Actual source code: dibtdc.c
slepc-3.10.1 2018-10-23
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2018, 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;

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

    111:   if (start < 1) {
    112:     *info = -1;
    113:   } else if (n < 3) {
    114:     *info = -2;
    115:   } else if (blkct < 3) {
    116:     *info = -3;
    117:   }
    118:   if (*info == 0) {
    119:     ksum = 0;
    120:     kchk = 0;
    121:     for (i = 0; i < blkct; ++i) {
    122:       ksk = bsizes[i];
    123:       ksum += ksk;
    124:       if (ksk < 1) kchk = 1;
    125:     }
    126:     if (ksum != n || kchk == 1) *info = -4;
    127:   }
    128:   if (*info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong argument %d in CUTLR",-(*info));

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

    132:   minrnk = n;
    133:   for (i = 0; i < blkct-1; ++i) {
    134:     if (ranks[i] < minrnk) minrnk = ranks[i];
    135:   }

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

    139:   nhalf = n / 2;
    140:   tmpsum = 0;
    141:   mindev = n;
    142:   for (i = 0; i < blkct; ++i) {
    143:     tmpsum += bsizes[i];
    144:     if (ranks[i] == minrnk) {

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

    148:       deviat = tmpsum - nhalf;
    149:       if (deviat<0) deviat = -deviat;

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

    153:       if (deviat < mindev) {
    154:         mindev = deviat;
    155:         *cut = start + i;
    156:         *lblks = i + 1;
    157:         *lsum = tmpsum;
    158:       }
    159:     }
    160:   }

    162:   if (*cut < start || *cut >= start + blkct - 1) {
    163:     *info = 6;
    164:   } else if (*lsum < 1 || *lsum >= n) {
    165:     *info = 7;
    166:   } else if (*lblks < 1 || *lblks >= blkct) {
    167:     *info = 8;
    168:   }
    169:   return(0);
    170: }

    172: PetscErrorCode BDC_dibtdc_(const char *jobz,PetscBLASInt n,PetscBLASInt nblks,
    173:         PetscBLASInt *ksizes,PetscReal *d,PetscBLASInt l1d,PetscBLASInt l2d,
    174:         PetscReal *e,PetscBLASInt *rank,PetscBLASInt l1e,PetscBLASInt l2e,
    175:         PetscReal tol,PetscReal *ev,PetscReal *z,PetscBLASInt ldz,PetscReal *work,
    176:         PetscBLASInt lwork,PetscBLASInt *iwork,PetscBLASInt liwork,
    177:         PetscBLASInt *info,PetscBLASInt jobz_len)
    178: {
    179: /*  -- Routine written in LAPACK Version 3.0 style -- */
    180: /* *************************************************** */
    181: /*     Written by */
    182: /*     Michael Moldaschl and Wilfried Gansterer */
    183: /*     University of Vienna */
    184: /*     last modification: March 16, 2014 */

    186: /*     Small adaptations of original code written by */
    187: /*     Wilfried Gansterer and Bob Ward, */
    188: /*     Department of Computer Science, University of Tennessee */
    189: /*     see https://doi.org/10.1137/S1064827501399432 */
    190: /* *************************************************** */

    192: /*  Purpose */
    193: /*  ======= */

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

    199: /*  Arguments */
    200: /*  ========= */

    202: /*  JOBZ    (input) CHARACTER*1 */
    203: /*          = 'N':  Compute eigenvalues only (not implemented); */
    204: /*          = 'D':  Compute eigenvalues and eigenvectors. */
    205: /*                  Eigenvectors are accumulated in the */
    206: /*                  divide-and-conquer process. */

    208: /*  N      (input) INTEGER */
    209: /*         The dimension of the symmetric irreducible block tridiagonal */
    210: /*         matrix.  N >= 2. */

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

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

    220: /*  D      (input) DOUBLE PRECISION array, dimension (L1D,L2D,NBLKS) */
    221: /*         The lower triangular elements of the symmetric diagonal */
    222: /*         blocks of the block tridiagonal matrix.  Elements of the top */
    223: /*         left diagonal block, which is of dimension KSIZES(1), are */
    224: /*         contained in D(*,*,1); the elements of the next diagonal */
    225: /*         block, which is of dimension KSIZES(2), are contained in */
    226: /*         D(*,*,2); etc. */

