GetFEM  5.4.3
gmm_solver_idgmres.h
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31 
32 /**@file gmm_solver_idgmres.h
33  @author Caroline Lecalvez <[email protected]>
34  @author Yves Renard <[email protected]>
35  @date October 6, 2003.
36  @brief Implicitly restarted and deflated Generalized Minimum Residual.
37 */
38 #ifndef GMM_IDGMRES_H
39 #define GMM_IDGMRES_H
40 
41 #include "gmm_kernel.h"
42 #include "gmm_iter.h"
43 #include "gmm_dense_sylvester.h"
44 
45 namespace gmm {
46 
47  template <typename T> compare_vp {
48  bool operator()(const std::pair<T, size_type> &a,
49  const std::pair<T, size_type> &b) const
50  { return (gmm::abs(a.first) > gmm::abs(b.first)); }
51  }
52 
53  struct idgmres_state {
54  size_type m, tb_deb, tb_def, p, k, nb_want, nb_unwant;
55  size_type nb_nolong, tb_deftot, tb_defwant, conv, nb_un, fin;
56  bool ok;
57 
58  idgmres_state(size_type mm, size_type pp, size_type kk)
59  : m(mm), tb_deb(1), tb_def(0), p(pp), k(kk), nb_want(0),
60  nb_unwant(0), nb_nolong(0), tb_deftot(0), tb_defwant(0),
61  conv(0), nb_un(0), fin(0), ok(false); {}
62  }
63 
64  idgmres_state(size_type mm, size_type pp, size_type kk)
65  : m(mm), tb_deb(1), tb_def(0), p(pp), k(kk), nb_want(0),
66  nb_unwant(0), nb_nolong(0), tb_deftot(0), tb_defwant(0),
67  conv(0), nb_un(0), fin(0), ok(false); {}
68 
69 
70  template <typename CONT, typename IND>
71  apply_permutation(CONT &cont, const IND &ind) {
72  size_type m = ind.end() - ind.begin();
73  std::vector<bool> sorted(m, false);
74 
75  for (size_type l = 0; l < m; ++l)
76  if (!sorted[l] && ind[l] != l) {
77 
78  typeid(cont[0]) aux = cont[l];
79  k = ind[l];
80  cont[l] = cont[k];
81  sorted[l] = true;
82 
83  for(k2 = ind[k]; k2 != l; k2 = ind[k]) {
84  cont[k] = cont[k2];
85  sorted[k] = true;
86  k = k2;
87  }
88  cont[k] = aux;
89  }
90  }
91 
92 
93  /** Implicitly restarted and deflated Generalized Minimum Residual
94 
95  See: C. Le Calvez, B. Molina, Implicitly restarted and deflated
96  FOM and GMRES, numerical applied mathematics,
97  (30) 2-3 (1999) pp191-212.
98 
99  @param A Real or complex unsymmetric matrix.
100  @param x initial guess vector and final result.
101  @param b right hand side
102  @param M preconditionner
103  @param m size of the subspace between two restarts
104  @param p number of converged ritz values seeked
105  @param k size of the remaining Krylov subspace when the p ritz values
106  have not yet converged 0 <= p <= k < m.
107  @param tol_vp : tolerance on the ritz values.
