1 /* 2 Include "petsctao.h" so that we can use TAO solvers. Note that this 3 file automatically includes libraries such as: 4 petsc.h - base PETSc routines petscvec.h - vectors 5 petscsys.h - system routines petscmat.h - matrices 6 petscis.h - index sets petscksp.h - Krylov subspace methods 7 petscviewer.h - viewers petscpc.h - preconditioners 8 9 This version tests correlated terms using both vector and listed forms 10 */ 11 12 #include <petsctao.h> 13 14 /* 15 Description: These data are the result of a NIST study involving 16 ultrasonic calibration. The response variable is 17 ultrasonic response, and the predictor variable is 18 metal distance. 19 20 Reference: Chwirut, D., NIST (197?). 21 Ultrasonic Reference Block Study. 22 */ 23 24 static char help[] = "Finds the nonlinear least-squares solution to the model \n\ 25 y = exp[-b1*x]/(b2+b3*x) + e \n"; 26 27 #define NOBSERVATIONS 214 28 #define NPARAMETERS 3 29 30 /* User-defined application context */ 31 typedef struct { 32 /* Working space */ 33 PetscReal t[NOBSERVATIONS]; /* array of independent variables of observation */ 34 PetscReal y[NOBSERVATIONS]; /* array of dependent variables */ 35 PetscReal j[NOBSERVATIONS][NPARAMETERS]; /* dense jacobian matrix array*/ 36 PetscInt idm[NOBSERVATIONS]; /* Matrix indices for jacobian */ 37 PetscInt idn[NPARAMETERS]; 38 } AppCtx; 39 40 /* User provided Routines */ 41 PetscErrorCode InitializeData(AppCtx *user); 42 PetscErrorCode FormStartingPoint(Vec); 43 PetscErrorCode EvaluateFunction(Tao, Vec, Vec, void *); 44 PetscErrorCode EvaluateJacobian(Tao, Vec, Mat, Mat, void *); 45 46 /*--------------------------------------------------------------------*/ 47 int main(int argc, char **argv) 48 { 49 PetscInt wtype = 0; 50 Vec x, f; /* solution, function */ 51 Vec w; /* weights */ 52 Mat J; /* Jacobian matrix */ 53 Tao tao; /* Tao solver context */ 54 PetscInt i; /* iteration information */ 55 PetscReal hist[100], resid[100]; 56 PetscInt lits[100]; 57 PetscInt w_row[NOBSERVATIONS]; /* explicit weights */ 58 PetscInt w_col[NOBSERVATIONS]; 59 PetscReal w_vals[NOBSERVATIONS]; 60 PetscBool flg; 61 AppCtx user; /* user-defined work context */ 62 63 PetscFunctionBeginUser; 64 PetscCall(PetscInitialize(&argc, &argv, (char *)0, help)); 65 PetscCall(PetscOptionsGetInt(NULL, NULL, "-wtype", &wtype, &flg)); 66 PetscCall(PetscPrintf(PETSC_COMM_WORLD, "wtype=%" PetscInt_FMT "\n", wtype)); 67 /* Allocate vectors */ 68 PetscCall(VecCreateSeq(MPI_COMM_SELF, NPARAMETERS, &x)); 69 PetscCall(VecCreateSeq(MPI_COMM_SELF, NOBSERVATIONS, &f)); 70 71 PetscCall(VecDuplicate(f, &w)); 72 73 /* no correlation, but set in different ways */ 74 PetscCall(VecSet(w, 1.0)); 75 for (i = 0; i < NOBSERVATIONS; i++) { 76 