1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-2020, 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: PetscErrorCode BDC_dsbtdc_(const char *jobz,const char *jobacc,PetscBLASInt n, 18: PetscBLASInt nblks,PetscBLASInt *ksizes,PetscReal *d,PetscBLASInt l1d, 19: PetscBLASInt l2d,PetscReal *e,PetscBLASInt l1e,PetscBLASInt l2e,PetscReal tol, 20: PetscReal tau1,PetscReal tau2,PetscReal *ev,PetscReal *z,PetscBLASInt ldz, 21: PetscReal *work,PetscBLASInt lwork,PetscBLASInt *iwork,PetscBLASInt liwork, 22: PetscReal *mingap,PetscBLASInt *mingapi,PetscBLASInt *info, 23: PetscBLASInt jobz_len,PetscBLASInt jobacc_len) 24: {
25: /* -- Routine written in LAPACK Version 3.0 style -- */
26: /* *************************************************** */
27: /* Written by */
28: /* Michael Moldaschl and Wilfried Gansterer */
29: /* University of Vienna */
30: /* last modification: March 28, 2014 */
32: /* Small adaptations of original code written by */
33: /* Wilfried Gansterer and Bob Ward, */
34: /* Department of Computer Science, University of Tennessee */
35: /* see https://doi.org/10.1137/S1064827501399432 */
36: /* *************************************************** */
38: /* Purpose */
39: /* ======= */
41: /* DSBTDC computes approximations to all eigenvalues and eigenvectors */
42: /* of a symmetric block tridiagonal matrix using the divide and */
43: /* conquer method with lower rank approximations to the subdiagonal blocks. */
45: /* This code makes very mild assumptions about floating point */
46: /* arithmetic. It will work on machines with a guard digit in */
47: /* add/subtract, or on those binary machines without guard digits */
48: /* which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. */
49: /* It could conceivably fail on hexadecimal or decimal machines */
50: /* without guard digits, but we know of none. See DLAED3M for details. */
52: /* Arguments */
53: /* ========= */
55: /* JOBZ (input) CHARACTER*1 */
56: /* = 'N': Compute eigenvalues only (not implemented); */
57: /* = 'D': Compute eigenvalues and eigenvectors. Eigenvectors */
58: /* are accumulated in the divide-and-conquer process. */
60: /* JOBACC (input) CHARACTER*1 */
61: /* = 'A' ("automatic"): The accuracy parameters TAU1 and TAU2 */
62: /* are determined automatically from the */
63: /* parameter TOL according to the analytical */
64: /* bounds. In that case the input values of */
65: /* TAU1 and TAU2 are irrelevant (ignored). */
66: /* = 'M' ("manual"): The input values of the accuracy parameters */
67: /* TAU1 and TAU2 are used. In that case the input */
68: /* value of the parameter TOL is irrelevant */
69: /* (ignored). */
71: /* N (input) INTEGER */
72: /* The dimension of the symmetric block tridiagonal matrix. */
73: /* N >= 1. */
75: /* NBLKS (input) INTEGER, 1 <= NBLKS <= N */
76: /* The number of diagonal blocks in the matrix. */
78: /* KSIZES (input) INTEGER array, dimension (NBLKS) */
79: /* The dimensions of the square diagonal blocks from top left */
80: /* to bottom right. KSIZES(I) >= 1 for all I, and the sum of */
81: /* KSIZES(I) for I = 1 to NBLKS has to be equal to N. */
83: /* D (input) DOUBLE PRECISION array, dimension (L1D,L2D,NBLKS) */
84: /* The lower triangular elements of the symmetric diagonal */
85: /* blocks of the block tridiagonal matrix. The elements of the top */
86: /* left diagonal block, which is of dimension KSIZES(1), have to */
87: /* be placed in D(*,*,1); the elements of the next diagonal */
88: /* block, which is of dimension KSIZES(2), have to be placed in */
89: /* D(*,*,2); etc. */
91: /* L1D (input) INTEGER */
92: /* The leading dimension of the array D. L1D >= max(3,KMAX), */