108  @param outer
109  @param KS
110  */
111  template < typename Mat, typename Vec, typename VecB, typename Precond,
112  typename Basis >
113  void idgmres(const Mat &A, Vec &x, const VecB &b, const Precond &M,
114  size_type m, size_type p, size_type k, double tol_vp,
115  iteration &outer, Basis& KS) {
116 
117  typedef typename linalg_traits<Mat>::value_type T;
118  typedef typename number_traits<T>::magnitude_type R;
119 
120  R a, beta;
121  idgmres_state st(m, p, k);
122 
123  std::vector<T> w(vect_size(x)), r(vect_size(x)), u(vect_size(x));
124  std::vector<T> c_rot(m+1), s_rot(m+1), s(m+1);
125  std::vector<T> y(m+1), ztest(m+1), gam(m+1);
126  std::vector<T> gamma(m+1);
127  gmm::dense_matrix<T> H(m+1, m), Hess(m+1, m),
128  Hobl(m+1, m), W(vect_size(x), m+1);
129  gmm::clear(H);
130 
131  outer.set_rhsnorm(gmm::vect_norm2(b));
132  if (outer.get_rhsnorm() == 0.0) { clear(x); return; }
133 
134  mult(A, scaled(x, -1.0), b, w);
135  mult(M, w, r);
136  beta = gmm::vect_norm2(r);
137 
138  iteration inner = outer;
139  inner.reduce_noisy();
140  inner.set_maxiter(m);
141  inner.set_name("GMRes inner iter");
142 
143  while (! outer.finished(beta)) {
144 
145  gmm::copy(gmm::scaled(r, 1.0/beta), KS[0]);
146  gmm::clear(s);
147  s[0] = beta;
148  gmm::copy(s, gamma);
149 
150  inner.set_maxiter(m - st.tb_deb + 1);
151  size_type i = st.tb_deb - 1; inner.init();
152 
153  do {
154  mult(A, KS[i], u);
155  mult(M, u, KS[i+1]);
156  orthogonalize_with_refinment(KS, mat_col(H, i), i);
157  H(i+1, i) = a = gmm::vect_norm2(KS[i+1]);
158  gmm::scale(KS[i+1], R(1) / a);
159 
160  gmm::copy(mat_col(H, i), mat_col(Hess, i));
161  gmm::copy(mat_col(H, i), mat_col(Hobl, i));
162 
163  for (size_type l = 0; l < i; ++l)
164  Apply_Givens_rotation_left(H(l,i), H(l+1,i), c_rot[l], s_rot[l]);
165 
166  Givens_rotation(H(i,i), H(i+1,i), c_rot[i], s_rot[i]);
167  Apply_Givens_rotation_left(H(i,i), H(i+1,i), c_rot[i], s_rot[i]);
168  H(i+1, i) = T(0);
169  Apply_Givens_rotation_left(s[i], s[i+1], c_rot[i], s_rot[i]);
170 
171  ++inner, ++outer, ++i;
172  } while (! inner.finished(gmm::abs(s[i])));
173 
174  if (inner.converged()) {
175  gmm::copy(s, y);
176  upper_tri_solve(H, y, i, false);
177  combine(KS, y, x, i);
178  mult(A, gmm::scaled(x, T(-1)), b, w);
179  mult(M, w, r);
180  beta = gmm::vect_norm2(r); // + verif sur beta ... à faire
181  break;
182  }
183 
184  gmm::clear(gam); gam[m] = s[i];
185  for (size_type l = m; l > 0; --l)
186  Apply_Givens_rotation_left(gam[l-1], gam[l], gmm::conj(c_rot[l-1]),
187  -s_rot[l-1]);
188 
189  mult(KS.mat(), gam, r);
190  beta = gmm::vect_norm2(r);
191 
192  mult(Hess, scaled(y, T(-1)), gamma, ztest);
193  // En fait, d'après Caroline qui s'y connait ztest et gam devrait
194  // être confondus
195  // Quand on aura vérifié que ça marche, il faudra utiliser gam à la
196  // place de ztest.
197  if (st.tb_def < p) {
198  T nss = H(m,m-1) / ztest[m];
199  nss /= gmm::abs(nss); // ns à calculer plus tard aussi
200  gmm::copy(KS.mat(), W); gmm::copy(scaled(r, nss /beta), mat_col(W, m));
201 
202  // Computation of the oblique matrix
203  sub_interval SUBI(0, m);
204  add(scaled(sub_vector(ztest, SUBI), -Hobl(m, m-1) / ztest[m]),
205  sub_vector(mat_col(Hobl, m-1), SUBI));
206  Hobl(m, m-1) *= nss * beta / ztest[m];
207 
208  /* **************************************************************** */
209  /* Locking */
210  /* **************************************************************** */
211 
212  // Computation of the Ritz eigenpairs.
213  std::vector<std::complex<R> > eval(m);
214  dense_matrix<T> YB(m-st.tb_def, m-st.tb_def);
215  std::vector<char> pure(m-st.tb_def, 0);
216  gmm::clear(YB);
217 
218  select_eval(Hobl, eval, YB, pure, st);
219 
220  if (st.conv != 0) {
221  // DEFLATION using the QR Factorization of YB
222 
223  T alpha = Lock(W, Hobl,
224  sub_matrix(YB, sub_interval(0, m-st.tb_def)),
225  sub_interval(st.tb_def, m-st.tb_def),
226  (st.tb_defwant < p));
227  // ns *= alpha; // à calculer plus tard ??
