Date: 2015-09-03T15:03:31+00:00

Actual source code: dibtdc.c
slepc-3.6.1 2015-09-03
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      1: /*
      2:    BDC - Block-divide and conquer (see description in README file).

      4:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      5:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      6:    Copyright (c) 2002-2015, Universitat Politecnica de Valencia, Spain

      8:    This file is part of SLEPc.

     10:    SLEPc is free software: you can redistribute it and/or modify it under  the
     11:    terms of version 3 of the GNU Lesser General Public License as published by
     12:    the Free Software Foundation.

     14:    SLEPc  is  distributed in the hope that it will be useful, but WITHOUT  ANY
     15:    WARRANTY;  without even the implied warranty of MERCHANTABILITY or  FITNESS
     16:    FOR  A  PARTICULAR PURPOSE. See the GNU Lesser General Public  License  for
     17:    more details.

     19:    You  should have received a copy of the GNU Lesser General  Public  License
     20:    along with SLEPc. If not, see <http://www.gnu.org/licenses/>.
     21:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     22: */

     24: #include <slepc/private/dsimpl.h>
     25: #include <slepcblaslapack.h>

     27: static PetscErrorCode cutlr_(PetscBLASInt start,PetscBLASInt n,PetscBLASInt blkct,
     28:         PetscBLASInt *bsizes,PetscBLASInt *ranks,PetscBLASInt *cut,
     29:         PetscBLASInt *lsum,PetscBLASInt *lblks,PetscBLASInt *info)
     30: {
     31: /*  -- Routine written in LAPACK Version 3.0 style -- */
     32: /* *************************************************** */
     33: /*     Written by */
     34: /*     Michael Moldaschl and Wilfried Gansterer */
     35: /*     University of Vienna */
     36: /*     last modification: March 16, 2014 */

     38: /*     Small adaptations of original code written by */
     39: /*     Wilfried Gansterer and Bob Ward, */
     40: /*     Department of Computer Science, University of Tennessee */
     41: /*     see http://dx.doi.org/10.1137/S1064827501399432 */
     42: /* *************************************************** */

     44: /*  Purpose */
     45: /*  ======= */

     47: /*  CUTLR computes the optimal cut in a sequence of BLKCT neighboring */
     48: /*  blocks whose sizes are given by the array BSIZES. */
     49: /*  The sum of all block sizes in the sequence considered is given by N. */
     50: /*  The cut is optimal in the sense that the difference of the sizes of */
     51: /*  the resulting two halves is minimum over all cuts with minimum ranks */
     52: /*  between blocks of the sequence considered. */

     54: /*  Arguments */
     55: /*  ========= */

     57: /*  START  (input) INTEGER */
     58: /*         In the original array KSIZES of the calling routine DIBTDC, */
     59: /*         the position where the sequence considered in this routine starts. */
     60: /*         START >= 1. */

     62: /*  N      (input) INTEGER */
     63: /*         The sum of all the block sizes of the sequence to be cut = */
     64: /*         = sum_{i=1}^{BLKCT} BSIZES( I ). */
     65: /*         N >= 3. */

     67: /*  BLKCT  (input) INTEGER */
     68: /*         The number of blocks in the sequence to be cut. */
     69: /*         BLKCT >= 3. */

     71: /*  BSIZES (input) INTEGER array, dimension (BLKCT) */
     72: /*         The dimensions of the (quadratic) blocks of the sequence to be */
     73: /*         cut. sum_{i=1}^{BLKCT} BSIZES( I ) = N. */

     75: /*  RANKS  (input) INTEGER array, dimension (BLKCT-1) */
     76: /*         The ranks determining the approximations of the off-diagonal */
     77: /*         blocks in the sequence considered. */

     79: /*  CUT    (output) INTEGER */
     80: /*         After the optimum cut has been determined, the position (in the */
     81: /*         overall problem as worked on in DIBTDC !) of the last block in */
     82: /*         the first half of the sequence to be cut. */
     83: /*         START <= CUT <= START+BLKCT-2. */

     85: /*  LSUM   (output) INTEGER */
     86: /*         After the optimum cut has been determined, the sum of the */
     87: /*         block sizes in the first half of the sequence to be cut. */
     88: /*         LSUM < N. */

     90: /*  LBLKS  (output) INTEGER */
     91: /*         After the optimum cut has been determined, the number of the */
     92: /*         blocks in the first half of the sequence to be cut. */
     93: /*         1 <= LBLKS < BLKCT. */

     95: /*  INFO   (output) INTEGER */
     96: /*          = 0:  successful exit. */
     97: /*          < 0:  illegal arguments. */
     98: /*                if INFO = -i, the i-th (input) argument had an illegal */
     99: /*                value. */
    100: /*          > 0:  illegal results. */
    101: /*                if INFO = i, the i-th (output) argument had an illegal */
    102: /*                value. */

