subtracting from row R-P |
M378 | RCL (I) | Subtract multiple of row R from row R-P |
M379 | RCL× T | |
M380 | STO− (J) | |
M381 | GTO M371 | Proceed to next column |
M382♦ | 1 | Point to next row to subtract from |
M383 | STO+ P | |
M384 | RCL P | |
M385 | RCL R | |
M386 | x≥y? | |
M387 | GTO M363 | Process next row to subtract from |
M388 | 1 | Point to previous row’s diagonal entry |
M389 | RCL+ N | |
M390 | STO− K | |
M391 | GTO M331 | Process previous row |
M392♦ | XEQ M396 | Call matrix multiplication routine |
M393 | SF 0 | Set flag 0 indicating we have A’ |
M394 | SF 2 | Set flag 2 indicating we have x |
M395 | RTN | |
M396♦ | RCL N | Quick solve by multplication A’b = x |
M397 | x^{2} | Estimate how long this will take (see note 3) |
M398 | 0.125 | |
M399 | x | |
M400 | STO T | |
M401 | VIEW T | Display estimate and make program stoppable |
M402♦ | XEQ M199 | Set up I, J, and E for multiplication |
M403 | RCL N | |
M404 | STO R | Start one past the last row |
M405♦ | CLx | |
M406 | STO T | Accumulate each dot product in T |
M407 | RCL R | |
M408 | 1 | |
M409 | − | |
M410 | STO I | Point to row R of A |
M411 | RCL E | |
M412 | RCL− N | |
M413 | STO J | Point to first row of b |
M414♦ | RCL (I) | Inner multiplication loop |
M415 | RCL× (J) | |
M416 | STO+ T | |
M417 | RCL N | |
M418 | STO+ I | Next column of A |
M419 | 1 | |
M420 | STO+ J | Next row of b |
M421 | RCL E | |
M422 | RCL J | |
M423 | x<y? | Are we past the end of memory (and thus b)? |
M424 | GTO M414 | Process next column of A and row of b |
M425 | RCL T | |
M426 | STO (I) | Store result in empty space between A and b |
M427 | DSE R | |
M428 | GTO M405 | Process next row of A |
M429 | RCL N | Move result into b |
M430 | STO− J | |
M431♦ | RCL (I) | |
M432 | STO (J) | |
M433 | 1 | Increment row counters |
M434 | STO+ I | |
M435 | STO+ J | |
M436 | RCL E | |
M437 | RCL J | |
M438 | x<y? | Are we past the end of memory (and thus b)? |
M439 | GTO M431 | If not, move next row |
M440 | SF 0 | Can once again undo the calculation |
M441 | SF 2 | |
M442 | RTN | Return to main menu |
Length: 1455
, Checksum: 9A4A
Note 1: The keystrokes to enter line M004
are: EQN 1 RCL A 2 L.R. 5 SPACE 3 RCL S BASE 7 RCL L 4 RCL U SUMS 1 5 × ENTER
Note 2: To enter the REGZ
instruction, press any function key that displays a menu of choices (such as FLAGS
), press the R
↓ key, and select z
.
Note 3: There is a bug in the HP 35s (and HP 33s) firmware that if a program last stopped to display an EQN
message, it will be uninterruptible until it stops for some other reason (a VIEW
, INPUT
, or STOP
instruction). By computing a time estimate, we have an excuse to use a VIEW
instruction (to display the time estimate) before entering into a potentially long-running computation.
Registers and Flags
Register | Use |
---|---|
A | Input/display of current entry of matrix A |
B | Input/display of current entry of vector b |
C | Column index (only during editing of A) |
D | Determinant |
E | Address of register beyond end of matrix memory |
I, J | General purpose indexing |
K | Loop control and diagonal traversal |
N | Matrix dimensions (NxN) |
P | Secondary row index |
R | Current row index |
T | Temporary accumulator and time estimate display |
0…N^{2}-1 | Storage for matrix A and inverse matrix A^{-1} |
N^{2}…2N^{2}-1 | Temporary matrix used for inversion |
2N^{2}…2N^{2}+N-1 | Column vector b and solution vector x |
2N^{2}+N | End of memory fence (to ensure memory stays allocated if b[N] = 0) |
Flag | Meaning |
---|---|
0 | Indicates memory contains A^{-1} instead of A. In conjunction with flag 2, indicates that undo is possible. |
1 | Set when editing, cleared when viewing |
2 | Indicates memory contains x instead of b. In conjunction with flag 0, indicates that undo is possible. |
Revision History
2007-Nov-07 — Initial release.
