Date: 2016-03-29T14:56:49+00:00

Actual source code: dsbtdc.c
slepc-3.6.3 2016-03-29
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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: PetscErrorCode BDC_dsbtdc_(const char *jobz,const char *jobacc,PetscBLASInt n,
     28:         PetscBLASInt nblks,PetscBLASInt *ksizes,PetscReal *d,PetscBLASInt l1d,
     29:         PetscBLASInt l2d,PetscReal *e,PetscBLASInt l1e,PetscBLASInt l2e,PetscReal tol,
     30:         PetscReal tau1,PetscReal tau2,PetscReal *ev,PetscReal *z,PetscBLASInt ldz,
     31:         PetscReal *work,PetscBLASInt lwork,PetscBLASInt *iwork,PetscBLASInt liwork,
     32:         PetscReal *mingap,PetscBLASInt *mingapi,PetscBLASInt *info,
     33:         PetscBLASInt jobz_len,PetscBLASInt jobacc_len)
     34: {
     35: /*  -- Routine written in LAPACK Version 3.0 style -- */
     36: /* *************************************************** */
     37: /*     Written by */
     38: /*     Michael Moldaschl and Wilfried Gansterer */
     39: /*     University of Vienna */
     40: /*     last modification: March 28, 2014 */

     42: /*     Small adaptations of original code written by */
     43: /*     Wilfried Gansterer and Bob Ward, */
     44: /*     Department of Computer Science, University of Tennessee */
     45: /*     see http://dx.doi.org/10.1137/S1064827501399432 */
     46: /* *************************************************** */

     48: /*  Purpose */
     49: /*  ======= */

     51: /*  DSBTDC computes approximations to all eigenvalues and eigenvectors */
     52: /*  of a symmetric block tridiagonal matrix using the divide and */
     53: /*  conquer method with lower rank approximations to the subdiagonal blocks. */

     55: /*  This code makes very mild assumptions about floating point */
     56: /*  arithmetic. It will work on machines with a guard digit in */
     57: /*  add/subtract, or on those binary machines without guard digits */
     58: /*  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. */
     59: /*  It could conceivably fail on hexadecimal or decimal machines */
     60: /*  without guard digits, but we know of none.  See DLAED3M for details. */

     62: /*  Arguments */
     63: /*  ========= */

     65: /*  JOBZ    (input) CHARACTER*1 */
     66: /*          = 'N':  Compute eigenvalues only (not implemented); */
     67: /*          = 'D':  Compute eigenvalues and eigenvectors. Eigenvectors */
     68: /*                  are accumulated in the divide-and-conquer process. */

     70: /*  JOBACC  (input) CHARACTER*1 */
     71: /*          = 'A' ("automatic"): The accuracy parameters TAU1 and TAU2 */
     72: /*                               are determined automatically from the */
     73: /*                               parameter TOL according to the analytical */
     74: /*                               bounds. In that case the input values of */
     75: /*                               TAU1 and TAU2 are irrelevant (ignored). */
     76: /*          = 'M' ("manual"): The input values of the accuracy parameters */
     77: /*                            TAU1 and TAU2 are used. In that case the input */
     78: /*                            value of the parameter TOL is irrelevant */
     79: /*                            (ignored). */

     81: /*  N       (input) INTEGER */
     82: /*          The dimension of the symmetric block tridiagonal matrix. */
     83: /*          N >= 1. */

     85: /*  NBLKS   (input) INTEGER, 1 <= NBLKS <= N */
     86: /*          The number of diagonal blocks in the matrix. */

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

     93: /*  D       (input) DOUBLE PRECISION array, dimension (L1D,L2D,NBLKS) */
     94: /*          The lower triangular elements of the symmetric diagonal */
     95: /*          blocks of the block tridiagonal matrix. The elements of the top */
     96: /*          left diagonal block, which is of dimension KSIZES(1), have to */
     97: /*          be placed in D(*,*,1); the elements of the next diagonal */
     98: /*          block, which is of dimension KSIZES(2), have to be placed in */
     99: /*          D(*,*,2); etc. */

