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 HP41 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
 XFunctions module on the HP41cv

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 HP41  ; \/ ; ; https://coertvonk.com/technology/hp41/complexmatrix4555 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 RP 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 PR 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 E3 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 E5 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 XY 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 E5 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 PR 439 R^ 440 R^ 441 PR 442 X<> T 443 X<>Y 444  445 X<> Z 446  447 RP 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 263265 

[3]  Handbook of electronic design and analysis procedures using programmable calculators Bruce K. Murdock 