2007-Nov-18 — Made the matrix dimensioning function and solver callable so they can be used by other programs I might write.
2007-Nov-20 — Compute and display time estimate before invoking solvers. This works around the HP 35s bug that programs are not interruptible if the last thing displayed was an EQN
message, which can be disconcerting during an 18 minute matrix inversion.
2008-Apr-03 — Made the matrix multiplication routine accessible from the main menu (5 R/S
), and also callable as a subroutine from other programs.
Calling Matrix Multi-Tool Subroutines from Other Programs
Several subroutines in the matrix multi-tool can be readily used from other programs. Note that the line numbers of these routines might change as I post future revisions to this program.
M085
: Set Up Matrix and Vector Storage
Sets up memory to hold an N×N matrix A, an N×N scratch matrix, and an N-element vector b. The dimension N is expected to be in register N
.
On return, register I
will contain 2N^{2}+N, the indirect address of the first register after the allocated memory, and that register will also contain its own address (to ensure that all the required memory remains allocated even if the last entries in b are zero).
M133
: Compute Index of A_{rc}
Given row and column indices in registers R
and C
respectively, compute the index of A_{rc} in the memory allocated above. Requires that register N
still contains the dimension, N, of matrix A and vector b.
M171
: Compute Index of b_{r}
Given a row index in register R
, compute the index of b_{r} in the memory allocated above. Requires that register N
still contains the dimension, N, of matrix A and vector b.
M222
: Solve Ax=b or Compute A^{-1} by Gaussian Elimination
Solves the system Ax=b for matrix A and vector b. The dimension must be in register N
. Matrix A is expected to be stored column-wise in indirect registers 0 through N^{2}-1. Vector b is expected to be stored in indirect registers 2N^{2} through 2N^{2}+N-1.
On return, A will have been replaced by its inverse (A^{-1}) and b will have been replaced by x. The determinant of A will be in register D
. This subroutine also uses registers E
, I
, J
, K
, P
, R
, and T
, and indirect registers N^{2} through 2N^{2}-1.
An alternate entry point for this subroutine is M215
, which will first display a time estimate (T=
…). The Gaussian elimination will commence only after the user presses R/S
. This works around an HP 35s firmware bug where a program is uninterruptible if the last thing displayed was an EQN
message.
M402
: Compute Matrix-Vector Product Ab
Computes the matrix-vector product Ab for matrix A and vector b. The dimension must be in register N
. Matrix A is expected to be stored column-wise in indirect registers 0 through N^{2}-1. Vector b is expected to be stored in indirect registers 2N^{2} through 2N^{2}+N-1.
On return, A will be unchanged and b will have been replaced by the product Ab. This subroutine also uses registers E
, I
, J
, R
, and T
, and indirect registers N^{2} through N^{2}+N-1.
An alternate entry point for this subroutine is M396
, which will first display a time estimate (T=
…). The matrix-vector multiplication will commence only after the user presses R/S
. Again, this is useful for working around the uninterruptible program bug.
Why I Wrote this Program
One of my first programmable calculators was a Texas Instruments SR-52 that I purchased at Eaton’s for $149 back in 1978 or so. At that time, the TI-59 had already been introduced and the store was clearing out the SR-52. This calculator had 224 steps of program memory, 20 registers, and a magnetic card reader. It was only slightly less powerful than the HP-67.
When I was in the 10th or 11th grade, we learned how to solve systems of linear equations using matrices, Gaussian elimination, and back substitution. After learning the method, it became clear to me that it would be easy to do on a computer, and I decided I’d try to program the SR-52 to do it. My teacher (who also taught computer science), was all in favour of the idea, and even said I could use the program during a test. He figured if I was smart enough to program it, especially in only 224 program steps, I obviously knew the algorithm inside out and backwards. Well, I managed to fit it onto the SR-52, although it was limited to 2×2 and 3×3 systems due to the limited number of storage registers. Of course with only 224 steps, there was no room for a user interface; I had to manually store the matrix elements into the appropriate registers.
