1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-2019, Universitat Politecnica de Valencia, Spain
6: This file is part of SLEPc.
7: SLEPc is distributed under a 2-clause BSD license (see LICENSE).
8: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
9: */
10: /*
11: BDC - Block-divide and conquer (see description in README file)
12: */
14: #include <slepc/private/dsimpl.h>
15: #include <slepcblaslapack.h>
17: static PetscErrorCode cutlr_(PetscBLASInt start,PetscBLASInt n,PetscBLASInt blkct, 18: PetscBLASInt *bsizes,PetscBLASInt *ranks,PetscBLASInt *cut, 19: PetscBLASInt *lsum,PetscBLASInt *lblks,PetscBLASInt *info) 20: {
21: /* -- Routine written in LAPACK Version 3.0 style -- */
22: /* *************************************************** */
23: /* Written by */
24: /* Michael Moldaschl and Wilfried Gansterer */
25: /* University of Vienna */
26: /* last modification: March 16, 2014 */
28: /* Small adaptations of original code written by */
29: /* Wilfried Gansterer and Bob Ward, */
30: /* Department of Computer Science, University of Tennessee */
31: /* see https://doi.org/10.1137/S1064827501399432 */
32: /* *************************************************** */
34: /* Purpose */
35: /* ======= */
37: /* CUTLR computes the optimal cut in a sequence of BLKCT neighboring */
38: /* blocks whose sizes are given by the array BSIZES. */
39: /* The sum of all block sizes in the sequence considered is given by N. */
40: /* The cut is optimal in the sense that the difference of the sizes of */
41: /* the resulting two halves is minimum over all cuts with minimum ranks */
42: /* between blocks of the sequence considered. */
44: /* Arguments */
45: /* ========= */
47: /* START (input) INTEGER */
48: /* In the original array KSIZES of the calling routine DIBTDC, */
49: /* the position where the sequence considered in this routine starts. */
50: /* START >= 1. */
52: /* N (input) INTEGER */
53: /* The sum of all the block sizes of the sequence to be cut = */
54: /* = sum_{i=1}^{BLKCT} BSIZES( I ). */
55: /* N >= 3. */
57: /* BLKCT (input) INTEGER */
58: /* The number of blocks in the sequence to be cut. */
59: /* BLKCT >= 3. */
61: /* BSIZES (input) INTEGER array, dimension (BLKCT) */
62: /* The dimensions of the (quadratic) blocks of the sequence to be */
63: /* cut. sum_{i=1}^{BLKCT} BSIZES( I ) = N. */
65: /* RANKS (input) INTEGER array, dimension (BLKCT-1) */
66: /* The ranks determining the approximations of the off-diagonal */
67: /* blocks in the sequence considered. */
69: /* CUT (output) INTEGER */
70: /* After the optimum cut has been determined, the position (in the */
71: /* overall problem as worked on in DIBTDC !) of the last block in */
72: /* the first half of the sequence to be cut. */
73: /* START <= CUT <= START+BLKCT-2. */
75: /* LSUM (output) INTEGER */
76: /* After the optimum cut has been determined, the sum of the */
77: /* block sizes in the first half of the sequence to be cut. */
78: /* LSUM < N. */
80: /* LBLKS (output) INTEGER */
81: /* After the optimum cut has been determined, the number of the */
82: /* blocks in the first half of the sequence to be cut. */
83: /* 1 <= LBLKS < BLKCT. */
85: /* INFO (output) INTEGER */
86: /* = 0: successful exit. */
87: /* < 0: illegal arguments. */
88: /* if INFO = -i, the i-th (input) argument had an illegal */
89: /* value. */
90: /* > 0: illegal results. */
91: /* if INFO = i, the i-th (output) argument had an illegal */
92: /* value. */
94: /* Further Details */
95: /* =============== */
97: /* Based on code written by */
98: /* Wilfried Gansterer and Bob Ward, */