    101: /*  L1D     (input) INTEGER */
    102: /*          The leading dimension of the array D.  L1D >= max(3,KMAX), */
    103: /*          where KMAX is the dimension of the largest diagonal block, */
    104: /*          i.e.,  KMAX = max_I ( KSIZES(I) ). */

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

    110: /*  E       (input) DOUBLE PRECISION array, dimension (L1E,L2E,NBLKS-1) */
    111: /*          The elements of the subdiagonal blocks of the */
    112: /*          block tridiagonal matrix. The elements of the top left */
    113: /*          subdiagonal block, which is KSIZES(2) x KSIZES(1), have to be */
    114: /*          placed in E(*,*,1); the elements of the next subdiagonal block, */
    115: /*          which is KSIZES(3) x KSIZES(2), have to be placed in E(*,*,2); etc. */
    116: /*          During runtime, the original contents of E(*,*,K) is */
    117: /*          overwritten by the singular vectors and singular values of */
    118: /*          the lower rank representation. */

    120: /*  L1E     (input) INTEGER */
    121: /*          The leading dimension of the array E.  L1E >= max(3,2*KMAX+1), */
    122: /*          where KMAX is as stated in L1D above. The size of L1E enables */
    123: /*          the storage of ALL singular vectors and singular values for */
    124: /*          the corresponding off-diagonal block in E(*,*,K) and therefore */
    125: /*          there are no restrictions on the rank of the approximation */
    126: /*          (only the "natural" restriction */
    127: /*          RANK( K ) .LE. MIN( KSIZES( K ),KSIZES( K+1 ) )). */

    129: /*  L2E     (input) INTEGER */
    130: /*          The second dimension of the array E.  L2E >= max(3,2*KMAX+1), */
    131: /*          where KMAX is as stated in L1D above. The size of L2E enables */
    132: /*          the storage of ALL singular vectors and singular values for */
    133: /*          the corresponding off-diagonal block in E(*,*,K) and therefore */
    134: /*          there are no restrictions on the rank of the approximation */
    135: /*          (only the "natural" restriction */
    136: /*          RANK( K ) .LE. MIN( KSIZES( K ),KSIZES( K+1 ) )). */

    138: /*  TOL     (input) DOUBLE PRECISION, TOL.LE.TOLMAX */
    139: /*          User specified tolerance for the residuals of the computed */
    140: /*          eigenpairs. If ( JOBACC.EQ.'A' ) then it is used to determine */
    141: /*          TAU1 and TAU2; ignored otherwise. */
    142: /*          If ( TOL.LT.40*EPS .AND. JOBACC.EQ.'A' ) then TAU1 is set to machine */
    143: /*          epsilon and TAU2 is set to the standard deflation tolerance from */
    144: /*          LAPACK. */

    146: /*  TAU1    (input) DOUBLE PRECISION, TAU1.LE.TOLMAX/2 */
    147: /*          User specified tolerance for determining the rank of the */
    148: /*          lower rank approximations to the off-diagonal blocks. */
    149: /*          The rank for each off-diagonal block is determined such that */
    150: /*          the resulting absolute eigenvalue error is less than or equal */
    151: /*          to TAU1. */
    152: /*          If ( JOBACC.EQ.'A' ) then TAU1 is determined automatically from */
    153: /*             TOL and the input value is ignored. */
    154: /*          If ( JOBACC.EQ.'M' .AND. TAU1.LT.20*EPS ) then TAU1 is set to */
    155: /*             machine epsilon. */

    157: /*  TAU2    (input) DOUBLE PRECISION, TAU2.LE.TOLMAX/2 */
    158: /*          User specified deflation tolerance for the routine DIBTDC. */
    159: /*          If ( 1.0D-1.GT.TAU2.GT.20*EPS ) then TAU2 is used as */
    160: /*          the deflation tolerance in DSRTDF (EPS is the machine epsilon). */
    161: /*          If ( TAU2.LE.20*EPS ) then the standard deflation tolerance from */
    162: /*          LAPACK is used as the deflation tolerance in DSRTDF. */
    163: /*          If ( JOBACC.EQ.'A' ) then TAU2 is determined automatically from */
    164: /*             TOL and the input value is ignored. */
    165: /*          If ( JOBACC.EQ.'M' .AND. TAU2.LT.20*EPS ) then TAU2 is set to */
    166: /*             the standard deflation tolerance from LAPACK. */

