1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-, 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: PetscReal c, s, t, eps, tau, tol, dmax, dmone = -1.;
174: PetscBLASInt i, j, i1, k2, n2, ct, nj, pj=0, js, iq1, iq2;
175: PetscBLASInt psm[4], imax, jmax, ctot[4], factmp=1, one=1;
177: *info = 0;
179: if (n < 0) *info = -2;
180: else if (n1 < PetscMin(1,n) || n1 > PetscMax(1,n)) *info = -3;
181: else if (ldq < PetscMax(1,n)) *info = -6;
184: /* Initialize deflation counters */
186: *dz = 0;
187: *de = 0;
189: /* **************************************************************************** */
191: /* Quick return if possible */
193: if (n == 0) return 0;
195: /* **************************************************************************** */
197: n2 = n - n1;
199: /* Modify Z so that RHO is positive. */
201: if (*rho < 0.) PetscCallBLAS("BLASscal",BLASscal_(&n2, &dmone, &z[n1], &one));
203: /* Normalize z so that norm2(z) = 1. Since z is the concatenation of */
204: /* two normalized vectors, norm2(z) = sqrt(2). (NOTE that this holds also */
205: /* here in the approximate block-tridiagonal D&C because the two vectors are */
206: /* singular vectors, but it would not necessarily hold in a D&C for a */
207: /* general banded matrix !) */
209: t = 1. / PETSC_SQRT2;
210: PetscCallBLAS("BLASscal",BLASscal_(&n, &t, z, &one));
212: /* NOTE: at this point the value of RHO is modified in order to */
213: /* normalize Z: RHO = ABS( norm2(z)**2 * RHO */
214: /* in our case: norm2(z) = sqrt(2), and therefore: */
216: *rho = PetscAbs(*rho * 2.);
218: /* Sort the eigenvalues into increasing order */
220: for (i = n1; i < n; ++i) indxq[i] += n1;
222: /* re-integrate the deflated parts from the last pass */
224: for (i = 0; i < n; ++i) dlamda[i] = d[indxq[i]-1];
225: PetscCallBLAS("LAPACKlamrg",LAPACKlamrg_(&n1, &n2, dlamda, &one, &one, indxc));
226: for (i = 0; i < n; ++i) indx[i] = indxq[indxc[i]-1];
227: for (i = 0; i < n; ++i) indxq[i] = 0;
229: /* Calculate the allowable deflation tolerance */
231: /* imax = BLASamax_(&n, z, &one); */
232: imax = 1;
233: dmax = PetscAbsReal(z[0]);
234: for (i=1;i<n;i++) {
235: if (PetscAbsReal(z[i])>dmax) {
236: imax = i+1;
237: dmax = PetscAbsReal(z[i]);
238: }
239: }
240: /* jmax = BLASamax_(&n, d, &one); */
241: jmax = 1;
242: dmax = PetscAbsReal(d[0]);
243: for (i=1;i<n;i++) {
244: if (PetscAbsReal(d[i])>dmax) {
245: jmax = i+1;
246: dmax = PetscAbsReal(d[i]);
247: }
248: }
250: eps = LAPACKlamch_("Epsilon");
251: if (reltol < eps * 20) {
252: /* use the standard deflation tolerance from the original LAPACK code */
253: tol = eps * 8. * PetscMax(PetscAbs(d[jmax-1]),PetscAbs(z[imax-1]));
254: } else {
255: /* otherwise set TOL to the input parameter RELTOL ! */
256: tol = reltol * PetscMax(PetscAbs(d[jmax-1]),PetscAbs(z[imax-1]));
257: }
259: /* If the rank-1 modifier is small enough, no more needs to be done */
260: /* except to reorganize Q so that its columns correspond with the */
261: /* elements in D. */
263: if (*rho * PetscAbs(z[imax-1]) <= tol) {
265: /* complete deflation because of small rank-one modifier */
