Actual source code: dsgnhep.c

      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-, Universitat Politecnica de Valencia, Spain

      6:    This file is part of SLEPc.
      7:    SLEPc is distributed under a 2-clause BSD license (see LICENSE).
      8:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      9: */

     11: #include <slepc/private/dsimpl.h>
     12: #include <slepcblaslapack.h>

     14: /*
     15:   1) Patterns of A and B
     16:       DS_STATE_RAW:       DS_STATE_INTERM/CONDENSED
     17:        0       n-1              0       n-1
     18:       -------------            -------------
     19:     0 |* * * * * *|          0 |* * * * * *|
     20:       |* * * * * *|            |  * * * * *|
     21:       |* * * * * *|            |    * * * *|
     22:       |* * * * * *|            |    * * * *|
     23:       |* * * * * *|            |        * *|
     24:   n-1 |* * * * * *|        n-1 |          *|
     25:       -------------            -------------

     27:   2) Moreover, P and Q are assumed to be the identity in DS_STATE_INTERMEDIATE.
     28: */

     30: static PetscErrorCode CleanDenseSchur(PetscInt n,PetscInt k,PetscScalar *S,PetscInt ldS,PetscScalar *T,PetscInt ldT,PetscScalar *X,PetscInt ldX,PetscScalar *Y,PetscInt ldY);

     32: static PetscErrorCode DSAllocate_GNHEP(DS ds,PetscInt ld)
     33: {
     34:   PetscFunctionBegin;
     35:   PetscCall(DSAllocateMat_Private(ds,DS_MAT_A));
     36:   PetscCall(DSAllocateMat_Private(ds,DS_MAT_B));
     37:   PetscCall(DSAllocateMat_Private(ds,DS_MAT_Z));
     38:   PetscCall(DSAllocateMat_Private(ds,DS_MAT_Q));
     39:   PetscCall(PetscFree(ds->perm));
     40:   PetscCall(PetscMalloc1(ld,&ds->perm));
     41:   PetscFunctionReturn(PETSC_SUCCESS);
     42: }

     44: static PetscErrorCode DSView_GNHEP(DS ds,PetscViewer viewer)
     45: {
     46:   PetscViewerFormat format;

     48:   PetscFunctionBegin;
     49:   PetscCall(PetscViewerGetFormat(viewer,&format));
     50:   if (format == PETSC_VIEWER_ASCII_INFO || format == PETSC_VIEWER_ASCII_INFO_DETAIL) PetscFunctionReturn(PETSC_SUCCESS);
     51:   PetscCall(DSViewMat(ds,viewer,DS_MAT_A));
     52:   PetscCall(DSViewMat(ds,viewer,DS_MAT_B));
     53:   if (ds->state>DS_STATE_INTERMEDIATE) {
     54:     PetscCall(DSViewMat(ds,viewer,DS_MAT_Z));
     55:     PetscCall(DSViewMat(ds,viewer,DS_MAT_Q));
     56:   }
     57:   if (ds->omat[DS_MAT_X]) PetscCall(DSViewMat(ds,viewer,DS_MAT_X));
     58:   if (ds->omat[DS_MAT_Y]) PetscCall(DSViewMat(ds,viewer,DS_MAT_Y));
     59:   PetscFunctionReturn(PETSC_SUCCESS);
     60: }

     62: static PetscErrorCode DSVectors_GNHEP_Eigen_Some(DS ds,PetscInt *k,PetscReal *rnorm,PetscBool left)
     63: {
     64:   PetscInt       i;
     65:   PetscBLASInt   n,ld,mout,info,*select,mm,inc=1,cols=1,zero=0;
     66:   PetscScalar    *X,*Y,*XY,*Z,*Q,*A,*B,fone=1.0,fzero=0.0;
     67:   PetscReal      norm,done=1.0;
     68:   PetscBool      iscomplex = PETSC_FALSE;
     69:   const char     *side;

