lemon-project-template-glpk: 33de93886c88 deps/glpk/src/glpscf.c

lemon-project-template-glpk

view deps/glpk/src/glpscf.c @ 9:33de93886c88

Import GLPK 4.47

author	Alpar Juttner <alpar@cs.elte.hu>
date	Sun, 06 Nov 2011 20:59:10 +0100
parents
children

line source

1 /* glpscf.c (Schur complement factorization) */

3 /***********************************************************************

4 * This code is part of GLPK (GNU Linear Programming Kit).

5 *

7 * 2009, 2010, 2011 Andrew Makhorin, Department for Applied Informatics,

9 * E-mail: <mao@gnu.org>.

10 *

11 * GLPK is free software: you can redistribute it and/or modify it

12 * under the terms of the GNU General Public License as published by

13 * the Free Software Foundation, either version 3 of the License, or

14 * (at your option) any later version.

15 *

16 * GLPK is distributed in the hope that it will be useful, but WITHOUT

17 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY

18 * or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public

19 * License for more details.

20 *

21 * You should have received a copy of the GNU General Public License

22 * along with GLPK. If not, see <http://www.gnu.org/licenses/>.

23 ***********************************************************************/

25 #include "glpenv.h"

26 #include "glpscf.h"

27 #define xfault xerror

29 #define _GLPSCF_DEBUG 0

31 #define eps 1e-10

33 /***********************************************************************

34 * NAME

35 *

36 * scf_create_it - create Schur complement factorization

37 *

38 * SYNOPSIS

39 *

40 * #include "glpscf.h"

41 * SCF *scf_create_it(int n_max);

42 *

43 * DESCRIPTION

44 *

45 * The routine scf_create_it creates the factorization of matrix C,

46 * which initially has no rows and columns.

47 *

48 * The parameter n_max specifies the maximal order of matrix C to be

49 * factorized, 1 <= n_max <= 32767.

50 *

51 * RETURNS

52 *

53 * The routine scf_create_it returns a pointer to the structure SCF,

54 * which defines the factorization. */

56 SCF *scf_create_it(int n_max)

57 { SCF *scf;

58 #if _GLPSCF_DEBUG

59 xprintf("scf_create_it: warning: debug mode enabled\n");

60 #endif

61 if (!(1 <= n_max && n_max <= 32767))

62 xfault("scf_create_it: n_max = %d; invalid parameter\n",

63 n_max);

64 scf = xmalloc(sizeof(SCF));

65 scf->n_max = n_max;

66 scf->n = 0;

67 scf->f = xcalloc(1 + n_max * n_max, sizeof(double));

68 scf->u = xcalloc(1 + n_max * (n_max + 1) / 2, sizeof(double));

69 scf->p = xcalloc(1 + n_max, sizeof(int));

70 scf->t_opt = SCF_TBG;

71 scf->rank = 0;

72 #if _GLPSCF_DEBUG

73 scf->c = xcalloc(1 + n_max * n_max, sizeof(double));

74 #else

75 scf->c = NULL;

76 #endif

77 scf->w = xcalloc(1 + n_max, sizeof(double));

78 return scf;

79 }

81 /***********************************************************************

82 * The routine f_loc determines location of matrix element F[i,j] in

83 * the one-dimensional array f. */

85 static int f_loc(SCF *scf, int i, int j)

86 { int n_max = scf->n_max;

87 int n = scf->n;

88 xassert(1 <= i && i <= n);

89 xassert(1 <= j && j <= n);

90 return (i - 1) * n_max + j;

91 }

93 /***********************************************************************

94 * The routine u_loc determines location of matrix element U[i,j] in

95 * the one-dimensional array u. */

97 static int u_loc(SCF *scf, int i, int j)

98 { int n_max = scf->n_max;

99 int n = scf->n;

100 xassert(1 <= i && i <= n);

101 xassert(i <= j && j <= n);

102 return (i - 1) * n_max + j - i * (i - 1) / 2;

103 }

104

105 /***********************************************************************

106 * The routine bg_transform applies Bartels-Golub version of gaussian

107 * elimination to restore triangular structure of matrix U.