w_row[i] = i; 77 w_col[i] = i; 78 w_vals[i] = 1.0; 79 } 80 81 /* Create the Jacobian matrix. */ 82 PetscCall(MatCreateSeqDense(MPI_COMM_SELF, NOBSERVATIONS, NPARAMETERS, NULL, &J)); 83 84 for (i = 0; i < NOBSERVATIONS; i++) user.idm[i] = i; 85 86 for (i = 0; i < NPARAMETERS; i++) user.idn[i] = i; 87 88 /* Create TAO solver and set desired solution method */ 89 PetscCall(TaoCreate(PETSC_COMM_SELF, &tao)); 90 PetscCall(TaoSetType(tao, TAOPOUNDERS)); 91 92 /* Set the function and Jacobian routines. */ 93 PetscCall(InitializeData(&user)); 94 PetscCall(FormStartingPoint(x)); 95 PetscCall(TaoSetSolution(tao, x)); 96 PetscCall(TaoSetResidualRoutine(tao, f, EvaluateFunction, (void *)&user)); 97 if (wtype == 1) { 98 PetscCall(TaoSetResidualWeights(tao, w, 0, NULL, NULL, NULL)); 99 } else if (wtype == 2) { 100 PetscCall(TaoSetResidualWeights(tao, NULL, NOBSERVATIONS, w_row, w_col, w_vals)); 101 } 102 PetscCall(TaoSetJacobianResidualRoutine(tao, J, J, EvaluateJacobian, (void *)&user)); 103 PetscCall(TaoSetTolerances(tao, 1e-5, 0.0, PETSC_DEFAULT)); 104 105 /* Check for any TAO command line arguments */ 106 PetscCall(TaoSetFromOptions(tao)); 107 108 PetscCall(TaoSetConvergenceHistory(tao, hist, resid, 0, lits, 100, PETSC_TRUE)); 109 /* Perform the Solve */ 110 PetscCall(TaoSolve(tao)); 111 112 /* Free TAO data structures */ 113 PetscCall(TaoDestroy(&tao)); 114 115 /* Free PETSc data structures */ 116 PetscCall(VecDestroy(&x)); 117 PetscCall(VecDestroy(&w)); 118 PetscCall(VecDestroy(&f)); 119 PetscCall(MatDestroy(&J)); 120 121 PetscCall(PetscFinalize()); 122 return 0; 123 } 124 125 /*--------------------------------------------------------------------*/ 126 PetscErrorCode EvaluateFunction(Tao tao, Vec X, Vec F, void *ptr) 127 { 128 AppCtx *user = (AppCtx *)ptr; 129 PetscInt i; 130 PetscReal *y = user->y, *f, *t = user->t; 131 const PetscReal *x; 132 133 PetscFunctionBegin; 134 PetscCall(VecGetArrayRead(X, &x)); 135 PetscCall(VecGetArray(F, &f)); 136 137 for (i = 0; i < NOBSERVATIONS; i++) f[i] = y[i] - PetscExpScalar(-x[0] * t[i]) / (x[1] + x[2] * t[i]); 138 PetscCall(VecRestoreArrayRead(X, &x)); 139 PetscCall(VecRestoreArray(F, &f)); 140 PetscCall(PetscLogFlops(6 * NOBSERVATIONS)); 141 PetscFunctionReturn(PETSC_SUCCESS); 142 } 143 144 /*------------------------------------------------------------*/ 145 /* J[i][j] = df[i]/dt[j] */ 146 PetscErrorCode EvaluateJacobian(Tao tao, Vec X, Mat J, Mat Jpre, void *ptr) 147 { 148 AppCtx *user = (AppCtx *)ptr; 149 PetscInt i; 150 PetscReal *t = user->t; 151 const PetscReal *x; 152 PetscReal base; 153 154 PetscFunctionBegin; 155 PetscCall(VecGetArrayRead(X, &x)); 156 for (i = 0; i < NOBSERVATIONS; i++) { 157 base = PetscExpScalar(-x[0] * t[i]) / (x[1] + x[2] * t[i]); 