    104: /*  Further Details */
    105: /*  =============== */

    107: /*  Based on code written by */
    108: /*     Wilfried Gansterer and Bob Ward, */
    109: /*     Department of Computer Science, University of Tennessee */

    111: /*  ===================================================================== */

    113:   PetscBLASInt i, ksk, kchk, ksum, nhalf, deviat, mindev, minrnk, tmpsum;

    116:   *info = 0;
    117:   *lblks = 1;
    118:   *lsum = 1;
    119:   *cut = start;

    121:   if (start < 1) {
    122:     *info = -1;
    123:   } else if (n < 3) {
    124:     *info = -2;
    125:   } else if (blkct < 3) {
    126:     *info = -3;
    127:   }
    128:   if (*info == 0) {
    129:     ksum = 0;
    130:     kchk = 0;
    131:     for (i = 0; i < blkct; ++i) {
    132:       ksk = bsizes[i];
    133:       ksum += ksk;
    134:       if (ksk < 1) kchk = 1;
    135:     }
    136:     if (ksum != n || kchk == 1) *info = -4;
    137:   }
    138:   if (*info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong argument %d in CUTLR",-(*info));

    140:   /* determine smallest rank in the range considered */

    142:   minrnk = n;
    143:   for (i = 0; i < blkct-1; ++i) {
    144:     if (ranks[i] < minrnk) minrnk = ranks[i];
    145:   }

    147:   /* determine best cut among those with smallest rank */

    149:   nhalf = n / 2;
    150:   tmpsum = 0;
    151:   mindev = n;
    152:   for (i = 0; i < blkct; ++i) {
    153:     tmpsum += bsizes[i];
    154:     if (ranks[i] == minrnk) {

    156:       /* determine deviation from "optimal" cut NHALF */

    158:       deviat = tmpsum - nhalf;
    159:       if (deviat<0) deviat = -deviat;

    161:       /* compare to best deviation so far */

    163:       if (deviat < mindev) {
    164:         mindev = deviat;
    165:         *cut = start + i;
    166:         *lblks = i + 1;
    167:         *lsum = tmpsum;
    168:       }
    169:     }
    170:   }

    172:   if (*cut < start || *cut >= start + blkct - 1) {
    173:     *info = 6;
    174:   } else if (*lsum < 1 || *lsum >= n) {
    175:     *info = 7;
    176:   } else if (*lblks < 1 || *lblks >= blkct) {
    177:     *info = 8;
    178:   }
    179:   return(0);
    180: }

    183: PetscErrorCode BDC_dibtdc_(const char *jobz,PetscBLASInt n,PetscBLASInt nblks,
    184:         PetscBLASInt *ksizes,PetscReal *d,PetscBLASInt l1d,PetscBLASInt l2d,
    185:         PetscReal *e,PetscBLASInt *rank,PetscBLASInt l1e,PetscBLASInt l2e,
    186:         PetscReal tol,PetscReal *ev,PetscReal *z,PetscBLASInt ldz,PetscReal *work,
    187:         PetscBLASInt lwork,PetscBLASInt *iwork,PetscBLASInt liwork,
    188:         PetscBLASInt *info,PetscBLASInt jobz_len)
    189: {
    190: /*  -- Routine written in LAPACK Version 3.0 style -- */
    191: /* *************************************************** */
    192: /*     Written by */
    193: /*     Michael Moldaschl and Wilfried Gansterer */
    194: /*     University of Vienna */
    195: /*     last modification: March 16, 2014 */

    197: /*     Small adaptations of original code written by */
    198: /*     Wilfried Gansterer and Bob Ward, */
    199: /*     Department of Computer Science, University of Tennessee */
    200: /*     see http://dx.doi.org/10.1137/S1064827501399432 */
    201: /* *************************************************** */

    203: /*  Purpose */
    204: /*  ======= */

    206: /*  DIBTDC computes all eigenvalues and corresponding eigenvectors of a */
    207: /*  symmetric irreducible block tridiagonal matrix with rank RANK matrices */
    208: /*  as the subdiagonal blocks using a block divide and conquer method. */

    210: /*  Arguments */
    211: /*  ========= */

    213: /*  JOBZ    (input) CHARACTER*1 */
    214: /*          = 'N':  Compute eigenvalues only (not implemented); */
    215: /*          = 'D':  Compute eigenvalues and eigenvectors. */
    216: /*                  Eigenvectors are accumulated in the */
    217: /*                  divide-and-conquer process. */

    219: /*  N      (input) INTEGER */
    220: /*         The dimension of the symmetric irreducible block tridiagonal */
    221: /*         matrix.  N >= 2. */

