This program transforms a complex matrix into row reduced echelon form. In addition it computes the inverse and the determinant of its left square part. To do so it uses a maximum pivot method which reverses the complex matrix on place.
History
The origin of this program is vague. My notes refer to a program section published under “Network Analysis” [1], but I can’t find such a reference. Perhaps it was from the book “Handbook of electronic design and analysis procedures using programmable calculators” [3]. Then again, maybe I wrote or at the very least, added an user interface in October 1986.
Instructions
Start with XEQ “CMA” and the program will walk you through the steps.
Method
The CMA program handles the 3 matrix problems simultaneously: 1) calculate the inverse of a complex square matrix; 2) calculate the determinant of a complex square matrix; 3) solve a system of complex equations or reduce a system of complex parameter equations.
Reversing the left square matrix on place requires almost no extra time over that required for just reducing a matrix. The latter method (when using the MA program) includes redundant computations known to give zero results. The CMA routine simply avoids this by replacing the pivots and the elements reduced to 0 by their matrix counterparts.
The elements of the input matrix will be stored row by row as sets of two numbers representing their polar form (magnitude first) by the input routine. The i/o mode (polar/rect) can be chosen by the user.
Consider the following matrix equations:
Exchanging two columns means exchanging two variables:
On the contrary, in the inverse of the matrix, two rows need to be exchanged for exchanging two variables. In the CMA program, when the largest elements is found out of the pivot candidates left, it is placed on the main diagonal of the square matrix by exchanging two columns (row index=column index); the exchange is saved in an extra vector. When the reducing process is finishes, the inverse of the matrix and the solutions can be rearranged to their appropriate form by reapplying the exchanges in reverse order, but this time on the rows.
This method allows us to reverse matrices on place with a maximum pivot algorithm.
Examples
-
As a first example, we will tackle problem 3 from the PPC ROM User Manual on page 263 [2]. Note that a real matrix can be written as a complex matrix with all angles 0.
Note that the HP-41 can handle complex numbers with negative magnitudes.
: -
Another example:
The last column gives the solution of the system of equations represented by the input matrix. The other columns are the corresponding elements of the inverse of the left square part of the input matrix. Running CMA on this matrix should transform it back into the input matrix. Because of rounding errors, this is not completely true.
In the above example, the matrix reads the same in SCI 7 mode. This gives a good indication of the accuracy to be expected from the maximum pivot method used. One warning however : do not trust the results of CMA when the determinant of the square matrix turns out to be unexpectedly small in magnitude in comparison with the magnitudes of the elements of the input matrix! In this case you are dealing with a so called “badly conditioned system”. You can always check this by running CMA twice on the input matrix, and by reading the Magn(Determinant) using the E key.
Listing
Available through the repository
-
Requires
- X-Functions module on the HP-41cv
-
Available as
- source code
- raw code for the V41 emulator
- bar code for the HP Wand
-
Size
- 783 bytes including END (785 if long jumps); 7 magnetic tracks
- Long Jumps: at line 103 and 304.
-
Registers used
- R04, magnitude of the determinant
- R05, angle of the determinant
- R06, scratch
- R07, starting address of the matrix (initiated to 16)
- R08, # of columns * 2
- R09, # of rows
- R10, ISG pointer to exchange pointer stack
- R11, DSE pointer to magnitudes
- R12, main row counter
- R13, column exchange ISG constant
- R14, scratch
- R15, scratch
- R M, scratch
- R O, scratch
- R N, save flag register d
; /-------------------------------------------------------------------\
; | C o m p l e x M a t r i x |
; | to Row Reduced Echelon Form |
; | |
; | for the HP-41 |
; \-------------------------------------------------------------------/
;
; https://coertvonk.com/technology/hp41/complex-matrix-4555
01 LBL "CMAT"
02 LBL A
03 "COMPL MATRIX"
04 AVIEW
05 TONE 89
06 PSE
07 3
08 STO 00
09 16
10 STO 07
11 "R^C ?"