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

    172: /*  Z       (output) DOUBLE PRECISION array, dimension (LDZ,N) */
    173: /*          If ( JOBZ.EQ.'D' .AND. INFO = 0 ) */
    174: /*          then Z contains the orthonormal eigenvectors of the symmetric */
    175: /*          block tridiagonal matrix computed by the routine DIBTDC */
    176: /*          (accumulated in the divide-and-conquer process). */
    177: /*          If ( -199 < INFO < -99 ) then Z contains the orthonormal */
    178: /*          eigenvectors of the symmetric block tridiagonal matrix, */
    179: /*          computed without divide-and-conquer (quick returns). */
    180: /*          Otherwise not referenced. */

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

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

    187: /*  LWORK   (input) INTEGER */
    188: /*          The dimension of the array WORK. */
    189: /*          If NBLKS.EQ.1, then LWORK has to be at least 2N^2+6N+1 */
    190: /*          (for the call of DSYEVD). */
    191: /*          If NBLKS.GE.2 and ( JOBZ.EQ.'D' ) then the absolute minimum */
    192: /*             required for DIBTDC is ( N**2 + 3*N ). This will not always */
    193: /*             suffice, though, the routine will return a corresponding */
    194: /*             error code and report how much work space was missing (see */
    195: /*             INFO). */
    196: /*          In order to guarantee correct results in all cases where */
    197: /*          NBLKS.GE.2, LWORK must be at least ( 2*N**2 + 3*N ). */

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

    201: /*  LIWORK  (input) INTEGER */
    202: /*          The dimension of the array IWORK. */
    203: /*          LIWORK must be at least ( 5*N + 5*NBLKS - 1 ) (for DIBTDC) */
    204: /*          Note that this should also suffice for the call of DSYEVD on a */
    205: /*          diagonal block which requires ( 5*KMAX + 3 ). */

    208: /*  MINGAP  (output) DOUBLE PRECISION */
    209: /*          The minimum "gap" between the approximate eigenvalues */
    210: /*          computed, i.e., MIN( ABS( EV( I+1 )-EV( I ) ) for I=1,2,..., N-1 */
    211: /*          IF ( MINGAP.LE.TOL/10 ) THEN a warning flag is returned in INFO, */
    212: /*          because the computed eigenvectors may be unreliable individually */
    213: /*          (only the subspaces spanned are approximated reliably). */

    215: /*  MINGAPI (output) INTEGER */
    216: /*          Index I where the minimum gap in the spectrum occurred. */

    218: /*  INFO    (output) INTEGER */
    219: /*          = 0:  successful exit, no special cases occurred. */
    220: /*          < -200: not enough workspace. Space for ABS(INFO + 200) */
    221: /*                numbers is required in addition to the workspace provided, */
    222: /*                otherwise some of the computed eigenvectors will be incorrect. */
    223: /*          < -99, > -199: successful exit, but quick returns. */
    224: /*                if INFO = -100, successful exit, but the input matrix */
    225: /*                                was the zero matrix and no */
    226: /*                                divide-and-conquer was performed */
    227: /*                if INFO = -101, successful exit, but N was 1 and no */
    228: /*                                divide-and-conquer was performed */
    229: /*                if INFO = -102, successful exit, but only a single */
    230: /*                                dense block. Standard dense solver */
    231: /*                                was called, no divide-and-conquer was */
    232: /*                                performed */
    233: /*                if INFO = -103, successful exit, but warning that */
    234: /*                                MINGAP.LE.TOL/10 and therefore the */
    235: /*                                eigenvectors corresponding to close */
    236: /*                                approximate eigenvalues may individually */
    237: /*                                be unreliable (although taken together they */
    238: /*                                do approximate the corresponding subspace to */
    239: /*                                the desired accuracy) */
    240: /*          = -99: error in the preprocessing in DIBTDC (when determining */
    241: /*                 the merging order). */
    242: /*          < 0, > -99: illegal arguments. */
    243: /*                if INFO = -i, the i-th argument had an illegal value. */
    244: /*          > 0:  The algorithm failed to compute an eigenvalue while */
    245: /*                working on the submatrix lying in rows and columns */
    246: /*                INFO/(N+1) through mod(INFO,N+1). */

