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: 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: #if defined(SLEPC_MISSING_LAPACK_SYEVD) || defined(PETSC_MISSING_LAPACK_GESVD) || defined(SLEPC_MISSING_LAPACK_LASET) || defined(SLEPC_MISSING_LAPACK_LASCL)
262: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"SYEVD/GESVD/LASET/LASCL - Lapack routine is unavailable");
263: #else
264: PetscBLASInt i, j, k, i1, iwspc, lwmin, start;
265: PetscBLASInt ii, ip, nk, rk, np, iu, rp1, ldu;
266: PetscBLASInt ksk, ivt, iend, kchk=0, kmax=0, one=1, zero=0;
267: PetscBLASInt ldvt, ksum=0, kskp1, spneed, nrblks, liwmin, isvals;
268: PetscReal p, d2, eps, dmax, emax, done = 1.0, dzero = 0.0;
269: PetscReal dnrm, tiny, anorm, exdnrm=0, dropsv, absdiff;
273: /* Determine machine epsilon. */
274: eps = LAPACKlamch_("Epsilon");
276: *info = 0;
278: if (*(unsigned char *)jobz != 'N' && *(unsigned char *)jobz != 'D') *info = -1;
279: else if (*(unsigned char *)jobacc != 'A' && *(unsigned char *)jobacc != 'M') *info = -2;
280: else if (n < 1) *info = -3;
281: else if (nblks < 1 || nblks > n) *info = -4;
282: if (*info == 0) {
283: for (k = 0; k < nblks; ++k) {
284: ksk = ksizes[k];
285: ksum += ksk;
286: if (ksk > kmax) kmax = ksk;
287: if (ksk < 1) kchk = 1;
288: }
289: if (nblks == 1) lwmin = 2*n*n + n*6 + 1;
290: else lwmin = n*n + n*3;
291: liwmin = n * 5 + nblks * 5 - 4;
292: if (ksum != n || kchk == 1) *info = -5;
293: else if (l1d < PetscMax(3,kmax)) *info = -7;
294: else if (l2d < PetscMax(3,kmax)) *info = -8;
295: else if (l1e < PetscMax(3,2*kmax+1)) *info = -10;
296: else if (l2e < PetscMax(3,2*kmax+1)) *info = -11;
297: else if (*(unsigned char *)jobacc == 'A' && tol > TOLMAX) *info = -12;
298: else if (*(unsigned char *)jobacc == 'M' && tau1 > TOLMAX/2) *info = -13;
299: else if (*(unsigned char *)jobacc == 'M' && tau2 > TOLMAX/2) *info = -14;
300: else if (ldz < PetscMax(1,n)) *info = -17;
301: else if (lwork < lwmin) *info = -19;
302: else if (liwork < liwmin) *info = -21;
303: }
305: if (*info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong argument %d in DSBTDC",-(*info));
307: /* Quick return if possible */
309: if (n == 1) {
310: ev[0] = d[0]; z[0] = 1.;
311: *info = -101;
312: return(0);
313: }
315: /* If NBLKS is equal to 1, then solve the problem with standard */
316: /* dense solver (in this case KSIZES(1) = N). */
318: if (nblks == 1) {
319: for (i = 0; i < n; ++i) {
320: for (j = 0; j <= i; ++j) {
321: z[i + j*ldz] = d[i + j*l1d];
322: }
323: }
324: PetscStackCallBLAS("LAPACKsyevd",LAPACKsyevd_("V", "L", &n, z, &ldz, ev, work, &lwork, iwork, &liwork, info));
325: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DSYEVD, info = %d",*info);
326: *info = -102;
327: return(0);
328: }
330: /* determine the accuracy parameters (if requested) */
332: if (*(unsigned char *)jobacc == 'A') {
333: tau1 = tol / 2;
334: if (tau1 < eps * 20) tau1 = eps;
335: tau2 = tol / 2;
336: }
338: /* Initialize Z as the identity matrix */
340: if (*(unsigned char *)jobz == 'D') {
341: PetscStackCallBLAS("LAPACKlaset",LAPACKlaset_("Full", &n, &n, &dzero, &done, z, &ldz));
342: }
344: /* Determine the off-diagonal ranks, form and store the lower rank */
345: /* approximations based on the tolerance parameters, the */
346: /* RANK( K ) largest singular values and the associated singular */
347: /* vectors of each subdiagonal block. Also find the maximum norm of */
348: /* the subdiagonal blocks (in EMAX). */
350: /* Compute SVDs of the subdiagonal blocks.... */
352: /* EMAX .... maximum norm of the off-diagonal blocks */