228  // V(:,m+1) = alpha*V(:, m+1); ça devait servir à qlq chose ...
229 
230 
231  // Clean the portions below the diagonal corresponding
232  // to the lock Schur vectors
233  for (size_type j = st.tb_def; j < st.tb_deftot; ++j) {
234  if ( pure[j-st.tb_def] == 0)
235  gmm::clear(sub_vector(mat_col(Hobl,j), sub_interval(j+1,m-j)));
236  else if (pure[j-st.tb_def] == 1) {
237  gmm::clear(sub_matrix(Hobl, sub_interval(j+2,m-j-1),
238  sub_interval(j, 2)));
239  ++j;
240  }
241  else GMM_ASSERT3(false, "internal error");
242  }
243 
244  if (!st.ok) {
245  // attention si m = 0;
246  size_type mm = std::min(k+st.nb_unwant+st.nb_nolong, m-1);
247 
248  if (eval_sort[m-mm-1].second != R(0)
249  && eval_sort[m-mm-1].second == -eval_sort[m-mm].second)
250  ++mm;
251 
252  std::vector<complex<R> > shifts(m-mm);
253  for (size_type i = 0; i < m-mm; ++i)
254  shifts[i] = eval_sort[i].second;
255 
256  apply_shift_to_Arnoldi_factorization(W, Hobl, shifts, mm,
257  m-mm, true);
258 
259  st.fin = mm;
260  }
261  else
262  st.fin = st.tb_deftot;
263 
264 
265  /* ************************************************************** */
266  /* Purge */
267  /* ************************************************************** */
268 
269  if (st.nb_nolong + st.nb_unwant > 0) {
270 
271  std::vector<std::complex<R> > eval(m);
272  dense_matrix<T> YB(st.fin, st.tb_deftot);
273  std::vector<char> pure(st.tb_deftot, 0);
274  gmm::clear(YB);
275  st.nb_un = st.nb_nolong + st.nb_unwant;
276 
277  select_eval_for_purging(Hobl, eval, YB, pure, st);
278 
279  T alpha = Lock(W, Hobl, YB, sub_interval(0, st.fin), ok);
280 
281  // Clean the portions below the diagonal corresponding
282  // to the unwanted lock Schur vectors
283  for (size_type j = 0; j < st.tb_deftot; ++j) {
284  if ( pure[j] == 0)
285  gmm::clear(sub_vector(mat_col(Hobl,j), sub_interval(j+1,m-j)));
286  else if (pure[j] == 1) {
287  gmm::clear(sub_matrix(Hobl, sub_interval(j+2,m-j-1),
288  sub_interval(j, 2)));
289  ++j;
290  }
291  else GMM_ASSERT3(false, "internal error");
292  }
293 
294  gmm::dense_matrix<T> z(st.nb_un, st.fin - st.nb_un);
295  sub_interval SUBI(0, st.nb_un), SUBJ(st.nb_un, st.fin - st.nb_un);
296  sylvester(sub_matrix(Hobl, SUBI),
297  sub_matrix(Hobl, SUBJ),
298  sub_matrix(gmm::scaled(Hobl, -T(1)), SUBI, SUBJ), z);
299  }
300  }
301  }
302  }
303  }
304 
305 
306  template < typename Mat, typename Vec, typename VecB, typename Precond >
307  void idgmres(const Mat &A, Vec &x, const VecB &b,
308  const Precond &M, size_type m, iteration& outer) {
309  typedef typename linalg_traits<Mat>::value_type T;
310  modified_gram_schmidt<T> orth(m, vect_size(x));
311  gmres(A, x, b, M, m, outer, orth);
312  }
313 
314 
315  // Lock stage of an implicit restarted Arnoldi process.
316  // 1- QR factorization of YB through Householder matrices
317  // Q(Rl) = YB
318  // (0 )
319  // 2- Update of the Arnoldi factorization.
320  // H <- Q*HQ, W <- WQ
321  // 3- Restore the Hessemberg form of H.
