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_dsrtdf_(PetscBLASInt *k,PetscBLASInt n,PetscBLASInt n1, 18: PetscReal *d,PetscReal *q,PetscBLASInt ldq,PetscBLASInt *indxq, 19: PetscReal *rho,PetscReal *z,PetscReal *dlamda,PetscReal *w, 20: PetscReal *q2,PetscBLASInt *indx,PetscBLASInt *indxc,PetscBLASInt *indxp, 21: PetscBLASInt *coltyp,PetscReal reltol,PetscBLASInt *dz,PetscBLASInt *de, 22: PetscBLASInt *info) 23: {
24: /* -- Routine written in LAPACK Version 3.0 style -- */
25: /* *************************************************** */
26: /* Written by */
27: /* Michael Moldaschl and Wilfried Gansterer */
28: /* University of Vienna */
29: /* last modification: March 16, 2014 */
31: /* Small adaptations of original code written by */
32: /* Wilfried Gansterer and Bob Ward, */
33: /* Department of Computer Science, University of Tennessee */
34: /* see https://doi.org/10.1137/S1064827501399432 */
35: /* *************************************************** */
37: /* Purpose */
38: /* ======= */
40: /* DSRTDF merges the two sets of eigenvalues of a rank-one modified */
41: /* diagonal matrix D+ZZ^T together into a single sorted set. Then it tries */
42: /* to deflate the size of the problem. */
43: /* There are two ways in which deflation can occur: when two or more */
44: /* eigenvalues of D are close together or if there is a tiny entry in the */
45: /* Z vector. For each such occurrence the order of the related secular */
46: /* equation problem is reduced by one. */
48: /* Arguments */
49: /* ========= */
51: /* K (output) INTEGER */
52: /* The number of non-deflated eigenvalues, and the order of the */
53: /* related secular equation. 0 <= K <=N. */
55: /* N (input) INTEGER */
56: /* The dimension of the diagonal matrix. N >= 0. */
58: /* N1 (input) INTEGER */
59: /* The location of the last eigenvalue in the leading sub-matrix. */
60: /* min(1,N) <= N1 <= max(1,N). */
62: /* D (input/output) DOUBLE PRECISION array, dimension (N) */
63: /* On entry, D contains the eigenvalues of the two submatrices to */
64: /* be combined. */
65: /* On exit, D contains the trailing (N-K) updated eigenvalues */
66: /* (those which were deflated) sorted into increasing order. */
68: /* Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N) */
69: /* On entry, Q contains the eigenvectors of two submatrices in */
70: /* the two square blocks with corners at (1,1), (N1,N1) */
71: /* and (N1+1, N1+1), (N,N). */
72: /* On exit, Q contains the trailing (N-K) updated eigenvectors */
73: /* (those which were deflated) in its last N-K columns. */
75: /* LDQ (input) INTEGER */
76: /* The leading dimension of the array Q. LDQ >= max(1,N). */
78: /* INDXQ (input/output) INTEGER array, dimension (N) */
79: /* The permutation which separately sorts the two sub-problems */
80: /* in D into ascending order. Note that elements in the second */
81: /* half of this permutation must first have N1 added to their */
82: /* values. Destroyed on exit. */
84: /* RHO (input/output) DOUBLE PRECISION */
85: /* On entry, the off-diagonal element associated with the rank-1 */
86: /* cut which originally split the two submatrices which are now */
87: /* being recombined. */
88: /* On exit, RHO has been modified to the value required by */
89: /* DLAED3M (made positive and multiplied by two, such that */