12 PROMPT
13 ST+ X
14 STO 08
15 X<>Y
16 STO 09
17 STO Z
18 *
19 +
20 15
21 +
22 "RESIZE="
23 CF 29
24 FIX 0
25 ARCL X
26 PROMPT
27 SF 09
28 GTO 12
29 LBL B
30 CF 09
31 LBL 12
32 CF 29
33 RCL 07
34 STO 04
35 RCL 08
36 RCL 09
37 *
38 STO 03
39 LBL 13
40 RCL 04
41 XEQ 18
42 ISG X
43 ""
44 FIX 0
45 2
46 ST/ Y
47 "("
48 ARCL Z
49 >","
50 ARCL Y
51 >")"
52 FC? 09
53 GTO 15
54 FS? 01
55 >"=I^R ?"
56 FC? 01
57 >"=M^< ?"
58 0
59 TONE 89
60 PROMPT
61 FC? 01
62 X<>Y
63 FS? 01
64 R-P
65 STO IND 04
66 X<>Y
67 ISG 04
68 ""
69 STO IND 04
70 GTO 17
71 LBL 15
72 ASTO 05
73 ENG IND 00
74 RCL IND 04
75 ISG 04
76 ""
77 RCL IND 04
78 X<>Y
79 FC? 01
80 GTO 28
81 P-R
82 "R"
83 LBL 28
84 FC? 01
85 "M"
86 ARCL 05
87 ARCL X
88 AVIEW
89 PSE
90 PSE
91 FS? 01
92 "I"
93 FC? 01
94 "<"
95 ARCL 05
96 ARCL Y
97 AVIEW
98 LBL 17
99 ISG 04
100 ""
101 DSE 03
102 DSE 03
103 GTO 13
104 RTN
105 LBL C
106 FIX 0
107 CF 29
108 2
109 ST* Y
110 RDN
111 XEQ 19
112 DSE X
113 STO 04
114 " R"
115 ARCL X
116 ISG X
117 ""
118 >"+"
119 ARCL X
120 AVIEW
121 2
122 STO 03
123 RDN
124 CF 09
125 GTO 13
126 RTN
127 LBL E
128 "D="
129 ARCL 04
130 AVIEW
131 RTN
132 LBL 24
133 X<>Y
134 STO O
135 X<>Y
136 MOD
137 ST- O
138 LASTX
139 ST/ O
140 CLX
141 X<> O
142 X<> Y
143 RTN
144 LBL 25
145 XEQ 20
146 X<>Y
147 LBL 21
148 ST* IND Y
149 ISG Y
150 GTO 21
151 RTN
152 LBL 26
153 RCL IND Y
154 X<> IND Y
155 STO IND Z
156 RDN
157 ISG X
158 ""
159 ISG Y
160 GTO 26
161 RTN
162 LBL 20
163 RCL 08
164 *
165 RCL 07
166 +
167 RCL X
168 RCL 08
169 ST- Z
170 SIGN
171 -
172 E3
173 /
174 +
175 RTN
176 LBL 18
177 RCL 07
178 -
179 RCL 08
180 XEQ 24
181 ISG Y
182 ""
183 ISG X
184 ""
185 RTN
186 LBL 19
187 X<> 08
188 ST- 08
189 *
190 ST+ 08
191 X<> L
192 X<> 08
193 1
194 -
195 RCL 07
196 +
197 RTN
198 LBL 27
199 SIGN
200 CLX
201 LBL 23
202 STO IND L
203 ISG L
204 GTO 23
205 RTN
206 LBL F
207 SF 01
208 "R"
209 AVIEW
210 RTN
211 LBL G
212 CF 01
213 "P"
