Date: 2024-10-28T09:14:38+00:00

Actual source code: dspriv.c
slepc-3.22.1 2024-10-28
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      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:    Private DS routines
     12: */

     14: #include <slepc/private/dsimpl.h>
     15: #include <slepcblaslapack.h>

     17: PetscErrorCode DSAllocateMat_Private(DS ds,DSMatType m)
     18: {
     19:   PetscInt       n,d,rows=0,cols;
     20:   PetscBool      ispep,isnep;

     22:   PetscFunctionBegin;
     23:   n = ds->ld;
     24:   PetscCall(PetscObjectTypeCompare((PetscObject)ds,DSPEP,&ispep));
     25:   if (ispep) {
     26:     if (m==DS_MAT_A || m==DS_MAT_B || m==DS_MAT_W || m==DS_MAT_U || m==DS_MAT_X || m==DS_MAT_Y) {
     27:       PetscCall(DSPEPGetDegree(ds,&d));
     28:       n = d*ds->ld;
     29:     }
     30:   } else {
     31:     PetscCall(PetscObjectTypeCompare((PetscObject)ds,DSNEP,&isnep));
     32:     if (isnep) {
     33:       if (m==DS_MAT_Q || m==DS_MAT_Z || m==DS_MAT_U || m==DS_MAT_V || m==DS_MAT_X || m==DS_MAT_Y) {
     34:         PetscCall(DSNEPGetMinimality(ds,&d));
     35:         n = d*ds->ld;
     36:       }
     37:     }
     38:   }
     39:   cols = n;

     41:   switch (m) {
     42:     case DS_MAT_T:
     43:       cols = PetscDefined(USE_COMPLEX)? 2: 3;
     44:       rows = n;
     45:       break;
     46:     case DS_MAT_D:
     47:       cols = 1;
     48:       rows = n;
     49:       break;
     50:     case DS_MAT_X:
     51:       rows = ds->ld;
     52:       break;
     53:     case DS_MAT_Y:
     54:       rows = ds->ld;
     55:       break;
     56:     default:
     57:       rows = n;
     58:   }
     59:   if (ds->omat[m]) PetscCall(MatZeroEntries(ds->omat[m]));
     60:   else {
     61:     PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,rows,cols,NULL,&ds->omat[m]));
     62:   }
     63:   PetscFunctionReturn(PETSC_SUCCESS);
     64: }

     66: PetscErrorCode DSAllocateWork_Private(DS ds,PetscInt s,PetscInt r,PetscInt i)
     67: {
     68:   PetscFunctionBegin;
     69:   if (s>ds->lwork) {
     70:     PetscCall(PetscFree(ds->work));
     71:     PetscCall(PetscMalloc1(s,&ds->work));
     72:     ds->lwork = s;
     73:   }
     74:   if (r>ds->lrwork) {
     75:     PetscCall(PetscFree(ds->rwork));
     76:     PetscCall(PetscMalloc1(r,&ds->rwork));
     77:     ds->lrwork = r;
     78:   }
     79:   if (i>ds->liwork) {
     80:     PetscCall(PetscFree(ds->iwork));
     81:     PetscCall(PetscMalloc1(i,&ds->iwork));
     82:     ds->liwork = i;
     83:   }
     84:   PetscFunctionReturn(PETSC_SUCCESS);
     85: }

     87: /*@
     88:    DSViewMat - Prints one of the internal DS matrices.

     90:    Collective

     92:    Input Parameters:
     93: +  ds     - the direct solver context
     94: .  viewer - visualization context
     95: -  m      - matrix to display

     97:    Note:
     98:    Works only for ascii viewers. Set the viewer in Matlab format if
     99:    want to paste into Matlab.

    101:    Level: developer

    103: .seealso: DSView()
    104: @*/
    105: PetscErrorCode DSViewMat(DS ds,PetscViewer viewer,DSMatType m)
    106: {
    107:   PetscInt          i,j,rows,cols;
    108:   const PetscScalar *M=NULL,*v;
    109:   PetscViewerFormat format;
    110: #if defined(PETSC_USE_COMPLEX)
    111:   PetscBool         allreal = PETSC_TRUE;
    112:   const PetscReal   *vr;
    113: #endif