    223: /*  NBLKS  (input) INTEGER, 2 <= NBLKS <= N */
    224: /*         The number of diagonal blocks in the matrix. */

    226: /*  KSIZES (input) INTEGER array, dimension (NBLKS) */
    227: /*         The dimension of the square diagonal blocks from top left */
    228: /*         to bottom right.  KSIZES(I) >= 1 for all I, and the sum of */
    229: /*         KSIZES(I) for I = 1 to NBLKS has to be equal to N. */

    231: /*  D      (input) DOUBLE PRECISION array, dimension (L1D,L2D,NBLKS) */
    232: /*         The lower triangular elements of the symmetric diagonal */
    233: /*         blocks of the block tridiagonal matrix.  Elements of the top */
    234: /*         left diagonal block, which is of dimension KSIZES(1), are */
    235: /*         contained in D(*,*,1); the elements of the next diagonal */
    236: /*         block, which is of dimension KSIZES(2), are contained in */
    237: /*         D(*,*,2); etc. */

    239: /*  L1D    (input) INTEGER */
    240: /*         The leading dimension of the array D.  L1D >= max(3,KMAX), */
    241: /*         where KMAX is the dimension of the largest diagonal block. */

    243: /*  L2D    (input) INTEGER */
    244: /*         The second dimension of the array D.  L2D >= max(3,KMAX), */
    245: /*         where KMAX is as stated in L1D above. */

    247: /*  E      (input) DOUBLE PRECISION array, dimension (L1E,L2E,NBLKS-1) */
    248: /*         Contains the elements of the scalars (singular values) and */
    249: /*         vectors (singular vectors) defining the rank RANK subdiagonal */
    250: /*         blocks of the matrix. */
    251: /*         E(1:RANK(K),RANK(K)+1,K) holds the RANK(K) scalars, */
    252: /*         E(:,1:RANK(K),K) holds the RANK(K) column vectors, and */
    253: /*         E(:,RANK(K)+2:2*RANK(K)+1,K) holds the row vectors for the K-th */
    254: /*         subdiagonal block. */

    256: /*  RANK   (input) INTEGER array, dimension (NBLKS-1). */
    257: /*         The ranks of all the subdiagonal blocks contained in the array E. */
    258: /*         RANK( K ) <= MIN( KSIZES( K ), KSIZES( K+1 ) ) */

    260: /*  L1E    (input) INTEGER */
    261: /*         The leading dimension of the array E.  L1E >= max(3,2*KMAX+1), */
    262: /*         where KMAX is as stated in L1D above. */

    264: /*  L2E    (input) INTEGER */
    265: /*         The second dimension of the array E.  L2E >= max(3,2*KMAX+1), */
    266: /*         where KMAX is as stated in L1D above. */

    268: /*  TOL    (input) DOUBLE PRECISION, TOL <= 1.0D-1 */
    269: /*         User specified deflation tolerance for the routine DMERG2. */
    270: /*         If ( 1.0D-1 >= TOL >= 20*EPS ) then TOL is used as */
    271: /*         the deflation tolerance in DSRTDF. */
    272: /*         If ( TOL < 20*EPS ) then the standard deflation tolerance from */
    273: /*         LAPACK is used as the deflation tolerance in DSRTDF. */

    275: /*  EV     (output) DOUBLE PRECISION array, dimension (N) */
    276: /*         If INFO = 0, then EV contains the eigenvalues of the */
    277: /*         symmetric block tridiagonal matrix in ascending order. */

    279: /*  Z      (input/output) DOUBLE PRECISION array, dimension (LDZ, N) */
    280: /*         On entry, Z will be the identity matrix. */
    281: /*         On exit, Z contains the eigenvectors of the block tridiagonal */
    282: /*         matrix. */

    284: /*  LDZ    (input) INTEGER */
    285: /*         The leading dimension of the array Z.  LDZ >= max(1,N). */

    287: /*  WORK   (workspace) DOUBLE PRECISION array, dimension (LWORK) */

    289: /*  LWORK   (input) INTEGER */
    290: /*          The dimension of the array WORK. */
    291: /*          In order to guarantee correct results in all cases, */
    292: /*          LWORK must be at least ( 2*N**2 + 3*N ). In many cases, */
    293: /*          less workspace is required. The absolute minimum required is */
    294: /*          ( N**2 + 3*N ). */
    295: /*          If the workspace provided is not sufficient, the routine will */
    296: /*          return a corresponding error code and report how much workspace */
    297: /*          was missing (see INFO). */

