Date: 2026-03-31T13:23:16+00:00

Actual source code: dibtdc.c

      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: static PetscErrorCode cutlr_(PetscBLASInt start,PetscBLASInt n,PetscBLASInt blkct,
     18:         PetscBLASInt *bsizes,PetscBLASInt *ranks,PetscBLASInt *cut,
     19:         PetscBLASInt *lsum,PetscBLASInt *lblks,PetscBLASInt *info)
     20: {
     21: /*  -- Routine written in LAPACK Version 3.0 style -- */
     22: /* *************************************************** */
     23: /*     Written by */
     24: /*     Michael Moldaschl and Wilfried Gansterer */
     25: /*     University of Vienna */
     26: /*     last modification: March 16, 2014 */

     28: /*     Small adaptations of original code written by */
     29: /*     Wilfried Gansterer and Bob Ward, */
     30: /*     Department of Computer Science, University of Tennessee */
     31: /*     see https://doi.org/10.1137/S1064827501399432 */
     32: /* *************************************************** */

     34: /*  Purpose */
     35: /*  ======= */

     37: /*  CUTLR computes the optimal cut in a sequence of BLKCT neighboring */
     38: /*  blocks whose sizes are given by the array BSIZES. */
     39: /*  The sum of all block sizes in the sequence considered is given by N. */
     40: /*  The cut is optimal in the sense that the difference of the sizes of */
     41: /*  the resulting two halves is minimum over all cuts with minimum ranks */
     42: /*  between blocks of the sequence considered. */

     44: /*  Arguments */
     45: /*  ========= */

     47: /*  START  (input) INTEGER */
     48: /*         In the original array KSIZES of the calling routine DIBTDC, */
     49: /*         the position where the sequence considered in this routine starts. */
     50: /*         START >= 1. */

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

     57: /*  BLKCT  (input) INTEGER */
     58: /*         The number of blocks in the sequence to be cut. */
     59: /*         BLKCT >= 3. */

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

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

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

     75: /*  LSUM   (output) INTEGER */
     76: /*         After the optimum cut has been determined, the sum of the */
     77: /*         block sizes in the first half of the sequence to be cut. */
     78: /*         LSUM < N. */

     80: /*  LBLKS  (output) INTEGER */
     81: /*         After the optimum cut has been determined, the number of the */
     82: /*         blocks in the first half of the sequence to be cut. */
     83: /*         1 <= LBLKS < BLKCT. */

     85: /*  INFO   (output) INTEGER */
     86: /*          = 0:  successful exit. */
     87: /*          < 0:  illegal arguments. */
     88: /*                if INFO = -i, the i-th (input) argument had an illegal */
     89: /*                value. */
     90: /*          > 0:  illegal results. */
     91: /*                if INFO = i, the i-th (output) argument had an illegal */
     92: /*                value. */

     94: /*  Further Details */
     95: /*  =============== */

     97: /*  Based on code written by */
     98: /*     Wilfried Gansterer and Bob Ward, */
     99: /*     Department of Computer Science, University of Tennessee */

    101: /*  ===================================================================== */

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

    105:   PetscFunctionBegin;
    106:   *info = 0;
    107:   *lblks = 1;
    108:   *lsum = 1;
    109:   *cut = start;

    111:   if (start < 1) *info = -1;
    112:   else if (n < 3) *info = -2;
    113:   else if (blkct < 3) *info = -3;
    114:   if (*info == 0) {
    115:     ksum = 0;
    116:     kchk = 0;
    117:     for (i = 0; i < blkct; ++i) {
    118:       ksk = bsizes[i];
    119:       ksum += ksk;
    120:       if (ksk < 1) kchk = 1;
    121:     }
    122:     if (ksum != n || kchk == 1) *info = -4;
    123:   }
    124:   PetscCheck(!*info,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong argument %" PetscBLASInt_FMT " in CUTLR",-(*info));

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

    128:   minrnk = n;
    129:   for (i = 0; i < blkct-1; ++i) {
    130:     if (ranks[i] < minrnk) minrnk = ranks[i];
    131:   }