    248: /*  Further Details */
    249: /*  =============== */

    251: /*  Small modifications of code written by */
    252: /*     Wilfried Gansterer and Bob Ward, */
    253: /*     Department of Computer Science, University of Tennessee */
    254: /*     see http://dx.doi.org/10.1137/S1064827501399432 */

    256: /*  Based on the design of the LAPACK code sstedc.f written by Jeff */
    257: /*  Rutter, Computer Science Division, University of California at */
    258: /*  Berkeley, and modified by Francoise Tisseur, University of Tennessee. */

    260: /*  ===================================================================== */

    262: /*     .. Parameters .. */

    264: #define TOLMAX 0.1

    266: /*        TOLMAX       .... upper bound for tolerances TOL, TAU1, TAU2 */
    267: /*                          NOTE: in the routine DIBTDC, the value */
    268: /*                                1.D-1 is hardcoded for TOLMAX ! */

    270: #if defined(SLEPC_MISSING_LAPACK_SYEVD) || defined(SLEPC_MISSING_LAPACK_GESVD) || defined(SLEPC_MISSING_LAPACK_LASET) || defined(SLEPC_MISSING_LAPACK_LASCL)
    272:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"SYEVD/GESVD/LASET/LASCL - Lapack routine is unavailable");
    273: #else
    274:   PetscBLASInt   i, j, k, i1, iwspc, lwmin, start;
    275:   PetscBLASInt   ii, ip, jp, nk, rk, np, iu, rp1, ldu;
    276:   PetscBLASInt   ksk, ivt, iend, kchk, kmax, one=1, zero=0;
    277:   PetscBLASInt   ldvt, ksum, kskp1, spneed, nrblks, liwmin, isvals;
    278:   PetscReal      p, d2, eps, dmax, emax, done = 1.0, dzero = 0.0;
    279:   PetscReal      dnrm, tiny, anorm, exdnrm=0, dropsv, absdiff;

    283:   /* Determine machine epsilon. */
    284:   eps = LAPACKlamch_("Epsilon");

    286:   *info = 0;

    288:   if (*(unsigned char *)jobz != 'N' && *(unsigned char *)jobz != 'D') {
    289:     *info = -1;
    290:   } else if (*(unsigned char *)jobacc != 'A' && *(unsigned char *)jobacc != 'M') {
    291:     *info = -2;
    292:   } else if (n < 1) {
    293:     *info = -3;
    294:   } else if (nblks < 1 || nblks > n) {
    295:     *info = -4;
    296:   }
    297:   if (*info == 0) {
    298:     ksum = 0;
    299:     kmax = 0;
    300:     kchk = 0;
    301:     for (k = 0; k < nblks; ++k) {
    302:       ksk = ksizes[k];
    303:       ksum += ksk;
    304:       if (ksk > kmax) kmax = ksk;
    305:       if (ksk < 1) kchk = 1;
    306:     }
    307:     if (nblks == 1) lwmin = 2*n*n + n*6 + 1;
    308:     else lwmin = n*n + n*3;
    309:     liwmin = n * 5 + nblks * 5 - 4;
    310:     if (ksum != n || kchk == 1) {
    311:       *info = -5;
    312:     } else if (l1d < PetscMax(3,kmax)) {
    313:       *info = -7;
    314:     } else if (l2d < PetscMax(3,kmax)) {
    315:       *info = -8;
    316:     } else if (l1e < PetscMax(3,2*kmax+1)) {
    317:       *info = -10;
    318:     } else if (l2e < PetscMax(3,2*kmax+1)) {
    319:       *info = -11;
    320:     } else if (*(unsigned char *)jobacc == 'A' && tol > TOLMAX) {
    321:       *info = -12;
    322:     } else if (*(unsigned char *)jobacc == 'M' && tau1 > TOLMAX/2) {
    323:       *info = -13;
    324:     } else if (*(unsigned char *)jobacc == 'M' && tau2 > TOLMAX/2) {
    325:       *info = -14;
    326:     } else if (ldz < PetscMax(1,n)) {
    327:       *info = -17;
    328:     } else if (lwork < lwmin) {
    329:       *info = -19;
    330:     } else if (liwork < liwmin) {
    331:       *info = -21;
    332:     }
    333:   }