Now, about 28 years later, an expanded version of this program came to mind as an ideal second project on the HP 35s, especially since my earlier HP calculators (41CX with Advantage Pac, and 42S) had this capability built in. This program will also prove useful as a subroutine in a polynomial curve fitting program I have planned for the future.
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Palmer O. Hanson, Jr.
January 17, 2008
The UNDO function of this program does NOT return the machine to the exact input values. The UNDO function in machines such as the HP-28S does return the machine to the exact input values.
Matrix A and the inverse matrix are stored in registers 0 through N^2 – 1 as indicated in the text. However, although the elements of the input are entered row by row the elements are stored column by column, e.g., for an nth order matrix A element A(2,1) will be stored in register 1, element A(n,1) will be stored in register n-1, element A(1,2) will be stored in register n, and so on.
Stefan Vorkoetter
January 17, 2008
Palmer, you are correct on both counts. The UNDO function re-inverts the inverse matrix, restoring a close approximation of the original matrix. This may not be the way a 28S does it, but it is the way the Advantage Pac matrix program works.
The matrix elements are stored in column order in memory because it made the index arithmetic simpler (and therefore faster). Although I’m a C programmer and am used to storing things in row order, there’s nothing conceptually different in storing them in column order (which, for example, FORTRAN does).
Palmer O. Hanson, Jr.
January 30, 2008
I need a routine for the HP-35s which will multiply an nxn matrix by a vector. It occurred to me that there must be such a routine buried in your matrix program which is used to multiply the inverse of the a matrix by the B vector. I found the notation at M383 of your program which states "Quick solve by multiplication A’b = x". So I installed my matrix in A and my vector in B using your input routine, but rather than solving I did XEQ M383. I got the first element of the product but I have not been able to find out how to get the remaining elements of the product. Can you help me? Even better, can you add to the capability of your program to provide this function as one of the options?
Stefan Vorkoetter
January 30, 2008
Palmer, the product replaces the original b vector, so you can use the method for viewing b to examine the result (i.e. select
B
from the main menu, and thenVIEW
from the B menu).That’s a good idea about adding it as a function, since all the code is already there anyway. I’ll do that as soon as I get a chance.
Stefan Vorkoetter
February 05, 2008
Palmer, I think I found the reason that a direct call to M383 isn’t working for you. Because the multiplication routine was designed as part of the whole program, it’s expecting certain values in a few registers, specifically:
N – Number of rows (or columns) in A and length of vector b.
E – End of memory pointer = 2N^{2}+N.
Palmer O. Hanson, Jr.
March 05, 2008
Stefan:
Your discussion of why you wrote this program states that your program for the SR-52 was limited to 2×2 and 3×3 systems by memory limitations. Back in the heyday of the SR-52 some of the "superprogrammers" managed to get 4×4 in place. The following paragraph from my history of the matrix manipulation "friendly" competition gives references where the old issues of 52 Notes can be seen on Viktor Toth’s site.
"The so-called friendly competition between users of HP and TI programmable calculators was announced for subscribers of 52 Notes in April 1977 (V2N4P1). The first problem which was agreed on was to mechanize a 5×5 determinant and inverse program where the competition would be between the SR-52/PC-100 and the HP-97. Work on calculating determinants and inverses had appeared in 52 Notes as early as August 1976 with the publication of a 4×4 determinant program by Dix Fulton (V1N3P4). The September 1976 issue published a program by Alan Trimble which would solve 4 simultaneous equations which would fit on one card (V1N4P2). The January 1977 issue presented a program by Rick Wenger which would solve 4 simultaneous equations, which was faster than that by Alan Trimble and which used fewer steps (V2N1P4/5). That issue also published a 4×4 determinant and inverse program by Barbara Osofsky. The February 1977 issue presented a program by Barbara Osofsy which would calculate the determinant and inverse of a 5×5 matrix, and also calculate a 4×4 determinant and solve a system of four simultaneous equations (V2N2P3/4). The May 1977 issue presented the latest version of Barbara Osofsky’s 5×5 matrix program (V2N5P5)."
