Date: 2016-03-29T14:56:44+00:00

Actual source code: dspriv.c
slepc-3.6.3 2016-03-29
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      1: /*
      2:    Private DS routines

      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>      /*I "slepcds.h" I*/
     25: #include <slepcblaslapack.h>

     29: PetscErrorCode DSAllocateMat_Private(DS ds,DSMatType m)
     30: {
     31:   size_t         sz;
     32:   PetscInt       n,d;
     33:   PetscBool      ispep;

     37:   PetscObjectTypeCompare((PetscObject)ds,DSPEP,&ispep);
     38:   if (ispep) {
     39:     DSPEPGetDegree(ds,&d);
     40:   }
     41:   if (ispep && (m==DS_MAT_A || m==DS_MAT_B || m==DS_MAT_W || m==DS_MAT_X)) n = d*ds->ld;
     42:   else n = ds->ld;
     43:   switch (m) {
     44:     case DS_MAT_T:
     45:       sz = 3*ds->ld*sizeof(PetscScalar);
     46:       break;
     47:     case DS_MAT_D:
     48:       sz = ds->ld*sizeof(PetscScalar);
     49:       break;
     50:     case DS_MAT_X:
     51:       sz = ds->ld*n*sizeof(PetscScalar);
     52:       break;
     53:     default:
     54:       sz = n*n*sizeof(PetscScalar);
     55:   }
     56:   if (ds->mat[m]) {
     57:     PetscFree(ds->mat[m]);
     58:   } else {
     59:     PetscLogObjectMemory((PetscObject)ds,sz);
     60:   }
     61:   PetscMalloc(sz,&ds->mat[m]);
     62:   PetscMemzero(ds->mat[m],sz);
     63:   return(0);
     64: }

     68: PetscErrorCode DSAllocateMatReal_Private(DS ds,DSMatType m)
     69: {
     70:   size_t         sz;

     74:   if (m==DS_MAT_T) sz = 3*ds->ld*sizeof(PetscReal);
     75:   else if (m==DS_MAT_D) sz = ds->ld*sizeof(PetscReal);
     76:   else sz = ds->ld*ds->ld*sizeof(PetscReal);
     77:   if (!ds->rmat[m]) {
     78:     PetscLogObjectMemory((PetscObject)ds,sz);
     79:     PetscMalloc(sz,&ds->rmat[m]);
     80:   }
     81:   PetscMemzero(ds->rmat[m],sz);
     82:   return(0);
     83: }

     87: PetscErrorCode DSAllocateWork_Private(DS ds,PetscInt s,PetscInt r,PetscInt i)
     88: {

     92:   if (s>ds->lwork) {
     93:     PetscFree(ds->work);
     94:     PetscMalloc1(s,&ds->work);
     95:     PetscLogObjectMemory((PetscObject)ds,(s-ds->lwork)*sizeof(PetscScalar));
     96:     ds->lwork = s;
     97:   }
     98:   if (r>ds->lrwork) {
     99:     PetscFree(ds->rwork);
    100:     PetscMalloc1(r,&ds->rwork);
    101:     PetscLogObjectMemory((PetscObject)ds,(r-ds->lrwork)*sizeof(PetscReal));
    102:     ds->lrwork = r;
    103:   }
    104:   if (i>ds->liwork) {
    105:     PetscFree(ds->iwork);
    106:     PetscMalloc1(i,&ds->iwork);
    107:     PetscLogObjectMemory((PetscObject)ds,(i-ds->liwork)*sizeof(PetscBLASInt));
    108:     ds->liwork = i;
    109:   }
    110:   return(0);
    111: }

    115: /*@C
    116:    DSViewMat - Prints one of the internal DS matrices.

    118:    Collective on DS

    120:    Input Parameters:
    121: +  ds     - the direct solver context
    122: .  viewer - visualization context
    123: -  m      - matrix to display

    125:    Note:
    126:    Works only for ascii viewers. Set the viewer in Matlab format if
    127:    want to paste into Matlab.