93: /* where KMAX is the dimension of the largest diagonal block, */
94: /* i.e., KMAX = max_I (KSIZES(I)). */
96: /* L2D (input) INTEGER */
97: /* The second dimension of the array D. L2D >= max(3,KMAX), */
98: /* where KMAX is as stated in L1D above. */
100: /* E (input) DOUBLE PRECISION array, dimension (L1E,L2E,NBLKS-1) */
101: /* The elements of the subdiagonal blocks of the */
102: /* block tridiagonal matrix. The elements of the top left */
103: /* subdiagonal block, which is KSIZES(2) x KSIZES(1), have to be */
104: /* placed in E(*,*,1); the elements of the next subdiagonal block, */
105: /* which is KSIZES(3) x KSIZES(2), have to be placed in E(*,*,2); etc. */
106: /* During runtime, the original contents of E(*,*,K) is */
107: /* overwritten by the singular vectors and singular values of */
108: /* the lower rank representation. */
110: /* L1E (input) INTEGER */
111: /* The leading dimension of the array E. L1E >= max(3,2*KMAX+1), */
112: /* where KMAX is as stated in L1D above. The size of L1E enables */
113: /* the storage of ALL singular vectors and singular values for */
114: /* the corresponding off-diagonal block in E(*,*,K) and therefore */
115: /* there are no restrictions on the rank of the approximation */
116: /* (only the "natural" restriction */
117: /* RANK(K) .LE. MIN(KSIZES(K),KSIZES(K+1))). */
119: /* L2E (input) INTEGER */
120: /* The second dimension of the array E. L2E >= max(3,2*KMAX+1), */
121: /* where KMAX is as stated in L1D above. The size of L2E enables */
122: /* the storage of ALL singular vectors and singular values for */
123: /* the corresponding off-diagonal block in E(*,*,K) and therefore */
124: /* there are no restrictions on the rank of the approximation */
125: /* (only the "natural" restriction */
126: /* RANK(K) .LE. MIN(KSIZES(K),KSIZES(K+1))). */
128: /* TOL (input) DOUBLE PRECISION, TOL.LE.TOLMAX */
129: /* User specified tolerance for the residuals of the computed */
130: /* eigenpairs. If (JOBACC.EQ.'A') then it is used to determine */
131: /* TAU1 and TAU2; ignored otherwise. */
132: /* If (TOL.LT.40*EPS .AND. JOBACC.EQ.'A') then TAU1 is set to machine */
133: /* epsilon and TAU2 is set to the standard deflation tolerance from */
134: /* LAPACK. */
136: /* TAU1 (input) DOUBLE PRECISION, TAU1.LE.TOLMAX/2 */
137: /* User specified tolerance for determining the rank of the */
138: /* lower rank approximations to the off-diagonal blocks. */
139: /* The rank for each off-diagonal block is determined such that */
140: /* the resulting absolute eigenvalue error is less than or equal */
141: /* to TAU1. */
142: /* If (JOBACC.EQ.'A') then TAU1 is determined automatically from */
143: /* TOL and the input value is ignored. */
144: /* If (JOBACC.EQ.'M' .AND. TAU1.LT.20*EPS) then TAU1 is set to */
145: /* machine epsilon. */
147: /* TAU2 (input) DOUBLE PRECISION, TAU2.LE.TOLMAX/2 */
148: /* User specified deflation tolerance for the routine DIBTDC. */
149: /* If (1.0D-1.GT.TAU2.GT.20*EPS) then TAU2 is used as */
150: /* the deflation tolerance in DSRTDF (EPS is the machine epsilon). */
151: /* If (TAU2.LE.20*EPS) then the standard deflation tolerance from */
152: /* LAPACK is used as the deflation tolerance in DSRTDF. */
153: /* If (JOBACC.EQ.'A') then TAU2 is determined automatically from */
154: /* TOL and the input value is ignored. */
155: /* If (JOBACC.EQ.'M' .AND. TAU2.LT.20*EPS) then TAU2 is set to */
156: /* the standard deflation tolerance from LAPACK. */
158: /* EV (output) DOUBLE PRECISION array, dimension (N) */
159: /* If INFO = 0, then EV contains the computed eigenvalues of the */
160: /* symmetric block tridiagonal matrix in ascending order. */