    115:   PetscFunctionBegin;
    118:   DSCheckValidMat(ds,m,3);
    119:   if (!viewer) PetscCall(PetscViewerASCIIGetStdout(PetscObjectComm((PetscObject)ds),&viewer));
    121:   PetscCheckSameComm(ds,1,viewer,2);
    122:   PetscCall(PetscViewerGetFormat(viewer,&format));
    123:   if (format == PETSC_VIEWER_ASCII_INFO || format == PETSC_VIEWER_ASCII_INFO_DETAIL) PetscFunctionReturn(PETSC_SUCCESS);
    124:   PetscCall(PetscViewerASCIIUseTabs(viewer,PETSC_FALSE));
    125:   PetscCall(DSMatGetSize(ds,m,&rows,&cols));
    126:   PetscCall(MatDenseGetArrayRead(ds->omat[m],&M));
    127: #if defined(PETSC_USE_COMPLEX)
    128:   /* determine if matrix has all real values */
    129:   if (m!=DS_MAT_T && m!=DS_MAT_D) {
    130:     /* determine if matrix has all real values */
    131:     v = M;
    132:     for (i=0;i<rows;i++)
    133:       for (j=0;j<cols;j++)
    134:         if (PetscImaginaryPart(v[i+j*ds->ld])) { allreal = PETSC_FALSE; break; }
    135:   }
    136:   if (m==DS_MAT_T) cols=3;
    137: #endif
    138:   if (format == PETSC_VIEWER_ASCII_MATLAB) {
    139:     PetscCall(PetscViewerASCIIPrintf(viewer,"%% Size = %" PetscInt_FMT " %" PetscInt_FMT "\n",rows,cols));
    140:     PetscCall(PetscViewerASCIIPrintf(viewer,"%s = [\n",DSMatName[m]));
    141:   } else PetscCall(PetscViewerASCIIPrintf(viewer,"Matrix %s =\n",DSMatName[m]));

    143:   for (i=0;i<rows;i++) {
    144:     v = M+i;
    145: #if defined(PETSC_USE_COMPLEX)
    146:     vr = (const PetscReal*)M+i;   /* handle compact storage, 2nd column is in imaginary part of 1st column */
    147: #endif
    148:     for (j=0;j<cols;j++) {
    149: #if defined(PETSC_USE_COMPLEX)
    150:       if (m!=DS_MAT_T && m!=DS_MAT_D) {
    151:         if (allreal) PetscCall(PetscViewerASCIIPrintf(viewer,"%18.16e ",(double)PetscRealPart(*v)));
    152:         else PetscCall(PetscViewerASCIIPrintf(viewer,"%18.16e%+18.16ei ",(double)PetscRealPart(*v),(double)PetscImaginaryPart(*v)));
    153:       } else PetscCall(PetscViewerASCIIPrintf(viewer,"%18.16e ",(double)*vr));
    154: #else
    155:       PetscCall(PetscViewerASCIIPrintf(viewer,"%18.16e ",(double)*v));
    156: #endif
    157:       v += ds->ld;
    158: #if defined(PETSC_USE_COMPLEX)
    159:       if (m==DS_MAT_T) vr += ds->ld;
    160: #endif
    161:     }
    162:     PetscCall(PetscViewerASCIIPrintf(viewer,"\n"));
    163:   }
    164:   PetscCall(MatDenseRestoreArrayRead(ds->omat[m],&M));

    166:   if (format == PETSC_VIEWER_ASCII_MATLAB) PetscCall(PetscViewerASCIIPrintf(viewer,"];\n"));
    167:   PetscCall(PetscViewerASCIIUseTabs(viewer,PETSC_TRUE));
    168:   PetscCall(PetscViewerFlush(viewer));
    169:   PetscFunctionReturn(PETSC_SUCCESS);
    170: }

    172: PetscErrorCode DSSortEigenvalues_Private(DS ds,PetscScalar *wr,PetscScalar *wi,PetscInt *perm,PetscBool isghiep)
    173: {
    174:   PetscScalar    re,im,wi0;
    175:   PetscInt       n,i,j,result,tmp1,tmp2=0,d=1;