108 *

109 * On entry matrix U has the following structure:

110 *

111 * 1 k n

112 * 1 * * * * * * * * * *

113 * . * * * * * * * * *

114 * . . * * * * * * * *

115 * . . . * * * * * * *

116 * k . . . . * * * * * *

117 * . . . . . * * * * *

118 * . . . . . . * * * *

119 * . . . . . . . * * *

120 * . . . . . . . . * *

121 * n . . . . # # # # # #

122 *

123 * where '#' is a row spike to be eliminated.

124 *

125 * Elements of n-th row are passed separately in locations un[k], ...,

126 * un[n]. On exit the content of the array un is destroyed.

127 *

128 * REFERENCES

129 *

130 * R.H.Bartels, G.H.Golub, "The Simplex Method of Linear Programming

131 * Using LU-decomposition", Comm. ACM, 12, pp. 266-68, 1969. */

132

133 static void bg_transform(SCF *scf, int k, double un[])

134 { int n = scf->n;

135 double *f = scf->f;

136 double *u = scf->u;

137 int j, k1, kj, kk, n1, nj;

138 double t;

139 xassert(1 <= k && k <= n);

140 /* main elimination loop */

141 for (k = k; k < n; k++)

142 { /* determine location of U[k,k] */

143 kk = u_loc(scf, k, k);

144 /* determine location of F[k,1] */

145 k1 = f_loc(scf, k, 1);

146 /* determine location of F[n,1] */

147 n1 = f_loc(scf, n, 1);

148 /* if |U[k,k]| < |U[n,k]|, interchange k-th and n-th rows to

149 provide |U[k,k]| >= |U[n,k]| */

150 if (fabs(u[kk]) < fabs(un[k]))

151 { /* interchange k-th and n-th rows of matrix U */

152 for (j = k, kj = kk; j <= n; j++, kj++)

153 t = u[kj], u[kj] = un[j], un[j] = t;

154 /* interchange k-th and n-th rows of matrix F to keep the

155 main equality F * C = U * P */

156 for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)

157 t = f[kj], f[kj] = f[nj], f[nj] = t;

158 }

159 /* now |U[k,k]| >= |U[n,k]| */

160 /* if U[k,k] is too small in the magnitude, replace U[k,k] and

161 U[n,k] by exact zero */

162 if (fabs(u[kk]) < eps) u[kk] = un[k] = 0.0;

163 /* if U[n,k] is already zero, elimination is not needed */

164 if (un[k] == 0.0) continue;

165 /* compute gaussian multiplier t = U[n,k] / U[k,k] */

166 t = un[k] / u[kk];

167 /* apply gaussian elimination to nullify U[n,k] */

168 /* (n-th row of U) := (n-th row of U) - t * (k-th row of U) */

169 for (j = k+1, kj = kk+1; j <= n; j++, kj++)

170 un[j] -= t * u[kj];

171 /* (n-th row of F) := (n-th row of F) - t * (k-th row of F)

172 to keep the main equality F * C = U * P */

173 for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)

174 f[nj] -= t * f[kj];

175 }

176 /* if U[n,n] is too small in the magnitude, replace it by exact

177 zero */

178 if (fabs(un[n]) < eps) un[n] = 0.0;

179 /* store U[n,n] in a proper location */

180 u[u_loc(scf, n, n)] = un[n];

181 return;

182 }

183

184 /***********************************************************************

185 * The routine givens computes the parameters of Givens plane rotation

186 * c = cos(teta) and s = sin(teta) such that:

187 *

188 * ( c -s ) ( a ) ( r )

189 * ( ) ( ) = ( ) ,

190 * ( s c ) ( b ) ( 0 )

191 *

192 * where a and b are given scalars.