354: emax = 0.;
355: for (k = 0; k < nblks-1; ++k) {
356: ksk = ksizes[k];
357: kskp1 = ksizes[k+1];
358: isvals = 0;
360: /* Note that min(KSKP1,KSK).LE.N/2 (equality possible for */
361: /* NBLKS=2), and therefore storing the singular values requires */
362: /* at most N/2 entries of the * array WORK. */
364: iu = isvals + n / 2;
365: ivt = isvals + n / 2;
367: /* Call of DGESVD: The space for U is not referenced, since */
368: /* JOBU='O' and therefore this portion of the array WORK */
369: /* is not referenced for U. */
371: ldu = kskp1;
372: ldvt = PetscMin(kskp1,ksk);
373: iwspc = ivt + n * n / 2;
375: /* Note that the minimum workspace required for this call */
376: /* of DGESVD is: N/2 for storing the singular values + N**2/2 for */
377: /* storing V^T + 5*N/2 workspace = N**2/2 + 3*N. */
379: i1 = lwork - iwspc;
380: PetscStackCallBLAS("LAPACKgesvd",LAPACKgesvd_("O", "S", &kskp1, &ksk,
381: &e[k*l1e*l2e], &l1e, &work[isvals],
382: &work[iu], &ldu, &work[ivt], &ldvt, &work[iwspc], &i1, info));
383: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DGESVD, info = %d",*info);
385: /* Note that after the return from DGESVD U is stored in */
386: /* E(*,*,K), and V^{\top} is stored in WORK( IVT, IVT+1, .... ) */
388: /* determine the ranks RANK() for the approximations */
390: rk = PetscMin(ksk,kskp1);
391: L8:392: dropsv = work[isvals - 1 + rk];
394: if (dropsv * 2. <= tau1) {
396: /* the error caused by dropping singular value RK is */
397: /* small enough, try to reduce the rank by one more */
399: if (--rk > 0) goto L8;
400: else iwork[k] = 0;
401: } else {
403: /* the error caused by dropping singular value RK is */
404: /* too large already, RK is the rank required to achieve the */
405: /* desired accuracy */
407: iwork[k] = rk;
408: }
410: /* ************************************************************************** */
412: /* Store the first RANK( K ) terms of the SVD of the current */
413: /* off-diagonal block. */
414: /* NOTE that here it is required that L1E, L2E >= 2*KMAX+1 in order */
415: /* to have enough space for storing singular vectors and values up */
416: /* to the full SVD of an off-diagonal block !!!! */
418: /* u1-u_RANK(K) is already contained in E(:,1:RANK(K),K) (as a */
419: /* result of the call of DGESVD !), the sigma1-sigmaK are to be */
420: /* stored in E(1:RANK(K),RANK(K)+1,K), and v1-v_RANK(K) are to be */
421: /* stored in E(:,RANK(K)+2:2*RANK(K)+1,K) */
423: rp1 = iwork[k];
424: for (j = 0; j < iwork[k]; ++j) {
426: /* store sigma_J in E( J,RANK( K )+1,K ) */
428: e[j + (rp1 + k*l2e)* l1e] = work[isvals + j];
430: /* update maximum norm of subdiagonal blocks */
432: if (e[j + (rp1 + k*l2e)*l1e] > emax) {
433: emax = e[j + (rp1 + k*l2e)*l1e];
434: }
436: /* store v_J in E( :,RANK( K )+1+J,K ) */
437: /* (note that WORK contains V^{\top} and therefore */
438: /* we need to read rowwise !) */
440: for (i = 1; i <= ksk; ++i) {
441: e[i-1 + (rp1+j+1 + k*l2e)*l1e] = work[ivt+j + (i-1)*ldvt];
442: }
443: }
445: }
447: /* Compute the maximum norm of diagonal blocks and store the norm */
448: /* of each diagonal block in E(RP1,RP1,K) (after the singular values); */
449: /* store the norm of the last diagonal block in EXDNRM. */
451: /* DMAX .... maximum one-norm of the diagonal blocks */
453: dmax = 0.;
454: for (k = 0; k < nblks; ++k) {
455: rp1 = iwork[k];
457: /* compute the one-norm of diagonal block K */
459: dnrm = LAPACKlansy_("1", "L", &ksizes[k], &d[k*l1d*l2d], &l1d, work);
460: if (k+1 == nblks) exdnrm = dnrm;
461: else e[rp1 + (rp1 + k*l2e)*l1e] = dnrm;