    177:   PetscFunctionBegin;
    178:   n = ds->t;   /* sort only first t pairs if truncated */
    179:   /* insertion sort */
    180:   i=ds->l+1;
    181: #if !defined(PETSC_USE_COMPLEX)
    182:   if (wi && wi[perm[i-1]]!=0.0) i++; /* initial value is complex */
    183: #else
    184:   if (isghiep && PetscImaginaryPart(wr[perm[i-1]])!=0.0) i++;
    185: #endif
    186:   for (;i<n;i+=d) {
    187:     re = wr[perm[i]];
    188:     if (wi) im = wi[perm[i]];
    189:     else im = 0.0;
    190:     tmp1 = perm[i];
    191: #if !defined(PETSC_USE_COMPLEX)
    192:     if (im!=0.0) { d = 2; tmp2 = perm[i+1]; }
    193:     else d = 1;
    194: #else
    195:     if (isghiep && PetscImaginaryPart(re)!=0.0) { d = 2; tmp2 = perm[i+1]; }
    196:     else d = 1;
    197: #endif
    198:     j = i-1;
    199:     if (wi) wi0 = wi[perm[j]];
    200:     else wi0 = 0.0;
    201:     PetscCall(SlepcSCCompare(ds->sc,re,im,wr[perm[j]],wi0,&result));
    202:     while (result<0 && j>=ds->l) {
    203:       perm[j+d] = perm[j];
    204:       j--;
    205: #if !defined(PETSC_USE_COMPLEX)
    206:       if (wi && wi[perm[j+1]]!=0)
    207: #else
    208:       if (isghiep && PetscImaginaryPart(wr[perm[j+1]])!=0)
    209: #endif
    210:         { perm[j+d] = perm[j]; j--; }

    212:       if (j>=ds->l) {
    213:         if (wi) wi0 = wi[perm[j]];
    214:         else wi0 = 0.0;
    215:         PetscCall(SlepcSCCompare(ds->sc,re,im,wr[perm[j]],wi0,&result));
    216:       }
    217:     }
    218:     perm[j+1] = tmp1;
    219:     if (d==2) perm[j+2] = tmp2;
    220:   }
    221:   PetscFunctionReturn(PETSC_SUCCESS);
    222: }

    224: PetscErrorCode DSSortEigenvaluesReal_Private(DS ds,PetscReal *eig,PetscInt *perm)
    225: {
    226:   PetscScalar    re;
    227:   PetscInt       i,j,result,tmp,l,n;

    229:   PetscFunctionBegin;
    230:   n = ds->t;   /* sort only first t pairs if truncated */
    231:   l = ds->l;
    232:   /* insertion sort */
    233:   for (i=l+1;i<n;i++) {
    234:     re = eig[perm[i]];
    235:     j = i-1;
    236:     PetscCall(SlepcSCCompare(ds->sc,re,0.0,eig[perm[j]],0.0,&result));
    237:     while (result<0 && j>=l) {
    238:       tmp = perm[j]; perm[j] = perm[j+1]; perm[j+1] = tmp; j--;
    239:       if (j>=l) PetscCall(SlepcSCCompare(ds->sc,re,0.0,eig[perm[j]],0.0,&result));
    240:     }
    241:   }
    242:   PetscFunctionReturn(PETSC_SUCCESS);
    243: }

    245: /*
    246:   Permute comumns [istart..iend-1] of [mat] according to perm. Columns have length n
    247:  */
    248: PetscErrorCode DSPermuteColumns_Private(DS ds,PetscInt istart,PetscInt iend,PetscInt n,DSMatType mat,PetscInt *perm)
    249: {
    250:   PetscInt    i,j,k,p,ld;
    251:   PetscScalar *Q,rtmp;

    253:   PetscFunctionBegin;
    254:   ld = ds->ld;
    255:   PetscCall(MatDenseGetArray(ds->omat[mat],&Q));
    256:   for (i=istart;i<iend;i++) {
    257:     p = perm[i];
    258:     if (p != i) {
    259:       j = i + 1;
    260:       while (perm[j] != i) j++;
    261:       perm[j] = p; perm[i] = i;
    262:       /* swap columns i and j */
    263:       for (k=0;k<n;k++) {
    264:         rtmp = Q[k+p*ld]; Q[k+p*ld] = Q[k+i*ld]; Q[k+i*ld] = rtmp;
    265:       }
    266:     }
    267:   }
    268:   PetscCall(MatDenseRestoreArray(ds->omat[mat],&Q));
    269:   PetscFunctionReturn(PETSC_SUCCESS);
    270: }