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

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

    236: /*  E      (input) DOUBLE PRECISION array, dimension (L1E,L2E,NBLKS-1) */
    237: /*         Contains the elements of the scalars (singular values) and */
    238: /*         vectors (singular vectors) defining the rank RANK subdiagonal */
    239: /*         blocks of the matrix. */
    240: /*         E(1:RANK(K),RANK(K)+1,K) holds the RANK(K) scalars, */
    241: /*         E(:,1:RANK(K),K) holds the RANK(K) column vectors, and */
    242: /*         E(:,RANK(K)+2:2*RANK(K)+1,K) holds the row vectors for the K-th */
    243: /*         subdiagonal block. */

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

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

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

    257: /*  TOL    (input) DOUBLE PRECISION, TOL <= 1.0D-1 */
    258: /*         User specified deflation tolerance for the routine DMERG2. */
    259: /*         If ( 1.0D-1 >= TOL >= 20*EPS ) then TOL is used as */
    260: /*         the deflation tolerance in DSRTDF. */
    261: /*         If ( TOL < 20*EPS ) then the standard deflation tolerance from */
    262: /*         LAPACK is used as the deflation tolerance in DSRTDF. */

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

    268: /*  Z      (input/output) DOUBLE PRECISION array, dimension (LDZ, N) */
    269: /*         On entry, Z will be the identity matrix. */
    270: /*         On exit, Z contains the eigenvectors of the block tridiagonal */
    271: /*         matrix. */

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

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

    278: /*  LWORK   (input) INTEGER */
    279: /*          The dimension of the array WORK. */
    280: /*          In order to guarantee correct results in all cases, */
    281: /*          LWORK must be at least ( 2*N**2 + 3*N ). In many cases, */
    282: /*          less workspace is required. The absolute minimum required is */
    283: /*          ( N**2 + 3*N ). */
    284: /*          If the workspace provided is not sufficient, the routine will */
    285: /*          return a corresponding error code and report how much workspace */
    286: /*          was missing (see INFO). */

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

    290: /*  LIWORK  (input) INTEGER */
    291: /*          The dimension of the array IWORK. */
    292: /*          LIWORK must be at least ( 5*N + 3 + 4*NBLKS - 4 ): */
    293: /*                 5*KMAX+3 for DSYEVD, 5*N for ????, */
    294: /*                 4*NBLKS-4 for the preprocessing (merging order) */
    295: /*          Summarizing, the minimum integer workspace needed is */
    296: /*          MAX( 5*N, 5*KMAX + 3 ) + 4*NBLKS - 4 */

    298: /*  INFO   (output) INTEGER */
    299: /*          = 0:  successful exit. */
    300: /*          < 0, > -99: illegal arguments. */
    301: /*                if INFO = -i, the i-th argument had an illegal value. */
    302: /*          = -99: error in the preprocessing (call of CUTLR). */
    303: /*          < -200: not enough workspace. Space for ABS(INFO + 200) */
    304: /*                numbers is required in addition to the workspace provided, */
    305: /*                otherwise some eigenvectors will be incorrect. */
    306: /*          > 0:  The algorithm failed to compute an eigenvalue while */
    307: /*                working on the submatrix lying in rows and columns */
    308: /*                INFO/(N+1) through mod(INFO,N+1). */

    310: /*  Further Details */
    311: /*  =============== */

    313: /*  Based on code written by */
    314: /*     Wilfried Gansterer and Bob Ward, */
    315: /*     Department of Computer Science, University of Tennessee */

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

    319: /*  ===================================================================== */