322 
323  template <typename T, typename MATYB>
324  void Lock(gmm::dense_matrix<T> &W, gmm::dense_matrix<T> &H,
325  const MATYB &YB, const sub_interval SUB,
326  bool restore, T &ns) {
327 
328  size_type n = mat_nrows(W), m = mat_ncols(W) - 1;
329  size_type ncols = mat_ncols(YB), nrows = mat_nrows(YB);
330  size_type begin = min(SUB); end = max(SUB) - 1;
331  sub_interval SUBR(0, nrows), SUBC(0, ncols);
332  T alpha(1);
333 
334  GMM_ASSERT2(((end-begin) == ncols) && (m == mat_nrows(H))
335  && (m+1 == mat_ncols(H)), "dimensions mismatch");
336 
337  // DEFLATION using the QR Factorization of YB
338 
339  dense_matrix<T> QR(n_rows, n_rows);
340  gmmm::copy(YB, sub_matrix(QR, SUBR, SUBC));
341  gmm::clear(submatrix(QR, SUBR, sub_interval(ncols, nrows-ncols)));
342  qr_factor(QR);
343 
344  apply_house_left(QR, sub_matrix(H, SUB));
345  apply_house_right(QR, sub_matrix(H, SUBR, SUB));
346  apply_house_right(QR, sub_matrix(W, sub_interval(0, n), SUB));
347 
348  // Restore to the initial block hessenberg form
349 
350  if (restore) {
351 
352  // verifier quand m = 0 ...
353  gmm::dense_matrix tab_p(end - st.tb_deftot, end - st.tb_deftot);
354  gmm::copy(identity_matrix(), tab_p);
355 
356  for (size_type j = end-1; j >= st.tb_deftot+2; --j) {
357 
358  size_type jm = j-1;
359  std::vector<T> v(jm - st.tb_deftot);
360  sub_interval SUBtot(st.tb_deftot, jm - st.tb_deftot);
361  sub_interval SUBtot2(st.tb_deftot, end - st.tb_deftot);
362  gmm::copy(sub_vector(mat_row(H, j), SUBtot), v);
363  house_vector_last(v);
364  w.resize(end);
365  col_house_update(sub_matrix(H, SUBI, SUBtot), v, w);
366  w.resize(end - st.tb_deftot);
367  row_house_update(sub_matrix(H, SUBtot, SUBtot2), v, w);
368  gmm::clear(sub_vector(mat_row(H, j),
369  sub_interval(st.tb_deftot, j-1-st.tb_deftot)));
370  w.resize(end - st.tb_deftot);
371  col_house_update(sub_matrix(tab_p, sub_interval(0, end-st.tb_deftot),
372  sub_interval(0, jm-st.tb_deftot)), v, w);
373  w.resize(n);
374  col_house_update(sub_matrix(W, sub_interval(0, n), SUBtot), v, w);
375  }
376 
377  // restore positive subdiagonal elements
378 
379  std::vector<T> d(fin-st.tb_deftot); d[0] = T(1);
380 
381  // We compute d[i+1] in order
382  // (d[i+1] * H(st.tb_deftot+i+1,st.tb_deftoti)) / d[i]
383  // be equal to |H(st.tb_deftot+i+1,st.tb_deftot+i))|.
384  for (size_type j = 0; j+1 < end-st.tb_deftot; ++j) {
385  T e = H(st.tb_deftot+j, st.tb_deftot+j-1);
386  d[j+1] = (e == T(0)) ? T(1) : d[j] * gmm::abs(e) / e;
387  scale(sub_vector(mat_row(H, st.tb_deftot+j+1),
388  sub_interval(st.tb_deftot, m-st.tb_deftot)), d[j+1]);
389  scale(mat_col(H, st.tb_deftot+j+1), T(1) / d[j+1]);
390  scale(mat_col(W, st.tb_deftot+j+1), T(1) / d[j+1]);
391  }
392 
393  alpha = tab_p(end-st.tb_deftot-1, end-st.tb_deftot-1) / d[end-st.tb_deftot-1];
394  alpha /= gmm::abs(alpha);
395  scale(mat_col(W, m), alpha);
396  }
397 
398  return alpha;
399  }
400 
401 
402  // Apply p implicit shifts to the Arnoldi factorization
403  // AV = VH+H(k+p+1,k+p) V(:,k+p+1) e_{k+p}*
404  // and produces the following new Arnoldi factorization
405  // A(VQ) = (VQ)(Q*HQ)+H(k+p+1,k+p) V(:,k+p+1) e_{k+p}* Q
406  // where only the first k columns are relevant.