    299: /*  IWORK  (workspace) INTEGER array, dimension (LIWORK) */

    301: /*  LIWORK  (input) INTEGER */
    302: /*          The dimension of the array IWORK. */
    303: /*          LIWORK must be at least ( 5*N + 3 + 4*NBLKS - 4 ): */
    304: /*                 5*KMAX+3 for DSYEVD, 5*N for ????, */
    305: /*                 4*NBLKS-4 for the preprocessing (merging order) */
    306: /*          Summarizing, the minimum integer workspace needed is */
    307: /*          MAX( 5*N, 5*KMAX + 3 ) + 4*NBLKS - 4 */

    309: /*  INFO   (output) INTEGER */
    310: /*          = 0:  successful exit. */
    311: /*          < 0, > -99: illegal arguments. */
    312: /*                if INFO = -i, the i-th argument had an illegal value. */
    313: /*          = -99: error in the preprocessing (call of CUTLR). */
    314: /*          < -200: not enough workspace. Space for ABS(INFO + 200) */
    315: /*                numbers is required in addition to the workspace provided, */
    316: /*                otherwise some eigenvectors will be incorrect. */
    317: /*          > 0:  The algorithm failed to compute an eigenvalue while */
    318: /*                working on the submatrix lying in rows and columns */
    319: /*                INFO/(N+1) through mod(INFO,N+1). */

    321: /*  Further Details */
    322: /*  =============== */

    324: /*  Based on code written by */
    325: /*     Wilfried Gansterer and Bob Ward, */
    326: /*     Department of Computer Science, University of Tennessee */

    328: /*  This routine is comparable to Dlaed0.f from LAPACK. */

    330: /*  ===================================================================== */

    332: #if defined(SLEPC_MISSING_LAPACK_LACPY) || defined(SLEPC_MISSING_LAPACK_SYEV)
    334:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LACPY/SYEV - Lapack routine is unavailable");
    335: #else
    336:   PetscBLASInt   i, j, k, np, rp1, ksk, one=1;
    337:   PetscBLASInt   cut, mat1, kchk, kbrk, blks, kmax, icut, size, ksum, lsum;
    338:   PetscBLASInt   lblks, rblks, isize, lwmin, ilsum;
    339:   PetscBLASInt   start, vstrt, istck1, istck2, istck3, merged;
    340:   PetscBLASInt   liwmin, matsiz, startp, istrtp;
    341:   PetscReal      rho, done=1.0, dmone=-1.0;

    345:   *info = 0;

    347:   if (*(unsigned char *)jobz != 'N' && *(unsigned char *)jobz != 'D') {
    348:     *info = -1;
    349:   } else if (n < 2) {
    350:     *info = -2;
    351:   } else if (nblks < 2 || nblks > n) {
    352:     *info = -3;
    353:   }
    354:   if (*info == 0) {
    355:     ksum = 0;
    356:     kmax = 0;
    357:     kchk = 0;
    358:     for (k = 0; k < nblks; ++k) {
    359:       ksk = ksizes[k];
    360:       ksum += ksk;
    361:       if (ksk > kmax) kmax = ksk;
    362:       if (ksk < 1) kchk = 1;
    363:     }
    364:     lwmin = n*n + n * 3;
    365:     liwmin = PetscMax(n * 5,kmax * 5 + 3) + 4*nblks - 4;
    366:     if (ksum != n || kchk == 1) {
    367:       *info = -4;
    368:     } else if (l1d < PetscMax(3,kmax)) {
    369:       *info = -6;
    370:     } else if (l2d < PetscMax(3,kmax)) {
    371:       *info = -7;
    372:     } else if (l1e < PetscMax(3,2*kmax + 1)) {
    373:       *info = -10;
    374:     } else if (l2e < PetscMax(3,2*kmax + 1)) {
    375:       *info = -11;
    376:     } else if (tol > .1) {
    377:       *info = -12;
    378:     } else if (ldz < PetscMax(1,n)) {
    379:       *info = -15;
    380:     } else if (lwork < lwmin) {
    381:       *info = -17;
    382:     } else if (liwork < liwmin) {
    383:       *info = -19;
    384:     }
    385:   }
    386:   if (*info == 0) {
    387:     for (k = 0; k < nblks-1; ++k) {
    388:       if (rank[k] > PetscMin(ksizes[k],ksizes[k+1]) || rank[k] < 1) *info = -9;
    389:     }
    390:   }

    392:   if (*info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong argument %d in DIBTDC",-(*info));

    394: /* **************************************************************************** */

    396:   /* ...Preprocessing..................................................... */
    397:   /*    Determine the optimal order for merging the subblocks and how much */
    398:   /*    workspace will be needed for the merging (determined by the last */
    399:   /*    merge). Cutpoints for the merging operations are determined and stored */
    400:   /*    in reverse chronological order (starting with the final merging */
    401:   /*    operation). */
    402:
    403:   /*    integer workspace requirements for the preprocessing: */
    404:   /*         4*(NBLKS-1) for merging history */
    405:   /*         at most 3*(NBLKS-1) for stack */