Palmer
Rick
April 01, 2008
Can the HP-33s calculator use this program?
Stefan Vorkoetter
April 03, 2008
Rick, the HP 33s doesn’t have the 801 indirectly-addressable registers that the 35s has. This program uses those registers to store the matrix in, so no, it won’t work on the 33s.
Carlos T.
February 01, 2009
Tried several times to insert program. Checked all lines twice and checksum doesn’t match the one indicated. Lines are OK. Matrix multiplication OK, but determinant when resolving Ax=b is always -1 or 1. Checked subrutine from line 222 and all are OK. Can you tell me if porobably there is one thing that I am doing incorrectly or there is an error in the list herein indicated. Many thanks for your help. Regards\\carlos
Stefan Vorkoetter
February 01, 2009
Unfortunately, the checksum isn’t all that useful. There’s a bug in the HP 35s firmware which makes the checksum inconsistent.
When you try the example, does it solve the system correctly? If not, then the problem with the determinant is probably the result of a larger problem. If it does solve correctly, then probably only the determinant code itself is wrong. Is line M319 correct in your calculator (i.e. "RCL (I)", not "RCL I"). Also double check lines M321 to M323.
I’m quite sure there are no problems in the program listing itself, since several people have entered the program from this listing, and had no problems.
Pepe Fdez
June 13, 2009
Documentation speak about flags 0, 1 & 2. Program uses 0, 2 & 10 ones. Program running very well for me but my checksum (DD37) don´t concorde. Its possible that fix checksum problem change flag 10 to 1?
Thanks & Regards.
Stefan Vorkoetter
June 14, 2009
Pepe, the program uses flags 0, 1, 2, and 10. For example, line M042 sets flag 1.
Flag 10 isn’t listed in the flags section because it’s not used to control the flow of the program. It’s just used to tell the calculator to display EQN messages instead of evaluating them (see the manual for details).
The checksum mismatch can occur even when the program has been entered correctly, due to a bug in the HP-35s firmware.
Dennis A
January 29, 2010
From the first step, my calculator says nonexistent. So I wonder if the program is right or my calculator is faulty.
Stefan Vorkoetter
January 29, 2010
It sounds as if you haven’t typed the program into the calculator yet.
juan
March 02, 2010
how put instruction m075 IP in a program
Stefan Vorkoetter
March 02, 2010
Left-shift, INTG, 6. It’s in the manual on page G-9.
mattia Tosi
May 12, 2010
I tried with N=2 and it works; with N=3,4 after the "time estimate", I press R/S and the program continues "RUNNING" and never ends…
Stefan Vorkoetter
May 12, 2010
It sounds like you’ve made a mistake entering the program. It works for me and many other people who have tried it.
Pablo
May 22, 2010
Hi Stefan, thanks for the program. You have made a good work.
You can visit my web page where there are some simple programs I have written for my new HP 35s.
blog.sortingthecomputer.com
Turbato Tomas
June 29, 2010
Thx for the program Stefanv!
I haven’t understand only one thing: i must use RPN mode for running the prgm…
i’ve tried in alg mode for fun but don’t run…(have i wrong the prgm or is right? i’ve read 1.000 times all
Stefan Vorkoetter
June 29, 2010
Yes, you need to use RPN mode.
John
July 17, 2010
After having Maple crunch the closed forms, I wrote a 2×2, 3×3, adjoint, determinant, inverse, and multiply programs for my 35s. It’s plug dimension N, coefs A,B,C… the formulas chug out answers. It works fine, but I can’t go any higher than N=3 as I ran out of labels for input/output, and they total over 500 lines. Also, it doesn’t directly solve systems. Then I stumbled upon this marvelous program of yours. You do it all in one program, using an index counter for Row,Col and indirect labels is so much better, and shorter. It gave me a base to write a different program, a big hat’s off and thanks.