462: if (dnrm > dmax) dmax = dnrm;
463: }
465: /* Check for zero matrix. */
467: if (emax == 0. && dmax == 0.) {
468: for (i = 0; i < n; ++i) ev[i] = 0.;
469: *info = -100;
470: return(0);
471: }
473: /* **************************************************************** */
475: /* ....Identify irreducible parts of the block tridiagonal matrix */
476: /* [while ( START <= NBLKS )].... */
478: start = 0;
479: np = 0;
480: L10:481: if (start < nblks) {
483: /* Let IEND be the number of the next subdiagonal block such that */
484: /* its RANK is 0 or IEND = NBLKS if no such subdiagonal exists. */
485: /* The matrix identified by the elements between the diagonal block START */
486: /* and the diagonal block IEND constitutes an independent (irreducible) */
487: /* sub-problem. */
489: iend = start;
491: L20:492: if (iend < nblks) {
493: rk = iwork[iend];
495: /* NOTE: if RANK( IEND ).EQ.0 then decoupling happens due to */
496: /* reduced accuracy requirements ! (because in this case */
497: /* we would not merge the corresponding two diagonal blocks) */
499: /* NOTE: seems like any combination may potentially happen: */
500: /* (i) RANK = 0 but no decoupling due to small norm of */
501: /* off-diagonal block (corresponding diagonal blocks */
502: /* also have small norm) as well as */
503: /* (ii) RANK > 0 but decoupling due to small norm of */
504: /* off-diagonal block (corresponding diagonal blocks */
505: /* have very large norm) */
506: /* case (i) is ruled out by checking for RANK = 0 above */
507: /* (we decide to decouple all the time when the rank */
508: /* of an off-diagonal block is zero, independently of */
509: /* the norms of the corresponding diagonal blocks. */
511: if (rk > 0) {
513: /* check for decoupling due to small norm of off-diagonal block */
514: /* (relative to the norms of the corresponding diagonal blocks) */
516: if (iend == nblks-2) {
517: d2 = PetscSqrtReal(exdnrm);
518: } else {
519: d2 = PetscSqrtReal(e[iwork[iend+1] + (iwork[iend+1] + (iend+1)*l2e)*l1e]);
520: }
522: /* this definition of TINY is analogous to the definition */
523: /* in the tridiagonal divide&conquer (dstedc) */
525: tiny = eps * PetscSqrtReal(e[iwork[iend] + (iwork[iend] + iend*l2e)*l1e])*d2;
526: if (e[(iwork[iend] + iend*l2e)*l1e] > tiny) {
528: /* no decoupling due to small norm of off-diagonal block */
530: ++iend;
531: goto L20;
532: }
533: }
534: }
536: /* ....(Sub) Problem determined: between diagonal blocks */
537: /* START and IEND. Compute its size and solve it.... */
539: nrblks = iend - start + 1;
540: if (nrblks == 1) {
542: /* Isolated problem is a single diagonal block */
544: nk = ksizes[start];
546: /* copy this isolated block into Z */
548: for (i = 0; i < nk; ++i) {
549: ip = np + i + 1;
550: for (j = 0; j <= i; ++j) z[ip + (np+j+1)*ldz] = d[i + (j + start*l2d)*l1d];
551: }
553: /* check whether there is enough workspace */
555: spneed = 2*nk*nk + nk * 6 + 1;
556: if (spneed > lwork) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: not enough workspace for DSYEVD, info = %d",lwork - 200 - spneed);
558: PetscStackCallBLAS("LAPACKsyevd",LAPACKsyevd_("V", "L", &nk,
559: &z[np + np*ldz], &ldz, &ev[np],
560: work, &lwork, &iwork[nblks-1], &liwork, info));
561: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DSYEVD, info = %d",*info);
562: start = iend + 1;
563: np += nk;
565: /* go to the next irreducible subproblem */
567: goto L10;
568: }
570: /* ....Isolated problem consists of more than one diagonal block. */
571: /* Start the divide and conquer algorithm.... */