193 *

194 * REFERENCES

195 *

196 * G.H.Golub, C.F.Van Loan, "Matrix Computations", 2nd ed. */

197

198 static void givens(double a, double b, double *c, double *s)

199 { double t;

200 if (b == 0.0)

201 (*c) = 1.0, (*s) = 0.0;

202 else if (fabs(a) <= fabs(b))

203 t = - a / b, (*s) = 1.0 / sqrt(1.0 + t * t), (*c) = (*s) * t;

204 else

205 t = - b / a, (*c) = 1.0 / sqrt(1.0 + t * t), (*s) = (*c) * t;

206 return;

207 }

208

209 /*----------------------------------------------------------------------

210 * The routine gr_transform applies Givens plane rotations to restore

211 * triangular structure of matrix U.

212 *

213 * On entry matrix U has the following structure:

214 *

215 * 1 k n

216 * 1 * * * * * * * * * *

217 * . * * * * * * * * *

218 * . . * * * * * * * *

219 * . . . * * * * * * *

220 * k . . . . * * * * * *

221 * . . . . . * * * * *

222 * . . . . . . * * * *

223 * . . . . . . . * * *

224 * . . . . . . . . * *

225 * n . . . . # # # # # #

226 *

227 * where '#' is a row spike to be eliminated.

228 *

229 * Elements of n-th row are passed separately in locations un[k], ...,

230 * un[n]. On exit the content of the array un is destroyed.

231 *

232 * REFERENCES

233 *

234 * R.H.Bartels, G.H.Golub, "The Simplex Method of Linear Programming

235 * Using LU-decomposition", Comm. ACM, 12, pp. 266-68, 1969. */

236

237 static void gr_transform(SCF *scf, int k, double un[])

238 { int n = scf->n;

239 double *f = scf->f;

240 double *u = scf->u;

241 int j, k1, kj, kk, n1, nj;

242 double c, s;

243 xassert(1 <= k && k <= n);

244 /* main elimination loop */

245 for (k = k; k < n; k++)

246 { /* determine location of U[k,k] */

247 kk = u_loc(scf, k, k);

248 /* determine location of F[k,1] */

249 k1 = f_loc(scf, k, 1);

250 /* determine location of F[n,1] */

251 n1 = f_loc(scf, n, 1);

252 /* if both U[k,k] and U[n,k] are too small in the magnitude,

253 replace them by exact zero */

254 if (fabs(u[kk]) < eps && fabs(un[k]) < eps)

255 u[kk] = un[k] = 0.0;

256 /* if U[n,k] is already zero, elimination is not needed */

257 if (un[k] == 0.0) continue;

258 /* compute the parameters of Givens plane rotation */

259 givens(u[kk], un[k], &c, &s);

260 /* apply Givens rotation to k-th and n-th rows of matrix U */

261 for (j = k, kj = kk; j <= n; j++, kj++)

262 { double ukj = u[kj], unj = un[j];

263 u[kj] = c * ukj - s * unj;

264 un[j] = s * ukj + c * unj;

265 }

266 /* apply Givens rotation to k-th and n-th rows of matrix F

267 to keep the main equality F * C = U * P */

268 for (j = 1, kj = k1, nj = n1; j <= n; j++, kj++, nj++)

269 { double fkj = f[kj], fnj = f[nj];

270 f[kj] = c * fkj - s * fnj;

271 f[nj] = s * fkj + c * fnj;

272 }

273 }

274 /* if U[n,n] is too small in the magnitude, replace it by exact

275 zero */

276 if (fabs(un[n]) < eps) un[n] = 0.0;

277 /* store U[n,n] in a proper location */

278 u[u_loc(scf, n, n)] = un[n];

279 return;

280 }

281

282 /***********************************************************************

283 * The routine transform restores triangular structure of matrix U.

284 * It is a driver to the routines bg_transform and gr_transform (see

285 * comments to these routines above). */

286

287 static void transform(SCF *scf, int k, double un[])

288 { switch (scf->t_opt)

289 { case SCF_TBG:

290 bg_transform(scf, k, un);

291 break;

292 case SCF_TGR:

293 gr_transform(scf, k, un);

294 break;

295 default:

296 xassert(scf != scf);

297 }

298 return;

299 }

300

301 /***********************************************************************

302 * The routine estimate_rank estimates the rank of matrix C.