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

    135:   nhalf = n / 2;
    136:   tmpsum = 0;
    137:   mindev = n;
    138:   for (i = 0; i < blkct; ++i) {
    139:     tmpsum += bsizes[i];
    140:     if (ranks[i] == minrnk) {

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

    144:       deviat = tmpsum - nhalf;
    145:       if (deviat<0) deviat = -deviat;

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

    149:       if (deviat < mindev) {
    150:         mindev = deviat;
    151:         *cut = start + i;
    152:         *lblks = i + 1;
    153:         *lsum = tmpsum;
    154:       }
    155:     }
    156:   }

    158:   if (*cut < start || *cut >= start + blkct - 1) *info = 6;
    159:   else if (*lsum < 1 || *lsum >= n) *info = 7;
    160:   else if (*lblks < 1 || *lblks >= blkct) *info = 8;
    161:   PetscFunctionReturn(PETSC_SUCCESS);
    162: }

    164: PetscErrorCode BDC_dibtdc_(const char *jobz,PetscBLASInt n,PetscBLASInt nblks,
    165:         PetscBLASInt *ksizes,PetscReal *d,PetscBLASInt l1d,PetscBLASInt l2d,
    166:         PetscReal *e,PetscBLASInt *rank,PetscBLASInt l1e,PetscBLASInt l2e,
    167:         PetscReal tol,PetscReal *ev,PetscReal *z,PetscBLASInt ldz,PetscReal *work,
    168:         PetscBLASInt lwork,PetscBLASInt *iwork,PetscBLASInt liwork,
    169:         PetscBLASInt *info,PetscBLASInt jobz_len)
    170: {
    171: /*  -- Routine written in LAPACK Version 3.0 style -- */
    172: /* *************************************************** */
    173: /*     Written by */
    174: /*     Michael Moldaschl and Wilfried Gansterer */
    175: /*     University of Vienna */
    176: /*     last modification: March 16, 2014 */

    178: /*     Small adaptations of original code written by */
    179: /*     Wilfried Gansterer and Bob Ward, */
    180: /*     Department of Computer Science, University of Tennessee */
    181: /*     see https://doi.org/10.1137/S1064827501399432 */
    182: /* *************************************************** */

    184: /*  Purpose */
    185: /*  ======= */

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

    191: /*  Arguments */
    192: /*  ========= */

    194: /*  JOBZ    (input) CHARACTER*1 */
    195: /*          = 'N':  Compute eigenvalues only (not implemented); */
    196: /*          = 'D':  Compute eigenvalues and eigenvectors. */
    197: /*                  Eigenvectors are accumulated in the */
    198: /*                  divide-and-conquer process. */

    200: /*  N      (input) INTEGER */
    201: /*         The dimension of the symmetric irreducible block tridiagonal */
    202: /*         matrix.  N >= 2. */

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

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

    212: /*  D      (input) DOUBLE PRECISION array, dimension (L1D,L2D,NBLKS) */
    213: /*         The lower triangular elements of the symmetric diagonal */
    214: /*         blocks of the block tridiagonal matrix.  Elements of the top */
    215: /*         left diagonal block, which is of dimension KSIZES(1), are */
    216: /*         contained in D(*,*,1); the elements of the next diagonal */
    217: /*         block, which is of dimension KSIZES(2), are contained in */
    218: /*         D(*,*,2); etc. */

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

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

    228: /*  E      (input) DOUBLE PRECISION array, dimension (L1E,L2E,NBLKS-1) */
    229: /*         Contains the elements of the scalars (singular values) and */
    230: /*         vectors (singular vectors) defining the rank RANK subdiagonal */
    231: /*         blocks of the matrix. */
    232: /*         E(1:RANK(K),RANK(K)+1,K) holds the RANK(K) scalars, */
    233: /*         E(:,1:RANK(K),K) holds the RANK(K) column vectors, and */
    234: /*         E(:,RANK(K)+2:2*RANK(K)+1,K) holds the row vectors for the K-th */
    235: /*         subdiagonal block. */