99: /* Department of Computer Science, University of Tennessee */
101: /* ===================================================================== */
103: PetscBLASInt i, ksk, kchk, ksum, nhalf, deviat, mindev, minrnk, tmpsum;
106: *info = 0;
107: *lblks = 1;
108: *lsum = 1;
109: *cut = start;
111: if (start < 1) *info = -1;
112: else if (n < 3) *info = -2;
113: else if (blkct < 3) *info = -3;
114: if (*info == 0) {
115: ksum = 0;
116: kchk = 0;
117: for (i = 0; i < blkct; ++i) {
118: ksk = bsizes[i];
119: ksum += ksk;
120: if (ksk < 1) kchk = 1;
121: }
122: if (ksum != n || kchk == 1) *info = -4;
123: }
124: if (*info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong argument %d in CUTLR",-(*info));
126: /* determine smallest rank in the range considered */
128: minrnk = n;
129: for (i = 0; i < blkct-1; ++i) {
130: if (ranks[i] < minrnk) minrnk = ranks[i];
131: }
133: /* determine best cut among those with smallest rank */
135: nhalf = n / 2;
136: tmpsum = 0;
137: mindev = n;
138: for (i = 0; i < blkct; ++i) {
139: tmpsum += bsizes[i];
140: if (ranks[i] == minrnk) {
142: /* determine deviation from "optimal" cut NHALF */
144: deviat = tmpsum - nhalf;
145: if (deviat<0) deviat = -deviat;
147: /* compare to best deviation so far */
149: if (deviat < mindev) {
150: mindev = deviat;
151: *cut = start + i;
152: *lblks = i + 1;
153: *lsum = tmpsum;
154: }
155: }
156: }
158: if (*cut < start || *cut >= start + blkct - 1) *info = 6;
159: else if (*lsum < 1 || *lsum >= n) *info = 7;
160: else if (*lblks < 1 || *lblks >= blkct) *info = 8;
161: return(0);
162: }
164: PetscErrorCode BDC_dibtdc_(const char *jobz,PetscBLASInt n,PetscBLASInt nblks,165: PetscBLASInt *ksizes,PetscReal *d,PetscBLASInt l1d,PetscBLASInt l2d,166: PetscReal *e,PetscBLASInt *rank,PetscBLASInt l1e,PetscBLASInt l2e,167: PetscReal tol,PetscReal *ev,PetscReal *z,PetscBLASInt ldz,PetscReal *work,168: PetscBLASInt lwork,PetscBLASInt *iwork,PetscBLASInt liwork,169: PetscBLASInt *info,PetscBLASInt jobz_len)170: {
171: /* -- Routine written in LAPACK Version 3.0 style -- */
172: /* *************************************************** */
173: /* Written by */
174: /* Michael Moldaschl and Wilfried Gansterer */
175: /* University of Vienna */
176: /* last modification: March 16, 2014 */
178: /* Small adaptations of original code written by */
179: /* Wilfried Gansterer and Bob Ward, */
180: /* Department of Computer Science, University of Tennessee */
181: /* see https://doi.org/10.1137/S1064827501399432 */
182: /* *************************************************** */
184: /* Purpose */
185: /* ======= */
187: /* DIBTDC computes all eigenvalues and corresponding eigenvectors of a */
188: /* symmetric irreducible block tridiagonal matrix with rank RANK matrices */
189: /* as the subdiagonal blocks using a block divide and conquer method. */
191: /* Arguments */
192: /* ========= */
194: /* JOBZ (input) CHARACTER*1 */
195: /* = 'N': Compute eigenvalues only (not implemented); */
196: /* = 'D': Compute eigenvalues and eigenvectors. */
197: /* Eigenvectors are accumulated in the */
198: /* divide-and-conquer process. */
200: /* N (input) INTEGER */
201: /* The dimension of the symmetric irreducible block tridiagonal */
202: /* matrix. N >= 2. */
204: /* NBLKS (input) INTEGER, 2 <= NBLKS <= N */
205: /* The number of diagonal blocks in the matrix. */
207: /* KSIZES (input) INTEGER array, dimension (NBLKS) */
208: /* The dimension of the square diagonal blocks from top left */
209: /* to bottom right. KSIZES(I) >= 1 for all I, and the sum of */