    272: /*
    273:   The same as DSPermuteColumns_Private but for two matrices [mat1] and [mat2]
    274:  */
    275: PetscErrorCode DSPermuteColumnsTwo_Private(DS ds,PetscInt istart,PetscInt iend,PetscInt n,DSMatType mat1,DSMatType mat2,PetscInt *perm)
    276: {
    277:   PetscInt    i,j,k,p,ld;
    278:   PetscScalar *Q,*Z,rtmp,rtmp2;

    280:   PetscFunctionBegin;
    281:   ld = ds->ld;
    282:   PetscCall(MatDenseGetArray(ds->omat[mat1],&Q));
    283:   PetscCall(MatDenseGetArray(ds->omat[mat2],&Z));
    284:   for (i=istart;i<iend;i++) {
    285:     p = perm[i];
    286:     if (p != i) {
    287:       j = i + 1;
    288:       while (perm[j] != i) j++;
    289:       perm[j] = p; perm[i] = i;
    290:       /* swap columns i and j */
    291:       for (k=0;k<n;k++) {
    292:         rtmp  = Q[k+p*ld]; Q[k+p*ld] = Q[k+i*ld]; Q[k+i*ld] = rtmp;
    293:         rtmp2 = Z[k+p*ld]; Z[k+p*ld] = Z[k+i*ld]; Z[k+i*ld] = rtmp2;
    294:       }
    295:     }
    296:   }
    297:   PetscCall(MatDenseRestoreArray(ds->omat[mat1],&Q));
    298:   PetscCall(MatDenseRestoreArray(ds->omat[mat2],&Z));
    299:   PetscFunctionReturn(PETSC_SUCCESS);
    300: }

    302: /*
    303:   Permute rows [istart..iend-1] of [mat] according to perm. Rows have length m
    304:  */
    305: PetscErrorCode DSPermuteRows_Private(DS ds,PetscInt istart,PetscInt iend,PetscInt m,DSMatType mat,PetscInt *perm)
    306: {
    307:   PetscInt    i,j,k,p,ld;
    308:   PetscScalar *Q,rtmp;

    310:   PetscFunctionBegin;
    311:   ld = ds->ld;
    312:   PetscCall(MatDenseGetArray(ds->omat[mat],&Q));
    313:   for (i=istart;i<iend;i++) {
    314:     p = perm[i];
    315:     if (p != i) {
    316:       j = i + 1;
    317:       while (perm[j] != i) j++;
    318:       perm[j] = p; perm[i] = i;
    319:       /* swap rows i and j */
    320:       for (k=0;k<m;k++) {
    321:         rtmp = Q[p+k*ld]; Q[p+k*ld] = Q[i+k*ld]; Q[i+k*ld] = rtmp;
    322:       }
    323:     }
    324:   }
    325:   PetscCall(MatDenseRestoreArray(ds->omat[mat],&Q));
    326:   PetscFunctionReturn(PETSC_SUCCESS);
    327: }

    329: /*
    330:   Permute columns [istart..iend-1] of [mat1] and [mat2] according to perm.
    331:   Columns of [mat1] have length n, columns of [mat2] have length m
    332:  */
    333: PetscErrorCode DSPermuteBoth_Private(DS ds,PetscInt istart,PetscInt iend,PetscInt n,PetscInt m,DSMatType mat1,DSMatType mat2,PetscInt *perm)
    334: {
    335:   PetscInt    i,j,k,p,ld;
    336:   PetscScalar *U,*V,rtmp;

    338:   PetscFunctionBegin;
    339:   ld = ds->ld;
    340:   PetscCall(MatDenseGetArray(ds->omat[mat1],&U));
    341:   PetscCall(MatDenseGetArray(ds->omat[mat2],&V));
    342:   for (i=istart;i<iend;i++) {
    343:     p = perm[i];
    344:     if (p != i) {
    345:       j = i + 1;
    346:       while (perm[j] != i) j++;
    347:       perm[j] = p; perm[i] = i;
    348:       /* swap columns i and j of U */
    349:       for (k=0;k<n;k++) {
    350:         rtmp = U[k+p*ld]; U[k+p*ld] = U[k+i*ld]; U[k+i*ld] = rtmp;
    351:       }
    352:       /* swap columns i and j of V */
    353:       for (k=0;k<m;k++) {
    354:         rtmp = V[k+p*ld]; V[k+p*ld] = V[k+i*ld]; V[k+i*ld] = rtmp;
    355:       }
    356:     }
    357:   }
    358:   PetscCall(MatDenseRestoreArray(ds->omat[mat1],&U));
    359:   PetscCall(MatDenseRestoreArray(ds->omat[mat2],&V));
    360:   PetscFunctionReturn(PETSC_SUCCESS);
    361: }

    363: /*@
    364:    DSSetIdentity - Copy the identity (a diagonal matrix with ones) on the
    365:    active part of a matrix.