Stefan Vorkoetter
July 17, 2010
What a coincidence! I actually used Maple during the development of this program. I first wrote it in Maple, then started to change the Maple code to use single letter variables, then hand-optimized the Maple code, until it was in a simple enough form to translate to HP-35s code. Of course, I do nearly everything in Maple, since I work for Maplesoft.
Umer Jamshaid
January 23, 2011
Hey Stefan, I was trying to code the program into my calculator when I came across line M093. I’m not sure how to enter that, and was hoping you could help me.
Thanks for the help, and thanks for the program!
Stefan Vorkoetter
January 26, 2011
Umer, press STO and then the (I) key (i.e. the zero key).
abel gonzalez
June 20, 2011
estas chido pero echemen la mano con un programa para mi hp 35s para calcular una poligonal abierta calculando puntos sobre un mismo azimut
Chris Copela
February 15, 2012
I’m a new owner of a 35s. I found your program and it looks great. I wanted to ask one question though. What effect would inserting CF 0 into line 2 have? I ask because there could be long stretches of time between uses of the program and I could forget to do it. (If it matters, I wouldn’t mind losing anything I keyed into the program before I turned the calculator off last.) Thanks very much.
Francesco Baldrighi
May 08, 2012
I think I found a bug.
If you want to multiply a matrix (not solve a system of equations)for a vector your program set the flags 0 and 2 (line M440 and M441) and so the program think A is the inverse matrix A’ but it’s not true.
A is the original matrix, untouched.
Now, if you want to solve a system of equations with the precedent matrix A the program executes the product thinking to use A’ because the flag 0 is on; obviously the solution is to clear the flag 0 manually before to solve the system or delete the two lines M440 and M441.
I hope it helps.
Thanks for your program.
luca
August 29, 2012
How put instruction M286 M287 and M288? Please
lucas
September 05, 2012
Is there a way to download this to my calculator so I don’t have to manually enter 442 lines of code?
Thanks for your help.
Bernardo
September 09, 2012
Lucas, you need to write all lines manually.
lucas
September 10, 2012
Yikes.. that’s what I was afraid of!
Thanks for getting back to me.
Joss
November 28, 2012
How to insert line M339
Joss
November 28, 2012
is the + sign really next to J and not to STO
Stefan Vorkoetter
November 28, 2012
Joss, no, it was just a space in the wrong place (not that it matters, since you don’t enter the space anyway).
Tom
January 06, 2013
Luca, regarding to M286, M287, M288 lines question:
Press:
“Right Shift” RCL 0
“Right Shift” RCL .
“Right Shirt” RCL 0
JKiD
January 16, 2013
I have a problem whenever I’m at the step to enter the data in the b matrix. When I press 2 R/S in the b matrix menu I’m being redirected to the same menu, it won’t let me write anything on the matrix I’m just stuck at the menu, I tried to view so I’m pressing 3 R/S in the b matrix menu, I have the same reaction as before: stuck in the b matrix menu.
Stefan Vorkoetter
January 16, 2013
Check that lines M061 through M073 of the program were entered correctly. It sounds like the code that’s supposed to send you to the editing routines isn’t doing so.
Philalethes
January 19, 2013
I’d like to be able to row reduce NxM matrices into RE form using Gauss-Jordan, per this page: http://vale.thus.ch/software/rref.html
I would be interested to know you think your program could easily be modified to do this. Thanks!
Dan Lewis
March 03, 2013
Very nice program. After I had checked it over a few times, I found that I had made some mistakes. After fixing them, it works like a charm! Even after fixing y mistakes, the checksums still didn’t match, but I guess there will always be some bugs. One thing I didn’t like about the 35s was its inability to work with matrices, and this program fits the bill. I can’t imagine ever having to solve an equation with 19 variables, but it’s nice to know that I could if the need ever arose. The undo feature is also a nice touch.
Thanks,
Dan
Stefan Vorkoetter
March 03, 2013
I’m glad you like the program Dan. Regarding the checksum, it’s commonly acknowledged (for example, in the forums at that the checksums can be inconsistent between calculators, so it doesn’t necessarily mean there’s something wrong with the program you entered (perhaps that’s what you meant by “bugs”).