    407:   start = 1;
    408:   size = n;
    409:   blks = nblks;
    410:   merged = 0;
    411:   k = 0;

    413:   /* integer workspace used for the stack is not needed any more after the */
    414:   /* preprocessing and therefore can use part of the 5*N */
    415:   /* integer workspace needed later on in the code */

    417:   istck1 = 0;
    418:   istck2 = istck1 + nblks;
    419:   istck3 = istck2 + nblks;

    421:   /* integer workspace used for storing the order of merges starts AFTER */
    422:   /* the integer workspace 5*N+3 which is needed later on in the code */
    423:   /* (5*KMAX+3 for DSYEVD, 4*N in DMERG2) */

    425:   istrtp = n * 5 + 4;
    426:   icut = istrtp + nblks - 1;
    427:   isize = icut + nblks - 1;
    428:   ilsum = isize + nblks - 1;

    430: L200:

    432:   if (nblks >= 3) {

    434:     /* Determine the cut point. Note that in the routine CUTLR it is */
    435:     /* chosen such that it yields the best balanced merging operation */
    436:     /* among all the rank modifications with minimum rank. */

    438:     cutlr_(start, size, blks, &ksizes[start-1], &rank[start-1], &cut,
    439:                   &lsum, &lblks, info);
    440:     if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: Error in cutlr, info = %d",*info);

    442:   } else {
    443:     cut = 1;
    444:     lsum = ksizes[0];
    445:     lblks = 1;
    446:   }

    448:   ++merged;
    449:   startp = 0;
    450:   for (i = 0; i < start-1; ++i) startp += ksizes[i];
    451:   iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    452:   iwork[icut + (nblks - 1) - merged-1] = cut;
    453:   iwork[isize + (nblks - 1) - merged-1] = size;
    454:   iwork[ilsum + (nblks - 1) - merged-1] = lsum;

    456:   if (lblks == 2) {

    458:     /* one merge in left branch, left branch done */
    459:     ++merged;
    460:     iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    461:     iwork[icut + (nblks - 1) - merged-1] = start;
    462:     iwork[isize + (nblks - 1) - merged-1] = lsum;
    463:     iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start-1];
    464:   }

    466:   if (lblks == 1 || lblks == 2) {

    468:     /* left branch done, continue on the right side */
    469:     start += lblks;
    470:     size -= lsum;
    471:     blks -= lblks;

    473:     if (blks <= 0) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: Error in preprocessing, blks = %d",blks);

    475:     if (blks == 2) {

    477:       /* one merge in right branch, right branch done */
    478:       ++merged;
    479:       startp += lsum;
    480:       iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    481:       iwork[icut + (nblks - 1) - merged-1] = start;
    482:       iwork[isize + (nblks - 1) - merged-1] = size;
    483:       iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start-1];
    484:     }

    486:     if (blks == 1 || blks == 2) {

    488:       /* get the next subproblem from the stack or finished */

    490:       if (k >= 1) {

    492:         /* something left on the stack */
    493:         start = iwork[istck1 + k-1];
    494:         size = iwork[istck2 + k-1];
    495:         blks = iwork[istck3 + k-1];
    496:         --k;
    497:         goto L200;
    498:       } else {

    500:         /* nothing left on the stack */
    501:         if (merged != nblks-1) SETERRQ(PETSC_COMM_SELF,1,"ERROR in preprocessing - not enough merges performed");

    503:         /* exit preprocessing */

    505:       }
    506:     } else {

    508:       /* BLKS.GE.3, and therefore analyze the right side */

    510:       goto L200;
    511:     }
    512:   } else {

    514:     /* LBLKS.GE.3, and therefore check the right side and */
    515:     /* put it on the stack if required */

    517:     rblks = blks - lblks;
    518:     if (rblks >= 3) {
    519:       ++k;
    520:       iwork[istck1 + k-1] = cut + 1;
    521:       iwork[istck2 + k-1] = size - lsum;
    522:       iwork[istck3 + k-1] = rblks;
    523:     } else if (rblks == 2) {

    525:       /* one merge in right branch, right branch done */
    526:       /* (note that nothing needs to be done if RBLKS.EQ.1 !) */

    528:       ++merged;
    529:       startp += lsum;
    530:       iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    531:       iwork[icut + (nblks - 1) - merged-1] = start + lblks;
    532:       iwork[isize + (nblks - 1) - merged-1] = size - lsum;
    533:       iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start + lblks-1];
    534:     }
    535:     if (rblks <= 0) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: ERROR in preprocessing - rblks = %d",rblks);