210: /* KSIZES(I) for I = 1 to NBLKS has to be equal to N. */
212: /* D (input) DOUBLE PRECISION array, dimension (L1D,L2D,NBLKS) */
213: /* The lower triangular elements of the symmetric diagonal */
214: /* blocks of the block tridiagonal matrix. Elements of the top */
215: /* left diagonal block, which is of dimension KSIZES(1), are */
216: /* contained in D(*,*,1); the elements of the next diagonal */
217: /* block, which is of dimension KSIZES(2), are contained in */
218: /* D(*,*,2); etc. */
220: /* L1D (input) INTEGER */
221: /* The leading dimension of the array D. L1D >= max(3,KMAX), */
222: /* where KMAX is the dimension of the largest diagonal block. */
224: /* L2D (input) INTEGER */
225: /* The second dimension of the array D. L2D >= max(3,KMAX), */
226: /* where KMAX is as stated in L1D above. */
228: /* E (input) DOUBLE PRECISION array, dimension (L1E,L2E,NBLKS-1) */
229: /* Contains the elements of the scalars (singular values) and */
230: /* vectors (singular vectors) defining the rank RANK subdiagonal */
231: /* blocks of the matrix. */
232: /* E(1:RANK(K),RANK(K)+1,K) holds the RANK(K) scalars, */
233: /* E(:,1:RANK(K),K) holds the RANK(K) column vectors, and */
234: /* E(:,RANK(K)+2:2*RANK(K)+1,K) holds the row vectors for the K-th */
235: /* subdiagonal block. */
237: /* RANK (input) INTEGER array, dimension (NBLKS-1). */
238: /* The ranks of all the subdiagonal blocks contained in the array E. */
239: /* RANK( K ) <= MIN( KSIZES( K ), KSIZES( K+1 ) ) */
241: /* L1E (input) INTEGER */
242: /* The leading dimension of the array E. L1E >= max(3,2*KMAX+1), */
243: /* where KMAX is as stated in L1D above. */
245: /* L2E (input) INTEGER */
246: /* The second dimension of the array E. L2E >= max(3,2*KMAX+1), */
247: /* where KMAX is as stated in L1D above. */
249: /* TOL (input) DOUBLE PRECISION, TOL <= 1.0D-1 */
250: /* User specified deflation tolerance for the routine DMERG2. */
251: /* If ( 1.0D-1 >= TOL >= 20*EPS ) then TOL is used as */
252: /* the deflation tolerance in DSRTDF. */
253: /* If ( TOL < 20*EPS ) then the standard deflation tolerance from */
254: /* LAPACK is used as the deflation tolerance in DSRTDF. */
256: /* EV (output) DOUBLE PRECISION array, dimension (N) */
257: /* If INFO = 0, then EV contains the eigenvalues of the */
258: /* symmetric block tridiagonal matrix in ascending order. */
260: /* Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N) */
261: /* On entry, Z will be the identity matrix. */
262: /* On exit, Z contains the eigenvectors of the block tridiagonal */
263: /* matrix. */
265: /* LDZ (input) INTEGER */
266: /* The leading dimension of the array Z. LDZ >= max(1,N). */
268: /* WORK (workspace) DOUBLE PRECISION array, dimension (LWORK) */
270: /* LWORK (input) INTEGER */
271: /* The dimension of the array WORK. */
272: /* In order to guarantee correct results in all cases, */
273: /* LWORK must be at least ( 2*N**2 + 3*N ). In many cases, */
274: /* less workspace is required. The absolute minimum required is */
275: /* ( N**2 + 3*N ). */
276: /* If the workspace provided is not sufficient, the routine will */
277: /* return a corresponding error code and report how much workspace */
278: /* was missing (see INFO). */
280: /* IWORK (workspace) INTEGER array, dimension (LIWORK) */
282: /* LIWORK (input) INTEGER */
283: /* The dimension of the array IWORK. */
284: /* LIWORK must be at least ( 5*N + 3 + 4*NBLKS - 4 ): */
285: /* 5*KMAX+3 for DSYEVD, 5*N for ????, */
286: /* 4*NBLKS-4 for the preprocessing (merging order) */
287: /* Summarizing, the minimum integer workspace needed is */