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

    337:   /* Quick return if possible */

    339:   if (n == 1) {
    340:     ev[0] = d[0];
    341:     z[0] = 1.;
    342:     *info = -101;
    343:     return(0);
    344:   }

    346:   /* If NBLKS is equal to 1, then solve the problem with standard */
    347:   /* dense solver (in this case KSIZES(1) = N). */

    349:   if (nblks == 1) {
    350:     PetscPrintf(PETSC_COMM_WORLD," dsbtdc: This branch still needs to be checked!\n");
    351:     for (i = 0; i < n; ++i) {
    352:       for (j = 0; j <= i; ++j) {
    353:         z[i + j*ldz] = d[i + j*l1d];
    354:       }
    355:     }
    356:     PetscStackCallBLAS("LAPACKsyevd",LAPACKsyevd_("V", "L", &n, z, &ldz, ev, work, &lwork, iwork, &liwork, info));
    357:     if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DSYEVD, info = %d",*info);
    358:     *info = -102;
    359:     return(0);
    360:   }

    362:   /* determine the accuracy parameters (if requested) */

    364:   if (*(unsigned char *)jobacc == 'A') {
    365:     tau1 = tol / 2;
    366:     if (tau1 < eps * 20) tau1 = eps;
    367:     tau2 = tol / 2;
    368:   }

    370:   /* Initialize Z as the identity matrix */

    372:   if (*(unsigned char *)jobz == 'D') {
    373:     PetscStackCallBLAS("LAPACKlaset",LAPACKlaset_("Full", &n, &n, &dzero, &done, z, &ldz));
    374:   }

    376:   /* Determine the off-diagonal ranks, form and store the lower rank */
    377:   /* approximations based on the tolerance parameters, the */
    378:   /* RANK( K ) largest singular values and the associated singular */
    379:   /* vectors of each subdiagonal block. Also find the maximum norm of */
    380:   /* the subdiagonal blocks (in EMAX). */
    381:
    382:   /* Compute SVDs of the subdiagonal blocks.... */

    384:   /* EMAX .... maximum norm of the off-diagonal blocks */

    386:   emax = 0.;
    387:   for (k = 0; k < nblks-1; ++k) {
    388:     ksk = ksizes[k];
    389:     kskp1 = ksizes[k+1];
    390:     isvals = 0;

    392:     /* Note that min(KSKP1,KSK).LE.N/2 (equality possible for */
    393:     /* NBLKS=2), and therefore storing the singular values requires */
    394:     /* at most N/2 entries of the *        array WORK. */

    396:     iu = isvals + n / 2;
    397:     ivt = isvals + n / 2;

    399:     /* Call of DGESVD: The space for U is not referenced, since */
    400:     /* JOBU='O' and therefore this portion of the array WORK */
    401:     /* is not referenced for U. */

    403:     ldu = kskp1;
    404:     ldvt = PetscMin(kskp1,ksk);
    405:     iwspc = ivt + n * n / 2;

    407:     /* Note that the minimum workspace required for this call */
    408:     /* of DGESVD is: N/2 for storing the singular values + N**2/2 for */
    409:     /* storing V^T + 5*N/2 workspace =  N**2/2 + 3*N. */

    411:     i1 = lwork - iwspc;
    412:     PetscStackCallBLAS("LAPACKgesvd",LAPACKgesvd_("O", "S", &kskp1, &ksk,
    413:             &e[k*l1e*l2e], &l1e, &work[isvals],
    414:             &work[iu], &ldu, &work[ivt], &ldvt, &work[iwspc], &i1, info));
    415:     if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DGESVD, info = %d",*info);

    417:     /* Note that after the return from DGESVD U is stored in */
    418:     /* E(*,*,K), and V^{\top} is stored in WORK( IVT, IVT+1, .... ) */

    420:     /* determine the ranks RANK() for the approximations */

    422:     rk = PetscMin(ksk,kskp1);
    423: L8:
    424:     dropsv = work[isvals - 1 + rk];