    537:     /* continue on the left side */

    539:     size = lsum;
    540:     blks = lblks;
    541:     goto L200;
    542:   }

    544:   /*  SIZE = IWORK( ISIZE+NBLKS-2 ) */
    545:   /*  MAT1 = IWORK( ILSUM+NBLKS-2 ) */
    546:
    547:   /* Note: after the dimensions SIZE and MAT1 of the last merging */
    548:   /* operation have been determined, an upper bound for the workspace */
    549:   /* requirements which is independent of how much deflation occurs in */
    550:   /* the last merging operation could be determined as follows */
    551:   /* (based on (3.15) and (3.19) from UT-CS-00-447): */
    552:
    553:   /*  IF( MAT1.LE.N/2 ) THEN */
    554:   /*     WSPREQ = 3*N + 3/2*( SIZE-MAT1 )**2 + N*N/2 + MAT1*MAT1 */
    555:   /*  ELSE */
    556:   /*     WSPREQ = 3*N + 3/2*MAT1*MAT1 + N*N/2 + ( SIZE-MAT1 )**2 */
    557:   /*  END IF */
    558:
    559:   /*  IF( LWORK-WSPREQ.LT.0 )THEN */
    560:   /*          not enough work space provided */
    561:   /*     INFO = -200 - ( WSPREQ-LWORK ) */
    562:   /*     RETURN */
    563:   /*  END IF */
    564:   /*  However, this is not really useful, since the actual check whether */
    565:   /*  enough workspace is provided happens in DMERG2.f ! */

    567: /* ************************************************************************* */

    569:   /* ...Solve subproblems................................... */
    570:
    571:   /* Divide the matrix into NBLKS submatrices using rank-r */
    572:   /* modifications (cuts) and solve for their eigenvalues and */
    573:   /* eigenvectors. Initialize index array to sort eigenvalues. */
    574:
    575:   /* first block: ...................................... */
    576:
    577:   /*    correction for block 1: D1 - V1 \Sigma1 V1^T */

    579:   ksk = ksizes[0];
    580:   rp1 = rank[0];

    582:   /* initialize the proper part of Z with the diagonal block D1 */
    583:   /* (the correction will be made in Z and then the call of DSYEVD will */
    584:   /*  overwrite it with the eigenvectors) */

    586:   PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, d, &l1d, z, &ldz));

    588:   /* copy D1 into WORK (in order to be able to restore it afterwards) */

    590:   PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, d, &l1d, work, &ksk));

    592:   /* copy V1 into the first RANK(1) columns of D1 and then */
    593:   /* multiply with \Sigma1 */

    595:   for (i = 0; i < rank[0]; ++i) {
    596:     PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(rp1 + i+1)*l1e], &one, &d[i*l1d], &one));
    597:     PetscStackCallBLAS("BLASscal",BLASscal_(&ksk, &e[i + rp1*l1e], &d[i*l1d], &one));
    598:   }

    600:   /* multiply the first RANK( 1 ) columns of D1 with V1^T and */
    601:   /* subtract the result from the proper part of Z (previously */
    602:   /* initialized with D1) */

    604:   PetscStackCallBLAS("BLASgemm",BLASgemm_("N", "T", &ksk, &ksk, rank, &dmone,
    605:           d, &l1d, &e[(rank[0]+1)*l1e], &l1e, &done, z, &ldz));

    607:   /* restore the original D1 from WORK */

    609:   PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, work, &ksk, d, &l1d));

    611:   /* eigenanalysis of block 1 (using DSYEVD) */

    613:   PetscStackCallBLAS("LAPACKsyev",LAPACKsyev_("V", "L", &ksk, z, &ldz, ev, work, &lwork, info));
    614:   if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: Error in DSYEVD for block 1, info = %d",*info);

    616:   /* EV( 1: ) contains the eigenvalues in ascending order */
    617:   /* (they are returned this way by DSYEVD) */

    619:   for (i = 0; i < ksk; ++i) iwork[i] = i+1;

    621:     /* intermediate blocks: .............................. */

    623:     np = ksk;

    625:     /* remaining number of blocks */

    627:     if (nblks > 2) {
    628:       for (k = 1; k < nblks-1; ++k) {

    630:       /* correction for block K: */
    631:       /* Dk - U(k-1) \Sigma(k-1) U(k-1)^T - Vk \Sigmak Vk^T */

    633:       ksk = ksizes[k];
    634:       rp1 = rank[k];

    636:       /* initialize the proper part of Z with the diagonal block Dk */
    637:       /* (the correction will be made in Z and then the call of DSYEVD will */
    638:       /*  overwrite it with the eigenvectors) */