288: /* MAX( 5*N, 5*KMAX + 3 ) + 4*NBLKS - 4 */
290: /* INFO (output) INTEGER */
291: /* = 0: successful exit. */
292: /* < 0, > -99: illegal arguments. */
293: /* if INFO = -i, the i-th argument had an illegal value. */
294: /* = -99: error in the preprocessing (call of CUTLR). */
295: /* < -200: not enough workspace. Space for ABS(INFO + 200) */
296: /* numbers is required in addition to the workspace provided, */
297: /* otherwise some eigenvectors will be incorrect. */
298: /* > 0: The algorithm failed to compute an eigenvalue while */
299: /* working on the submatrix lying in rows and columns */
300: /* INFO/(N+1) through mod(INFO,N+1). */
302: /* Further Details */
303: /* =============== */
305: /* Based on code written by */
306: /* Wilfried Gansterer and Bob Ward, */
307: /* Department of Computer Science, University of Tennessee */
309: /* This routine is comparable to Dlaed0.f from LAPACK. */
311: /* ===================================================================== */
313: #if defined(SLEPC_MISSING_LAPACK_LACPY) || defined(PETSC_MISSING_LAPACK_SYEV)
315: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LACPY/SYEV - Lapack routine is unavailable");
316: #else
317: PetscBLASInt i, j, k, np, rp1, ksk, one=1;
318: PetscBLASInt cut, mat1, kchk, kbrk, blks, kmax, icut, size, ksum, lsum;
319: PetscBLASInt lblks, rblks, isize, lwmin, ilsum;
320: PetscBLASInt start, vstrt, istck1, istck2, istck3, merged;
321: PetscBLASInt liwmin, matsiz, startp, istrtp;
322: PetscReal rho, done=1.0, dmone=-1.0;
326: *info = 0;
328: if (*(unsigned char *)jobz != 'N' && *(unsigned char *)jobz != 'D') *info = -1;
329: else if (n < 2) *info = -2;
330: else if (nblks < 2 || nblks > n) *info = -3;
331: if (*info == 0) {
332: ksum = 0;
333: kmax = 0;
334: kchk = 0;
335: for (k = 0; k < nblks; ++k) {
336: ksk = ksizes[k];
337: ksum += ksk;
338: if (ksk > kmax) kmax = ksk;
339: if (ksk < 1) kchk = 1;
340: }
341: lwmin = n*n + n * 3;
342: liwmin = PetscMax(n * 5,kmax * 5 + 3) + 4*nblks - 4;
343: if (ksum != n || kchk == 1) *info = -4;
344: else if (l1d < PetscMax(3,kmax)) *info = -6;
345: else if (l2d < PetscMax(3,kmax)) *info = -7;
346: else if (l1e < PetscMax(3,2*kmax + 1)) *info = -10;
347: else if (l2e < PetscMax(3,2*kmax + 1)) *info = -11;
348: else if (tol > .1) *info = -12;
349: else if (ldz < PetscMax(1,n)) *info = -15;
350: else if (lwork < lwmin) *info = -17;
351: else if (liwork < liwmin) *info = -19;
352: }
353: if (*info == 0) {
354: for (k = 0; k < nblks-1; ++k) {
355: if (rank[k] > PetscMin(ksizes[k],ksizes[k+1]) || rank[k] < 1) *info = -9;
356: }
357: }
359: if (*info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong argument %d in DIBTDC",-(*info));
361: /* **************************************************************************** */
363: /* ...Preprocessing..................................................... */
364: /* Determine the optimal order for merging the subblocks and how much */
365: /* workspace will be needed for the merging (determined by the last */
366: /* merge). Cutpoints for the merging operations are determined and stored */
367: /* in reverse chronological order (starting with the final merging */
368: /* operation). */
370: /* integer workspace requirements for the preprocessing: */
371: /* 4*(NBLKS-1) for merging history */
372: /* at most 3*(NBLKS-1) for stack */
374: start = 1;
375: size = n;
376: blks = nblks;
377: merged = 0;
378: k = 0;
380: /* integer workspace used for the stack is not needed any more after the */
381: /* preprocessing and therefore can use part of the 5*N */