     71:   PetscFunctionBegin;
     72:   PetscCall(PetscBLASIntCast(ds->n,&n));
     73:   PetscCall(PetscBLASIntCast(ds->ld,&ld));
     74:   if (left) {
     75:     X = NULL;
     76:     PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Y],&Y));
     77:     side = "L";
     78:   } else {
     79:     PetscCall(MatDenseGetArray(ds->omat[DS_MAT_X],&X));
     80:     Y = NULL;
     81:     side = "R";
     82:   }
     83:   XY = left? Y: X;
     84:   PetscCall(DSAllocateWork_Private(ds,0,0,ld));
     85:   select = ds->iwork;
     86:   for (i=0;i<n;i++) select[i] = (PetscBLASInt)PETSC_FALSE;
     87:   if (ds->state <= DS_STATE_INTERMEDIATE) {
     88:     PetscCall(DSSetIdentity(ds,DS_MAT_Q));
     89:     PetscCall(DSSetIdentity(ds,DS_MAT_Z));
     90:   }
     91:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_A],&A));
     92:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_B],&B));
     93:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Q],&Q));
     94:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Z],&Z));
     95:   PetscCall(CleanDenseSchur(n,0,A,ld,B,ld,Q,ld,Z,ld));
     96:   if (ds->state < DS_STATE_CONDENSED) PetscCall(DSSetState(ds,DS_STATE_CONDENSED));

     98:   /* compute k-th eigenvector */
     99:   select[*k] = (PetscBLASInt)PETSC_TRUE;
    100: #if defined(PETSC_USE_COMPLEX)
    101:   mm = 1;
    102:   PetscCall(DSAllocateWork_Private(ds,2*ld,2*ld,0));
    103:   PetscCallBLAS("LAPACKtgevc",LAPACKtgevc_(side,"S",select,&n,A,&ld,B,&ld,PetscSafePointerPlusOffset(Y,(*k)*ld),&ld,PetscSafePointerPlusOffset(X,(*k)*ld),&ld,&mm,&mout,ds->work,ds->rwork,&info));
    104: #else
    105:   if ((*k)<n-1 && (A[ld*(*k)+(*k)+1] != 0.0 || B[ld*(*k)+(*k)+1] != 0.0)) iscomplex = PETSC_TRUE;
    106:   mm = iscomplex? 2: 1;
    107:   if (iscomplex) select[(*k)+1] = (PetscBLASInt)PETSC_TRUE;
    108:   PetscCall(DSAllocateWork_Private(ds,6*ld,0,0));
    109:   PetscCallBLAS("LAPACKtgevc",LAPACKtgevc_(side,"S",select,&n,A,&ld,B,&ld,PetscSafePointerPlusOffset(Y,(*k)*ld),&ld,PetscSafePointerPlusOffset(X,(*k)*ld),&ld,&mm,&mout,ds->work,&info));
    110: #endif
    111:   SlepcCheckLapackInfo("tgevc",info);
    112:   PetscCheck(select[*k] && mout==mm,PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong arguments in call to Lapack xTGEVC");
    113:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_A],&A));
    114:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_B],&B));

    116:   /* accumulate and normalize eigenvectors */
    117:   PetscCall(PetscArraycpy(ds->work,XY+(*k)*ld,mm*ld));
    118:   PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&mm,&n,&fone,left?Z:Q,&ld,ds->work,&ld,&fzero,XY+(*k)*ld,&ld));
    119:   norm = BLASnrm2_(&n,XY+(*k)*ld,&inc);
    120: #if !defined(PETSC_USE_COMPLEX)
    121:   if (iscomplex) {
    122:     norm = SlepcAbsEigenvalue(norm,BLASnrm2_(&n,XY+(*k+1)*ld,&inc));
    123:     cols = 2;
    124:   }
    125: #endif
    126:   PetscCallBLAS("LAPACKlascl",LAPACKlascl_("G",&zero,&zero,&norm,&done,&n,&cols,XY+(*k)*ld,&ld,&info));
    127:   SlepcCheckLapackInfo("lascl",info);
    128:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Q],&Q));
    129:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Z],&Z));