266: /* NOTE: in this case we need N*N space in the array Q2 ! */
268: *dz = n; *k = 0;
269: iq2 = 1;
270: for (j = 0; j < n; ++j) {
271: i = indx[j]; indxq[j] = i;
272: PetscCallBLAS("BLAScopy",BLAScopy_(&n, &q[(i-1)*ldq], &one, &q2[iq2-1], &one));
273: dlamda[j] = d[i-1];
274: iq2 += n;
275: }
276: for (j=0;j<n;j++) for (i=0;i<n;i++) q[i+j*ldq] = q2[i+j*n];
277: PetscCallBLAS("BLAScopy",BLAScopy_(&n, dlamda, &one, d, &one));
278: return 0;
279: }
281: /* If there are multiple eigenvalues then the problem deflates. Here */
282: /* the number of equal eigenvalues is found. As each equal */
283: /* eigenvalue is found, an elementary reflector is computed to rotate */
284: /* the corresponding eigensubspace so that the corresponding */
285: /* components of Z are zero in this new basis. */
287: /* initialize the column types */
289: /* first N1 columns are initially (before deflation) of type 1 */
290: for (i = 0; i < n1; ++i) coltyp[i] = 1;
291: /* columns N1+1 to N are initially (before deflation) of type 3 */
292: for (i = n1; i < n; ++i) coltyp[i] = 3;
294: *k = 0;
295: k2 = n + 1;
296: for (j = 0; j < n; ++j) {
297: nj = indx[j]-1;
298: if (*rho * PetscAbs(z[nj]) <= tol) {
300: /* Deflate due to small z component. */
301: ++(*dz);
302: --k2;
304: /* deflated eigenpair, therefore column type 4 */
305: coltyp[nj] = 4;
306: indxp[k2-1] = nj+1;
307: indxq[k2-1] = nj+1;
308: if (j+1 == n) goto L100;
309: } else {
311: /* not deflated */
312: pj = nj;
313: goto L80;
314: }
315: }
316: factmp = 1;
317: L80:318: ++j;
319: nj = indx[j]-1;
320: if (j+1 > n) goto L100;
321: if (*rho * PetscAbs(z[nj]) <= tol) {
323: /* Deflate due to small z component. */
324: ++(*dz);
325: --k2;
326: coltyp[nj] = 4;
327: indxp[k2-1] = nj+1;
328: indxq[k2-1] = nj+1;
329: } else {
331: /* Check if eigenvalues are close enough to allow deflation. */
332: s = z[pj];
333: c = z[nj];
335: /* Find sqrt(a**2+b**2) without overflow or */
336: /* destructive underflow. */
338: tau = SlepcAbs(c, s);
339: t = d[nj] - d[pj];
340: c /= tau;
341: s = -s / tau;
342: if (PetscAbs(t * c * s) <= tol) {
344: /* Deflate due to close eigenvalues. */
345: ++(*de);
346: z[nj] = tau;
347: z[pj] = 0.;
348: if (coltyp[nj] != coltyp[pj]) coltyp[nj] = 2;
350: /* if deflation happens across the two eigenvector blocks */
351: /* (eigenvalues corresponding to different blocks), then */
352: /* on column in the eigenvector matrix fills up completely */
353: /* (changes from type 1 or 3 to type 2) */
355: /* eigenpair PJ is deflated, therefore the column type changes */
356: /* to 4 */
358: coltyp[pj] = 4;
359: PetscCallBLAS("BLASrot",BLASrot_(&n, &q[pj*ldq], &one, &q[nj*ldq], &one, &c, &s));
360: t = d[pj] * (c * c) + d[nj] * (s * s);
361: d[nj] = d[pj] * (s * s) + d[nj] * (c * c);
362: d[pj] = t;
363: --k2;
364: i = 1;
365: L90:366: if (k2 + i <= n) {
367: if (d[pj] < d[indxp[k2 + i-1]-1]) {
368: indxp[k2 + i - 2] = indxp[k2 + i - 1];
369: indxq[k2 + i - 2] = indxq[k2 + i - 1];
370: indxp[k2 + i - 1] = pj + 1;
371: indxq[k2 + i - 2] = pj + 1;
372: ++i;
373: goto L90;
374: } else {
375: indxp[k2 + i - 2] = pj + 1;
376: indxq[k2 + i - 2] = pj + 1;
377: }