407  //
408  // Dan Sorensen and Richard J. Radke, 11/95
409  template<typename T, typename C>
410  apply_shift_to_Arnoldi_factorization(dense_matrix<T> V, dense_matrix<T> H,
411  std::vector<C> Lambda, size_type &k,
412  size_type p, bool true_shift = false) {
413 
414  size_type k1 = 0, num = 0, kend = k+p, kp1 = k + 1;
415  bool mark = false;
416  T c, s, x, y, z;
417 
418  dense_matrix<T> q(1, kend);
419  gmm::clear(q); q(0,kend-1) = T(1);
420  std::vector<T> hv(3), w(std::max(kend, mat_nrows(V)));
421 
422  for(size_type jj = 0; jj < p; ++jj) {
423  // compute and apply a bulge chase sweep initiated by the
424  // implicit shift held in w(jj)
425 
426  if (abs(Lambda[jj].real()) == 0.0) {
427  // apply a real shift using 2 by 2 Givens rotations
428 
429  for (size_type k1 = 0, k2 = 0; k2 != kend-1; k1 = k2+1) {
430  k2 = k1;
431  while (h(k2+1, k2) != T(0) && k2 < kend-1)
432  ++k2;
433 
434  Givens_rotation(H(k1, k1) - Lambda[jj], H(k1+1, k1), c, s);
435 
436  for (i = k1; i <= k2; ++i) {
437  if (i > k1)
438  Givens_rotation(H(i, i-1), H(i+1, i-1), c, s);
439 
440  // Ne pas oublier de nettoyer H(i+1,i-1) (le mettre à zéro).
441  // Vérifier qu'au final H(i+1,i) est bien un réel positif.
442 
443  // apply rotation from left to rows of H
444  row_rot(sub_matrix(H, sub_interval(i,2), sub_interval(i, kend-i)),
445  c, s, 0, 0);
446 
447  // apply rotation from right to columns of H
448  size_type ip2 = std::min(i+2, kend);
449  col_rot(sub_matrix(H, sub_interval(0, ip2), sub_interval(i, 2))
450  c, s, 0, 0);
451 
452  // apply rotation from right to columns of V
453  col_rot(V, c, s, i, i+1);
454 
455  // accumulate e' Q so residual can be updated k+p
456  Apply_Givens_rotation_left(q(0,i), q(0,i+1), c, s);
457  // peut être que nous utilisons G au lieu de G* et que
458  // nous allons trop loin en k2.
459  }
460  }
461 
462  num = num + 1;
463  }
464  else {
465 
466  // Apply a double complex shift using 3 by 3 Householder
467  // transformations
468 
469  if (jj == p || mark)
470  mark = false; // skip application of conjugate shift
471  else {
472  num = num + 2; // mark that a complex conjugate
473  mark = true; // pair has been applied
474 
475  // Indices de fin de boucle à surveiller... de près !
476  for (size_type k1 = 0, k3 = 0; k3 != kend-2; k1 = k3+1) {
477  k3 = k1;
478  while (h(k3+1, k3) != T(0) && k3 < kend-2)
479  ++k3;
480  size_type k2 = k1+1;
481 
482 
483  x = H(k1,k1) * H(k1,k1) + H(k1,k2) * H(k2,k1)
484  - 2.0*Lambda[jj].real() * H(k1,k1) + gmm::abs_sqr(Lambda[jj]);
485  y = H(k2,k1) * (H(k1,k1) + H(k2,k2) - 2.0*Lambda[jj].real());
486  z = H(k2+1,k2) * H(k2,k1);
487 
488  for (size_type i = k1; i <= k3; ++i) {
489  if (i > k1) {
490  x = H(i, i-1);
491  y = H(i+1, i-1);
492  z = H(i+2, i-1);
493  // Ne pas oublier de nettoyer H(i+1,i-1) et H(i+2,i-1)
494  // (les mettre à zéro).