    321: #if defined(SLEPC_MISSING_LAPACK_LACPY) || defined(PETSC_MISSING_LAPACK_SYEV)
    323:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LACPY/SYEV - Lapack routine is unavailable");
    324: #else
    325:   PetscBLASInt   i, j, k, np, rp1, ksk, one=1;
    326:   PetscBLASInt   cut, mat1, kchk, kbrk, blks, kmax, icut, size, ksum, lsum;
    327:   PetscBLASInt   lblks, rblks, isize, lwmin, ilsum;
    328:   PetscBLASInt   start, vstrt, istck1, istck2, istck3, merged;
    329:   PetscBLASInt   liwmin, matsiz, startp, istrtp;
    330:   PetscReal      rho, done=1.0, dmone=-1.0;

    334:   *info = 0;

    336:   if (*(unsigned char *)jobz != 'N' && *(unsigned char *)jobz != 'D') {
    337:     *info = -1;
    338:   } else if (n < 2) {
    339:     *info = -2;
    340:   } else if (nblks < 2 || nblks > n) {
    341:     *info = -3;
    342:   }
    343:   if (*info == 0) {
    344:     ksum = 0;
    345:     kmax = 0;
    346:     kchk = 0;
    347:     for (k = 0; k < nblks; ++k) {
    348:       ksk = ksizes[k];
    349:       ksum += ksk;
    350:       if (ksk > kmax) kmax = ksk;
    351:       if (ksk < 1) kchk = 1;
    352:     }
    353:     lwmin = n*n + n * 3;
    354:     liwmin = PetscMax(n * 5,kmax * 5 + 3) + 4*nblks - 4;
    355:     if (ksum != n || kchk == 1) {
    356:       *info = -4;
    357:     } else if (l1d < PetscMax(3,kmax)) {
    358:       *info = -6;
    359:     } else if (l2d < PetscMax(3,kmax)) {
    360:       *info = -7;
    361:     } else if (l1e < PetscMax(3,2*kmax + 1)) {
    362:       *info = -10;
    363:     } else if (l2e < PetscMax(3,2*kmax + 1)) {
    364:       *info = -11;
    365:     } else if (tol > .1) {
    366:       *info = -12;
    367:     } else if (ldz < PetscMax(1,n)) {
    368:       *info = -15;
    369:     } else if (lwork < lwmin) {
    370:       *info = -17;
    371:     } else if (liwork < liwmin) {
    372:       *info = -19;
    373:     }
    374:   }
    375:   if (*info == 0) {
    376:     for (k = 0; k < nblks-1; ++k) {
    377:       if (rank[k] > PetscMin(ksizes[k],ksizes[k+1]) || rank[k] < 1) *info = -9;
    378:     }
    379:   }

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

    383: /* **************************************************************************** */

    385:   /* ...Preprocessing..................................................... */
    386:   /*    Determine the optimal order for merging the subblocks and how much */
    387:   /*    workspace will be needed for the merging (determined by the last */
    388:   /*    merge). Cutpoints for the merging operations are determined and stored */
    389:   /*    in reverse chronological order (starting with the final merging */
    390:   /*    operation). */

    392:   /*    integer workspace requirements for the preprocessing: */
    393:   /*         4*(NBLKS-1) for merging history */
    394:   /*         at most 3*(NBLKS-1) for stack */

    396:   start = 1;
    397:   size = n;
    398:   blks = nblks;
    399:   merged = 0;
    400:   k = 0;

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

    406:   istck1 = 0;
    407:   istck2 = istck1 + nblks;
    408:   istck3 = istck2 + nblks;

    410:   /* integer workspace used for storing the order of merges starts AFTER */
    411:   /* the integer workspace 5*N+3 which is needed later on in the code */
    412:   /* (5*KMAX+3 for DSYEVD, 4*N in DMERG2) */

    414:   istrtp = n * 5 + 4;
    415:   icut = istrtp + nblks - 1;
    416:   isize = icut + nblks - 1;
    417:   ilsum = isize + nblks - 1;

    419: L200:

    421:   if (nblks >= 3) {

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

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

    431:   } else {
    432:     cut = 1;
    433:     lsum = ksizes[0];
    434:     lblks = 1;
    435:   }

    437:   ++merged;
    438:   startp = 0;
    439:   for (i = 0; i < start-1; ++i) startp += ksizes[i];
    440:   iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    441:   iwork[icut + (nblks - 1) - merged-1] = cut;
    442:   iwork[isize + (nblks - 1) - merged-1] = size;
    443:   iwork[ilsum + (nblks - 1) - merged-1] = lsum;