303 *

304 * Since all transformations applied to matrix F are non-singular,

305 * and F is assumed to be well conditioned, from the main equaility

306 * F * C = U * P it follows that rank(C) = rank(U), where rank(U) is

307 * estimated as the number of non-zero diagonal elements of U. */

308

309 static int estimate_rank(SCF *scf)

310 { int n_max = scf->n_max;

311 int n = scf->n;

312 double *u = scf->u;

313 int i, ii, inc, rank = 0;

314 for (i = 1, ii = u_loc(scf, i, i), inc = n_max; i <= n;

315 i++, ii += inc, inc--)

316 if (u[ii] != 0.0) rank++;

317 return rank;

318 }

319

320 #if _GLPSCF_DEBUG

321 /***********************************************************************

322 * The routine check_error computes the maximal relative error between

323 * left- and right-hand sides of the main equality F * C = U * P. (This

324 * routine is intended only for debugging.) */

325

326 static void check_error(SCF *scf, const char *func)

327 { int n = scf->n;

328 double *f = scf->f;

329 double *u = scf->u;

330 int *p = scf->p;

331 double *c = scf->c;

332 int i, j, k;

333 double d, dmax = 0.0, s, t;

334 xassert(c != NULL);

335 for (i = 1; i <= n; i++)

336 { for (j = 1; j <= n; j++)

337 { /* compute element (i,j) of product F * C */

338 s = 0.0;

339 for (k = 1; k <= n; k++)

340 s += f[f_loc(scf, i, k)] * c[f_loc(scf, k, j)];

341 /* compute element (i,j) of product U * P */

342 k = p[j];

343 t = (i <= k ? u[u_loc(scf, i, k)] : 0.0);

344 /* compute the maximal relative error */

345 d = fabs(s - t) / (1.0 + fabs(t));

346 if (dmax < d) dmax = d;

347 }

348 }

349 if (dmax > 1e-8)

350 xprintf("%s: dmax = %g; relative error too large\n", func,

351 dmax);

352 return;

353 }

354 #endif

355

356 /***********************************************************************

357 * NAME

358 *

359 * scf_update_exp - update factorization on expanding C

360 *

361 * SYNOPSIS

362 *

363 * #include "glpscf.h"

364 * int scf_update_exp(SCF *scf, const double x[], const double y[],

365 * double z);

366 *

367 * DESCRIPTION

368 *

369 * The routine scf_update_exp updates the factorization of matrix C on

370 * expanding it by adding a new row and column as follows:

371 *

372 * ( C x )

373 * new C = ( )

374 * ( y' z )

375 *

376 * where x[1,...,n] is a new column, y[1,...,n] is a new row, and z is

377 * a new diagonal element.

378 *

379 * If on entry the factorization is empty, the parameters x and y can

380 * be specified as NULL.

381 *

382 * RETURNS

383 *

384 * 0 The factorization has been successfully updated.

385 *

386 * SCF_ESING

387 * The factorization has been successfully updated, however, new

388 * matrix C is singular within working precision. Note that the new

389 * factorization remains valid.

390 *

391 * SCF_ELIMIT

392 * There is not enough room to expand the factorization, because

393 * n = n_max. The factorization remains unchanged.

394 *

395 * ALGORITHM

396 *

397 * We can see that:

398 *

399 * ( F 0 ) ( C x ) ( FC Fx ) ( UP Fx )

400 * ( ) ( ) = ( ) = ( ) =

401 * ( 0 1 ) ( y' z ) ( y' z ) ( y' z )

402 *

403 * ( U Fx ) ( P 0 )

404 * = ( ) ( ),

405 * ( y'P' z ) ( 0 1 )

406 *

407 * therefore to keep the main equality F * C = U * P we can take:

408 *

409 * ( F 0 ) ( U Fx ) ( P 0 )

410 * new F = ( ), new U = ( ), new P = ( ),

411 * ( 0 1 ) ( y'P' z ) ( 0 1 )