    367:    Logically Collective

    369:    Input Parameters:
    370: +  ds  - the direct solver context
    371: -  mat - the matrix to modify

    373:    Level: intermediate

    375: .seealso: DSGetMat()
    376: @*/
    377: PetscErrorCode DSSetIdentity(DS ds,DSMatType mat)
    378: {
    379:   PetscScalar    *A;
    380:   PetscInt       i,ld,n,l;

    382:   PetscFunctionBegin;
    385:   DSCheckValidMat(ds,mat,2);

    387:   PetscCall(DSGetDimensions(ds,&n,&l,NULL,NULL));
    388:   PetscCall(DSGetLeadingDimension(ds,&ld));
    389:   PetscCall(PetscLogEventBegin(DS_Other,ds,0,0,0));
    390:   PetscCall(MatDenseGetArray(ds->omat[mat],&A));
    391:   PetscCall(PetscArrayzero(A+l*ld,ld*(n-l)));
    392:   for (i=l;i<n;i++) A[i+i*ld] = 1.0;
    393:   PetscCall(MatDenseRestoreArray(ds->omat[mat],&A));
    394:   PetscCall(PetscLogEventEnd(DS_Other,ds,0,0,0));
    395:   PetscFunctionReturn(PETSC_SUCCESS);
    396: }

    398: /*@
    399:    DSOrthogonalize - Orthogonalize the columns of a matrix.

    401:    Logically Collective

    403:    Input Parameters:
    404: +  ds   - the direct solver context
    405: .  mat  - a matrix
    406: -  cols - number of columns to orthogonalize (starting from column zero)

    408:    Output Parameter:
    409: .  lindcols - (optional) number of linearly independent columns of the matrix

    411:    Level: developer

    413: .seealso: DSPseudoOrthogonalize()
    414: @*/
    415: PetscErrorCode DSOrthogonalize(DS ds,DSMatType mat,PetscInt cols,PetscInt *lindcols)
    416: {
    417:   PetscInt       n,l,ld;
    418:   PetscBLASInt   ld_,rA,cA,info,ltau,lw;
    419:   PetscScalar    *A,*tau,*w,saux,dummy;

    421:   PetscFunctionBegin;
    423:   DSCheckAlloc(ds,1);
    425:   DSCheckValidMat(ds,mat,2);

    428:   PetscCall(DSGetDimensions(ds,&n,&l,NULL,NULL));
    429:   PetscCall(DSGetLeadingDimension(ds,&ld));
    430:   n = n - l;
    431:   PetscCheck(cols<=n,PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_WRONG,"Invalid number of columns");
    432:   if (n == 0 || cols == 0) PetscFunctionReturn(PETSC_SUCCESS);

    434:   PetscCall(PetscLogEventBegin(DS_Other,ds,0,0,0));
    435:   PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
    436:   PetscCall(MatDenseGetArray(ds->omat[mat],&A));
    437:   PetscCall(PetscBLASIntCast(PetscMin(cols,n),&ltau));
    438:   PetscCall(PetscBLASIntCast(ld,&ld_));
    439:   PetscCall(PetscBLASIntCast(n,&rA));
    440:   PetscCall(PetscBLASIntCast(cols,&cA));
    441:   lw = -1;
    442:   PetscCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&rA,&cA,A,&ld_,&dummy,&saux,&lw,&info));
    443:   SlepcCheckLapackInfo("geqrf",info);
    444:   lw = (PetscBLASInt)PetscRealPart(saux);
    445:   PetscCall(DSAllocateWork_Private(ds,lw+ltau,0,0));
    446:   tau = ds->work;
    447:   w = &tau[ltau];
    448:   PetscCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&rA,&cA,&A[ld*l+l],&ld_,tau,w,&lw,&info));
    449:   SlepcCheckLapackInfo("geqrf",info);
    450:   PetscCallBLAS("LAPACKorgqr",LAPACKorgqr_(&rA,&ltau,&ltau,&A[ld*l+l],&ld_,tau,w,&lw,&info));
    451:   SlepcCheckLapackInfo("orgqr",info);
    452:   if (lindcols) *lindcols = ltau;
    453:   PetscCall(PetscFPTrapPop());
    454:   PetscCall(MatDenseRestoreArray(ds->omat[mat],&A));
    455:   PetscCall(PetscLogEventEnd(DS_Other,ds,0,0,0));
    456:   PetscCall(PetscObjectStateIncrease((PetscObject)ds));
    457:   PetscFunctionReturn(PETSC_SUCCESS);
    458: }