    131:   /* set output arguments */
    132:   if (rnorm) {
    133:     if (iscomplex) *rnorm = SlepcAbsEigenvalue(XY[n-1+(*k)*ld],XY[n-1+(*k+1)*ld]);
    134:     else *rnorm = PetscAbsScalar(XY[n-1+(*k)*ld]);
    135:   }
    136:   if (iscomplex) (*k)++;
    137:   PetscCall(MatDenseRestoreArray(ds->omat[left?DS_MAT_Y:DS_MAT_X],&XY));
    138:   PetscFunctionReturn(PETSC_SUCCESS);
    139: }

    141: static PetscErrorCode DSVectors_GNHEP_Eigen_All(DS ds,PetscBool left)
    142: {
    143:   PetscInt       i;
    144:   PetscBLASInt   n,ld,mout,info,inc = 1;
    145:   PetscBool      iscomplex;
    146:   PetscScalar    *X,*Y,*XY,*Q,*Z,*A,*B,tmp;
    147:   PetscReal      norm;
    148:   const char     *side,*back;

    150:   PetscFunctionBegin;
    151:   PetscCall(PetscBLASIntCast(ds->n,&n));
    152:   PetscCall(PetscBLASIntCast(ds->ld,&ld));
    153:   if (left) {
    154:     X = NULL;
    155:     PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Y],&Y));
    156:     side = "L";
    157:   } else {
    158:     PetscCall(MatDenseGetArray(ds->omat[DS_MAT_X],&X));
    159:     Y = NULL;
    160:     side = "R";
    161:   }
    162:   XY = left? Y: X;
    163:   if (ds->state <= DS_STATE_INTERMEDIATE) {
    164:     PetscCall(DSSetIdentity(ds,DS_MAT_Q));
    165:     PetscCall(DSSetIdentity(ds,DS_MAT_Z));
    166:   }
    167:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_A],&A));
    168:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_B],&B));
    169:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Q],&Q));
    170:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Z],&Z));
    171:   PetscCall(CleanDenseSchur(n,0,A,ld,B,ld,Q,ld,Z,ld));
    172:   if (ds->state>=DS_STATE_CONDENSED) {
    173:     /* DSSolve() has been called, backtransform with matrix Q */
    174:     back = "B";
    175:     PetscCall(PetscArraycpy(left?Y:X,left?Z:Q,ld*ld));
    176:   } else {
    177:     back = "A";
    178:     PetscCall(DSSetState(ds,DS_STATE_CONDENSED));
    179:   }
    180: #if defined(PETSC_USE_COMPLEX)
    181:   PetscCall(DSAllocateWork_Private(ds,2*ld,2*ld,0));
    182:   PetscCallBLAS("LAPACKtgevc",LAPACKtgevc_(side,back,NULL,&n,A,&ld,B,&ld,Y,&ld,X,&ld,&n,&mout,ds->work,ds->rwork,&info));
    183: #else
    184:   PetscCall(DSAllocateWork_Private(ds,6*ld,0,0));
    185:   PetscCallBLAS("LAPACKtgevc",LAPACKtgevc_(side,back,NULL,&n,A,&ld,B,&ld,Y,&ld,X,&ld,&n,&mout,ds->work,&info));
    186: #endif
    187:   SlepcCheckLapackInfo("tgevc",info);
    188:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Q],&Q));
    189:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Z],&Z));