158 159 user->j[i][0] = t[i] * base; 160 user->j[i][1] = base / (x[1] + x[2] * t[i]); 161 user->j[i][2] = base * t[i] / (x[1] + x[2] * t[i]); 162 } 163 164 /* Assemble the matrix */ 165 PetscCall(MatSetValues(J, NOBSERVATIONS, user->idm, NPARAMETERS, user->idn, (PetscReal *)user->j, INSERT_VALUES)); 166 PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY)); 167 PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY)); 168 169 PetscCall(VecRestoreArrayRead(X, &x)); 170 PetscCall(PetscLogFlops(NOBSERVATIONS * 13)); 171 PetscFunctionReturn(PETSC_SUCCESS); 172 } 173 174 /* ------------------------------------------------------------ */ 175 PetscErrorCode FormStartingPoint(Vec X) 176 { 177 PetscReal *x; 178 179 PetscFunctionBegin; 180 PetscCall(VecGetArray(X, &x)); 181 x[0] = 1.19; 182 x[1] = -1.86; 183 x[2] = 1.08; 184 PetscCall(VecRestoreArray(X, &x)); 185 PetscFunctionReturn(PETSC_SUCCESS); 186 } 187 188 /* ---------------------------------------------------------------------- */ 189 PetscErrorCode InitializeData(AppCtx *user) 190 { 191 PetscReal *t = user->t, *y = user->y; 192 PetscInt i = 0; 193 194 PetscFunctionBegin; 195 y[i] = 92.9000; 196 t[i++] = 0.5000; 197 y[i] = 78.7000; 198 t[i++] = 0.6250; 199 y[i] = 64.2000; 200 t[i++] = 0.7500; 201 y[i] = 64.9000; 202 t[i++] = 0.8750; 203 y[i] = 57.1000; 204 t[i++] = 1.0000; 205 y[i] = 43.3000; 206 t[i++] = 1.2500; 207 y[i] = 31.1000; 208 t[i++] = 1.7500; 209 y[i] = 23.6000; 210 t[i++] = 2.2500; 211 y[i] = 31.0500; 212 t[i++] = 1.7500; 213 y[i] = 23.7750; 214 t[i++] = 2.2500; 215 y[i] = 17.7375; 216 t[i++] = 2.7500; 217 y[i] = 13.8000; 218 t[i++] = 3.2500; 219 y[i] = 11.5875; 220 t[i++] = 3.7500; 221 y[i] = 9.4125; 222 t[i++] = 4.2500; 223 y[i] = 7.7250; 224 t[i++] = 4.7500; 225 y[i] = 7.3500; 226 t[i++] = 5.2500; 227 y[i] = 8.0250; 228 t[i++] = 5.7500; 229 y[i] = 90.6000; 230 t[i++] = 0.5000; 231 y[i] = 76.9000; 232 t[i++] = 0.6250; 233 y[i] = 71.6000; 234 t[i++] = 0.7500; 235 y[i] = 63.6000; 236 t[i++] = 0.8750; 237 y[i] = 54.0000; 238 t[i++] = 1.0000; 239 y[i] = 39.2000; 240 t[i++] = 1.2500; 241 y[i] = 29.3000; 242 t[i++] = 1.7500; 243 y[i] = 21.4000; 244 t[i++] = 2.2500; 245 y[i] = 29.1750; 246 t[i++] = 1.7500; 247 y[i] = 22.1250; 248 t[i++] = 2.2500; 249 y[i] = 17.5125; 250 t[i++] = 2.7500; 251 y[i] = 14.2500; 252 t[i++] = 3.2500; 253 y[i] = 9.4500; 254 t[i++] = 3.7500; 255 y[i] = 9.1500; 256 t[i++] = 4.2500; 257 y[i] = 7.9125; 258 t[i++] = 4.7500; 259 y[i] = 8.4750; 260 t[i++] = 5.2500; 261 y[i] = 6.1125; 262 t[i++] = 5.7500; 263 y[i] = 80.0000; 264 t[i++] = 0.5000; 265 y[i] = 79.0000; 266 t[i++] = 0.6250; 267 y[i] = 63.8000; 268 t[i++] = 0.7500; 269 y[i] = 57.2000; 270 t[i++] = 0.8750; 271 y[i] = 