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

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

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

    249: /*  TOL    (input) DOUBLE PRECISION, TOL <= 1.0D-1 */
    250: /*         User specified deflation tolerance for the routine DMERG2. */
    251: /*         If (1.0D-1 >= TOL >= 20*EPS) then TOL is used as */
    252: /*         the deflation tolerance in DSRTDF. */
    253: /*         If (TOL < 20*EPS) then the standard deflation tolerance from */
    254: /*         LAPACK is used as the deflation tolerance in DSRTDF. */

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

    260: /*  Z      (input/output) DOUBLE PRECISION array, dimension (LDZ, N) */
    261: /*         On entry, Z will be the identity matrix. */
    262: /*         On exit, Z contains the eigenvectors of the block tridiagonal */
    263: /*         matrix. */

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

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

    270: /*  LWORK   (input) INTEGER */
    271: /*          The dimension of the array WORK. */
    272: /*          In order to guarantee correct results in all cases, */
    273: /*          LWORK must be at least (2*N**2 + 3*N). In many cases, */
    274: /*          less workspace is required. The absolute minimum required is */
    275: /*          (N**2 + 3*N). */
    276: /*          If the workspace provided is not sufficient, the routine will */
    277: /*          return a corresponding error code and report how much workspace */
    278: /*          was missing (see INFO). */

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

    282: /*  LIWORK  (input) INTEGER */
    283: /*          The dimension of the array IWORK. */
    284: /*          LIWORK must be at least (5*N + 3 + 4*NBLKS - 4): */
    285: /*                 5*KMAX+3 for DSYEVD, 5*N for ????, */
    286: /*                 4*NBLKS-4 for the preprocessing (merging order) */
    287: /*          Summarizing, the minimum integer workspace needed is */
    288: /*          MAX(5*N, 5*KMAX + 3) + 4*NBLKS - 4 */

    290: /*  INFO   (output) INTEGER */
    291: /*          = 0:  successful exit. */
    292: /*          < 0, > -99: illegal arguments. */
    293: /*                if INFO = -i, the i-th argument had an illegal value. */
    294: /*          = -99: error in the preprocessing (call of CUTLR). */
    295: /*          < -200: not enough workspace. Space for ABS(INFO + 200) */
    296: /*                numbers is required in addition to the workspace provided, */
    297: /*                otherwise some eigenvectors will be incorrect. */
    298: /*          > 0:  The algorithm failed to compute an eigenvalue while */
    299: /*                working on the submatrix lying in rows and columns */
    300: /*                INFO/(N+1) through mod(INFO,N+1). */

    302: /*  Further Details */
    303: /*  =============== */

    305: /*  Based on code written by */
    306: /*     Wilfried Gansterer and Bob Ward, */
    307: /*     Department of Computer Science, University of Tennessee */

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

    311: /*  ===================================================================== */

    313:   PetscBLASInt   i, j, k, np, rp1, ksk, one=1;
    314:   PetscBLASInt   cut, mat1, kchk, kbrk, blks, kmax, icut, size, ksum, lsum;
    315:   PetscBLASInt   lblks, rblks, isize, lwmin, ilsum;
    316:   PetscBLASInt   start, istck1, istck2, istck3, merged;
    317:   PetscBLASInt   liwmin, matsiz, startp, istrtp;
    318:   PetscReal      rho, done=1.0, dmone=-1.0;

    320:   PetscFunctionBegin;
    321:   *info = 0;