382: /* integer workspace needed later on in the code */
384: istck1 = 0;
385: istck2 = istck1 + nblks;
386: istck3 = istck2 + nblks;
388: /* integer workspace used for storing the order of merges starts AFTER */
389: /* the integer workspace 5*N+3 which is needed later on in the code */
390: /* (5*KMAX+3 for DSYEVD, 4*N in DMERG2) */
392: istrtp = n * 5 + 4;
393: icut = istrtp + nblks - 1;
394: isize = icut + nblks - 1;
395: ilsum = isize + nblks - 1;
397: L200:399: if (nblks >= 3) {
401: /* Determine the cut point. Note that in the routine CUTLR it is */
402: /* chosen such that it yields the best balanced merging operation */
403: /* among all the rank modifications with minimum rank. */
405: cutlr_(start, size, blks, &ksizes[start-1], &rank[start-1], &cut,
406: &lsum, &lblks, info);
407: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: Error in cutlr, info = %d",*info);
409: } else {
410: cut = 1;
411: lsum = ksizes[0];
412: lblks = 1;
413: }
415: ++merged;
416: startp = 0;
417: for (i = 0; i < start-1; ++i) startp += ksizes[i];
418: iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
419: iwork[icut + (nblks - 1) - merged-1] = cut;
420: iwork[isize + (nblks - 1) - merged-1] = size;
421: iwork[ilsum + (nblks - 1) - merged-1] = lsum;
423: if (lblks == 2) {
425: /* one merge in left branch, left branch done */
426: ++merged;
427: iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
428: iwork[icut + (nblks - 1) - merged-1] = start;
429: iwork[isize + (nblks - 1) - merged-1] = lsum;
430: iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start-1];
431: }
433: if (lblks == 1 || lblks == 2) {
435: /* left branch done, continue on the right side */
436: start += lblks;
437: size -= lsum;
438: blks -= lblks;
440: if (blks <= 0) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: Error in preprocessing, blks = %d",blks);
442: if (blks == 2) {
444: /* one merge in right branch, right branch done */
445: ++merged;
446: startp += lsum;
447: iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
448: iwork[icut + (nblks - 1) - merged-1] = start;
449: iwork[isize + (nblks - 1) - merged-1] = size;
450: iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start-1];
451: }
453: if (blks == 1 || blks == 2) {
455: /* get the next subproblem from the stack or finished */
457: if (k >= 1) {
459: /* something left on the stack */
460: start = iwork[istck1 + k-1];
461: size = iwork[istck2 + k-1];
462: blks = iwork[istck3 + k-1];
463: --k;
464: goto L200;
465: } else {
467: /* nothing left on the stack */
468: if (merged != nblks-1) SETERRQ(PETSC_COMM_SELF,1,"ERROR in preprocessing - not enough merges performed");
470: /* exit preprocessing */
472: }
473: } else {
475: /* BLKS.GE.3, and therefore analyze the right side */
477: goto L200;
478: }
479: } else {
481: /* LBLKS.GE.3, and therefore check the right side and */
482: /* put it on the stack if required */
484: rblks = blks - lblks;
485: if (rblks >= 3) {
486: ++k;
487: iwork[istck1 + k-1] = cut + 1;
488: iwork[istck2 + k-1] = size - lsum;
489: iwork[istck3 + k-1] = rblks;
490: } else if (rblks == 2) {
492: /* one merge in right branch, right branch done */
493: /* (note that nothing needs to be done if RBLKS.EQ.1 !) */
495: ++merged;
496: startp += lsum;
497: iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
498: iwork[icut + (nblks - 1) - merged-1] = start + lblks;
499: iwork[isize + (nblks - 1) - merged-1] = size - lsum;
500: iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start + lblks-1];
501: }