495  }
496 
497  hv[0] = x; hv[1] = y; hv[2] = z;
498  house_vector(v);
499 
500  // Vérifier qu'au final H(i+1,i) est bien un réel positif
501 
502  // apply transformation from left to rows of H
503  w.resize(kend-i);
504  row_house_update(sub_matrix(H, sub_interval(i, 2),
505  sub_interval(i, kend-i)),
506  v, w);
507 
508  // apply transformation from right to columns of H
509 
510  size_type ip3 = std::min(kend, i + 3);
511  w.resize(ip3);
512  col_house_update(sub_matrix(H, sub_interval(0, ip3),
513  sub_interval(i, 2)),
514  v, w);
515 
516  // apply transformation from right to columns of V
517 
518  w.resize(mat_nrows(V));
519  col_house_update(sub_matrix(V, sub_interval(0, mat_nrows(V)),
520  sub_interval(i, 2)),
521  v, w);
522 
523  // accumulate e' Q so residual can be updated k+p
524  w.resize(1);
525  col_house_update(sub_matrix(q, sub_interval(0,1),
526  sub_interval(i,2)),
527  v, w);
528  }
529  }
530 
531  // clean up step with Givens rotation
532 
533  i = kend-2;
534  c = x;
535  s = y;
536  if (i > k1)
537  Givens_rotation(H(i, i-1), H(i+1, i-1), c, s);
538 
539  // Ne pas oublier de nettoyer H(i+1,i-1) (le mettre à zéro).
540  // Vérifier qu'au final H(i+1,i) est bien un réel positif.
541 
542  // apply rotation from left to rows of H
543  row_rot(sub_matrix(H, sub_interval(i,2), sub_interval(i, kend-i)),
544  c, s, 0, 0);
545 
546  // apply rotation from right to columns of H
547  size_type ip2 = std::min(i+2, kend);
548  col_rot(sub_matrix(H, sub_interval(0, ip2), sub_interval(i, 2))
549  c, s, 0, 0);
550 
551  // apply rotation from right to columns of V
552  col_rot(V, c, s, i, i+1);
553 
554  // accumulate e' Q so residual can be updated k+p
555  Apply_Givens_rotation_left(q(0,i), q(0,i+1), c, s);
556  }
557  }
558  }
559 
560  // update residual and store in the k+1 -st column of v
561 
562  k = kend - num;
563  scale(mat_col(V, kend), q(0, k));
564 
565  if (k < mat_nrows(H)) {
566  if (true_shift)
567  gmm::copy(mat_col(V, kend), mat_col(V, k));
568  else
569  // v(:,k+1) = v(:,kend+1) + v(:,k+1)*h(k+1,k);
570  // v(:,k+1) = v(:,kend+1) ;
571  gmm::add(scaled(mat_col(V, kend), H(kend, kend-1)),
572  scaled(mat_col(V, k), H(k, k-1)), mat_col(V, k));
573  }
574 
575  H(k, k-1) = vect_norm2(mat_col(V, k));
576  scale(mat_col(V, kend), T(1) / H(k, k-1));
577  }
578 
579 
580 
581  template<typename MAT, typename EVAL, typename PURE>
582  void select_eval(const MAT &Hobl, EVAL &eval, MAT &YB, PURE &pure,
583  idgmres_state &st) {
584 
585  typedef typename linalg_traits<MAT>::value_type T;
586  typedef typename number_traits<T>::magnitude_type R;
587  size_type m = st.m;
588 
589  // Computation of the Ritz eigenpairs.
590 
591  col_matrix< std::vector<T> > evect(m-st.tb_def, m-st.tb_def);
592  // std::vector<std::complex<R> > eval(m);
593  std::vector<R> ritznew(m, T(-1));
594 
595  // dense_matrix<T> evect_lock(st.tb_def, st.tb_def);
596 
597  sub_interval SUB1(st.tb_def, m-st.tb_def);
598  implicit_qr_algorithm(sub_matrix(Hobl, SUB1),
599  sub_vector(eval, SUB1), evect);
600  sub_interval SUB2(0, st.tb_def);
601  implicit_qr_algorithm(sub_matrix(Hobl, SUB2),
602  sub_vector(eval, SUB2), /* evect_lock */);
603 
604  for (size_type l = st.tb_def; l < m; ++l)
605  ritznew[l] = gmm::abs(evect(m-st.tb_def-1, l-st.tb_def) * Hobl(m, m-1));
606 
607  std::vector< std::pair<T, size_type> > eval_sort(m);
608  for (size_type l = 0; l < m; ++l)
609  eval_sort[l] = std::pair<T, size_type>(eval[l], l);
610  std::sort(eval_sort.begin(), eval_sort.end(), compare_vp());
611 
612  std::vector<size_type> index(m);
613  for (size_type l = 0; l < m; ++l) index[l] = eval_sort[l].second;
614 
615  std::vector<bool> kept(m, false);
616  std::fill(kept.begin(), kept.begin()+st.tb_def, true);
617 
618  apply_permutation(eval, index);
619  apply_permutation(evect, index);
620  apply_permutation(ritznew, index);
621  apply_permutation(kept, index);
622 
623  // Which are the eigenvalues that converged ?