    445:   if (lblks == 2) {

    447:     /* one merge in left branch, left branch done */
    448:     ++merged;
    449:     iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    450:     iwork[icut + (nblks - 1) - merged-1] = start;
    451:     iwork[isize + (nblks - 1) - merged-1] = lsum;
    452:     iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start-1];
    453:   }

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

    457:     /* left branch done, continue on the right side */
    458:     start += lblks;
    459:     size -= lsum;
    460:     blks -= lblks;

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

    464:     if (blks == 2) {

    466:       /* one merge in right branch, right branch done */
    467:       ++merged;
    468:       startp += lsum;
    469:       iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    470:       iwork[icut + (nblks - 1) - merged-1] = start;
    471:       iwork[isize + (nblks - 1) - merged-1] = size;
    472:       iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start-1];
    473:     }

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

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

    479:       if (k >= 1) {

    481:         /* something left on the stack */
    482:         start = iwork[istck1 + k-1];
    483:         size = iwork[istck2 + k-1];
    484:         blks = iwork[istck3 + k-1];
    485:         --k;
    486:         goto L200;
    487:       } else {

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

    492:         /* exit preprocessing */

    494:       }
    495:     } else {

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

    499:       goto L200;
    500:     }
    501:   } else {

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

    506:     rblks = blks - lblks;
    507:     if (rblks >= 3) {
    508:       ++k;
    509:       iwork[istck1 + k-1] = cut + 1;
    510:       iwork[istck2 + k-1] = size - lsum;
    511:       iwork[istck3 + k-1] = rblks;
    512:     } else if (rblks == 2) {

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

    517:       ++merged;
    518:       startp += lsum;
    519:       iwork[istrtp + (nblks - 1) - merged-1] = startp + 1;
    520:       iwork[icut + (nblks - 1) - merged-1] = start + lblks;
    521:       iwork[isize + (nblks - 1) - merged-1] = size - lsum;
    522:       iwork[ilsum + (nblks - 1) - merged-1] = ksizes[start + lblks-1];
    523:     }
    524:     if (rblks <= 0) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: ERROR in preprocessing - rblks = %d",rblks);

    526:     /* continue on the left side */

    528:     size = lsum;
    529:     blks = lblks;
    530:     goto L200;
    531:   }

    533:   /*  SIZE = IWORK( ISIZE+NBLKS-2 ) */
    534:   /*  MAT1 = IWORK( ILSUM+NBLKS-2 ) */

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

    542:   /*  IF( MAT1.LE.N/2 ) THEN */
    543:   /*     WSPREQ = 3*N + 3/2*( SIZE-MAT1 )**2 + N*N/2 + MAT1*MAT1 */
    544:   /*  ELSE */
    545:   /*     WSPREQ = 3*N + 3/2*MAT1*MAT1 + N*N/2 + ( SIZE-MAT1 )**2 */
    546:   /*  END IF */

    548:   /*  IF( LWORK-WSPREQ.LT.0 )THEN */
    549:   /*          not enough work space provided */
    550:   /*     INFO = -200 - ( WSPREQ-LWORK ) */
    551:   /*     RETURN */
    552:   /*  END IF */
    553:   /*  However, this is not really useful, since the actual check whether */
    554:   /*  enough workspace is provided happens in DMERG2.f ! */

    556: /* ************************************************************************* */

    558:   /* ...Solve subproblems................................... */

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

    564:   /* first block: ...................................... */

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

    568:   ksk = ksizes[0];
    569:   rp1 = rank[0];

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

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

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

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

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

    584:   for (i = 0; i < rank[0]; ++i) {
    585:     PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(rp1 + i+1)*l1e], &one, &d[i*l1d], &one));
    586:     PetscStackCallBLAS("BLASscal",BLASscal_(&ksk, &e[i + rp1*l1e], &d[i*l1d], &one));
    587:   }