502: if (rblks <= 0) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: ERROR in preprocessing - rblks = %d",rblks);
504: /* continue on the left side */
506: size = lsum;
507: blks = lblks;
508: goto L200;
509: }
511: /* SIZE = IWORK( ISIZE+NBLKS-2 ) */
512: /* MAT1 = IWORK( ILSUM+NBLKS-2 ) */
514: /* Note: after the dimensions SIZE and MAT1 of the last merging */
515: /* operation have been determined, an upper bound for the workspace */
516: /* requirements which is independent of how much deflation occurs in */
517: /* the last merging operation could be determined as follows */
518: /* (based on (3.15) and (3.19) from UT-CS-00-447): */
520: /* IF( MAT1.LE.N/2 ) THEN */
521: /* WSPREQ = 3*N + 3/2*( SIZE-MAT1 )**2 + N*N/2 + MAT1*MAT1 */
522: /* ELSE */
523: /* WSPREQ = 3*N + 3/2*MAT1*MAT1 + N*N/2 + ( SIZE-MAT1 )**2 */
524: /* END IF */
526: /* IF( LWORK-WSPREQ.LT.0 )THEN */
527: /* not enough work space provided */
528: /* INFO = -200 - ( WSPREQ-LWORK ) */
529: /* RETURN */
530: /* END IF */
531: /* However, this is not really useful, since the actual check whether */
532: /* enough workspace is provided happens in DMERG2.f ! */
534: /* ************************************************************************* */
536: /* ...Solve subproblems................................... */
538: /* Divide the matrix into NBLKS submatrices using rank-r */
539: /* modifications (cuts) and solve for their eigenvalues and */
540: /* eigenvectors. Initialize index array to sort eigenvalues. */
542: /* first block: ...................................... */
544: /* correction for block 1: D1 - V1 \Sigma1 V1^T */
546: ksk = ksizes[0];
547: rp1 = rank[0];
549: /* initialize the proper part of Z with the diagonal block D1 */
550: /* (the correction will be made in Z and then the call of DSYEVD will */
551: /* overwrite it with the eigenvectors) */
553: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, d, &l1d, z, &ldz));
555: /* copy D1 into WORK (in order to be able to restore it afterwards) */
557: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, d, &l1d, work, &ksk));
559: /* copy V1 into the first RANK(1) columns of D1 and then */
560: /* multiply with \Sigma1 */
562: for (i = 0; i < rank[0]; ++i) {
563: PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(rp1 + i+1)*l1e], &one, &d[i*l1d], &one));
564: PetscStackCallBLAS("BLASscal",BLASscal_(&ksk, &e[i + rp1*l1e], &d[i*l1d], &one));
565: }
567: /* multiply the first RANK( 1 ) columns of D1 with V1^T and */
568: /* subtract the result from the proper part of Z (previously */
569: /* initialized with D1) */
571: PetscStackCallBLAS("BLASgemm",BLASgemm_("N", "T", &ksk, &ksk, rank, &dmone,
572: d, &l1d, &e[(rank[0]+1)*l1e], &l1e, &done, z, &ldz));
574: /* restore the original D1 from WORK */
576: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, work, &ksk, d, &l1d));
578: /* eigenanalysis of block 1 (using DSYEVD) */
580: PetscStackCallBLAS("LAPACKsyev",LAPACKsyev_("V", "L", &ksk, z, &ldz, ev, work, &lwork, info));
581: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: Error in DSYEVD for block 1, info = %d",*info);
583: /* EV( 1: ) contains the eigenvalues in ascending order */
584: /* (they are returned this way by DSYEVD) */
586: for (i = 0; i < ksk; ++i) iwork[i] = i+1;
588: /* intermediate blocks: .............................. */
590: np = ksk;
592: /* remaining number of blocks */
594: if (nblks > 2) {
595: for (k = 1; k < nblks-1; ++k) {
597: /* correction for block K: */
598: /* Dk - U(k-1) \Sigma(k-1) U(k-1)^T - Vk \Sigmak Vk^T */
600: ksk = ksizes[k];
601: rp1 = rank[k];