    460: /*
    461:   Compute C <- a*A*B + b*C, where
    462:     ldC, the leading dimension of C,
    463:     ldA, the leading dimension of A,
    464:     rA, cA, rows and columns of A,
    465:     At, if true use the transpose of A instead,
    466:     ldB, the leading dimension of B,
    467:     rB, cB, rows and columns of B,
    468:     Bt, if true use the transpose of B instead
    469: */
    470: 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)
    471: {
    472:   PetscInt       tmp;
    473:   PetscBLASInt   m, n, k, ldA, ldB, ldC;
    474:   const char     *N = "N", *T = "C", *qA = N, *qB = N;

    476:   PetscFunctionBegin;
    477:   if ((rA == 0) || (cB == 0)) PetscFunctionReturn(PETSC_SUCCESS);
    478:   PetscAssertPointer(C,1);
    479:   PetscAssertPointer(A,5);
    480:   PetscAssertPointer(B,10);
    481:   PetscCall(PetscBLASIntCast(_ldA,&ldA));
    482:   PetscCall(PetscBLASIntCast(_ldB,&ldB));
    483:   PetscCall(PetscBLASIntCast(_ldC,&ldC));

    485:   /* Transpose if needed */
    486:   if (At) tmp = rA, rA = cA, cA = tmp, qA = T;
    487:   if (Bt) tmp = rB, rB = cB, cB = tmp, qB = T;

    489:   /* Check size */
    490:   PetscCheck(cA==rB,PETSC_COMM_SELF,PETSC_ERR_PLIB,"Matrix dimensions do not match");

    492:   /* Do stub */
    493:   if ((rA == 1) && (cA == 1) && (cB == 1)) {
    494:     if (!At && !Bt) *C = *A * *B;
    495:     else if (At && !Bt) *C = PetscConj(*A) * *B;
    496:     else if (!At && Bt) *C = *A * PetscConj(*B);
    497:     else *C = PetscConj(*A) * PetscConj(*B);
    498:     m = n = k = 1;
    499:   } else {
    500:     PetscCall(PetscBLASIntCast(rA,&m));
    501:     PetscCall(PetscBLASIntCast(cB,&n));
    502:     PetscCall(PetscBLASIntCast(cA,&k));
    503:     PetscCallBLAS("BLASgemm",BLASgemm_(qA,qB,&m,&n,&k,&a,(PetscScalar*)A,&ldA,(PetscScalar*)B,&ldB,&b,C,&ldC));
    504:   }

    506:   PetscCall(PetscLogFlops(2.0*m*n*k));
    507:   PetscFunctionReturn(PETSC_SUCCESS);
    508: }

    510: /*@
    511:    DSPseudoOrthogonalize - Orthogonalize the columns of a matrix with Modified
    512:    Gram-Schmidt in an indefinite inner product space defined by a signature.

    514:    Logically Collective

    516:    Input Parameters:
    517: +  ds   - the direct solver context
    518: .  mat  - the matrix
    519: .  cols - number of columns to orthogonalize (starting from column zero)
    520: -  s    - the signature that defines the inner product

    522:    Output Parameters:
    523: +  lindcols - (optional) linearly independent columns of the matrix
    524: -  ns   - (optional) the new signature of the vectors

    526:    Note:
    527:    After the call the matrix satisfies A'*s*A = ns.