162: /* Z (output) DOUBLE PRECISION array, dimension (LDZ,N) */
163: /* If (JOBZ.EQ.'D' .AND. INFO = 0) */
164: /* then Z contains the orthonormal eigenvectors of the symmetric */
165: /* block tridiagonal matrix computed by the routine DIBTDC */
166: /* (accumulated in the divide-and-conquer process). */
167: /* If (-199 < INFO < -99) then Z contains the orthonormal */
168: /* eigenvectors of the symmetric block tridiagonal matrix, */
169: /* computed without divide-and-conquer (quick returns). */
170: /* Otherwise not referenced. */
172: /* LDZ (input) INTEGER */
173: /* The leading dimension of the array Z. LDZ >= max(1,N). */
175: /* WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK) */
177: /* LWORK (input) INTEGER */
178: /* The dimension of the array WORK. */
179: /* If NBLKS.EQ.1, then LWORK has to be at least 2N^2+6N+1 */
180: /* (for the call of DSYEVD). */
181: /* If NBLKS.GE.2 and (JOBZ.EQ.'D') then the absolute minimum */
182: /* required for DIBTDC is (N**2 + 3*N). This will not always */
183: /* suffice, though, the routine will return a corresponding */
184: /* error code and report how much work space was missing (see */
185: /* INFO). */
186: /* In order to guarantee correct results in all cases where */
187: /* NBLKS.GE.2, LWORK must be at least (2*N**2 + 3*N). */
189: /* IWORK (workspace/output) INTEGER array, dimension (LIWORK) */
191: /* LIWORK (input) INTEGER */
192: /* The dimension of the array IWORK. */
193: /* LIWORK must be at least (5*N + 5*NBLKS - 1) (for DIBTDC) */
194: /* Note that this should also suffice for the call of DSYEVD on a */
195: /* diagonal block which requires (5*KMAX + 3). */
198: /* MINGAP (output) DOUBLE PRECISION */
199: /* The minimum "gap" between the approximate eigenvalues */
200: /* computed, i.e., MIN( ABS(EV(I+1)-EV(I)) for I=1,2,..., N-1 */
201: /* IF (MINGAP.LE.TOL/10) THEN a warning flag is returned in INFO, */
202: /* because the computed eigenvectors may be unreliable individually */
203: /* (only the subspaces spanned are approximated reliably). */
205: /* MINGAPI (output) INTEGER */
206: /* Index I where the minimum gap in the spectrum occurred. */
208: /* INFO (output) INTEGER */
209: /* = 0: successful exit, no special cases occurred. */
210: /* < -200: not enough workspace. Space for ABS(INFO + 200) */
211: /* numbers is required in addition to the workspace provided, */
212: /* otherwise some of the computed eigenvectors will be incorrect. */
213: /* < -99, > -199: successful exit, but quick returns. */
214: /* if INFO = -100, successful exit, but the input matrix */
215: /* was the zero matrix and no */
216: /* divide-and-conquer was performed */
217: /* if INFO = -101, successful exit, but N was 1 and no */
218: /* divide-and-conquer was performed */
219: /* if INFO = -102, successful exit, but only a single */
220: /* dense block. Standard dense solver */
221: /* was called, no divide-and-conquer was */
222: /* performed */
223: /* if INFO = -103, successful exit, but warning that */
224: /* MINGAP.LE.TOL/10 and therefore the */
225: /* eigenvectors corresponding to close */
226: /* approximate eigenvalues may individually */
227: /* be unreliable (although taken together they */
228: /* do approximate the corresponding subspace to */
229: /* the desired accuracy) */
230: /* = -99: error in the preprocessing in DIBTDC (when determining */
231: /* the merging order). */
232: /* < 0, > -99: illegal arguments. */
233: /* if INFO = -i, the i-th argument had an illegal value. */
234: /* > 0: The algorithm failed to compute an eigenvalue while */
235: /* working on the submatrix lying in rows and columns */
236: /* INFO/(N+1) through mod(INFO,N+1). */
238: /* Further Details */