603: /* initialize the proper part of Z with the diagonal block Dk */
604: /* (the correction will be made in Z and then the call of DSYEVD will */
605: /* overwrite it with the eigenvectors) */
607: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, &d[k*l1d*l2d], &l1d, &z[np+np*ldz], &ldz));
609: /* copy Dk into WORK (in order to be able to restore it afterwards) */
611: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, &d[k*l1d*l2d], &l1d, work, &ksk));
613: /* copy U(K-1) into the first RANK(K-1) columns of Dk and then */
614: /* multiply with \Sigma(K-1) */
616: for (i = 0; i < rank[k-1]; ++i) {
617: PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(i+(k-1)*l2e)*l1e], &one, &d[(i+k*l2d)*l1d], &one));
618: PetscStackCallBLAS("BLASscal",BLASscal_(&ksk, &e[i+(rank[k-1]+(k-1)*l2e)*l1e], &d[(i+k*l2d)*l1d], &one));
619: }
621: /* multiply the first RANK(K-1) columns of Dk with U(k-1)^T and */
622: /* subtract the result from the proper part of Z (previously */
623: /* initialized with Dk) */
625: PetscStackCallBLAS("BLASgemm",BLASgemm_("N", "T", &ksk, &ksk, &rank[k-1],
626: &dmone, &d[k*l1d*l2d],
627: &l1d, &e[(k-1)*l1e*l2e], &l1e, &done, &z[np+np*ldz], &ldz));
629: /* copy Vk into the first RANK(K) columns of Dk and then */
630: /* multiply with \Sigmak */
632: for (i = 0; i < rank[k]; ++i) {
633: PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(rp1+i+1 + k*l2e)*l1e], &one, &d[(i + k*l2d)*l1d], &one));
634: PetscStackCallBLAS("BLASscal",BLASscal_(&ksk, &e[i + (rp1 + k*l2e)*l1e], &d[(i + k*l2d)*l1d], &one));
635: }
637: /* multiply the first RANK(K) columns of Dk with Vk^T and */
638: /* subtract the result from the proper part of Z (previously */
639: /* updated with [- U(k-1) \Sigma(k-1) U(k-1)^T] ) */
641: PetscStackCallBLAS("BLASgemm",BLASgemm_("N", "T", &ksk, &ksk, &rank[k],
642: &dmone, &d[k*l1d*l2d], &l1d,
643: &e[(rank[k]+1 + k*l2e)*l1e], &l1e, &done, &z[np+np*ldz], &ldz));
645: /* restore the original Dk from WORK */
647: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, work, &ksk, &d[k*l1d*l2d], &l1d));
649: /* eigenanalysis of block K (using dsyevd) */
651: PetscStackCallBLAS("LAPACKsyev",LAPACKsyev_("V", "L", &ksk, &z[np+np*ldz],
652: &ldz, &ev[np], work, &lwork, info));
653: if (*info) SETERRQ2(PETSC_COMM_SELF,1,"dibtdc: Error in DSYEVD for block %d, info = %d",k,*info);
655: /* EV( NPP1: ) contains the eigenvalues in ascending order */
656: /* (they are returned this way by DSYEVD) */
658: for (i = 0; i < ksk; ++i) iwork[np + i] = i+1;
660: /* update NP */
661: np += ksk;
662: }
663: }
665: /* last block: ....................................... */
667: /* correction for block NBLKS: */
668: /* D(nblks) - U(nblks-1) \Sigma(nblks-1) U(nblks-1)^T */
670: ksk = ksizes[nblks-1];
672: /* initialize the proper part of Z with the diagonal block D(nblks) */
673: /* (the correction will be made in Z and then the call of DSYEVD will */
674: /* overwrite it with the eigenvectors) */
676: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, &d[(nblks-1)*l1d*l2d], &l1d, &z[np+np*ldz], &ldz));
678: /* copy D(nblks) into WORK (in order to be able to restore it afterwards) */
680: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, &d[(nblks-1)*l1d*l2d], &l1d, work, &ksk));
682: /* copy U(nblks-1) into the first RANK(nblks-1) columns of D(nblks) and then */
683: /* multiply with \Sigma(nblks-1) */
685: for (i = 0; i < rank[nblks-2]; ++i) {
686: PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(i + (nblks-2)*l2e)*l1e],
687: &one, &d[(i + (nblks-1)*l2d)*l1d], &one));
688: PetscStackCallBLAS("BLASscal",BLASscal_(&ksk,