    129:    Level: developer

    131: .seealso: DSView()
    132: @*/
    133: PetscErrorCode DSViewMat(DS ds,PetscViewer viewer,DSMatType m)
    134: {
    135:   PetscErrorCode    ierr;
    136:   PetscInt          i,j,rows,cols;
    137:   PetscScalar       *v;
    138:   PetscViewerFormat format;
    139: #if defined(PETSC_USE_COMPLEX)
    140:   PetscBool         allreal = PETSC_TRUE;
    141: #endif

    144:   if (m>=DS_NUM_MAT) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_WRONG,"Invalid matrix");
    145:   if (!ds->mat[m]) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_WRONGSTATE,"Requested matrix was not created in this DS");
    146:   PetscViewerGetFormat(viewer,&format);
    147:   if (format == PETSC_VIEWER_ASCII_INFO || format == PETSC_VIEWER_ASCII_INFO_DETAIL) return(0);
    148:   PetscViewerASCIIUseTabs(viewer,PETSC_FALSE);
    149:   if (ds->state==DS_STATE_TRUNCATED && m>=DS_MAT_Q) rows = ds->t;
    150:   else rows = (m==DS_MAT_A && ds->extrarow)? ds->n+1: ds->n;
    151:   cols = (ds->m!=0)? ds->m: ds->n;
    152: #if defined(PETSC_USE_COMPLEX)
    153:   /* determine if matrix has all real values */
    154:   v = ds->mat[m];
    155:   for (i=0;i<rows;i++)
    156:     for (j=0;j<cols;j++)
    157:       if (PetscImaginaryPart(v[i+j*ds->ld])) { allreal = PETSC_FALSE; break; }
    158: #endif
    159:   if (format == PETSC_VIEWER_ASCII_MATLAB) {
    160:     PetscViewerASCIIPrintf(viewer,"%% Size = %D %D\n",rows,cols);
    161:     PetscViewerASCIIPrintf(viewer,"%s = [\n",DSMatName[m]);
    162:   } else {
    163:     PetscViewerASCIIPrintf(viewer,"Matrix %s =\n",DSMatName[m]);
    164:   }

    166:   for (i=0;i<rows;i++) {
    167:     v = ds->mat[m]+i;
    168:     for (j=0;j<cols;j++) {
    169: #if defined(PETSC_USE_COMPLEX)
    170:       if (allreal) {
    171:         PetscViewerASCIIPrintf(viewer,"%18.16e ",PetscRealPart(*v));
    172:       } else {
    173:         PetscViewerASCIIPrintf(viewer,"%18.16e%+18.16ei ",PetscRealPart(*v),PetscImaginaryPart(*v));
    174:       }
    175: #else
    176:       PetscViewerASCIIPrintf(viewer,"%18.16e ",*v);
    177: #endif
    178:       v += ds->ld;
    179:     }
    180:     PetscViewerASCIIPrintf(viewer,"\n");
    181:   }

    183:   if (format == PETSC_VIEWER_ASCII_MATLAB) {
    184:     PetscViewerASCIIPrintf(viewer,"];\n");
    185:   }
    186:   PetscViewerASCIIUseTabs(viewer,PETSC_TRUE);
    187:   PetscViewerFlush(viewer);
    188:   return(0);
    189: }

    193: PetscErrorCode DSSortEigenvalues_Private(DS ds,PetscScalar *wr,PetscScalar *wi,PetscInt *perm,PetscBool isghiep)
    194: {
    196:   PetscScalar    re,im,wi0;
    197:   PetscInt       n,i,j,result,tmp1,tmp2=0,d=1;

    200:   n = ds->t;   /* sort only first t pairs if truncated */
    201:   /* insertion sort */
    202:   i=ds->l+1;
    203: #if !defined(PETSC_USE_COMPLEX)
    204:   if (wi && wi[perm[i-1]]!=0.0) i++; /* initial value is complex */
    205: #else
    206:   if (isghiep && PetscImaginaryPart(wr[perm[i-1]])!=0.0) i++;
    207: #endif
    208:   for (;i<n;i+=d) {
    209:     re = wr[perm[i]];
    210:     if (wi) im = wi[perm[i]];
    211:     else im = 0.0;
    212:     tmp1 = perm[i];
    213: #if !defined(PETSC_USE_COMPLEX)
    214:     if (im!=0.0) { d = 2; tmp2 = perm[i+1]; }
    215:     else d = 1;
    216: #else
    217:     if (isghiep && PetscImaginaryPart(re)!=0.0) { d = 2; tmp2 = perm[i+1]; }
    218:     else d = 1;
    219: #endif
    220:     j = i-1;
    221:     if (wi) wi0 = wi[perm[j]];
    222:     else wi0 = 0.0;
    223:     SlepcSCCompare(ds->sc,re,im,wr[perm[j]],wi0,&result);
    224:     while (result<0 && j>=ds->l) {
    225:       perm[j+d] = perm[j];
    226:       j--;
    227: #if !defined(PETSC_USE_COMPLEX)
    228:       if (wi && wi[perm[j+1]]!=0)
    229: #else
    230:       if (isghiep && PetscImaginaryPart(wr[perm[j+1]])!=0)
    231: #endif
    232:         { perm[j+d] = perm[j]; j--; }