239: /* =============== */
241: /* Small modifications of code written by */
242: /* Wilfried Gansterer and Bob Ward, */
243: /* Department of Computer Science, University of Tennessee */
244: /* see https://doi.org/10.1137/S1064827501399432 */
246: /* Based on the design of the LAPACK code sstedc.f written by Jeff */
247: /* Rutter, Computer Science Division, University of California at */
248: /* Berkeley, and modified by Francoise Tisseur, University of Tennessee. */
250: /* ===================================================================== */
252: /* .. Parameters .. */
254: #define TOLMAX 0.1256: /* TOLMAX .... upper bound for tolerances TOL, TAU1, TAU2 */
257: /* NOTE: in the routine DIBTDC, the value */
258: /* 1.D-1 is hardcoded for TOLMAX ! */
260: PetscBLASInt i, j, k, i1, iwspc, lwmin, start;
261: PetscBLASInt ii, ip, nk, rk, np, iu, rp1, ldu;
262: PetscBLASInt ksk, ivt, iend, kchk=0, kmax=0, one=1, zero=0;
263: PetscBLASInt ldvt, ksum=0, kskp1, spneed, nrblks, liwmin, isvals;
264: PetscReal p, d2, eps, dmax, emax, done = 1.0;
265: PetscReal dnrm, tiny, anorm, exdnrm=0, dropsv, absdiff;
269: /* Determine machine epsilon. */
270: eps = LAPACKlamch_("Epsilon");
272: *info = 0;
274: if (*(unsigned char *)jobz != 'N' && *(unsigned char *)jobz != 'D') *info = -1;
275: else if (*(unsigned char *)jobacc != 'A' && *(unsigned char *)jobacc != 'M') *info = -2;
276: else if (n < 1) *info = -3;
277: else if (nblks < 1 || nblks > n) *info = -4;
278: if (*info == 0) {
279: for (k = 0; k < nblks; ++k) {
280: ksk = ksizes[k];
281: ksum += ksk;
282: if (ksk > kmax) kmax = ksk;
283: if (ksk < 1) kchk = 1;
284: }
285: if (nblks == 1) lwmin = 2*n*n + n*6 + 1;
286: else lwmin = n*n + n*3;
287: liwmin = n * 5 + nblks * 5 - 4;
288: if (ksum != n || kchk == 1) *info = -5;
289: else if (l1d < PetscMax(3,kmax)) *info = -7;
290: else if (l2d < PetscMax(3,kmax)) *info = -8;
291: else if (l1e < PetscMax(3,2*kmax+1)) *info = -10;
292: else if (l2e < PetscMax(3,2*kmax+1)) *info = -11;
293: else if (*(unsigned char *)jobacc == 'A' && tol > TOLMAX) *info = -12;
294: else if (*(unsigned char *)jobacc == 'M' && tau1 > TOLMAX/2) *info = -13;
295: else if (*(unsigned char *)jobacc == 'M' && tau2 > TOLMAX/2) *info = -14;
296: else if (ldz < PetscMax(1,n)) *info = -17;
297: else if (lwork < lwmin) *info = -19;
298: else if (liwork < liwmin) *info = -21;
299: }
301: if (*info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong argument %d in DSBTDC",-(*info));
303: /* Quick return if possible */
305: if (n == 1) {
306: ev[0] = d[0]; z[0] = 1.;
307: *info = -101;
308: return(0);
309: }
311: /* If NBLKS is equal to 1, then solve the problem with standard */
312: /* dense solver (in this case KSIZES(1) = N). */
314: if (nblks == 1) {
315: for (i = 0; i < n; ++i) {
316: for (j = 0; j <= i; ++j) {
317: z[i + j*ldz] = d[i + j*l1d];
318: }
319: }
320: PetscStackCallBLAS("LAPACKsyevd",LAPACKsyevd_("V", "L", &n, z, &ldz, ev, work, &lwork, iwork, &liwork, info));
321: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DSYEVD, info = %d",*info);
322: *info = -102;
323: return(0);
324: }
326: /* determine the accuracy parameters (if requested) */
328: if (*(unsigned char *)jobacc == 'A') {
329: tau1 = tol / 2;
330: if (tau1 < eps * 20) tau1 = eps;
331: tau2 = tol / 2;
332: }
334: /* Initialize Z as the identity matrix */
336: if (*(unsigned char *)jobz == 'D') {
337: for (j=0;j<n;j++) for (i=0;i<n;i++) z[i+j*ldz] = 0.0;
338: for (i=0;i<n;i++) z[i+i*ldz] = 1.0;
339: }
341: /* Determine the off-diagonal ranks, form and store the lower rank */
342: /* approximations based on the tolerance parameters, the */
343: /* RANK(K) largest singular values and the associated singular */
