Date: 2020-01-13T11:59:11+00:00

Actual source code: dsrtdf.c
slepc-3.12.2 2020-01-13
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      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: }