    234:       if (j>=ds->l) {
    235:         if (wi) wi0 = wi[perm[j]];
    236:         else wi0 = 0.0;
    237:         SlepcSCCompare(ds->sc,re,im,wr[perm[j]],wi0,&result);
    238:       }
    239:     }
    240:     perm[j+1] = tmp1;
    241:     if (d==2) perm[j+2] = tmp2;
    242:   }
    243:   return(0);
    244: }

    248: PetscErrorCode DSSortEigenvaluesReal_Private(DS ds,PetscReal *eig,PetscInt *perm)
    249: {
    251:   PetscScalar    re;
    252:   PetscInt       i,j,result,tmp,l,n;

    255:   n = ds->t;   /* sort only first t pairs if truncated */
    256:   l = ds->l;
    257:   /* insertion sort */
    258:   for (i=l+1;i<n;i++) {
    259:     re = eig[perm[i]];
    260:     j = i-1;
    261:     SlepcSCCompare(ds->sc,re,0.0,eig[perm[j]],0.0,&result);
    262:     while (result<0 && j>=l) {
    263:       tmp = perm[j]; perm[j] = perm[j+1]; perm[j+1] = tmp; j--;
    264:       if (j>=l) {
    265:         SlepcSCCompare(ds->sc,re,0.0,eig[perm[j]],0.0,&result);
    266:       }
    267:     }
    268:   }
    269:   return(0);
    270: }

    274: /*
    275:   DSCopyMatrix_Private - Copies the trailing block of a matrix (from
    276:   rows/columns l to n).
    277: */
    278: PetscErrorCode DSCopyMatrix_Private(DS ds,DSMatType dst,DSMatType src)
    279: {
    281:   PetscInt    j,m,off,ld;
    282:   PetscScalar *S,*D;

    285:   ld  = ds->ld;
    286:   m   = ds->n-ds->l;
    287:   off = ds->l+ds->l*ld;
    288:   S   = ds->mat[src];
    289:   D   = ds->mat[dst];
    290:   for (j=0;j<m;j++) {
    291:     PetscMemcpy(D+off+j*ld,S+off+j*ld,m*sizeof(PetscScalar));
    292:   }
    293:   return(0);
    294: }

    298: PetscErrorCode DSPermuteColumns_Private(DS ds,PetscInt l,PetscInt n,DSMatType mat,PetscInt *perm)
    299: {
    300:   PetscInt    i,j,k,p,ld;
    301:   PetscScalar *Q,rtmp;

    304:   ld = ds->ld;
    305:   Q  = ds->mat[mat];
    306:   for (i=l;i<n;i++) {
    307:     p = perm[i];
    308:     if (p != i) {
    309:       j = i + 1;
    310:       while (perm[j] != i) j++;
    311:       perm[j] = p; perm[i] = i;
    312:       /* swap columns i and j */
    313:       for (k=0;k<n;k++) {
    314:         rtmp = Q[k+p*ld]; Q[k+p*ld] = Q[k+i*ld]; Q[k+i*ld] = rtmp;
    315:       }
    316:     }
    317:   }
    318:   return(0);
    319: }

    323: PetscErrorCode DSPermuteRows_Private(DS ds,PetscInt l,PetscInt n,DSMatType mat,PetscInt *perm)
    324: {
    325:   PetscInt    i,j,m=ds->m,k,p,ld;
    326:   PetscScalar *Q,rtmp;