    426:     if (dropsv * 2. <= tau1) {

    428:       /* the error caused by dropping singular value RK is */
    429:       /* small enough, try to reduce the rank by one more */

    431:       --rk;
    432:       if (rk > 0) goto L8;
    433:       else iwork[k] = 0;
    434:     } else {

    436:       /* the error caused by dropping singular value RK is */
    437:       /* too large already, RK is the rank required to achieve the */
    438:       /* desired accuracy */

    440:       iwork[k] = rk;
    441:     }

    443: /* ************************************************************************** */

    445:     /* Store the first RANK( K ) terms of the SVD of the current */
    446:     /* off-diagonal block. */
    447:     /* NOTE that here it is required that L1E, L2E >= 2*KMAX+1 in order */
    448:     /* to have enough space for storing singular vectors and values up */
    449:     /* to the full SVD of an off-diagonal block !!!! */
    450:
    451:     /* u1-u_RANK(K) is already contained in E(:,1:RANK(K),K) (as a */
    452:     /* result of the call of DGESVD !), the sigma1-sigmaK are to be */
    453:     /* stored in E(1:RANK(K),RANK(K)+1,K),  and v1-v_RANK(K) are to be */
    454:     /* stored in E(:,RANK(K)+2:2*RANK(K)+1,K) */

    456:     rp1 = iwork[k];
    457:     for (j = 0; j < iwork[k]; ++j) {

    459:       /* store sigma_J in E( J,RANK( K )+1,K ) */

    461:       e[j + (rp1 + k*l2e)* l1e] = work[isvals + j];

    463:       /* update maximum norm of subdiagonal blocks */

    465:       if (e[j + (rp1 + k*l2e)*l1e] > emax) {
    466:         emax = e[j + (rp1 + k*l2e)*l1e];
    467:       }

    469:       /* store v_J in E( :,RANK( K )+1+J,K ) */
    470:       /* (note that WORK contains V^{\top} and therefore */
    471:       /* we need to read rowwise !) */

    473:       for (i = 1; i <= ksk; ++i) {
    474:         e[i-1 + (rp1+j+1 + k*l2e)*l1e] = work[ivt+j + (i-1)*ldvt];
    475:       }
    476:     }

    478:   }

    480:   /* Compute the maximum norm of diagonal blocks and store the norm */
    481:   /* of each diagonal block in E(RP1,RP1,K) (after the singular values); */
    482:   /* store the norm of the last diagonal block in EXDNRM. */
    483:
    484:   /* DMAX .... maximum one-norm of the diagonal blocks */

    486:   dmax = 0.;
    487:   for (k = 0; k < nblks; ++k) {
    488:     rp1 = iwork[k];

    490:     /* compute the one-norm of diagonal block K */

    492:     dnrm = LAPACKlansy_("1", "L", &ksizes[k], &d[k*l1d*l2d], &l1d, work);
    493:     if (k+1 == nblks) exdnrm = dnrm;
    494:     else e[rp1 + (rp1 + k*l2e)*l1e] = dnrm;
    495:     if (dnrm > dmax) dmax = dnrm;
    496:   }

    498:   /* Check for zero matrix. */

    500:   if (emax == 0. && dmax == 0.) {
    501:     for (i = 0; i < n; ++i) ev[i] = 0.;
    502:     *info = -100;
    503:     return(0);
    504:   }

    506: /* **************************************************************** */

    508:   /* ....Identify irreducible parts of the block tridiagonal matrix */
    509:   /* [while ( START <= NBLKS )].... */

    511:   start = 0;
    512:   np = 0;
    513: L10:
    514:   if (start < nblks) {

    516:     /* Let IEND be the number of the next subdiagonal block such that */
    517:     /* its RANK is 0 or IEND = NBLKS if no such subdiagonal exists. */
    518:     /* The matrix identified by the elements between the diagonal block START */
    519:     /* and the diagonal block IEND constitutes an independent (irreducible) */
    520:     /* sub-problem. */

    522:     iend = start;

    524: L20:
    525:     if (iend < nblks) {
    526:       rk = iwork[iend];