53.2000; 272 t[i++] = 1.0000; 273 y[i] = 42.5000; 274 t[i++] = 1.2500; 275 y[i] = 26.8000; 276 t[i++] = 1.7500; 277 y[i] = 20.4000; 278 t[i++] = 2.2500; 279 y[i] = 26.8500; 280 t[i++] = 1.7500; 281 y[i] = 21.0000; 282 t[i++] = 2.2500; 283 y[i] = 16.4625; 284 t[i++] = 2.7500; 285 y[i] = 12.5250; 286 t[i++] = 3.2500; 287 y[i] = 10.5375; 288 t[i++] = 3.7500; 289 y[i] = 8.5875; 290 t[i++] = 4.2500; 291 y[i] = 7.1250; 292 t[i++] = 4.7500; 293 y[i] = 6.1125; 294 t[i++] = 5.2500; 295 y[i] = 5.9625; 296 t[i++] = 5.7500; 297 y[i] = 74.1000; 298 t[i++] = 0.5000; 299 y[i] = 67.3000; 300 t[i++] = 0.6250; 301 y[i] = 60.8000; 302 t[i++] = 0.7500; 303 y[i] = 55.5000; 304 t[i++] = 0.8750; 305 y[i] = 50.3000; 306 t[i++] = 1.0000; 307 y[i] = 41.0000; 308 t[i++] = 1.2500; 309 y[i] = 29.4000; 310 t[i++] = 1.7500; 311 y[i] = 20.4000; 312 t[i++] = 2.2500; 313 y[i] = 29.3625; 314 t[i++] = 1.7500; 315 y[i] = 21.1500; 316 t[i++] = 2.2500; 317 y[i] = 16.7625; 318 t[i++] = 2.7500; 319 y[i] = 13.2000; 320 t[i++] = 3.2500; 321 y[i] = 10.8750; 322 t[i++] = 3.7500; 323 y[i] = 8.1750; 324 t[i++] = 4.2500; 325 y[i] = 7.3500; 326 t[i++] = 4.7500; 327 y[i] = 5.9625; 328 t[i++] = 5.2500; 329 y[i] = 5.6250; 330 t[i++] = 5.7500; 331 y[i] = 81.5000; 332 t[i++] = .5000; 333 y[i] = 62.4000; 334 t[i++] = .7500; 335 y[i] = 32.5000; 336 t[i++] = 1.5000; 337 y[i] = 12.4100; 338 t[i++] = 3.0000; 339 y[i] = 13.1200; 340 t[i++] = 3.0000; 341 y[i] = 15.5600; 342 t[i++] = 3.0000; 343 y[i] = 5.6300; 344 t[i++] = 6.0000; 345 y[i] = 78.0000; 346 t[i++] = .5000; 347 y[i] = 59.9000; 348 t[i++] = .7500; 349 y[i] = 33.2000; 350 t[i++] = 1.5000; 351 y[i] = 13.8400; 352 t[i++] = 3.0000; 353 y[i] = 12.7500; 354 t[i++] = 3.0000; 355 y[i] = 14.6200; 356 t[i++] = 3.0000; 357 y[i] = 3.9400; 358 t[i++] = 6.0000; 359 y[i] = 76.8000; 360 t[i++] = .5000; 361 y[i] = 61.0000; 362 t[i++] = .7500; 363 y[i] = 32.9000; 364 t[i++] = 1.5000; 365 y[i] = 13.8700; 366 t[i++] = 3.0000; 367 y[i] = 11.8100; 368 t[i++] = 3.0000; 369 y[i] = 13.3100; 370 t[i++] = 3.0000; 371 y[i] = 5.4400; 372 t[i++] = 6.0000; 373 y[i] = 78.0000; 374 t[i++] = .5000; 375 y[i] = 63.5000; 376 t[i++] = .7500; 377 y[i] = 33.8000; 378 t[i++] = 1.5000; 379 y[i] = 12.5600; 380 t[i++] = 3.0000; 381 y[i] = 5.6300; 382 t[i++] = 6.0000; 383 y[i] = 12.7500; 384 t[i++] = 3.0000; 385 y[i] = 13.1200; 386 t[i++] = 3.0000; 387 y[i] = 5.4400; 388 t[i++] = 6.0000; 389 y[i] = 76.8000; 390 t[i++] = .5000; 391 y[i] = 60.0000; 392 t[i++] = .7500; 393 y[i] = 47.8000; 394 t[i++] = 1.0000; 395 y[i] = 32.0000; 396 t[i++] = 1.5000; 397 y[i] = 22.2000; 398 t[i++] = 2.0000; 399 y[i] = 22.5700; 400 t[i++] = 2.0000; 401 y[i] = 18.8200; 402 t[i++] = 2.5000; 403 y[i] = 13.9500; 404 t[i++] = 3.0000; 405 y[i] = 