90: /* the modification vector has norm one). */
92: /* Z (input/output) DOUBLE PRECISION array, dimension (N) */
93: /* On entry, Z contains the updating vector. */
94: /* On exit, the contents of Z has been destroyed by the updating */
95: /* process. */
97: /* DLAMDA (output) DOUBLE PRECISION array, dimension (N) */
98: /* A copy of the first K non-deflated eigenvalues which */
99: /* will be used by DLAED3M to form the secular equation. */
101: /* W (output) DOUBLE PRECISION array, dimension (N) */
102: /* The first K values of the final deflation-altered z-vector */
103: /* which will be passed to DLAED3M. */
105: /* Q2 (output) DOUBLE PRECISION array, dimension */
106: /* ( N*N ) (if everything is deflated) or */
107: /* ( N1*(COLTYP(1)+COLTYP(2)) + (N-N1)*(COLTYP(2)+COLTYP(3)) ) */
108: /* (if not everything is deflated) */
109: /* If everything is deflated, then N*N intermediate workspace */
110: /* is needed in Q2. */
111: /* If not everything is deflated, Q2 contains on exit a copy of the */
112: /* first K non-deflated eigenvectors which will be used by */
113: /* DLAED3M in a matrix multiply (DGEMM) to accumulate the new */
114: /* eigenvectors, followed by the N-K deflated eigenvectors. */
116: /* INDX (workspace) INTEGER array, dimension (N) */
117: /* The permutation used to sort the contents of DLAMDA into */
118: /* ascending order. */
120: /* INDXC (output) INTEGER array, dimension (N) */
121: /* The permutation used to arrange the columns of the deflated */
122: /* Q matrix into three groups: columns in the first group contain */
123: /* non-zero elements only at and above N1 (type 1), columns in the */
124: /* second group are dense (type 2), and columns in the third group */
125: /* contain non-zero elements only below N1 (type 3). */
127: /* INDXP (workspace) INTEGER array, dimension (N) */
128: /* The permutation used to place deflated values of D at the end */
129: /* of the array. INDXP(1:K) points to the nondeflated D-values */
130: /* and INDXP(K+1:N) points to the deflated eigenvalues. */
132: /* COLTYP (workspace/output) INTEGER array, dimension (N) */
133: /* During execution, a label which will indicate which of the */
134: /* following types a column in the Q2 matrix is: */
135: /* 1 : non-zero in the upper half only; */
136: /* 2 : dense; */
137: /* 3 : non-zero in the lower half only; */
138: /* 4 : deflated. */
139: /* On exit, COLTYP(i) is the number of columns of type i, */
140: /* for i=1 to 4 only. */
142: /* RELTOL (input) DOUBLE PRECISION */
143: /* User specified deflation tolerance. If RELTOL.LT.20*EPS, then */
144: /* the standard value used in the original LAPACK routines is used. */
146: /* DZ (output) INTEGER, DZ.GE.0 */
147: /* counts the deflation due to a small component in the modification */
148: /* vector. */
150: /* DE (output) INTEGER, DE.GE.0 */
151: /* counts the deflation due to close eigenvalues. */
153: /* INFO (output) INTEGER */
154: /* = 0: successful exit. */
155: /* < 0: if INFO = -i, the i-th argument had an illegal value. */
157: /* Further Details */
158: /* =============== */
160: /* Based on code written by */
161: /* Wilfried Gansterer and Bob Ward, */
162: /* Department of Computer Science, University of Tennessee */
164: /* Based on the design of the LAPACK code DLAED2 with modifications */
165: /* to allow a block divide and conquer scheme */