    640:       PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, &d[k*l1d*l2d], &l1d, &z[np+np*ldz], &ldz));

    642:       /* copy Dk into WORK (in order to be able to restore it afterwards) */

    644:       PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, &d[k*l1d*l2d], &l1d, work, &ksk));

    646:       /* copy U(K-1) into the first RANK(K-1) columns of Dk and then */
    647:       /* multiply with \Sigma(K-1) */

    649:       for (i = 0; i < rank[k-1]; ++i) {
    650:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(i+(k-1)*l2e)*l1e], &one, &d[(i+k*l2d)*l1d], &one));
    651:         PetscStackCallBLAS("BLASscal",BLASscal_(&ksk, &e[i+(rank[k-1]+(k-1)*l2e)*l1e], &d[(i+k*l2d)*l1d], &one));
    652:       }

    654:       /* multiply the first RANK(K-1) columns of Dk with U(k-1)^T and */
    655:       /* subtract the result from the proper part of Z (previously */
    656:       /* initialized with Dk) */

    658:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N", "T", &ksk, &ksk, &rank[k-1],
    659:                     &dmone, &d[k*l1d*l2d],
    660:                     &l1d, &e[(k-1)*l1e*l2e], &l1e, &done, &z[np+np*ldz], &ldz));

    662:       /* copy Vk into the first RANK(K) columns of Dk and then */
    663:       /* multiply with \Sigmak */

    665:       for (i = 0; i < rank[k]; ++i) {
    666:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(rp1+i+1 + k*l2e)*l1e], &one, &d[(i + k*l2d)*l1d], &one));
    667:         PetscStackCallBLAS("BLASscal",BLASscal_(&ksk, &e[i + (rp1 + k*l2e)*l1e], &d[(i + k*l2d)*l1d], &one));
    668:       }

    670:       /* multiply the first RANK(K) columns of Dk with Vk^T and */
    671:       /* subtract the result from the proper part of Z (previously */
    672:       /* updated with [- U(k-1) \Sigma(k-1) U(k-1)^T] ) */

    674:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N", "T", &ksk, &ksk, &rank[k],
    675:                     &dmone, &d[k*l1d*l2d], &l1d,
    676:                     &e[(rank[k]+1 + k*l2e)*l1e], &l1e, &done, &z[np+np*ldz], &ldz));

    678:       /* restore the original Dk from WORK */

    680:       PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, work, &ksk, &d[k*l1d*l2d], &l1d));

    682:       /* eigenanalysis of block K (using dsyevd) */

    684:       PetscStackCallBLAS("LAPACKsyev",LAPACKsyev_("V", "L", &ksk, &z[np+np*ldz],
    685:                      &ldz, &ev[np], work, &lwork, info));
    686:       if (*info) SETERRQ2(PETSC_COMM_SELF,1,"dibtdc: Error in DSYEVD for block %d, info = %d",k,*info);

    688:       /* EV( NPP1: ) contains the eigenvalues in ascending order */
    689:       /* (they are returned this way by DSYEVD) */

    691:       for (i = 0; i < ksk; ++i) iwork[np + i] = i+1;

    693:       /* update NP */
    694:       np += ksk;
    695:     }
    696:   }

    698:   /* last block: ....................................... */
    699:
    700:   /*    correction for block NBLKS: */
    701:   /*    D(nblks) - U(nblks-1) \Sigma(nblks-1) U(nblks-1)^T */

    703:   ksk = ksizes[nblks-1];

    705:   /* initialize the proper part of Z with the diagonal block D(nblks) */
    706:   /* (the correction will be made in Z and then the call of DSYEVD will */
    707:   /* overwrite it with the eigenvectors) */

    709:   PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, &d[(nblks-1)*l1d*l2d], &l1d, &z[np+np*ldz], &ldz));

    711:   /* copy D(nblks) into WORK (in order to be able to restore it afterwards) */

    713:   PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, &d[(nblks-1)*l1d*l2d], &l1d, work, &ksk));

    715:   /* copy U(nblks-1) into the first RANK(nblks-1) columns of D(nblks) and then */
    716:   /* multiply with \Sigma(nblks-1) */

    718:   for (i = 0; i < rank[nblks-2]; ++i) {
    719:     PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(i + (nblks-2)*l2e)*l1e],
    720:               &one, &d[(i + (nblks-1)*l2d)*l1d], &one));
    721:     PetscStackCallBLAS("BLASscal",BLASscal_(&ksk,
    722:               &e[i + (rank[nblks-2] + (nblks-2)*l2e)*l1e],
    723:               &d[(i + (nblks-1)*l2d)*l1d], &one));
    724:   }