11.2500; 406 t[i++] = 4.0000; 407 y[i] = 9.0000; 408 t[i++] = 5.0000; 409 y[i] = 6.6700; 410 t[i++] = 6.0000; 411 y[i] = 75.8000; 412 t[i++] = .5000; 413 y[i] = 62.0000; 414 t[i++] = .7500; 415 y[i] = 48.8000; 416 t[i++] = 1.0000; 417 y[i] = 35.2000; 418 t[i++] = 1.5000; 419 y[i] = 20.0000; 420 t[i++] = 2.0000; 421 y[i] = 20.3200; 422 t[i++] = 2.0000; 423 y[i] = 19.3100; 424 t[i++] = 2.5000; 425 y[i] = 12.7500; 426 t[i++] = 3.0000; 427 y[i] = 10.4200; 428 t[i++] = 4.0000; 429 y[i] = 7.3100; 430 t[i++] = 5.0000; 431 y[i] = 7.4200; 432 t[i++] = 6.0000; 433 y[i] = 70.5000; 434 t[i++] = .5000; 435 y[i] = 59.5000; 436 t[i++] = .7500; 437 y[i] = 48.5000; 438 t[i++] = 1.0000; 439 y[i] = 35.8000; 440 t[i++] = 1.5000; 441 y[i] = 21.0000; 442 t[i++] = 2.0000; 443 y[i] = 21.6700; 444 t[i++] = 2.0000; 445 y[i] = 21.0000; 446 t[i++] = 2.5000; 447 y[i] = 15.6400; 448 t[i++] = 3.0000; 449 y[i] = 8.1700; 450 t[i++] = 4.0000; 451 y[i] = 8.5500; 452 t[i++] = 5.0000; 453 y[i] = 10.1200; 454 t[i++] = 6.0000; 455 y[i] = 78.0000; 456 t[i++] = .5000; 457 y[i] = 66.0000; 458 t[i++] = .6250; 459 y[i] = 62.0000; 460 t[i++] = .7500; 461 y[i] = 58.0000; 462 t[i++] = .8750; 463 y[i] = 47.7000; 464 t[i++] = 1.0000; 465 y[i] = 37.8000; 466 t[i++] = 1.2500; 467 y[i] = 20.2000; 468 t[i++] = 2.2500; 469 y[i] = 21.0700; 470 t[i++] = 2.2500; 471 y[i] = 13.8700; 472 t[i++] = 2.7500; 473 y[i] = 9.6700; 474 t[i++] = 3.2500; 475 y[i] = 7.7600; 476 t[i++] = 3.7500; 477 y[i] = 5.4400; 478 t[i++] = 4.2500; 479 y[i] = 4.8700; 480 t[i++] = 4.7500; 481 y[i] = 4.0100; 482 t[i++] = 5.2500; 483 y[i] = 3.7500; 484 t[i++] = 5.7500; 485 y[i] = 24.1900; 486 t[i++] = 3.0000; 487 y[i] = 25.7600; 488 t[i++] = 3.0000; 489 y[i] = 18.0700; 490 t[i++] = 3.0000; 491 y[i] = 11.8100; 492 t[i++] = 3.0000; 493 y[i] = 12.0700; 494 t[i++] = 3.0000; 495 y[i] = 16.1200; 496 t[i++] = 3.0000; 497 y[i] = 70.8000; 498 t[i++] = .5000; 499 y[i] = 54.7000; 500 t[i++] = .7500; 501 y[i] = 48.0000; 502 t[i++] = 1.0000; 503 y[i] = 39.8000; 504 t[i++] = 1.5000; 505 y[i] = 29.8000; 506 t[i++] = 2.0000; 507 y[i] = 23.7000; 508 t[i++] = 2.5000; 509 y[i] = 29.6200; 510 t[i++] = 2.0000; 511 y[i] = 23.8100; 512 t[i++] = 2.5000; 513 y[i] = 17.7000; 514 t[i++] = 3.0000; 515 y[i] = 11.5500; 516 t[i++] = 4.0000; 517 y[i] = 12.0700; 518 t[i++] = 5.0000; 519 y[i] = 8.7400; 520 t[i++] = 6.0000; 521 y[i] = 80.7000; 522 t[i++] = .5000; 523 y[i] = 61.3000; 524 t[i++] = .7500; 525 y[i] = 47.5000; 526 t[i++] = 1.0000; 527 y[i] = 29.0000; 528 t[i++] = 1.5000; 529 y[i] = 24.0000; 530 t[i++] = 2.0000; 531 y[i] = 17.7000; 532 t[i++] = 2.5000; 533 y[i] = 24.5600; 534 t[i++] = 2.0000; 535 y[i] = 18.6700; 536 t[i++] = 2.5000; 537 y[i] = 16.2400; 538 t[i++] = 3.0000; 539 y[i] = 8.7400; 