    528:       /* NOTE: if RANK( IEND ).EQ.0 then decoupling happens due to */
    529:       /*       reduced accuracy requirements ! (because in this case */
    530:       /*       we would not merge the corresponding two diagonal blocks) */
    531:
    532:       /* NOTE: seems like any combination may potentially happen: */
    533:       /*       (i) RANK = 0 but no decoupling due to small norm of */
    534:       /*           off-diagonal block (corresponding diagonal blocks */
    535:       /*           also have small norm) as well as */
    536:       /*       (ii) RANK > 0 but decoupling due to small norm of */
    537:       /*           off-diagonal block (corresponding diagonal blocks */
    538:       /*           have very large norm) */
    539:       /*       case (i) is ruled out by checking for RANK = 0 above */
    540:       /*       (we decide to decouple all the time when the rank */
    541:       /*       of an off-diagonal block is zero, independently of */
    542:       /*       the norms of the corresponding diagonal blocks. */

    544:       if (rk > 0) {

    546:         /* check for decoupling due to small norm of off-diagonal block */
    547:         /* (relative to the norms of the corresponding diagonal blocks) */

    549:         if (iend == nblks-2) {
    550:           d2 = PetscSqrtReal(exdnrm);
    551:         } else {
    552:           d2 = PetscSqrtReal(e[iwork[iend+1] + (iwork[iend+1] + (iend+1)*l2e)*l1e]);
    553:         }

    555:         /* this definition of TINY is analogous to the definition */
    556:         /* in the tridiagonal divide&conquer (dstedc) */

    558:         tiny = eps * PetscSqrtReal(e[iwork[iend] + (iwork[iend] + iend*l2e)*l1e])*d2;
    559:         if (e[(iwork[iend] + iend*l2e)*l1e] > tiny) {

    561:           /* no decoupling due to small norm of off-diagonal block */

    563:           ++iend;
    564:           goto L20;
    565:         }
    566:       }
    567:     }

    569:     /* ....(Sub) Problem determined: between diagonal blocks */
    570:     /*     START and IEND. Compute its size and solve it.... */

    572:     nrblks = iend - start + 1;
    573:     if (nrblks == 1) {

    575:       /* Isolated problem is a single diagonal block */

    577:       nk = ksizes[start];

    579:       /* copy this isolated block into Z */

    581:       for (i = 0; i < nk; ++i) {
    582:         ip = np + i + 1;
    583:         for (j = 0; j <= i; ++j) {
    584:           jp = np + j + 1;
    585:           z[ip + jp*ldz] = d[i + (j + start*l2d)*l1d];
    586:         }
    587:       }

    589:       /* check whether there is enough workspace */

    591:       spneed = 2*nk*nk + nk * 6 + 1;
    592:       if (spneed > lwork) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: not enough workspace for DSYEVD, info = %d",lwork - 200 - spneed);

    594:       PetscStackCallBLAS("LAPACKsyevd",LAPACKsyevd_("V", "L", &nk,
    595:                     &z[np + np*ldz], &ldz, &ev[np],
    596:                     work, &lwork, &iwork[nblks-1], &liwork, info));
    597:       if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DSYEVD, info = %d",*info);
    598:       start = iend + 1;
    599:       np += nk;

    601:       /* go to the next irreducible subproblem */

    603:       goto L10;
    604:     }

    606:     /* ....Isolated problem consists of more than one diagonal block. */
    607:     /*     Start the divide and conquer algorithm.... */
    608:
    609:     /* Scale: Divide by the maximum of all norms of diagonal blocks */
    610:     /*        and singular values of the subdiagonal blocks */
    611:
    612:     /* ....determine maximum of the norms of all diagonal and subdiagonal */
    613:     /*     blocks.... */

    615:     if (iend == nblks-1) anorm = exdnrm;
    616:     else anorm = e[iwork[iend] + (iwork[iend] + iend*l2e)*l1e];
    617:     for (k = start; k < iend; ++k) {
    618:       rp1 = iwork[k];

    620:       /* norm of diagonal block */
    621:       anorm = PetscMax(anorm,e[rp1 + (rp1 + k*l2e)*l1e]);

    623:       /* singular value of subdiagonal block */
    624:       anorm = PetscMax(anorm,e[(rp1 + k*l2e)*l1e]);
    625:     }