    726:   /* multiply the first RANK(nblks-1) columns of D(nblks) with U(nblks-1)^T */
    727:   /* and subtract the result from the proper part of Z (previously */
    728:   /* initialized with D(nblks) ) */

    730:   PetscStackCallBLAS("BLASgemm",BLASgemm_("N", "T", &ksk, &ksk, &rank[nblks - 2],
    731:           &dmone, &d[(nblks-1)*l1d*l2d], &l1d,
    732:           &e[(nblks-2)*l1e*l2e], &l1e, &done, &z[np+np*ldz], &ldz));

    734:   /* restore the original D(nblks) from WORK */

    736:   PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("L", &ksk, &ksk, work, &ksk, &d[(nblks-1)*l1d*l2d], &l1d));

    738:   /* eigenanalysis of block NBLKS (using dsyevd) */

    740:   PetscStackCallBLAS("LAPACKsyev",LAPACKsyev_("V", "L", &ksk, &z[np+np*ldz], &ldz, &ev[np], work, &lwork, info));
    741:   if (*info) SETERRQ2(PETSC_COMM_SELF,1,"dibtdc: Error in DSYEVD for block %d, info = %d",nblks,*info);

    743:   /* EV( NPP1: ) contains the eigenvalues in ascending order */
    744:   /* (they are returned this way by DSYEVD) */

    746:   for (i = 0; i < ksk; ++i) iwork[np + i] = i+1;

    748:   /* note that from here on the entire workspace is available again */

    751:   /* Perform all the merging operations. */

    753:   vstrt = 0;
    754:   for (i = 0; i < nblks-1; ++i) {

    756:     /* MATSIZ = total size of the current rank RANK modification problem */

    758:     matsiz = iwork[isize + i - 1];
    759:     np = iwork[istrtp + i - 1];
    760:     kbrk = iwork[icut + i - 1];
    761:     mat1 = iwork[ilsum + i - 1];
    762:     vstrt += np;

    764:     for (j = 0; j < rank[kbrk-1]; ++j) {

    766:       /* NOTE: The parameter RHO in DMERG2 is modified in DSRTDF */
    767:       /*       (multiplied by 2) ! In order not to change the */
    768:       /*       singular value stored in E( :, RANK( KBRK )+1, KBRK ), */
    769:       /*       we do not pass on this variable as an argument to DMERG2, */
    770:       /*       but we assign a separate variable RHO here which is passed */
    771:       /*       on to DMERG2. */
    772:       /*       Alternative solution in F90: */
    773:       /*       pass E( :,RANK( KBRK )+1,KBRK ) to an INTENT( IN ) parameter */
    774:       /*       in DMERG2. */

    776:       rho = e[j + (rank[kbrk-1] + (kbrk-1)*l2e)*l1e];

    778:       /* eigenvectors are accumulated ( JOBZ.EQ.'D' ) */

    780:       BDC_dmerg2_(jobz, j+1, matsiz, &ev[np-1], &z[np-1+(np-1)*ldz],
    781:                     ldz, &iwork[np-1], &rho, &e[(j + (kbrk-1)*l2e)*l1e],
    782:                     ksizes[kbrk], &e[(rank[kbrk-1]+j+1 + (kbrk-1)*l2e)*l1e],
    783:                     ksizes[kbrk-1], mat1, work, lwork, &iwork[n], tol, info, 1);
    784:
    785:       if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: Error in dmerg2, info = %d",*info);
    786:     }

    788:     /* at this point all RANK( KBRK ) rank-one modifications corresponding */
    789:     /* to the current off-diagonal block are finished. */
    790:     /* Move on to the next off-diagonal block. */

    792:   }

    794:   /* Re-merge the eigenvalues/vectors which were deflated at the final */
    795:   /* merging step by sorting all eigenvalues and eigenvectors according */
    796:   /* to the permutation stored in IWORK. */
    797:
    798:   /* copy eigenvalues and eigenvectors in ordered form into WORK */
    799:   /* (eigenvalues into WORK( 1:N ), eigenvectors into WORK( N+1:N+1+N^2 ) ) */

    801:   for (i = 0; i < n; ++i) {
    802:     j = iwork[i];
    803:     work[i] = ev[j-1];
    804:     PetscStackCallBLAS("BLAScopy",BLAScopy_(&n, &z[(j-1)*ldz], &one, &work[n*(i+1)], &one));
    805:   }

    807:   /* copy ordered eigenvalues back from WORK( 1:N ) into EV */

    809:   PetscStackCallBLAS("BLAScopy",BLAScopy_(&n, work, &one, ev, &one));

    811:   /* copy ordered eigenvectors back from WORK( N+1:N+1+N^2 ) into Z */

    813:   PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("A", &n, &n, &work[n], &n, z, &ldz));
    814:   return(0);
    815: #endif
    816: }