412 *

413 * and eliminate the row spike y'P' in the last row of new U to restore

414 * its upper triangular structure. */

415

416 int scf_update_exp(SCF *scf, const double x[], const double y[],

417 double z)

418 { int n_max = scf->n_max;

419 int n = scf->n;

420 double *f = scf->f;

421 double *u = scf->u;

422 int *p = scf->p;

423 #if _GLPSCF_DEBUG

424 double *c = scf->c;

425 #endif

426 double *un = scf->w;

427 int i, ij, in, j, k, nj, ret = 0;

428 double t;

429 /* check if the factorization can be expanded */

430 if (n == n_max)

431 { /* there is not enough room */

432 ret = SCF_ELIMIT;

433 goto done;

434 }

435 /* increase the order of the factorization */

436 scf->n = ++n;

437 /* fill new zero column of matrix F */

438 for (i = 1, in = f_loc(scf, i, n); i < n; i++, in += n_max)

439 f[in] = 0.0;

440 /* fill new zero row of matrix F */

441 for (j = 1, nj = f_loc(scf, n, j); j < n; j++, nj++)

442 f[nj] = 0.0;

443 /* fill new unity diagonal element of matrix F */

444 f[f_loc(scf, n, n)] = 1.0;

445 /* compute new column of matrix U, which is (old F) * x */

446 for (i = 1; i < n; i++)

447 { /* u[i,n] := (i-th row of old F) * x */

448 t = 0.0;

449 for (j = 1, ij = f_loc(scf, i, 1); j < n; j++, ij++)

450 t += f[ij] * x[j];

451 u[u_loc(scf, i, n)] = t;

452 }

453 /* compute new (spiked) row of matrix U, which is (old P) * y */

454 for (j = 1; j < n; j++) un[j] = y[p[j]];

455 /* store new diagonal element of matrix U, which is z */

456 un[n] = z;

457 /* expand matrix P */

458 p[n] = n;

459 #if _GLPSCF_DEBUG

460 /* expand matrix C */

461 /* fill its new column, which is x */

462 for (i = 1, in = f_loc(scf, i, n); i < n; i++, in += n_max)

463 c[in] = x[i];

464 /* fill its new row, which is y */

465 for (j = 1, nj = f_loc(scf, n, j); j < n; j++, nj++)

466 c[nj] = y[j];

467 /* fill its new diagonal element, which is z */

468 c[f_loc(scf, n, n)] = z;

469 #endif

470 /* restore upper triangular structure of matrix U */

471 for (k = 1; k < n; k++)

472 if (un[k] != 0.0) break;

473 transform(scf, k, un);

474 /* estimate the rank of matrices C and U */

475 scf->rank = estimate_rank(scf);

476 if (scf->rank != n) ret = SCF_ESING;

477 #if _GLPSCF_DEBUG

478 /* check that the factorization is accurate enough */

479 check_error(scf, "scf_update_exp");

480 #endif

481 done: return ret;

482 }

483

484 /***********************************************************************

485 * The routine solve solves the system C * x = b.

486 *

487 * From the main equation F * C = U * P it follows that:

488 *

489 * C * x = b => F * C * x = F * b => U * P * x = F * b =>

490 *

491 * P * x = inv(U) * F * b => x = P' * inv(U) * F * b.

492 *

493 * On entry the array x contains right-hand side vector b. On exit this

494 * array contains solution vector x. */

495

496 static void solve(SCF *scf, double x[])

497 { int n = scf->n;

498 double *f = scf->f;

499 double *u = scf->u;

500 int *p = scf->p;

501 double *y = scf->w;

502 int i, j, ij;

503 double t;

504 /* y := F * b */

505 for (i = 1; i <= n; i++)

506 { /* y[i] = (i-th row of F) * b */

507 t = 0.0;

508 for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++)

509 t += f[ij] * x[j];

510 y[i] = t;

511 }

512 /* y := inv(U) * y */

513 for (i = n; i >= 1; i--)

514 { t = y[i];

515 for (j = n, ij = u_loc(scf, i, n); j > i; j--, ij--)

516 t -= u[ij] * y[j];

517 y[i] = t / u[ij];

518 }

519 /* x := P' * y */

520 for (i = 1; i <= n; i++) x[p[i]] = y[i];

521 return;

522 }

523

524 /***********************************************************************

525 * The routine tsolve solves the transposed system C' * x = b.