167: /* DLAED2 was written by Jeff Rutter, Computer Science Division, University */
168: /* of California at Berkeley, USA, and modified by Francoise Tisseur, */
169: /* University of Tennessee. */
171: /* ===================================================================== */
173: #if defined(SLEPC_MISSING_LAPACK_LAMRG) || defined(SLEPC_MISSING_LAPACK_LACPY)
175: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LAMRG/LACPY - Lapack routine is unavailable");
176: #else
177: PetscReal c, s, t, eps, tau, tol, dmax, dmone = -1.;
178: PetscBLASInt i, j, i1, k2, n2, ct, nj, pj=0, js, iq1, iq2;
179: PetscBLASInt psm[4], imax, jmax, ctot[4], factmp=1, one=1;
182: *info = 0;
184: if (n < 0) *info = -2;
185: else if (n1 < PetscMin(1,n) || n1 > PetscMax(1,n)) *info = -3;
186: else if (ldq < PetscMax(1,n)) *info = -6;
187: if (*info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong argument %d in DSRTDF",-(*info));
189: /* Initialize deflation counters */
191: *dz = 0;
192: *de = 0;
194: /* **************************************************************************** */
196: /* Quick return if possible */
198: if (n == 0) return(0);
200: /* **************************************************************************** */
202: n2 = n - n1;
204: /* Modify Z so that RHO is positive. */
206: if (*rho < 0.) PetscStackCallBLAS("BLASscal",BLASscal_(&n2, &dmone, &z[n1], &one));
208: /* Normalize z so that norm2(z) = 1. Since z is the concatenation of */
209: /* two normalized vectors, norm2(z) = sqrt(2). (NOTE that this holds also */
210: /* here in the approximate block-tridiagonal D&C because the two vectors are */
211: /* singular vectors, but it would not necessarily hold in a D&C for a */
212: /* general banded matrix !) */
214: t = 1. / PetscSqrtReal(2.);
215: PetscStackCallBLAS("BLASscal",BLASscal_(&n, &t, z, &one));
217: /* NOTE: at this point the value of RHO is modified in order to */
218: /* normalize Z: RHO = ABS( norm2(z)**2 * RHO */
219: /* in our case: norm2(z) = sqrt(2), and therefore: */
221: *rho = PetscAbs(*rho * 2.);
223: /* Sort the eigenvalues into increasing order */
225: for (i = n1; i < n; ++i) indxq[i] += n1;
227: /* re-integrate the deflated parts from the last pass */
229: for (i = 0; i < n; ++i) dlamda[i] = d[indxq[i]-1];
230: PetscStackCallBLAS("LAPACKlamrg",LAPACKlamrg_(&n1, &n2, dlamda, &one, &one, indxc));
231: for (i = 0; i < n; ++i) indx[i] = indxq[indxc[i]-1];
232: for (i = 0; i < n; ++i) indxq[i] = 0;
234: /* Calculate the allowable deflation tolerance */
236: /* imax = BLASamax_(&n, z, &one); */
237: imax = 1;
238: dmax = PetscAbsReal(z[0]);
239: for (i=1;i<n;i++) {
240: if (PetscAbsReal(z[i])>dmax) {
241: imax = i+1;
242: dmax = PetscAbsReal(z[i]);
243: }
244: }
245: /* jmax = BLASamax_(&n, d, &one); */
246: jmax = 1;
247: dmax = PetscAbsReal(d[0]);
248: for (i=1;i<n;i++) {
249: if (PetscAbsReal(d[i])>dmax) {
250: jmax = i+1;
251: dmax = PetscAbsReal(d[i]);
252: }
253: }
255: eps = LAPACKlamch_("Epsilon");
256: if (reltol < eps * 20) {
257: /* use the standard deflation tolerance from the original LAPACK code */
258: tol = eps * 8. * PetscMax(PetscAbs(d[jmax-1]),PetscAbs(z[imax-1]));
259: } else {
260: /* otherwise set TOL to the input parameter RELTOL ! */
261: tol = reltol * PetscMax(PetscAbs(d[jmax-1]),PetscAbs(z[imax-1]));
262: }