    323:   if (*(unsigned char *)jobz != 'N' && *(unsigned char *)jobz != 'D') *info = -1;
    324:   else if (n < 2) *info = -2;
    325:   else if (nblks < 2 || nblks > n) *info = -3;
    326:   if (*info == 0) {
    327:     ksum = 0;
    328:     kmax = 0;
    329:     kchk = 0;
    330:     for (k = 0; k < nblks; ++k) {
    331:       ksk = ksizes[k];
    332:       ksum += ksk;
    333:       if (ksk > kmax) kmax = ksk;
    334:       if (ksk < 1) kchk = 1;
    335:     }
    336:     lwmin = n*n + n * 3;
    337:     liwmin = PetscMax(n * 5,kmax * 5 + 3) + 4*nblks - 4;
    338:     if (ksum != n || kchk == 1) *info = -4;
    339:     else if (l1d < PetscMax(3,kmax)) *info = -6;
    340:     else if (l2d < PetscMax(3,kmax)) *info = -7;
    341:     else if (l1e < PetscMax(3,2*kmax + 1)) *info = -10;
    342:     else if (l2e < PetscMax(3,2*kmax + 1)) *info = -11;
    343:     else if (tol > .1) *info = -12;
    344:     else if (ldz < PetscMax(1,n)) *info = -15;
    345:     else if (lwork < lwmin) *info = -17;
    346:     else if (liwork < liwmin) *info = -19;
    347:   }
    348:   if (*info == 0) {
    349:     for (k = 0; k < nblks-1; ++k) {
    350:       if (rank[k] > PetscMin(ksizes[k],ksizes[k+1]) || rank[k] < 1) *info = -9;
    351:     }
    352:   }

    354:   PetscCheck(!*info,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong argument %" PetscBLASInt_FMT " in DIBTDC",-(*info));

    356: /* **************************************************************************** */

    358:   /* ...Preprocessing..................................................... */
    359:   /*    Determine the optimal order for merging the subblocks and how much */
    360:   /*    workspace will be needed for the merging (determined by the last */
    361:   /*    merge). Cutpoints for the merging operations are determined and stored */
    362:   /*    in reverse chronological order (starting with the final merging */
    363:   /*    operation). */

    365:   /*    integer workspace requirements for the preprocessing: */
    366:   /*         4*(NBLKS-1) for merging history */
    367:   /*         at most 3*(NBLKS-1) for stack */

    369:   start = 1;
    370:   size = n;
    371:   blks = nblks;
    372:   merged = 0;
    373:   k = 0;

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

    379:   istck1 = 0;
    380:   istck2 = istck1 + nblks;
    381:   istck3 = istck2 + nblks;

    383:   /* integer workspace used for storing the order of merges starts AFTER */
    384:   /* the integer workspace 5*N+3 which is needed later on in the code */
    385:   /* (5*KMAX+3 for DSYEVD, 4*N in DMERG2) */

    387:   istrtp = n * 5 + 4;
    388:   icut = istrtp + nblks - 1;
    389:   isize = icut + nblks - 1;
    390:   ilsum = isize + nblks - 1;

    392: L200:

    394:   if (nblks >= 3) {

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

    400:     PetscCall(cutlr_(start, size, blks, &ksizes[start-1], &rank[start-1], &cut, &lsum, &lblks, info));
    401:     PetscCheck(!*info,PETSC_COMM_SELF,PETSC_ERR_PLIB,"dibtdc: Error in cutlr, info = %" PetscBLASInt_FMT,*info);

    403:   } else {
    404:     cut = 1;
    405:     lsum = ksizes[0];
    406:     lblks = 1;
    407:   }

    409:   ++merged;
    410:   startp = 0;
    411:   for (i = 0; i < start-1; ++i) startp += ksizes[i];
    412:   iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    413:   iwork[icut + (nblks - 1) - merged-1] = cut;
    414:   iwork[isize + (nblks - 1) - merged-1] = size;
    415:   iwork[ilsum + (nblks - 1) - merged-1] = lsum;

    417:   if (lblks == 2) {

    419:     /* one merge in left branch, left branch done */
    420:     ++merged;
    421:     iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    422:     iwork[icut + (nblks - 1) - merged-1] = start;
    423:     iwork[isize + (nblks - 1) - merged-1] = lsum;
    424:     iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start-1];
    425:   }

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

    429:     /* left branch done, continue on the right side */
    430:     start += lblks;
    431:     size -= lsum;
    432:     blks -= lblks;