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

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

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

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

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

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

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

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

    610:     /* intermediate blocks: .............................. */

    612:     np = ksk;

    614:     /* remaining number of blocks */

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

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

    622:       ksk = ksizes[k];
    623:       rp1 = rank[k];

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

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

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

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

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

    638:       for (i = 0; i < rank[k-1]; ++i) {
    639:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(i+(k-1)*l2e)*l1e], &one, &d[(i+k*l2d)*l1d], &one));
    640:         PetscStackCallBLAS("BLASscal",BLASscal_(&ksk, &e[i+(rank[k-1]+(k-1)*l2e)*l1e], &d[(i+k*l2d)*l1d], &one));
    641:       }

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

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

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

    654:       for (i = 0; i < rank[k]; ++i) {
    655:         PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(rp1+i+1 + k*l2e)*l1e], &one, &d[(i + k*l2d)*l1d], &one));
    656:         PetscStackCallBLAS("BLASscal",BLASscal_(&ksk, &e[i + (rp1 + k*l2e)*l1e], &d[(i + k*l2d)*l1d], &one));
    657:       }

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

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

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

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

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

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

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

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

    682:       /* update NP */
    683:       np += ksk;
    684:     }
    685:   }

    687:   /* last block: ....................................... */

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

    692:   ksk = ksizes[nblks-1];

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

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

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

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

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

    707:   for (i = 0; i < rank[nblks-2]; ++i) {
    708:     PetscStackCallBLAS("BLAScopy",BLAScopy_(&ksk, &e[(i + (nblks-2)*l2e)*l1e],
    709:               &one, &d[(i + (nblks-1)*l2d)*l1d], &one));
    710:     PetscStackCallBLAS("BLASscal",BLASscal_(&ksk,
    711:               &e[i + (rank[nblks-2] + (nblks-2)*l2e)*l1e],
    712:               &d[(i + (nblks-1)*l2d)*l1d], &one));
    713:   }

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

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

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

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

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

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

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

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

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

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

    742:   vstrt = 0;
    743:   for (i = 0; i < nblks-1; ++i) {

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

    747:     matsiz = iwork[isize + i - 1];
    748:     np = iwork[istrtp + i - 1];
    749:     kbrk = iwork[icut + i - 1];
    750:     mat1 = iwork[ilsum + i - 1];
    751:     vstrt += np;

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

    755:       /* NOTE: The parameter RHO in DMERG2 is modified in DSRTDF */
    756:       /*       (multiplied by 2) ! In order not to change the */
    757:       /*       singular value stored in E( :, RANK( KBRK )+1, KBRK ), */
    758:       /*       we do not pass on this variable as an argument to DMERG2, */
    759:       /*       but we assign a separate variable RHO here which is passed */
    760:       /*       on to DMERG2. */
    761:       /*       Alternative solution in F90: */
    762:       /*       pass E( :,RANK( KBRK )+1,KBRK ) to an INTENT( IN ) parameter */
    763:       /*       in DMERG2. */

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

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

    769:       BDC_dmerg2_(jobz, j+1, matsiz, &ev[np-1], &z[np-1+(np-1)*ldz],
    770:                     ldz, &iwork[np-1], &rho, &e[(j + (kbrk-1)*l2e)*l1e],
    771:                     ksizes[kbrk], &e[(rank[kbrk-1]+j+1 + (kbrk-1)*l2e)*l1e],
    772:                     ksizes[kbrk-1], mat1, work, lwork, &iwork[n], tol, info, 1);
    773:
    774:       if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dibtdc: Error in dmerg2, info = %d",*info);
    775:     }

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

    781:   }

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

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

    790:   for (i = 0; i < n; ++i) {
    791:     j = iwork[i];
    792:     work[i] = ev[j-1];
    793:     PetscStackCallBLAS("BLAScopy",BLAScopy_(&n, &z[(j-1)*ldz], &one, &work[n*(i+1)], &one));
    794:   }

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

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

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

    802:   PetscStackCallBLAS("LAPACKlacpy",LAPACKlacpy_("A", &n, &n, &work[n], &n, z, &ldz));
    803:   return(0);
    804: #endif
    805: }