264: /* If the rank-1 modifier is small enough, no more needs to be done */
265: /* except to reorganize Q so that its columns correspond with the */
266: /* elements in D. */
268: if (*rho * PetscAbs(z[imax-1]) <= tol) {
270: /* complete deflation because of small rank-one modifier */
271: /* NOTE: in this case we need N*N space in the array Q2 ! */
273: *dz = n; *k = 0;
274: iq2 = 1;
275: for (j = 0; j < n; ++j) {
276: i = indx[j]; indxq[j] = i;
277: PetscStackCallBLAS("BLAScopy",BLAScopy_(&n, &q[(i-1)*ldq], &one, &q2[iq2-1], &one));
278: dlamda[j] = d[i-1];
279: iq2 += n;
280: }
281: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("A", &n, &n, q2, &n, q, &ldq));
282: PetscStackCallBLAS("BLAScopy",BLAScopy_(&n, dlamda, &one, d, &one));
283: return(0);
284: }
286: /* If there are multiple eigenvalues then the problem deflates. Here */
287: /* the number of equal eigenvalues is found. As each equal */
288: /* eigenvalue is found, an elementary reflector is computed to rotate */
289: /* the corresponding eigensubspace so that the corresponding */
290: /* components of Z are zero in this new basis. */
292: /* initialize the column types */
294: /* first N1 columns are initially (before deflation) of type 1 */
295: for (i = 0; i < n1; ++i) coltyp[i] = 1;
296: /* columns N1+1 to N are initially (before deflation) of type 3 */
297: for (i = n1; i < n; ++i) coltyp[i] = 3;
299: *k = 0;
300: k2 = n + 1;
301: for (j = 0; j < n; ++j) {
302: nj = indx[j]-1;
303: if (*rho * PetscAbs(z[nj]) <= tol) {
305: /* Deflate due to small z component. */
306: ++(*dz);
307: --k2;
309: /* deflated eigenpair, therefore column type 4 */
310: coltyp[nj] = 4;
311: indxp[k2-1] = nj+1;
312: indxq[k2-1] = nj+1;
313: if (j+1 == n) goto L100;
314: } else {
316: /* not deflated */
317: pj = nj;
318: goto L80;
319: }
320: }
321: factmp = 1;
322: L80:323: ++j;
324: nj = indx[j]-1;
325: if (j+1 > n) goto L100;
326: if (*rho * PetscAbs(z[nj]) <= tol) {
328: /* Deflate due to small z component. */
329: ++(*dz);
330: --k2;
331: coltyp[nj] = 4;
332: indxp[k2-1] = nj+1;
333: indxq[k2-1] = nj+1;
334: } else {
336: /* Check if eigenvalues are close enough to allow deflation. */
337: s = z[pj];
338: c = z[nj];
340: /* Find sqrt(a**2+b**2) without overflow or */
341: /* destructive underflow. */
343: tau = LAPACKlapy2_(&c, &s);
344: t = d[nj] - d[pj];
345: c /= tau;
346: s = -s / tau;
347: if (PetscAbs(t * c * s) <= tol) {
349: /* Deflate due to close eigenvalues. */
350: ++(*de);
351: z[nj] = tau;
352: z[pj] = 0.;
353: if (coltyp[nj] != coltyp[pj]) coltyp[nj] = 2;
355: /* if deflation happens across the two eigenvector blocks */
356: /* (eigenvalues corresponding to different blocks), then */
357: /* on column in the eigenvector matrix fills up completely */
358: /* (changes from type 1 or 3 to type 2) */
360: /* eigenpair PJ is deflated, therefore the column type changes */
361: /* to 4 */
363: coltyp[pj] = 4;
364: PetscStackCallBLAS("BLASrot",BLASrot_(&n, &q[pj*ldq], &one, &q[nj*ldq], &one, &c, &s));
365: t = d[pj] * (c * c) + d[nj] * (s * s);
366: d[nj] = d[pj] * (s * s) + d[nj] * (c * c);
367: d[pj] = t;
368: --k2;
369: i = 1;
370: L90:371: if (k2 + i <= n) {
372: if (d[pj] < d[indxp[k2 + i-1]-1]) {
373: indxp[k2 + i - 2] = indxp[k2 + i - 1];
374: indxq[k2 + i - 2] = indxq[k2 + i - 1];
375: indxp[k2 + i - 1] = pj + 1;