378: } else {
379: indxp[k2 + i - 2] = pj + 1;
380: indxq[k2 + i - 2] = pj + 1;
381: }
382: pj = nj;
383: factmp = -1;
384: } else {
386: /* do not deflate */
387: ++(*k);
388: dlamda[*k-1] = d[pj];
389: w[*k-1] = z[pj];
390: indxp[*k-1] = pj + 1;
391: indxq[*k-1] = pj + 1;
392: factmp = 1;
393: pj = nj;
394: }
395: }
396: goto L80;
397: L100:399: /* Record the last eigenvalue. */
400: ++(*k);
401: dlamda[*k-1] = d[pj];
402: w[*k-1] = z[pj];
403: indxp[*k-1] = pj+1;
404: indxq[*k-1] = (pj+1) * factmp;
406: /* Count up the total number of the various types of columns, then */
407: /* form a permutation which positions the four column types into */
408: /* four uniform groups (although one or more of these groups may be */
409: /* empty). */
411: for (j = 0; j < 4; ++j) ctot[j] = 0;
412: for (j = 0; j < n; ++j) {
413: ct = coltyp[j];
414: ++ctot[ct-1];
415: }
417: /* PSM(*) = Position in SubMatrix (of types 1 through 4) */
418: psm[0] = 1;
419: psm[1] = ctot[0] + 1;
420: psm[2] = psm[1] + ctot[1];
421: psm[3] = psm[2] + ctot[2];
422: *k = n - ctot[3];
424: /* Fill out the INDXC array so that the permutation which it induces */
425: /* will place all type-1 columns first, all type-2 columns next, */
426: /* then all type-3's, and finally all type-4's. */
428: for (j = 0; j < n; ++j) {
429: js = indxp[j];
430: ct = coltyp[js-1];
431: indx[psm[ct - 1]-1] = js;
432: indxc[psm[ct - 1]-1] = j+1;
433: ++psm[ct - 1];
434: }
436: /* Sort the eigenvalues and corresponding eigenvectors into DLAMDA */
437: /* and Q2 respectively. The eigenvalues/vectors which were not */
438: /* deflated go into the first K slots of DLAMDA and Q2 respectively, */
439: /* while those which were deflated go into the last N - K slots. */
441: i = 0;
442: iq1 = 0;
443: iq2 = (ctot[0] + ctot[1]) * n1;
444: for (j = 0; j < ctot[0]; ++j) {
445: js = indx[i];
446: PetscCallBLAS("BLAScopy",BLAScopy_(&n1, &q[(js-1)*ldq], &one, &q2[iq1], &one));
447: z[i] = d[js-1];
448: ++i;
449: iq1 += n1;
450: }
452: for (j = 0; j < ctot[1]; ++j) {
453: js = indx[i];
454: PetscCallBLAS("BLAScopy",BLAScopy_(&n1, &q[(js-1)*ldq], &one, &q2[iq1], &one));
455: PetscCallBLAS("BLAScopy",BLAScopy_(&n2, &q[n1+(js-1)*ldq], &one, &q2[iq2], &one));
456: z[i] = d[js-1];
457: ++i;
458: iq1 += n1;
459: iq2 += n2;
460: }
462: for (j = 0; j < ctot[2]; ++j) {
463: js = indx[i];
464: PetscCallBLAS("BLAScopy",BLAScopy_(&n2, &q[n1+(js-1)*ldq], &one, &q2[iq2], &one));
465: z[i] = d[js-1];
466: ++i;
467: iq2 += n2;
468: }
470: iq1 = iq2;
471: for (j = 0; j < ctot[3]; ++j) {
472: js = indx[i];
473: PetscCallBLAS("BLAScopy",BLAScopy_(&n, &q[(js-1)*ldq], &one, &q2[iq2], &one));
474: iq2 += n;
475: z[i] = d[js-1];
476: ++i;
477: }
479: /* The deflated eigenvalues and their corresponding vectors go back */
480: /* into the last N - K slots of D and Q respectively. */
482: for (j=0;j<ctot[3];j++) for (i=0;i<n;i++) q[i+(j+*k)*ldq] = q2[iq1+i+j*n];
483: i1 = n - *k;
484: PetscCallBLAS("BLAScopy",BLAScopy_(&i1, &z[*k], &one, &d[*k], &one));
486: /* Copy CTOT into COLTYP for referencing in DLAED3M. */
488: for (j = 0; j < 4; ++j) coltyp[j] = ctot[j];
489: return 0;
490: }