    434:     PetscCheck(blks>0,PETSC_COMM_SELF,PETSC_ERR_PLIB,"dibtdc: Error in preprocessing, blks = %" PetscBLASInt_FMT,blks);

    436:     if (blks == 2) {

    438:       /* one merge in right branch, right branch done */
    439:       ++merged;
    440:       startp += lsum;
    441:       iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    442:       iwork[icut + (nblks - 1) - merged-1] = start;
    443:       iwork[isize + (nblks - 1) - merged-1] = size;
    444:       iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start-1];
    445:     }

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

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

    451:       if (k >= 1) {

    453:         /* something left on the stack */
    454:         start = iwork[istck1 + k-1];
    455:         size = iwork[istck2 + k-1];
    456:         blks = iwork[istck3 + k-1];
    457:         --k;
    458:         goto L200;
    459:       } else {

    461:         /* nothing left on the stack */
    462:         PetscCheck(merged==nblks-1,PETSC_COMM_SELF,PETSC_ERR_PLIB,"ERROR in preprocessing - not enough merges performed");

    464:         /* exit preprocessing */

    466:       }
    467:     } else {

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

    471:       goto L200;
    472:     }
    473:   } else {

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

    478:     rblks = blks - lblks;
    479:     if (rblks >= 3) {
    480:       ++k;
    481:       iwork[istck1 + k-1] = cut + 1;
    482:       iwork[istck2 + k-1] = size - lsum;
    483:       iwork[istck3 + k-1] = rblks;
    484:     } else if (rblks == 2) {

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

    489:       ++merged;
    490:       startp += lsum;
    491:       iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    492:       iwork[icut + (nblks - 1) - merged-1] = start + lblks;
    493:       iwork[isize + (nblks - 1) - merged-1] = size - lsum;
    494:       iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start + lblks-1];
    495:     }
    496:     PetscCheck(rblks>0,PETSC_COMM_SELF,PETSC_ERR_PLIB,"dibtdc: ERROR in preprocessing - rblks = %" PetscBLASInt_FMT,rblks);

    498:     /* continue on the left side */

    500:     size = lsum;
    501:     blks = lblks;
    502:     goto L200;
    503:   }

    505:   /*  SIZE = IWORK(ISIZE+NBLKS-2) */
    506:   /*  MAT1 = IWORK(ILSUM+NBLKS-2) */

    508:   /* Note: after the dimensions SIZE and MAT1 of the last merging */
    509:   /* operation have been determined, an upper bound for the workspace */
    510:   /* requirements which is independent of how much deflation occurs in */
    511:   /* the last merging operation could be determined as follows */
    512:   /* (based on (3.15) and (3.19) from UT-CS-00-447): */

    514:   /*  IF(MAT1.LE.N/2) THEN */
    515:   /*     WSPREQ = 3*N + 3/2*(SIZE-MAT1)**2 + N*N/2 + MAT1*MAT1 */
    516:   /*  ELSE */
    517:   /*     WSPREQ = 3*N + 3/2*MAT1*MAT1 + N*N/2 + (SIZE-MAT1)**2 */
    518:   /*  END IF */

    520:   /*  IF(LWORK-WSPREQ.LT.0)THEN */
    521:   /*          not enough work space provided */
    522:   /*     INFO = -200 - (WSPREQ-LWORK) */
    523:   /*     RETURN */
    524:   /*  END IF */
    525:   /*  However, this is not really useful, since the actual check whether */
    526:   /*  enough workspace is provided happens in DMERG2.f ! */

    528: /* ************************************************************************* */

    530:   /* ...Solve subproblems................................... */

    532:   /* Divide the matrix into NBLKS submatrices using rank-r */
    533:   /* modifications (cuts) and solve for their eigenvalues and */
    534:   /* eigenvectors. Initialize index array to sort eigenvalues. */

    536:   /* first block: ...................................... */

    538:   /*    correction for block 1: D1 - V1 \Sigma1 V1^T */

    540:   ksk = ksizes[0];
    541:   rp1 = rank[0];