689: &e[i + (rank[nblks-2] + (nblks-2)*l2e)*l1e],
690: &d[(i + (nblks-1)*l2d)*l1d], &one));
691: }
693: /* multiply the first RANK(nblks-1) columns of D(nblks) with U(nblks-1)^T */
694: /* and subtract the result from the proper part of Z (previously */
695: /* initialized with D(nblks) ) */
697: PetscStackCallBLAS("BLASgemm",BLASgemm_("N", "T", &ksk, &ksk, &rank[nblks - 2],
698: &dmone, &d[(nblks-1)*l1d*l2d], &l1d,
699: &e[(nblks-2)*l1e*l2e], &l1e, &done, &z[np+np*ldz], &ldz));
701: /* restore the original D(nblks) from WORK */
703: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, work, &ksk, &d[(nblks-1)*l1d*l2d], &l1d));
705: /* eigenanalysis of block NBLKS (using dsyevd) */
707: PetscStackCallBLAS("LAPACKsyev",LAPACKsyev_("V", "L", &ksk, &z[np+np*ldz], &ldz, &ev[np], work, &lwork, info));
708: if (*info) SETERRQ2(PETSC_COMM_SELF,1,"dibtdc: Error in DSYEVD for block %d, info = %d",nblks,*info);
710: /* EV( NPP1: ) contains the eigenvalues in ascending order */
711: /* (they are returned this way by DSYEVD) */
713: for (i = 0; i < ksk; ++i) iwork[np + i] = i+1;
715: /* note that from here on the entire workspace is available again */
718: /* Perform all the merging operations. */
720: vstrt = 0;
721: for (i = 0; i < nblks-1; ++i) {
723: /* MATSIZ = total size of the current rank RANK modification problem */
725: matsiz = iwork[isize + i - 1];
726: np = iwork[istrtp + i - 1];
727: kbrk = iwork[icut + i - 1];
728: mat1 = iwork[ilsum + i - 1];
729: vstrt += np;
731: for (j = 0; j < rank[kbrk-1]; ++j) {
733: /* NOTE: The parameter RHO in DMERG2 is modified in DSRTDF */
734: /* (multiplied by 2) ! In order not to change the */
735: /* singular value stored in E( :, RANK( KBRK )+1, KBRK ), */
736: /* we do not pass on this variable as an argument to DMERG2, */
737: /* but we assign a separate variable RHO here which is passed */
738: /* on to DMERG2. */
739: /* Alternative solution in F90: */
740: /* pass E( :,RANK( KBRK )+1,KBRK ) to an INTENT( IN ) parameter */
741: /* in DMERG2. */
743: rho = e[j + (rank[kbrk-1] + (kbrk-1)*l2e)*l1e];
745: /* eigenvectors are accumulated ( JOBZ.EQ.'D' ) */
747: BDC_dmerg2_(jobz, j+1, matsiz, &ev[np-1], &z[np-1+(np-1)*ldz],
748: ldz, &iwork[np-1], &rho, &e[(j + (kbrk-1)*l2e)*l1e],
749: ksizes[kbrk], &e[(rank[kbrk-1]+j+1 + (kbrk-1)*l2e)*l1e],
750: ksizes[kbrk-1], mat1, work, lwork, &iwork[n], tol, info, 1);
751: 752: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: Error in dmerg2, info = %d",*info);
753: }
755: /* at this point all RANK( KBRK ) rank-one modifications corresponding */
756: /* to the current off-diagonal block are finished. */
757: /* Move on to the next off-diagonal block. */
759: }
761: /* Re-merge the eigenvalues/vectors which were deflated at the final */
762: /* merging step by sorting all eigenvalues and eigenvectors according */
763: /* to the permutation stored in IWORK. */
765: /* copy eigenvalues and eigenvectors in ordered form into WORK */
766: /* (eigenvalues into WORK( 1:N ), eigenvectors into WORK( N+1:N+1+N^2 ) ) */
768: for (i = 0; i < n; ++i) {
769: j = iwork[i];
770: work[i] = ev[j-1];
771: PetscStackCallBLAS("BLAScopy",BLAScopy_(&n, &z[(j-1)*ldz], &one, &work[n*(i+1)], &one));
772: }
774: /* copy ordered eigenvalues back from WORK( 1:N ) into EV */
776: PetscStackCallBLAS("BLAScopy",BLAScopy_(&n, work, &one, ev, &one));
778: /* copy ordered eigenvectors back from WORK( N+1:N+1+N^2 ) into Z */
780: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("A", &n, &n, &work[n], &n, z, &ldz));
781: return(0);
782: #endif
783: }