344: /* vectors of each subdiagonal block. Also find the maximum norm of */
345: /* the subdiagonal blocks (in EMAX). */
347: /* Compute SVDs of the subdiagonal blocks.... */
349: /* EMAX .... maximum norm of the off-diagonal blocks */
351: emax = 0.;
352: for (k = 0; k < nblks-1; ++k) {
353: ksk = ksizes[k];
354: kskp1 = ksizes[k+1];
355: isvals = 0;
357: /* Note that min(KSKP1,KSK).LE.N/2 (equality possible for */
358: /* NBLKS=2), and therefore storing the singular values requires */
359: /* at most N/2 entries of the * array WORK. */
361: iu = isvals + n / 2;
362: ivt = isvals + n / 2;
364: /* Call of DGESVD: The space for U is not referenced, since */
365: /* JOBU='O' and therefore this portion of the array WORK */
366: /* is not referenced for U. */
368: ldu = kskp1;
369: ldvt = PetscMin(kskp1,ksk);
370: iwspc = ivt + n * n / 2;
372: /* Note that the minimum workspace required for this call */
373: /* of DGESVD is: N/2 for storing the singular values + N**2/2 for */
374: /* storing V^T + 5*N/2 workspace = N**2/2 + 3*N. */
376: i1 = lwork - iwspc;
377: PetscStackCallBLAS("LAPACKgesvd",LAPACKgesvd_("O", "S", &kskp1, &ksk,
378: &e[k*l1e*l2e], &l1e, &work[isvals],
379: &work[iu], &ldu, &work[ivt], &ldvt, &work[iwspc], &i1, info));
380: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DGESVD, info = %d",*info);
382: /* Note that after the return from DGESVD U is stored in */
383: /* E(*,*,K), and V^{\top} is stored in WORK(IVT, IVT+1, ....) */
385: /* determine the ranks RANK() for the approximations */
387: rk = PetscMin(ksk,kskp1);
388: L8:389: dropsv = work[isvals - 1 + rk];
391: if (dropsv * 2. <= tau1) {
393: /* the error caused by dropping singular value RK is */
394: /* small enough, try to reduce the rank by one more */
396: if (--rk > 0) goto L8;
397: else iwork[k] = 0;
398: } else {
400: /* the error caused by dropping singular value RK is */
401: /* too large already, RK is the rank required to achieve the */
402: /* desired accuracy */
404: iwork[k] = rk;
405: }
407: /* ************************************************************************** */
409: /* Store the first RANK(K) terms of the SVD of the current */
410: /* off-diagonal block. */
411: /* NOTE that here it is required that L1E, L2E >= 2*KMAX+1 in order */
412: /* to have enough space for storing singular vectors and values up */
413: /* to the full SVD of an off-diagonal block !!!! */
415: /* u1-u_RANK(K) is already contained in E(:,1:RANK(K),K) (as a */
416: /* result of the call of DGESVD !), the sigma1-sigmaK are to be */
417: /* stored in E(1:RANK(K),RANK(K)+1,K), and v1-v_RANK(K) are to be */
418: /* stored in E(:,RANK(K)+2:2*RANK(K)+1,K) */
420: rp1 = iwork[k];
421: for (j = 0; j < iwork[k]; ++j) {
423: /* store sigma_J in E(J,RANK(K)+1,K) */
425: e[j + (rp1 + k*l2e)* l1e] = work[isvals + j];
427: /* update maximum norm of subdiagonal blocks */
429: if (e[j + (rp1 + k*l2e)*l1e] > emax) {
430: emax = e[j + (rp1 + k*l2e)*l1e];
431: }
433: /* store v_J in E(:,RANK(K)+1+J,K) */
434: /* (note that WORK contains V^{\top} and therefore */
435: /* we need to read rowwise !) */
437: for (i = 1; i <= ksk; ++i) {
438: e[i-1 + (rp1+j+1 + k*l2e)*l1e] = work[ivt+j + (i-1)*ldvt];
439: }
440: }
442: }
444: /* Compute the maximum norm of diagonal blocks and store the norm */
445: /* of each diagonal block in E(RP1,RP1,K) (after the singular values); */
446: /* store the norm of the last diagonal block in EXDNRM. */
448: /* DMAX .... maximum one-norm of the diagonal blocks */
450: dmax = 0.;
451: for (k = 0; k < nblks; ++k) {
452: rp1 = iwork[k];
454: /* compute the one-norm of diagonal block K */