214 AVIEW
215 RTN
216 LBL D
217 CLA
218 E-3
219 RCL 07
220 RCL 08
221 RCL 09
222 STO 12
223 *
224 +
225 STO 11
226 *
227 -2
228 LASTX
229 +
230 X<> 11
231 X<> Z
232 -
233 +
234 STO 10
235 LASTX
236 RCL 07
237 1
238 STO 04
239 -
240 +
241 RCL 08
242 E-5
243 ST+ 11
244 ST+ 11
245 *
246 +
247 STO 13
248 0
249 STO 05
250 RCL 08
251 2 E3
252 /
253 RCL 09
254 +
255 XEQ 14
256 GTO 00
257 LBL 06
258 SF IND X
259 LBL 00
260 ISG X
261 GTO 06
262 LBL 16
263 RCL 11
264 STO O
265 RCL 09
266 STO 14
267 CLX
268 LBL 02
269 RCL 08
270 2
271 /
272 STO M
273 LBL 03
274 RDN
275 FC? IND 14
276 GTO 04
277 RCL 08
278 ST- O
279 DSE 14
280 GTO 03
281 RDN
282 GTO 05
283 LBL 04
284 FS? IND M
285 GTO 01
286 RCL IND O
287 ABS
288 X<Y?
289 GTO 00
290 RCL O
291 STO 15
292 LBL 00
293 X<>Y
294 LBL 01
295 DSE O
296 DSE M
297 GTO 04
298 DSE 14
299 GTO 02
300 LBL 05
301 X#0?
302 GTO 00
303 STO 04
304 GTO 14
305 LBL 00
306 RCL 15
307 INT
308 STO 15
309 XEQ 18
310 STO M
311 X<>Y
312 STO 06
313 1
314 X<> IND 15
315 ST* 04
316 1/X
317 STO 14
318 ISG 15
319 ""
320 0
321 X<> IND 15
322 ST+ 05
323 STO 15
324 R^
325 RCL 13
326 +
327 RCL 06
328 SF IND X
329 ST+ X
330 DSE X
331 LASTX
332 +
333 STO O
334 X=Y?
335 GTO 00
336 XEQ 26
337 RCL M
338 1
339 +
340 RCL 13
341 +
342 RCL O
343 -1
344 ST* 04
345 -
346 XEQ 26
347 RCL M
348 E
349 +
350 2
351 /
352 E
353 XEQ 19
354 E3
355 /
356 RCL 06
357 E
358 XEQ 19
359 +
360 RCL 08
361 E6
362 /
363 +
364 STO IND 10
365 ISG 10
366 LBL 00
367 2 E-5
368 ST+ 07
369 RCL 14
370 RCL 06
371 XEQ 25
372 CLX
373 RCL 08
374 -
375 1
376 +
377 RCL 15
378 LBL 09
379 ST- IND Y
380 ISG Y
381 GTO 09
382 RCL 07
383 INT
384 X<> 07
385 STO 14
386 1
387 +
388 STO 15
389 RCL 13
390 INT
391 RCL 06
392 RCL 08
393 *
394 +
395 1.001
396 *
397 RCL 08
398 -
399 1
400 +
401 STO 06
402 XEQ 14
403 LBL 07
404 RCL 14
405 INT
406 RCL 06
407 STO M
408 INT
409 X#Y?
410 GTO 00
411 RCL 08
412 ST+ 14
413 ST+ 15
414 GTO 11
415 LBL 00
416 RCL O
417 1
418 +
419 0
420 X<> IND Y
421 0
422 X<> IND O
423 LBL 10
424 RCL IND M
425 X<>Y
426 *
427 LASTX
428 X<> IND 14
429 X<> Z
430 ISG M
431 RCL IND M
432 X<>Y
433 +
434 X<>Y
435 LASTX
436 X<> IND 15
437 R^
438 P-R
439 R^
440 R^
441 P-R
442 X<> T
443 X<>Y
444 -
445 X<> Z
446 -
447 R-P
448 X<> IND 14
449 X<>Y
450 X<> IND 15
451 X<>Y
452 ISG 14
453 ""
454 ISG 15
455 ""
456 ISG M
457 GTO 10
458 LBL 11
459 ISG O
460 GTO 07
461 XEQ 14
462 DSE 12
463 GTO 16
464 GTO 00
465 LBL 08
466 RCL IND 10
467 REGSWAP
468 LBL 00
469 DSE 10
470 GTO 08
471 LBL 14
472 X<> N
473 X<> d
474 X<> N
475 END
References
| [1] | Origin unknown Perhaps a section from “Network Analysis” |
|
| [2] | Application program RRM for M1
John Kennedy, December 1981 PPC ROM User’s Manual, page 263-265 |
|
| [3] | Handbook of electronic design and analysis procedures using programmable calculators Bruce K. Murdock |