    329:   if (!m) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_WRONG,"m was not set");
    330:   ld = ds->ld;
    331:   Q  = ds->mat[mat];
    332:   for (i=l;i<n;i++) {
    333:     p = perm[i];
    334:     if (p != i) {
    335:       j = i + 1;
    336:       while (perm[j] != i) j++;
    337:       perm[j] = p; perm[i] = i;
    338:       /* swap rows i and j */
    339:       for (k=0;k<m;k++) {
    340:         rtmp = Q[p+k*ld]; Q[p+k*ld] = Q[i+k*ld]; Q[i+k*ld] = rtmp;
    341:       }
    342:     }
    343:   }
    344:   return(0);
    345: }

    349: PetscErrorCode DSPermuteBoth_Private(DS ds,PetscInt l,PetscInt n,DSMatType mat1,DSMatType mat2,PetscInt *perm)
    350: {
    351:   PetscInt    i,j,m=ds->m,k,p,ld;
    352:   PetscScalar *U,*VT,rtmp;

    355:   if (!m) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_WRONG,"m was not set");
    356:   ld = ds->ld;
    357:   U  = ds->mat[mat1];
    358:   VT = ds->mat[mat2];
    359:   for (i=l;i<n;i++) {
    360:     p = perm[i];
    361:     if (p != i) {
    362:       j = i + 1;
    363:       while (perm[j] != i) j++;
    364:       perm[j] = p; perm[i] = i;
    365:       /* swap columns i and j of U */
    366:       for (k=0;k<n;k++) {
    367:         rtmp = U[k+p*ld]; U[k+p*ld] = U[k+i*ld]; U[k+i*ld] = rtmp;
    368:       }
    369:       /* swap rows i and j of VT */
    370:       for (k=0;k<m;k++) {
    371:         rtmp = VT[p+k*ld]; VT[p+k*ld] = VT[i+k*ld]; VT[i+k*ld] = rtmp;
    372:       }
    373:     }
    374:   }
    375:   return(0);
    376: }

    380: /*@
    381:    DSSetIdentity - Copy the identity (a diagonal matrix with ones) on the
    382:    active part of a matrix.

    384:    Logically Collective on DS

    386:    Input Parameters:
    387: +  ds     - the direct solver context
    388: -  mat    - the matrix to modify

    390:    Level: intermediate
    391: @*/
    392: PetscErrorCode DSSetIdentity(DS ds,DSMatType mat)
    393: {
    395:   PetscScalar    *x;
    396:   PetscInt       i,ld,n,l;

    399:   DSGetDimensions(ds,&n,NULL,&l,NULL,NULL);
    400:   DSGetLeadingDimension(ds,&ld);
    401:   PetscLogEventBegin(DS_Other,ds,0,0,0);
    402:   DSGetArray(ds,mat,&x);
    403:   PetscMemzero(&x[ld*l],ld*(n-l)*sizeof(PetscScalar));
    404:   for (i=l;i<n;i++) {
    405:     x[ld*i+i] = 1.0;
    406:   }
    407:   DSRestoreArray(ds,mat,&x);
    408:   PetscLogEventEnd(DS_Other,ds,0,0,0);
    409:   return(0);
    410: }

    414: /*
    415:    DSOrthogonalize - Orthogonalize the columns of a matrix.

    417:    Input Parameters:
    418: +  ds   - the direct solver context
    419: .  mat  - a matrix
    420: -  cols - number of columns to orthogonalize (starting from the column zero)

    422:    Output Parameter:
    423: .  lindcols - number of linearly independent columns of the matrix (can be NULL)
    424: */
    425: PetscErrorCode DSOrthogonalize(DS ds,DSMatType mat,PetscInt cols,PetscInt *lindcols)
    426: {
    427: #if defined(PETSC_MISSING_LAPACK_GEQRF) || defined(SLEPC_MISSING_LAPACK_ORGQR)
    429:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"GEQRF/ORGQR - Lapack routine is unavailable");
    430: #else
    431:   PetscErrorCode  ierr;
    432:   PetscInt        n,l,ld;
    433:   PetscBLASInt    ld_,rA,cA,info,ltau,lw;
    434:   PetscScalar     *A,*tau,*w,saux;