456: dnrm = LAPACKlansy_("1", "L", &ksizes[k], &d[k*l1d*l2d], &l1d, work);
457: if (k+1 == nblks) exdnrm = dnrm;
458: else e[rp1 + (rp1 + k*l2e)*l1e] = dnrm;
459: if (dnrm > dmax) dmax = dnrm;
460: }
462: /* Check for zero matrix. */
464: if (emax == 0. && dmax == 0.) {
465: for (i = 0; i < n; ++i) ev[i] = 0.;
466: *info = -100;
467: return(0);
468: }
470: /* **************************************************************** */
472: /* ....Identify irreducible parts of the block tridiagonal matrix */
473: /* [while (START <= NBLKS)].... */
475: start = 0;
476: np = 0;
477: L10:478: if (start < nblks) {
480: /* Let IEND be the number of the next subdiagonal block such that */
481: /* its RANK is 0 or IEND = NBLKS if no such subdiagonal exists. */
482: /* The matrix identified by the elements between the diagonal block START */
483: /* and the diagonal block IEND constitutes an independent (irreducible) */
484: /* sub-problem. */
486: iend = start;
488: L20:489: if (iend < nblks) {
490: rk = iwork[iend];
492: /* NOTE: if RANK(IEND).EQ.0 then decoupling happens due to */
493: /* reduced accuracy requirements ! (because in this case */
494: /* we would not merge the corresponding two diagonal blocks) */
496: /* NOTE: seems like any combination may potentially happen: */
497: /* (i) RANK = 0 but no decoupling due to small norm of */
498: /* off-diagonal block (corresponding diagonal blocks */
499: /* also have small norm) as well as */
500: /* (ii) RANK > 0 but decoupling due to small norm of */
501: /* off-diagonal block (corresponding diagonal blocks */
502: /* have very large norm) */
503: /* case (i) is ruled out by checking for RANK = 0 above */
504: /* (we decide to decouple all the time when the rank */
505: /* of an off-diagonal block is zero, independently of */
506: /* the norms of the corresponding diagonal blocks. */
508: if (rk > 0) {
510: /* check for decoupling due to small norm of off-diagonal block */
511: /* (relative to the norms of the corresponding diagonal blocks) */
513: if (iend == nblks-2) {
514: d2 = PetscSqrtReal(exdnrm);
515: } else {
516: d2 = PetscSqrtReal(e[iwork[iend+1] + (iwork[iend+1] + (iend+1)*l2e)*l1e]);
517: }
519: /* this definition of TINY is analogous to the definition */
520: /* in the tridiagonal divide&conquer (dstedc) */
522: tiny = eps * PetscSqrtReal(e[iwork[iend] + (iwork[iend] + iend*l2e)*l1e])*d2;
523: if (e[(iwork[iend] + iend*l2e)*l1e] > tiny) {
525: /* no decoupling due to small norm of off-diagonal block */
527: ++iend;
528: goto L20;
529: }
530: }
531: }
533: /* ....(Sub) Problem determined: between diagonal blocks */
534: /* START and IEND. Compute its size and solve it.... */
536: nrblks = iend - start + 1;
537: if (nrblks == 1) {
539: /* Isolated problem is a single diagonal block */
541: nk = ksizes[start];
543: /* copy this isolated block into Z */
545: for (i = 0; i < nk; ++i) {
546: ip = np + i + 1;
547: for (j = 0; j <= i; ++j) z[ip + (np+j+1)*ldz] = d[i + (j + start*l2d)*l1d];
548: }
550: /* check whether there is enough workspace */
552: spneed = 2*nk*nk + nk * 6 + 1;
553: if (spneed > lwork) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: not enough workspace for DSYEVD, info = %d",lwork - 200 - spneed);
555: PetscStackCallBLAS("LAPACKsyevd",LAPACKsyevd_("V", "L", &nk,
556: &z[np + np*ldz], &ldz, &ev[np],
557: work, &lwork, &iwork[nblks-1], &liwork, info));
558: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DSYEVD, info = %d",*info);
559: start = iend + 1;
560: np += nk;
562: /* go to the next irreducible subproblem */
564: goto L10;
565: }
567: /* ....Isolated problem consists of more than one diagonal block. */
568: /* Start the divide and conquer algorithm.... */