573: /* Scale: Divide by the maximum of all norms of diagonal blocks */
574: /* and singular values of the subdiagonal blocks */
576: /* ....determine maximum of the norms of all diagonal and subdiagonal */
577: /* blocks.... */
579: if (iend == nblks-1) anorm = exdnrm;
580: else anorm = e[iwork[iend] + (iwork[iend] + iend*l2e)*l1e];
581: for (k = start; k < iend; ++k) {
582: rp1 = iwork[k];
584: /* norm of diagonal block */
585: anorm = PetscMax(anorm,e[rp1 + (rp1 + k*l2e)*l1e]);
587: /* singular value of subdiagonal block */
588: anorm = PetscMax(anorm,e[(rp1 + k*l2e)*l1e]);
589: }
591: nk = 0;
592: for (k = start; k < iend+1; ++k) {
593: ksk = ksizes[k];
594: nk += ksk;
596: /* scale the diagonal block */
597: PetscStackCallBLAS("LAPACKlascl",LAPACKlascl_("L", &zero, &zero,
598: &anorm, &done, &ksk, &ksk, &d[k*l2d*l1d], &l1d, info));
599: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DLASCL, info = %d",*info);
601: /* scale the (approximated) off-diagonal block by dividing its */
602: /* singular values */
604: if (k != iend) {
606: /* the last subdiagonal block has index IEND-1 !!!! */
607: for (i = 0; i < iwork[k]; ++i) {
608: e[i + (iwork[k] + k*l2e)*l1e] /= anorm;
609: }
610: }
611: }
613: /* call the block-tridiagonal divide-and-conquer on the */
614: /* irreducible subproblem which has been identified */
616: BDC_dibtdc_(jobz, nk, nrblks, &ksizes[start], &d[start*l1d*l2d], l1d, l2d,
617: &e[start*l2e*l1e], &iwork[start], l1e, l2e, tau2, &ev[np],
618: &z[np + np*ldz], ldz, work, lwork, &iwork[nblks-1], liwork, info, 1);
619: 620: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DIBTDC, info = %d",*info);
622: /* ************************************************************************** */
624: /* Scale back the computed eigenvalues. */
626: PetscStackCallBLAS("LAPACKlascl",LAPACKlascl_("G", &zero, &zero, &done,
627: &anorm, &nk, &one, &ev[np], &nk, info));
628: if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DLASCL, info = %d",*info);
630: start = iend + 1;
631: np += nk;
633: /* Go to the next irreducible subproblem. */
635: goto L10;
636: }
638: /* ....If the problem split any number of times, then the eigenvalues */
639: /* will not be properly ordered. Here we permute the eigenvalues */
640: /* (and the associated eigenvectors) across the irreducible parts */
641: /* into ascending order.... */
643: /* IF( NRBLKS.LT.NBLKS )THEN */
645: /* Use Selection Sort to minimize swaps of eigenvectors */
647: for (ii = 1; ii < n; ++ii) {
648: i = ii;
649: k = i;
650: p = ev[i];
651: for (j = ii; j < n; ++j) {
652: if (ev[j] < p) {
653: k = j;
654: p = ev[j];
655: }
656: }
657: if (k != i) {
658: ev[k] = ev[i];
659: ev[i] = p;
660: PetscStackCallBLAS("BLASswap",BLASswap_(&n, &z[i*ldz], &one, &z[k*ldz], &one));
661: }
662: }
664: /* ...Compute MINGAP (minimum difference between neighboring eigenvalue */
665: /* approximations).............................................. */
667: *mingap = ev[1] - ev[0];
668: if (*mingap < 0.) SETERRQ2(PETSC_COMM_SELF,1,"dsbtdc: Eigenvalue approximations are not ordered properly. Approximation %d is larger than approximation %d.",1,2);
669: *mingapi = 1;
670: for (i = 2; i < n; ++i) {
671: absdiff = ev[i] - ev[i-1];
672: 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);
673: else if (absdiff < *mingap) {
674: *mingap = absdiff;
675: *mingapi = i;
676: }
677: }
679: /* check whether the minimum gap between eigenvalue approximations */
680: /* may indicate severe inaccuracies in the eigenvector approximations */
682: if (*mingap <= tol / 10) *info = -103;
683: return(0);
684: #endif
685: }