    440:   DSGetDimensions(ds,&n,NULL,&l,NULL,NULL);
    441:   DSGetLeadingDimension(ds,&ld);
    442:   n = n - l;
    443:   if (cols > n) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_WRONG,"Invalid number of columns");
    444:   if (n == 0 || cols == 0) return(0);
    445:   DSGetArray(ds,mat,&A);
    446:   PetscBLASIntCast(PetscMin(cols,n),&ltau);
    447:   PetscBLASIntCast(ld,&ld_);
    448:   PetscBLASIntCast(n,&rA);
    449:   PetscBLASIntCast(cols,&cA);
    450:   lw = -1;
    451:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&rA,&cA,A,&ld_,NULL,&saux,&lw,&info));
    452:   if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error in Lapack xGEQRF %d",info);
    453:   lw = (PetscBLASInt)PetscRealPart(saux);
    454:   DSAllocateWork_Private(ds,lw+ltau,0,0);
    455:   tau = ds->work;
    456:   w = &tau[ltau];
    457:   PetscLogEventBegin(DS_Other,ds,0,0,0);
    458:   PetscFPTrapPush(PETSC_FP_TRAP_OFF);
    459:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&rA,&cA,&A[ld*l+l],&ld_,tau,w,&lw,&info));
    460:   if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error in Lapack xGEQRF %d",info);
    461:   PetscStackCallBLAS("LAPACKorgqr",LAPACKorgqr_(&rA,&ltau,&ltau,&A[ld*l+l],&ld_,tau,w,&lw,&info));
    462:   if (info) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"Error in Lapack xORGQR %d",info);
    463:   PetscFPTrapPop();
    464:   PetscLogEventEnd(DS_Other,ds,0,0,0);
    465:   DSRestoreArray(ds,mat,&A);
    466:   PetscObjectStateIncrease((PetscObject)ds);
    467:   if (lindcols) *lindcols = ltau;
    468:   return(0);
    469: #endif
    470: }

    474: /*
    475:   Compute C <- a*A*B + b*C, where
    476:     ldC, the leading dimension of C,
    477:     ldA, the leading dimension of A,
    478:     rA, cA, rows and columns of A,
    479:     At, if true use the transpose of A instead,
    480:     ldB, the leading dimension of B,
    481:     rB, cB, rows and columns of B,
    482:     Bt, if true use the transpose of B instead
    483: */
    484: static PetscErrorCode SlepcMatDenseMult(PetscScalar *C,PetscInt _ldC,PetscScalar b,PetscScalar a,const PetscScalar *A,PetscInt _ldA,PetscInt rA,PetscInt cA,PetscBool At,const PetscScalar *B,PetscInt _ldB,PetscInt rB,PetscInt cB,PetscBool Bt)
    485: {
    487:   PetscInt       tmp;
    488:   PetscBLASInt   m, n, k, ldA = _ldA, ldB = _ldB, ldC = _ldC;
    489:   const char     *N = "N", *T = "C", *qA = N, *qB = N;

    492:   if ((rA == 0) || (cB == 0)) return(0);

    497:   /* Transpose if needed */
    498:   if (At) tmp = rA, rA = cA, cA = tmp, qA = T;
    499:   if (Bt) tmp = rB, rB = cB, cB = tmp, qB = T;

    501:   /* Check size */
    502:   if (cA != rB) SETERRQ(PETSC_COMM_SELF,1, "Matrix dimensions do not match");

    504:   /* Do stub */
    505:   if ((rA == 1) && (cA == 1) && (cB == 1)) {
    506:     if (!At && !Bt) *C = *A * *B;
    507:     else if (At && !Bt) *C = PetscConj(*A) * *B;
    508:     else if (!At && Bt) *C = *A * PetscConj(*B);
    509:     else *C = PetscConj(*A) * PetscConj(*B);
    510:     m = n = k = 1;
    511:   } else {
    512:     m = rA; n = cB; k = cA;
    513:     PetscStackCallBLAS("BLASgemm",BLASgemm_(qA,qB,&m,&n,&k,&a,(PetscScalar*)A,&ldA,(PetscScalar*)B,&ldB,&b,C,&ldC));
    514:   }

    516:   PetscLogFlops(m*n*2*k);
    517:   return(0);
    518: }

    522: /*
    523:    DSPseudoOrthogonalize - Orthogonalize the columns of a matrix with Modified
    524:    Gram-Schmidt in an indefinite inner product space defined by a signature.