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

    547:   for (j=0;j<ksk;j++) for (i=j;i<ksk;i++) z[i+j*ldz] = d[i+j*l1d];

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

    551:   for (j=0;j<ksk;j++) for (i=j;i<ksk;i++) work[i+j*ksk] = d[i+j*l1d];

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

    556:   for (i = 0; i < rank[0]; ++i) {
    557:     PetscCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(rp1 + i+1)*l1e], &one, &d[i*l1d], &one));
    558:     PetscCallBLAS("BLASscal",BLASscal_(&ksk, &e[i + rp1*l1e], &d[i*l1d], &one));
    559:   }

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

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

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

    570:   for (j=0;j<ksk;j++) for (i=j;i<ksk;i++) d[i+j*l1d] = work[i+j*ksk];

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

    574:   PetscCallBLAS("LAPACKsyev",LAPACKsyev_("V", "L", &ksk, z, &ldz, ev, work, &lwork, info));
    575:   SlepcCheckLapackInfo("syev",*info);

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

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

    582:     /* intermediate blocks: .............................. */

    584:     np = ksk;

    586:     /* remaining number of blocks */

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

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

    594:       ksk = ksizes[k];
    595:       rp1 = rank[k];

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

    601:       for (j=0;j<ksk;j++) for (i=j;i<ksk;i++) z[np+i+(np+j)*ldz] = d[i+(j+k*l2d)*l1d];

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

    605:       for (j=0;j<ksk;j++) for (i=j;i<ksk;i++) work[i+j*ksk] = d[i+(j+k*l2d)*l1d];

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

    610:       for (i = 0; i < rank[k-1]; ++i) {
    611:         PetscCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(i+(k-1)*l2e)*l1e], &one, &d[(i+k*l2d)*l1d], &one));
    612:         PetscCallBLAS("BLASscal",BLASscal_(&ksk, &e[i+(rank[k-1]+(k-1)*l2e)*l1e], &d[(i+k*l2d)*l1d], &one));
    613:       }

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

    619:       PetscCallBLAS("BLASgemm",BLASgemm_("N", "T", &ksk, &ksk, &rank[k-1],
    620:                     &dmone, &d[k*l1d*l2d],
    621:                     &l1d, &e[(k-1)*l1e*l2e], &l1e, &done, &z[np+np*ldz], &ldz));

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

    626:       for (i = 0; i < rank[k]; ++i) {
    627:         PetscCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(rp1+i+1 + k*l2e)*l1e], &one, &d[(i + k*l2d)*l1d], &one));
    628:         PetscCallBLAS("BLASscal",BLASscal_(&ksk, &e[i + (rp1 + k*l2e)*l1e], &d[(i + k*l2d)*l1d], &one));
    629:       }

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

    635:       PetscCallBLAS("BLASgemm",BLASgemm_("N", "T", &ksk, &ksk, &rank[k],
    636:                     &dmone, &d[k*l1d*l2d], &l1d,
    637:                     &e[(rank[k]+1 + k*l2e)*l1e], &l1e, &done, &z[np+np*ldz], &ldz));

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

    641:       for (j=0;j<ksk;j++) for (i=j;i<ksk;i++) d[i+(j+k*l2d)*l1d] = work[i+j*ksk];

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

    645:       PetscCallBLAS("LAPACKsyev",LAPACKsyev_("V", "L", &ksk, &z[np+np*ldz],
    646:                      &ldz, &ev[np], work, &lwork, info));
    647:       SlepcCheckLapackInfo("syev",*info);

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

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

    654:       /* update NP */
    655:       np += ksk;
    656:     }
    657:   }

    659:   /* last block: ....................................... */

    661:   /*    correction for block NBLKS: */
    662:   /*    D(nblks) - U(nblks-1) \Sigma(nblks-1) U(nblks-1)^T */

    664:   ksk = ksizes[nblks-1];

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

    670:   for (j=0;j<ksk;j++) for (i=j;i<ksk;i++) z[np+i+(np+j)*ldz] = d[i+(j+(nblks-1)*l2d)*l1d];

