COIN-OR::LEMON - Graph Library

source: glpk-cmake/src/glpscf.c @ 1:c445c931472f

Last change on this file since 1:c445c931472f was 1:c445c931472f, checked in by Alpar Juttner <alpar@…>, 10 years ago

Import glpk-4.45

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1/* glpscf.c (Schur complement factorization) */
2
3/***********************************************************************
4*  This code is part of GLPK (GNU Linear Programming Kit).
5*
6*  Copyright (C) 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008,
7*  2009, 2010 Andrew Makhorin, Department for Applied Informatics,
8*  Moscow Aviation Institute, Moscow, Russia. All rights reserved.
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***********************************************************************/
24
25#include "glpenv.h"
26#include "glpscf.h"
27#define xfault xerror
28
29#define _GLPSCF_DEBUG 0
30
31#define eps 1e-10
32
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. */
55
56SCF *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}
80
81/***********************************************************************
82*  The routine f_loc determines location of matrix element F[i,j] in
83*  the one-dimensional array f. */
84
85static 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}
92
93/***********************************************************************
94*  The routine u_loc determines location of matrix element U[i,j] in
95*  the one-dimensional array u. */
96
97static 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
133static 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
198static 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
237static 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
287static 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
309static 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
326static 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
416int 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
481done: 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
496static 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
542static 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
591void 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
601void 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
622void 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 */
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