    627:     nk = 0;
    628:     for (k = start; k < iend+1; ++k) {
    629:       ksk = ksizes[k];
    630:       nk += ksk;

    632:       /* scale the diagonal block */
    633:       PetscStackCallBLAS("LAPACKlascl",LAPACKlascl_("L", &zero, &zero,
    634:                     &anorm, &done, &ksk, &ksk, &d[k*l2d*l1d], &l1d, info));
    635:       if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DLASCL, info = %d",*info);

    637:       /* scale the (approximated) off-diagonal block by dividing its */
    638:       /* singular values */

    640:       if (k != iend) {

    642:         /* the last subdiagonal block has index IEND-1 !!!! */
    643:         for (i = 0; i < iwork[k]; ++i) {
    644:           e[i + (iwork[k] + k*l2e)*l1e] /= anorm;
    645:         }
    646:       }
    647:     }

    649:     /* call the block-tridiagonal divide-and-conquer on the */
    650:     /* irreducible subproblem which has been identified */

    652:     BDC_dibtdc_(jobz, nk, nrblks, &ksizes[start], &d[start*l1d*l2d], l1d, l2d,
    653:                 &e[start*l2e*l1e], &iwork[start], l1e, l2e, tau2, &ev[np],
    654:                 &z[np + np*ldz], ldz, work, lwork, &iwork[nblks-1], liwork, info, 1);
    655:
    656:     if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DIBTDC, info = %d",*info);

    658: /* ************************************************************************** */

    660:     /* Scale back the computed eigenvalues. */

    662:     PetscStackCallBLAS("LAPACKlascl",LAPACKlascl_("G", &zero, &zero, &done,
    663:             &anorm, &nk, &one, &ev[np], &nk, info));
    664:     if (*info) SETERRQ1(PETSC_COMM_SELF,1,"dsbtdc: Error in DLASCL, info = %d",*info);

    666:     start = iend + 1;
    667:     np += nk;

    669:     /* Go to the next irreducible subproblem. */

    671:     goto L10;
    672:   }

    674:   /* ....If the problem split any number of times, then the eigenvalues */
    675:   /* will not be properly ordered. Here we permute the eigenvalues */
    676:   /* (and the associated eigenvectors) across the irreducible parts */
    677:   /* into ascending order.... */
    678:
    679:   /*  IF( NRBLKS.LT.NBLKS )THEN */
    680:
    681:   /*    Use Selection Sort to minimize swaps of eigenvectors */

    683:   for (ii = 1; ii < n; ++ii) {
    684:     i = ii;
    685:     k = i;
    686:     p = ev[i];
    687:     for (j = ii; j < n; ++j) {
    688:       if (ev[j] < p) {
    689:         k = j;
    690:         p = ev[j];
    691:       }
    692:     }
    693:     if (k != i) {
    694:       ev[k] = ev[i];
    695:       ev[i] = p;
    696:       PetscStackCallBLAS("BLASswap",BLASswap_(&n, &z[i*ldz], &one, &z[k*ldz], &one));
    697:     }
    698:   }

    700:   /* ...Compute MINGAP (minimum difference between neighboring eigenvalue */
    701:   /*    approximations).............................................. */

    703:   *mingap = ev[1] - ev[0];
    704:   if (*mingap < 0.) SETERRQ2(PETSC_COMM_SELF,1,"dsbtdc: Eigenvalue approximations are not ordered properly. Approximation %d is larger than approximation %d.",1,2);
    705:   *mingapi = 1;
    706:   for (i = 2; i < n; ++i) {
    707:     absdiff = ev[i] - ev[i-1];
    708:     if (absdiff < 0.) {
    709:       SETERRQ2(PETSC_COMM_SELF,1,"dsbtdc: Eigenvalue approximations are not ordered properly. Approximation %d is larger than approximation %d.",i,i+1);
    710:     } else if (absdiff < *mingap) {
    711:       *mingap = absdiff;
    712:       *mingapi = i;
    713:     }
    714:   }

    716:   /* check whether the minimum gap between eigenvalue approximations */
    717:   /* may indicate severe inaccuracies in the eigenvector approximations */

    719:   if (*mingap <= tol / 10) *info = -103;
    720:   return(0);
    721: #endif
    722: }