526 *

527 * From the main equation F * C = U * P it follows that:

528 *

529 * C' * F' = P' * U',

530 *

531 * therefore:

532 *

533 * C' * x = b => C' * F' * inv(F') * x = b =>

534 *

535 * P' * U' * inv(F') * x = b => U' * inv(F') * x = P * b =>

536 *

537 * inv(F') * x = inv(U') * P * b => x = F' * inv(U') * P * b.

538 *

539 * On entry the array x contains right-hand side vector b. On exit this

540 * array contains solution vector x. */

541

542 static void tsolve(SCF *scf, double x[])

543 { int n = scf->n;

544 double *f = scf->f;

545 double *u = scf->u;

546 int *p = scf->p;

547 double *y = scf->w;

548 int i, j, ij;

549 double t;

550 /* y := P * b */

551 for (i = 1; i <= n; i++) y[i] = x[p[i]];

552 /* y := inv(U') * y */

553 for (i = 1; i <= n; i++)

554 { /* compute y[i] */

555 ij = u_loc(scf, i, i);

556 t = (y[i] /= u[ij]);

557 /* substitute y[i] in other equations */

558 for (j = i+1, ij++; j <= n; j++, ij++)

559 y[j] -= u[ij] * t;

560 }

561 /* x := F' * y (computed as linear combination of rows of F) */

562 for (j = 1; j <= n; j++) x[j] = 0.0;

563 for (i = 1; i <= n; i++)

564 { t = y[i]; /* coefficient of linear combination */

565 for (j = 1, ij = f_loc(scf, i, 1); j <= n; j++, ij++)

566 x[j] += f[ij] * t;

567 }

568 return;

569 }

570

571 /***********************************************************************

572 * NAME

573 *

574 * scf_solve_it - solve either system C * x = b or C' * x = b

575 *

576 * SYNOPSIS

577 *

578 * #include "glpscf.h"

579 * void scf_solve_it(SCF *scf, int tr, double x[]);

580 *

581 * DESCRIPTION

582 *

583 * The routine scf_solve_it solves either the system C * x = b (if tr

584 * is zero) or the system C' * x = b, where C' is a matrix transposed

585 * to C (if tr is non-zero). C is assumed to be non-singular.

586 *

587 * On entry the array x should contain the right-hand side vector b in

588 * locations x[1], ..., x[n], where n is the order of matrix C. On exit

589 * the array x contains the solution vector x in the same locations. */

590

591 void scf_solve_it(SCF *scf, int tr, double x[])

592 { if (scf->rank < scf->n)

593 xfault("scf_solve_it: singular matrix\n");

594 if (!tr)

595 solve(scf, x);

596 else

597 tsolve(scf, x);

598 return;

599 }

600

601 void scf_reset_it(SCF *scf)

602 { /* reset factorization for empty matrix C */

603 scf->n = scf->rank = 0;

604 return;

605 }

606

607 /***********************************************************************

608 * NAME

609 *

610 * scf_delete_it - delete Schur complement factorization

611 *

612 * SYNOPSIS

613 *

614 * #include "glpscf.h"

615 * void scf_delete_it(SCF *scf);

616 *

617 * DESCRIPTION

618 *

619 * The routine scf_delete_it deletes the specified factorization and

620 * frees all the memory allocated to this object. */

621

622 void scf_delete_it(SCF *scf)

623 { xfree(scf->f);

624 xfree(scf->u);

625 xfree(scf->p);

626 #if _GLPSCF_DEBUG

627 xfree(scf->c);

628 #endif

629 xfree(scf->w);

630 xfree(scf);

631 return;

632 }

633

634 /* eof */