Dan Lewis
April 12, 2013
Furthermore,
I’m sure you know what mesh and nodal analysis are. This involves multiplying the inverse of a matrix by a vector. So, just fake solve a system of linear equations (to get the inverse of the matrix), then go back and put in the vector and voila! I had to use these analyses on the Fundamentals of Engineering Exam (on which the only HP allowed is the 35s) and I would have had to solve them by hand had I not had your program.
Thanks again,
Dan
al penero
December 14, 2013
Mr stefanv could you please give an example on how to enter the program for N=3.tnx for help
Stefan Vorkoetter
December 14, 2013
Al, the program is the same for all N. The value of N is specified when you run the program. Please look at the “Using the Program” section.
jason foose
April 01, 2014
Great programming Stefan. A great addition to the 35s. Can you build a rectangular / polar conversion key too 😉 Thanks for sharing.
Jason Foose
April 04, 2014
Stefan,
I am writing an article for land surveyors regarding surface modeling. I would like to reference your Matrix multi tool in the article. Do you have any objections providing the link?
I also am working to modify MMT (matrix multi tool) to accomdate the six term surface modeling equation and matrix found at http://www.ahinson.com/algorithms_general/Sections/InterpolationRegression/FitPolynomialSurface.pdf
I have played with this to include more data points which requires a rectangular “A” matrix of “n” rows x 6 columns. The two modifications I would like to make are 1.) input program for x,y,z coordiantes that arranges the data in the six term format (i.e. X^2 XY Y^2 ect) then stores the refined data in the appropriate matrix designated registers. I think I can pull this off with my skill level. ANy thoughts or comments??? Thanks again for the wonderful tool. I really opens the door for the 35s.
Marco
June 18, 2014
You did a great job. Thanks for that nice little and handy program. 🙂
Carolina
August 15, 2014
Thanks 😀
Martin
September 03, 2014
Stefan,
nice work, reviving my HP calculator enthusiasm. Talking about Maple – wouldn’t it be a nice hobby project for you to create a CodeGeneration function which outputs HP 35s or better RPL code for HP48/49/50 calculators … ? Would this be feasible?
Martin
Mandla Skhosana
October 06, 2014
What are the keystrokes for the following? (M038, M059,M082, M180, M189 and M104)
Stefan Vorkoetter
October 06, 2014
See page 13-16 of the HP 35s manual, “Using Equations to Display Messages”.
Mandla Skhosana
October 07, 2014
Thanks Stefan, it took me almost 20hrs to programme, test and recheck the programm. Enventually it worked, its magic.My worry is that during exams they sometimes reset all the prommable calculators and I will loose the programme at the time i need. Is there a way this cannot be deleted
Stefan Vorkoetter
October 08, 2014
Unfortunately, there’s no way to prevent a program from being deleted.
mandla Skhosana
October 08, 2014
is the a way I can load an engineering equation like
CTerror = Ie/(I’+Ie)
Gerard Guerra
February 26, 2015
Stefan
Thanks for the hard the program,the code is new to me I am using the manual as a reference, but still have a couple of questions. In M075 IP, what is the appropriate input for this, also are the EQN titles space sensitive? Is there a space key on my calculator? Thanks for your help.
GGUERR
Pedro Daniel Leiva
October 31, 2015
Dear Gerard:
To insert IP in step M075, just press yellow keys; left arrow INTG 6
To get space during equation typing press (blue keys): right arrow 0
I hope it helps
Pedro
Raymond
June 20, 2017
Hi, Im going by the instructions. Xeq m, 1 rs, 1 rs, I choose the dimension, enter my matrix, after that when I go to 3 rs to view the inverse, I can see my first entry, after that its shows “invalid (I)”, any suggestion?
Raymond
June 20, 2017
When I go to programing, it takes me to M093 = STO(I)
Stefan Vorkoetter
June 20, 2017
All I can suggest is that you double check that the program was entered correctly.
Stefan Vorkoetter
June 20, 2017
Somehow you’ve ended up back in the matrix entry routine with the index set up incorrectly, which suggests that the code branched to the wrong place from somewhere. Again, check that the program was entered correctly.