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

    674:   for (j=0;j<ksk;j++) for (i=j;i<ksk;i++) work[i+j*ksk] = d[i+(j+(nblks-1)*l2d)*l1d];

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

    679:   for (i = 0; i < rank[nblks-2]; ++i) {
    680:     PetscCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(i + (nblks-2)*l2e)*l1e],
    681:               &one, &d[(i + (nblks-1)*l2d)*l1d], &one));
    682:     PetscCallBLAS("BLASscal",BLASscal_(&ksk,
    683:               &e[i + (rank[nblks-2] + (nblks-2)*l2e)*l1e],
    684:               &d[(i + (nblks-1)*l2d)*l1d], &one));
    685:   }

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

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

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

    697:   for (j=0;j<ksk;j++) for (i=j;i<ksk;i++) d[i+(j+(nblks-1)*l2d)*l1d] = work[i+j*ksk];

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

    701:   PetscCallBLAS("LAPACKsyev",LAPACKsyev_("V", "L", &ksk, &z[np+np*ldz], &ldz, &ev[np], work, &lwork, info));
    702:   SlepcCheckLapackInfo("syev",*info);

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

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

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

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

    713:   for (i = 0; i < nblks-1; ++i) {

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

    717:     matsiz = iwork[isize + i - 1];
    718:     np = iwork[istrtp + i - 1];
    719:     kbrk = iwork[icut + i - 1];
    720:     mat1 = iwork[ilsum + i - 1];

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

    724:       /* NOTE: The parameter RHO in DMERG2 is modified in DSRTDF */
    725:       /*       (multiplied by 2) ! In order not to change the */
    726:       /*       singular value stored in E(:, RANK(KBRK)+1, KBRK), */
    727:       /*       we do not pass on this variable as an argument to DMERG2, */
    728:       /*       but we assign a separate variable RHO here which is passed */
    729:       /*       on to DMERG2. */
    730:       /*       Alternative solution in F90: */
    731:       /*       pass E(:,RANK(KBRK)+1,KBRK) to an INTENT(IN) parameter */
    732:       /*       in DMERG2. */

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

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

    738:       PetscCall(BDC_dmerg2_(jobz, j+1, matsiz, &ev[np-1], &z[np-1+(np-1)*ldz],
    739:                     ldz, &iwork[np-1], &rho, &e[(j + (kbrk-1)*l2e)*l1e],
    740:                     ksizes[kbrk], &e[(rank[kbrk-1]+j+1 + (kbrk-1)*l2e)*l1e],
    741:                     ksizes[kbrk-1], mat1, work, lwork, &iwork[n], tol, info, 1));
    742:       PetscCheck(!*info,PETSC_COMM_SELF,PETSC_ERR_PLIB,"dibtdc: Error in dmerg2, info = %" PetscBLASInt_FMT,*info);
    743:     }

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

    749:   }

    751:   /* Re-merge the eigenvalues/vectors which were deflated at the final */
    752:   /* merging step by sorting all eigenvalues and eigenvectors according */
    753:   /* to the permutation stored in IWORK. */

    755:   /* copy eigenvalues and eigenvectors in ordered form into WORK */
    756:   /* (eigenvalues into WORK(1:N), eigenvectors into WORK(N+1:N+1+N^2)) */

    758:   for (i = 0; i < n; ++i) {
    759:     j = iwork[i];
    760:     work[i] = ev[j-1];
    761:     PetscCallBLAS("BLAScopy",BLAScopy_(&n, &z[(j-1)*ldz], &one, &work[n*(i+1)], &one));
    762:   }

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

    766:   PetscCallBLAS("BLAScopy",BLAScopy_(&n, work, &one, ev, &one));

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

    770:   for (j=0;j<n;j++) for (i=0;i<n;i++) z[i+j*ldz] = work[i+(j+1)*n];
    771:   PetscFunctionReturn(PETSC_SUCCESS);
    772: }