    526:    Input Parameters:
    527: +  ds   - the direct solver context
    528: .  mat  - the matrix
    529: .  cols - number of columns to orthogonalize (starting from the column zero)
    530: -  s    - the signature that defines the inner product

    532:    Output Parameter:
    533: +  lindcols - linear independent columns of the matrix (can be NULL)
    534: -  ns - the new norm of the vectors (can be NULL)
    535: */
    536: PetscErrorCode DSPseudoOrthogonalize(DS ds,DSMatType mat,PetscInt cols,PetscReal *s,PetscInt *lindcols,PetscReal *ns)
    537: {
    538:   PetscErrorCode  ierr;
    539:   PetscInt        i,j,k,l,n,ld;
    540:   PetscBLASInt    one=1,rA_;
    541:   PetscScalar     alpha,*A,*A_,*m,*h,nr0;
    542:   PetscReal       nr_o,nr,*ns_;

    550:   DSGetDimensions(ds,&n,NULL,&l,NULL,NULL);
    551:   DSGetLeadingDimension(ds,&ld);
    552:   n = n - l;
    553:   if (cols > n) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_WRONG,"Invalid number of columns");
    554:   if (n == 0 || cols == 0) return(0);
    555:   PetscBLASIntCast(n,&rA_);
    556:   DSGetArray(ds,mat,&A_);
    557:   A = &A_[ld*l+l];
    558:   DSAllocateWork_Private(ds,n+cols,ns?0:cols,0);
    559:   m = ds->work;
    560:   h = &m[n];
    561:   ns_ = ns ? ns : ds->rwork;
    562:   PetscLogEventBegin(DS_Other,ds,0,0,0);
    563:   for (i=0; i<cols; i++) {
    564:     /* m <- diag(s)*A[i] */
    565:     for (k=0; k<n; k++) m[k] = s[k]*A[k+i*ld];
    566:     /* nr_o <- mynorm(A[i]'*m), mynorm(x) = sign(x)*sqrt(|x|) */
    567:     SlepcMatDenseMult(&nr0,1,0.0,1.0,&A[ld*i],ld,n,1,PETSC_TRUE,m,n,n,1,PETSC_FALSE);
    568:     nr = nr_o = PetscSign(PetscRealPart(nr0))*PetscSqrtReal(PetscAbsScalar(nr0));
    569:     for (j=0; j<3 && i>0; j++) {
    570:       /* h <- A[0:i-1]'*m */
    571:       SlepcMatDenseMult(h,i,0.0,1.0,A,ld,n,i,PETSC_TRUE,m,n,n,1,PETSC_FALSE);
    572:       /* h <- diag(ns)*h */
    573:       for (k=0; k<i; k++) h[k] *= ns_[k];
    574:       /* A[i] <- A[i] - A[0:i-1]*h */
    575:       SlepcMatDenseMult(&A[ld*i],ld,1.0,-1.0,A,ld,n,i,PETSC_FALSE,h,i,i,1,PETSC_FALSE);
    576:       /* m <- diag(s)*A[i] */
    577:       for (k=0; k<n; k++) m[k] = s[k]*A[k+i*ld];
    578:       /* nr_o <- mynorm(A[i]'*m) */
    579:       SlepcMatDenseMult(&nr0,1,0.0,1.0,&A[ld*i],ld,n,1,PETSC_TRUE,m,n,n,1,PETSC_FALSE);
    580:       nr = PetscSign(PetscRealPart(nr0))*PetscSqrtReal(PetscAbsScalar(nr0));
    581:       if (PetscAbs(nr) < PETSC_MACHINE_EPSILON) SETERRQ(PETSC_COMM_SELF,1,"Linear dependency detected");
    582:       if (PetscAbs(nr) > 0.7*PetscAbs(nr_o)) break;
    583:       nr_o = nr;
    584:     }
    585:     ns_[i] = PetscSign(nr);
    586:     /* A[i] <- A[i]/|nr| */
    587:     alpha = 1.0/PetscAbs(nr);
    588:     PetscStackCallBLAS("BLASscal",BLASscal_(&rA_,&alpha,&A[i*ld],&one));
    589:   }
    590:   PetscLogEventEnd(DS_Other,ds,0,0,0);
    591:   DSRestoreArray(ds,mat,&A_);
    592:   PetscObjectStateIncrease((PetscObject)ds);
    593:   if (lindcols) *lindcols = cols;
    594:   return(0);
    595: }