540 t[i++] = 4.0000; 541 y[i] = 7.8700; 542 t[i++] = 5.0000; 543 y[i] = 8.5100; 544 t[i++] = 6.0000; 545 y[i] = 66.7000; 546 t[i++] = .5000; 547 y[i] = 59.2000; 548 t[i++] = .7500; 549 y[i] = 40.8000; 550 t[i++] = 1.0000; 551 y[i] = 30.7000; 552 t[i++] = 1.5000; 553 y[i] = 25.7000; 554 t[i++] = 2.0000; 555 y[i] = 16.3000; 556 t[i++] = 2.5000; 557 y[i] = 25.9900; 558 t[i++] = 2.0000; 559 y[i] = 16.9500; 560 t[i++] = 2.5000; 561 y[i] = 13.3500; 562 t[i++] = 3.0000; 563 y[i] = 8.6200; 564 t[i++] = 4.0000; 565 y[i] = 7.2000; 566 t[i++] = 5.0000; 567 y[i] = 6.6400; 568 t[i++] = 6.0000; 569 y[i] = 13.6900; 570 t[i++] = 3.0000; 571 y[i] = 81.0000; 572 t[i++] = .5000; 573 y[i] = 64.5000; 574 t[i++] = .7500; 575 y[i] = 35.5000; 576 t[i++] = 1.5000; 577 y[i] = 13.3100; 578 t[i++] = 3.0000; 579 y[i] = 4.8700; 580 t[i++] = 6.0000; 581 y[i] = 12.9400; 582 t[i++] = 3.0000; 583 y[i] = 5.0600; 584 t[i++] = 6.0000; 585 y[i] = 15.1900; 586 t[i++] = 3.0000; 587 y[i] = 14.6200; 588 t[i++] = 3.0000; 589 y[i] = 15.6400; 590 t[i++] = 3.0000; 591 y[i] = 25.5000; 592 t[i++] = 1.7500; 593 y[i] = 25.9500; 594 t[i++] = 1.7500; 595 y[i] = 81.7000; 596 t[i++] = .5000; 597 y[i] = 61.6000; 598 t[i++] = .7500; 599 y[i] = 29.8000; 600 t[i++] = 1.7500; 601 y[i] = 29.8100; 602 t[i++] = 1.7500; 603 y[i] = 17.1700; 604 t[i++] = 2.7500; 605 y[i] = 10.3900; 606 t[i++] = 3.7500; 607 y[i] = 28.4000; 608 t[i++] = 1.7500; 609 y[i] = 28.6900; 610 t[i++] = 1.7500; 611 y[i] = 81.3000; 612 t[i++] = .5000; 613 y[i] = 60.9000; 614 t[i++] = .7500; 615 y[i] = 16.6500; 616 t[i++] = 2.7500; 617 y[i] = 10.0500; 618 t[i++] = 3.7500; 619 y[i] = 28.9000; 620 t[i++] = 1.7500; 621 y[i] = 28.9500; 622 t[i++] = 1.7500; 623 PetscFunctionReturn(PETSC_SUCCESS); 624 } 625 626 /*TEST 627 628 build: 629 requires: !complex 630 631 test: 632 args: -tao_monitor_short -tao_max_it 100 -tao_type pounders -tao_pounders_delta 0.05 -tao_gatol 1.e-5 633 requires: !single 634 TODO: produces different output for many different systems 635 636 test: 637 suffix: 2 638 args: -tao_monitor_short -tao_max_it 100 -wtype 1 -tao_type pounders -tao_pounders_delta 0.05 -tao_gatol 1.e-5 639 requires: !single 640 TODO: produces different output for many different systems 641 642 test: 643 suffix: 3 644 args: -tao_monitor_short -tao_max_it 100 -wtype 2 -tao_type pounders -tao_pounders_delta 0.05 -tao_gatol 1.e-5 645 requires: !single 646 TODO: produces different output for many different systems 647 648 test: 649 suffix: 4 650 args: -tao_monitor_short -tao_max_it 100 -tao_type pounders -tao_pounders_delta 0.05 -pounders_subsolver_tao_type blmvm -tao_gatol 1.e-5 651 requires: !single 652 TODO: produces different output for many different systems 653 654 TEST*/ 655