624  //
625  // nb_want is the number of eigenvalues of
626  // Hess(tb_def+1:n,tb_def+1:n) that converged and are WANTED
627  //
628  // nb_unwant is the number of eigenvalues of
629  // Hess(tb_def+1:n,tb_def+1:n) that converged and are UNWANTED
630  //
631  // nb_nolong is the number of eigenvalues of
632  // Hess(1:tb_def,1:tb_def) that are NO LONGER WANTED.
633  //
634  // tb_deftot is the number of the deflated eigenvalues
635  // that is tb_def + nb_want + nb_unwant
636  //
637  // tb_defwant is the number of the wanted deflated eigenvalues
638  // that is tb_def + nb_want - nb_nolong
639 
640  st.nb_want = 0, st.nb_unwant = 0, st.nb_nolong = 0;
641  size_type j, ind;
642 
643  for (j = 0, ind = 0; j < m-p; ++j) {
644  if (ritznew[j] == R(-1)) {
645  if (std::imag(eval[j]) != R(0)) {
646  st.nb_nolong += 2; ++j; // à adapter dans le cas complexe ...
647  } else
648  st.nb_nolong++;
649  } else {
650  if (ritznew[j] < tol_vp * gmm::abs(eval[j])) {
651 
652  for (size_type l = 0, l < m-st.tb_def; ++l)
653  YB(l, ind) = std::real(evect(l, j));
654  kept[j] = true;
655  ++j; ++st.nb_unwant; ind++;
656 
657  if (std::imag(eval[j]) != R(0)) {
658  for (size_type l = 0, l < m-st.tb_def; ++l)
659  YB(l, ind) = std::imag(evect(l, j));
660  pure[ind-1] = 1;
661  pure[ind] = 2;
662 
663  kept[j] = true;
664 
665  st.nb_unwant++;
666  ++ind;
667  }
668  }
669  }
670  }
671 
672 
673  for (; j < m; ++j) {
674  if (ritznew[j] != R(-1)) {
675 
676  for (size_type l = 0, l < m-st.tb_def; ++l)
677  YB(l, ind) = std::real(evect(l, j));
678  pure[ind] = 1;
679  ++ind;
680  kept[j] = true;
681  ++st.nb_want;
682 
683  if (ritznew[j] < tol_vp * gmm::abs(eval[j])) {
684  for (size_type l = 0, l < m-st.tb_def; ++l)
685  YB(l, ind) = std::imag(evect(l, j));
686  pure[ind] = 2;
687 
688  j++;
689  kept[j] = true;
690 
691  st.nb_want++;
692  ++ind;
693  }
694  }
695  }
696 
697  std::vector<T> shift(m - st.tb_def - st.nb_want - st.nb_unwant);
698  for (size_type j = 0, i = 0; j < m; ++j)
699  if (!kept[j])
700  shift[i++] = eval[j];
701 
702  // st.conv (st.nb_want+st.nb_unwant) is the number of eigenpairs that
703  // have just converged.
704  // st.tb_deftot is the total number of eigenpairs that have converged.
705 
706  size_type st.conv = ind;
707  size_type st.tb_deftot = st.tb_def + st.conv;
708  size_type st.tb_defwant = st.tb_def + st.nb_want - st.nb_nolong;
709 
710  sub_interval SUBYB(0, st.conv);
711 
712  if ( st.tb_defwant >= p ) { // An invariant subspace has been found.