    529:    Level: developer

    531: .seealso: DSOrthogonalize()
    532: @*/
    533: PetscErrorCode DSPseudoOrthogonalize(DS ds,DSMatType mat,PetscInt cols,PetscReal s[],PetscInt *lindcols,PetscReal ns[])
    534: {
    535:   PetscInt       i,j,k,l,n,ld;
    536:   PetscBLASInt   info,one=1,zero=0,rA_,ld_;
    537:   PetscScalar    *A,*A_,*m,*h,nr0;
    538:   PetscReal      nr_o,nr,nr_abs,*ns_,done=1.0;

    540:   PetscFunctionBegin;
    542:   DSCheckAlloc(ds,1);
    544:   DSCheckValidMat(ds,mat,2);
    546:   PetscAssertPointer(s,4);
    547:   PetscCall(DSGetDimensions(ds,&n,&l,NULL,NULL));
    548:   PetscCall(DSGetLeadingDimension(ds,&ld));
    549:   n = n - l;
    550:   PetscCheck(cols<=n,PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_WRONG,"Invalid number of columns");
    551:   if (n == 0 || cols == 0) PetscFunctionReturn(PETSC_SUCCESS);
    552:   PetscCall(PetscLogEventBegin(DS_Other,ds,0,0,0));
    553:   PetscCall(PetscBLASIntCast(n,&rA_));
    554:   PetscCall(PetscBLASIntCast(ld,&ld_));
    555:   PetscCall(MatDenseGetArray(ds->omat[mat],&A_));
    556:   A = &A_[ld*l+l];
    557:   PetscCall(DSAllocateWork_Private(ds,n+cols,ns?0:cols,0));
    558:   m = ds->work;
    559:   h = &m[n];
    560:   ns_ = ns ? ns : ds->rwork;
    561:   for (i=0; i<cols; i++) {
    562:     /* m <- diag(s)*A[i] */
    563:     for (k=0; k<n; k++) m[k] = s[k]*A[k+i*ld];
    564:     /* nr_o <- mynorm(A[i]'*m), mynorm(x) = sign(x)*sqrt(|x|) */
    565:     PetscCall(SlepcMatDenseMult(&nr0,1,0.0,1.0,&A[ld*i],ld,n,1,PETSC_TRUE,m,n,n,1,PETSC_FALSE));
    566:     nr = nr_o = PetscSign(PetscRealPart(nr0))*PetscSqrtReal(PetscAbsScalar(nr0));
    567:     for (j=0; j<3 && i>0; j++) {
    568:       /* h <- A[0:i-1]'*m */
    569:       PetscCall(SlepcMatDenseMult(h,i,0.0,1.0,A,ld,n,i,PETSC_TRUE,m,n,n,1,PETSC_FALSE));
    570:       /* h <- diag(ns)*h */
    571:       for (k=0; k<i; k++) h[k] *= ns_[k];
    572:       /* A[i] <- A[i] - A[0:i-1]*h */
    573:       PetscCall(SlepcMatDenseMult(&A[ld*i],ld,1.0,-1.0,A,ld,n,i,PETSC_FALSE,h,i,i,1,PETSC_FALSE));
    574:       /* m <- diag(s)*A[i] */
    575:       for (k=0; k<n; k++) m[k] = s[k]*A[k+i*ld];
    576:       /* nr_o <- mynorm(A[i]'*m) */
    577:       PetscCall(SlepcMatDenseMult(&nr0,1,0.0,1.0,&A[ld*i],ld,n,1,PETSC_TRUE,m,n,n,1,PETSC_FALSE));
    578:       nr = PetscSign(PetscRealPart(nr0))*PetscSqrtReal(PetscAbsScalar(nr0));
    579:       PetscCheck(PetscAbs(nr)>PETSC_MACHINE_EPSILON,PETSC_COMM_SELF,PETSC_ERR_CONV_FAILED,"Linear dependency detected");
    580:       if (PetscAbs(nr) > 0.7*PetscAbs(nr_o)) break;
    581:       nr_o = nr;
    582:     }
    583:     ns_[i] = PetscSign(nr);
    584:     /* A[i] <- A[i]/|nr| */
    585:     nr_abs = PetscAbs(nr);
    586:     PetscCallBLAS("LAPACKlascl",LAPACKlascl_("G",&zero,&zero,&nr_abs,&done,&rA_,&one,A+i*ld,&ld_,&info));
    587:     SlepcCheckLapackInfo("lascl",info);
    588:   }
    589:   PetscCall(MatDenseRestoreArray(ds->omat[mat],&A_));
    590:   PetscCall(PetscLogEventEnd(DS_Other,ds,0,0,0));
    591:   PetscCall(PetscObjectStateIncrease((PetscObject)ds));
    592:   if (lindcols) *lindcols = cols;
    593:   PetscFunctionReturn(PETSC_SUCCESS);
    594: }