570: /* Scale: Divide by the maximum of all norms of diagonal blocks */
571: /* and singular values of the subdiagonal blocks */
573: /* ....determine maximum of the norms of all diagonal and subdiagonal */
574: /* blocks.... */
576: if (iend == nblks-1) anorm = exdnrm;
577: else anorm = e[iwork[iend] + (iwork[iend] + iend*l2e)*l1e];
578: for (k = start; k < iend; ++k) {
579: rp1 = iwork[k];
581: /* norm of diagonal block */
582: anorm = PetscMax(anorm,e[rp1 + (rp1 + k*l2e)*l1e]);
584: /* singular value of subdiagonal block */
585: anorm = PetscMax(anorm,e[(rp1 + k*l2e)*l1e]);
586: }
588: nk = 0;
589: for (k = start; k < iend+1; ++k) {
590: ksk = ksizes[k];
591: nk += ksk;
593: /* scale the diagonal block */
594: PetscStackCallBLAS("LAPACKlascl",LAPACKlascl_("L", &zero, &zero,
595: &anorm, &done, &ksk, &ksk, &d[k*l2d*l1d], &l1d, info));
596: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DLASCL, info = %d",*info);
598: /* scale the (approximated) off-diagonal block by dividing its */
599: /* singular values */
601: if (k != iend) {
603: /* the last subdiagonal block has index IEND-1 !!!! */
604: for (i = 0; i < iwork[k]; ++i) {
605: e[i + (iwork[k] + k*l2e)*l1e] /= anorm;
606: }
607: }
608: }
610: /* call the block-tridiagonal divide-and-conquer on the */
611: /* irreducible subproblem which has been identified */
613: BDC_dibtdc_(jobz, nk, nrblks, &ksizes[start], &d[start*l1d*l2d], l1d, l2d,
614: &e[start*l2e*l1e], &iwork[start], l1e, l2e, tau2, &ev[np],
615: &z[np + np*ldz], ldz, work, lwork, &iwork[nblks-1], liwork, info, 1);
616: 617: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DIBTDC, info = %d",*info);
619: /* ************************************************************************** */
621: /* Scale back the computed eigenvalues. */
623: PetscStackCallBLAS("LAPACKlascl",LAPACKlascl_("G", &zero, &zero, &done,
624: &anorm, &nk, &one, &ev[np], &nk, info));
625: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DLASCL, info = %d",*info);
627: start = iend + 1;
628: np += nk;
630: /* Go to the next irreducible subproblem. */
632: goto L10;
633: }
635: /* ....If the problem split any number of times, then the eigenvalues */
636: /* will not be properly ordered. Here we permute the eigenvalues */
637: /* (and the associated eigenvectors) across the irreducible parts */
638: /* into ascending order.... */
640: /* IF(NRBLKS.LT.NBLKS)THEN */
642: /* Use Selection Sort to minimize swaps of eigenvectors */
644: for (ii = 1; ii < n; ++ii) {
645: i = ii;
646: k = i;
647: p = ev[i];
648: for (j = ii; j < n; ++j) {
649: if (ev[j] < p) {
650: k = j;
651: p = ev[j];
652: }
653: }
654: if (k != i) {
655: ev[k] = ev[i];
656: ev[i] = p;
657: PetscStackCallBLAS("BLASswap",BLASswap_(&n, &z[i*ldz], &one, &z[k*ldz], &one));
658: }
659: }
661: /* ...Compute MINGAP (minimum difference between neighboring eigenvalue */
662: /* approximations).............................................. */
664: *mingap = ev[1] - ev[0];
665: if (*mingap < 0.) SETERRQ2(PETSC_COMM_SELF,1,"dsbtdc: Eigenvalue approximations are not ordered properly. Approximation %d is larger than approximation %d.",1,2);
666: *mingapi = 1;
667: for (i = 2; i < n; ++i) {
668: absdiff = ev[i] - ev[i-1];
669: if (absdiff < 0.) SETERRQ2(PETSC_COMM_SELF,1,"dsbtdc: Eigenvalue approximations are not ordered properly. Approximation %d is larger than approximation %d.",i,i+1);
670: else if (absdiff < *mingap) {
671: *mingap = absdiff;
672: *mingapi = i;
673: }
674: }
676: /* check whether the minimum gap between eigenvalue approximations */
677: /* may indicate severe inaccuracies in the eigenvector approximations */
679: if (*mingap <= tol / 10) *info = -103;
680: return(0);
681: }