713 
714  st.nb_unwant = 0;
715  st.nb_want = p + st.nb_nolong - st.tb_def;
716  st.tb_defwant = p;
717 
718  if ( pure[st.conv - st.nb_want + 1] == 2 ) {
719  ++st.nb_want; st.tb_defwant = ++p;// il faudrait que ce soit un p local
720  }
721 
722  SUBYB = sub_interval(st.conv - st.nb_want, st.nb_want);
723  // YB = YB(:, st.conv-st.nb_want+1 : st.conv); // On laisse en suspend ..
724  // pure = pure(st.conv-st.nb_want+1 : st.conv,1); // On laisse suspend ..
725  st.conv = st.nb_want;
726  st.tb_deftot = st.tb_def + st.conv;
727  st.ok = true;
728  }
729  }
730 
731 
732 
733  template<typename MAT, typename EVAL, typename PURE>
734  void select_eval_for_purging(const MAT &Hobl, EVAL &eval, MAT &YB,
735  PURE &pure, idgmres_state &st) {
736 
737  typedef typename linalg_traits<MAT>::value_type T;
738  typedef typename number_traits<T>::magnitude_type R;
739  size_type m = st.m;
740 
741  // Computation of the Ritz eigenpairs.
742 
743  col_matrix< std::vector<T> > evect(st.tb_deftot, st.tb_deftot);
744 
745  sub_interval SUB1(0, st.tb_deftot);
746  implicit_qr_algorithm(sub_matrix(Hobl, SUB1),
747  sub_vector(eval, SUB1), evect);
748  std::fill(eval.begin() + st.tb_deftot, eval.end(), std::complex<R>(0));
749 
750  std::vector< std::pair<T, size_type> > eval_sort(m);
751  for (size_type l = 0; l < m; ++l)
752  eval_sort[l] = std::pair<T, size_type>(eval[l], l);
753  std::sort(eval_sort.begin(), eval_sort.end(), compare_vp());
754 
755  std::vector<bool> sorted(m);
756  std::fill(sorted.begin(), sorted.end(), false);
757 
758  std::vector<size_type> ind(m);
759  for (size_type l = 0; l < m; ++l) ind[l] = eval_sort[l].second;
760 
761  std::vector<bool> kept(m, false);
762  std::fill(kept.begin(), kept.begin()+st.tb_def, true);
763 
764  apply_permutation(eval, ind);
765  apply_permutation(evect, ind);
766 
767  size_type j;
768  for (j = 0; j < st.tb_deftot; ++j) {
769 
770  for (size_type l = 0, l < st.tb_deftot; ++l)
771  YB(l, j) = std::real(evect(l, j));
772 
773  if (std::imag(eval[j]) != R(0)) {
774  for (size_type l = 0, l < m-st.tb_def; ++l)
775  YB(l, j+1) = std::imag(evect(l, j));
776  pure[j] = 1;
777  pure[j+1] = 2;
778 
779  j += 2;
780  }
781  else ++j;
782  }
783  }
784 
785 
786 }
787 
788 #endif
gmm::copy
void copy(const L1 &l1, L2 &l2)
*‍/
Definition: gmm_blas.h:978
gmm::vect_norm2
number_traits< typename linalg_traits< V >::value_type >::magnitude_type vect_norm2(const V &v)
Euclidean norm of a vector.
Definition: gmm_blas.h:558
gmm::clear
void clear(L &l)
clear (fill with zeros) a vector or matrix.
Definition: gmm_blas.h:59
gmm::mult
void mult(const L1 &l1, const L2 &l2, L3 &l3)
*‍/
Definition: gmm_blas.h:1664
gmm::add
void add(const L1 &l1, L2 &l2)
*‍/
Definition: gmm_blas.h:1277
gmm::qr_factor
void qr_factor(const MAT1 &A_)
QR factorization using Householder method (complex and real version).
Definition: gmm_dense_qr.h:49
gmm_dense_sylvester.h
Sylvester equation solver.
gmm_iter.h
Iteration object.
gmm_kernel.h
Include the base gmm files.
gmm::gmres
void gmres(const Mat &A, Vec &x, const VecB &b, const Precond &M, int restart, iteration &outer, Basis &KS)
Generalized Minimum Residual.
Definition: gmm_solver_gmres.h:90
bgeot::size_type
size_t size_type
used as the common size type in the library
Definition: bgeot_poly.h:49
bgeot::alpha
size_type alpha(short_type n, short_type d)
Return the value of which is the number of monomials of a polynomial of variables and degree .
Definition: bgeot_poly.cc:47