376: indxq[k2 + i - 2] = pj + 1;
377: ++i;
378: goto L90;
379: } else {
380: indxp[k2 + i - 2] = pj + 1;
381: indxq[k2 + i - 2] = pj + 1;
382: }
383: } else {
384: indxp[k2 + i - 2] = pj + 1;
385: indxq[k2 + i - 2] = pj + 1;
386: }
387: pj = nj;
388: factmp = -1;
389: } else {
391: /* do not deflate */
392: ++(*k);
393: dlamda[*k-1] = d[pj];
394: w[*k-1] = z[pj];
395: indxp[*k-1] = pj + 1;
396: indxq[*k-1] = pj + 1;
397: factmp = 1;
398: pj = nj;
399: }
400: }
401: goto L80;
402: L100:404: /* Record the last eigenvalue. */
405: ++(*k);
406: dlamda[*k-1] = d[pj];
407: w[*k-1] = z[pj];
408: indxp[*k-1] = pj+1;
409: indxq[*k-1] = (pj+1) * factmp;
411: /* Count up the total number of the various types of columns, then */
412: /* form a permutation which positions the four column types into */
413: /* four uniform groups (although one or more of these groups may be */
414: /* empty). */
416: for (j = 0; j < 4; ++j) ctot[j] = 0;
417: for (j = 0; j < n; ++j) {
418: ct = coltyp[j];
419: ++ctot[ct-1];
420: }
422: /* PSM(*) = Position in SubMatrix (of types 1 through 4) */
423: psm[0] = 1;
424: psm[1] = ctot[0] + 1;
425: psm[2] = psm[1] + ctot[1];
426: psm[3] = psm[2] + ctot[2];
427: *k = n - ctot[3];
429: /* Fill out the INDXC array so that the permutation which it induces */
430: /* will place all type-1 columns first, all type-2 columns next, */
431: /* then all type-3's, and finally all type-4's. */
433: for (j = 0; j < n; ++j) {
434: js = indxp[j];
435: ct = coltyp[js-1];
436: indx[psm[ct - 1]-1] = js;
437: indxc[psm[ct - 1]-1] = j+1;
438: ++psm[ct - 1];
439: }
441: /* Sort the eigenvalues and corresponding eigenvectors into DLAMDA */
442: /* and Q2 respectively. The eigenvalues/vectors which were not */
443: /* deflated go into the first K slots of DLAMDA and Q2 respectively, */
444: /* while those which were deflated go into the last N - K slots. */
446: i = 0;
447: iq1 = 0;
448: iq2 = (ctot[0] + ctot[1]) * n1;
449: for (j = 0; j < ctot[0]; ++j) {
450: js = indx[i];
451: PetscStackCallBLAS("BLAScopy",BLAScopy_(&n1, &q[(js-1)*ldq], &one, &q2[iq1], &one));
452: z[i] = d[js-1];
453: ++i;
454: iq1 += n1;
455: }
457: for (j = 0; j < ctot[1]; ++j) {
458: js = indx[i];
459: PetscStackCallBLAS("BLAScopy",BLAScopy_(&n1, &q[(js-1)*ldq], &one, &q2[iq1], &one));
460: PetscStackCallBLAS("BLAScopy",BLAScopy_(&n2, &q[n1+(js-1)*ldq], &one, &q2[iq2], &one));
461: z[i] = d[js-1];
462: ++i;
463: iq1 += n1;
464: iq2 += n2;
465: }
467: for (j = 0; j < ctot[2]; ++j) {
468: js = indx[i];
469: PetscStackCallBLAS("BLAScopy",BLAScopy_(&n2, &q[n1+(js-1)*ldq], &one, &q2[iq2], &one));
470: z[i] = d[js-1];
471: ++i;
472: iq2 += n2;
473: }
475: iq1 = iq2;
476: for (j = 0; j < ctot[3]; ++j) {
477: js = indx[i];
478: PetscStackCallBLAS("BLAScopy",BLAScopy_(&n, &q[(js-1)*ldq], &one, &q2[iq2], &one));
479: iq2 += n;
480: z[i] = d[js-1];
481: ++i;
482: }
484: /* The deflated eigenvalues and their corresponding vectors go back */
485: /* into the last N - K slots of D and Q respectively. */
487: PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("A", &n, &ctot[3], &q2[iq1], &n, &q[*k*ldq], &ldq));
488: i1 = n - *k;
489: PetscStackCallBLAS("BLAScopy",BLAScopy_(&i1, &z[*k], &one, &d[*k], &one));
491: /* Copy CTOT into COLTYP for referencing in DLAED3M. */
493: for (j = 0; j < 4; ++j) coltyp[j] = ctot[j];
494: return(0);
495: #endif
496: }