    191:   /* normalize eigenvectors */
    192:   for (i=0;i<n;i++) {
    193:     iscomplex = (i<n-1 && (A[i+1+i*ld]!=0.0 || B[i+1+i*ld]!=0.0))? PETSC_TRUE: PETSC_FALSE;
    194:     norm = BLASnrm2_(&n,XY+i*ld,&inc);
    195: #if !defined(PETSC_USE_COMPLEX)
    196:     if (iscomplex) {
    197:       tmp = BLASnrm2_(&n,XY+(i+1)*ld,&inc);
    198:       norm = SlepcAbsEigenvalue(norm,tmp);
    199:     }
    200: #endif
    201:     tmp = 1.0 / norm;
    202:     PetscCallBLAS("BLASscal",BLASscal_(&n,&tmp,XY+i*ld,&inc));
    203: #if !defined(PETSC_USE_COMPLEX)
    204:     if (iscomplex) PetscCallBLAS("BLASscal",BLASscal_(&n,&tmp,XY+(i+1)*ld,&inc));
    205: #endif
    206:     if (iscomplex) i++;
    207:   }
    208:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_A],&A));
    209:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_B],&B));
    210:   PetscCall(MatDenseRestoreArray(ds->omat[left?DS_MAT_Y:DS_MAT_X],&XY));
    211:   PetscFunctionReturn(PETSC_SUCCESS);
    212: }

    214: static PetscErrorCode DSVectors_GNHEP(DS ds,DSMatType mat,PetscInt *k,PetscReal *rnorm)
    215: {
    216:   PetscFunctionBegin;
    217:   switch (mat) {
    218:     case DS_MAT_X:
    219:     case DS_MAT_Y:
    220:       if (k) PetscCall(DSVectors_GNHEP_Eigen_Some(ds,k,rnorm,mat == DS_MAT_Y?PETSC_TRUE:PETSC_FALSE));
    221:       else PetscCall(DSVectors_GNHEP_Eigen_All(ds,mat == DS_MAT_Y?PETSC_TRUE:PETSC_FALSE));
    222:       break;
    223:     default:
    224:       SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_OUTOFRANGE,"Invalid mat parameter");
    225:   }
    226:   PetscFunctionReturn(PETSC_SUCCESS);
    227: }

    229: static PetscErrorCode DSSort_GNHEP_Arbitrary(DS ds,PetscScalar *wr,PetscScalar *wi,PetscScalar *rr,PetscScalar *ri,PetscInt *k)
    230: {
    231:   PetscInt       i;
    232:   PetscBLASInt   info,n,ld,mout,lwork,liwork,*iwork,*selection,zero_=0,true_=1;
    233:   PetscScalar    *S,*T,*Q,*Z,*work,*beta;

    235:   PetscFunctionBegin;
    236:   if (!ds->sc) PetscFunctionReturn(PETSC_SUCCESS);
    237:   PetscCall(PetscBLASIntCast(ds->n,&n));
    238:   PetscCall(PetscBLASIntCast(ds->ld,&ld));
    239: #if !defined(PETSC_USE_COMPLEX)
    240:   lwork = 4*n+16;
    241: #else
    242:   lwork = 1;
    243: #endif
    244:   liwork = 1;
    245:   PetscCall(DSAllocateWork_Private(ds,lwork+2*n,0,liwork+n));
    246:   beta      = ds->work;
    247:   work      = ds->work + n;
    248:   PetscCall(PetscBLASIntCast(ds->lwork-n,&lwork));
    249:   selection = ds->iwork;
    250:   iwork     = ds->iwork + n;
    251:   PetscCall(PetscBLASIntCast(ds->liwork-n,&liwork));
    252:   /* Compute the selected eigenvalue to be in the leading position */
    253:   PetscCall(DSSortEigenvalues_Private(ds,rr,ri,ds->perm,PETSC_FALSE));
    254:   PetscCall(PetscArrayzero(selection,n));
    255:   for (i=0; i<*k; i++) selection[ds->perm[i]] = 1;
    256:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_A],&S));
    257:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_B],&T));
    258:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Q],&Q));
    259:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Z],&Z));
    260: #if !defined(PETSC_USE_COMPLEX)
    261:   PetscCallBLAS("LAPACKtgsen",LAPACKtgsen_(&zero_,&true_,&true_,selection,&n,S,&ld,T,&ld,wr,wi,beta,Z,&ld,Q,&ld,&mout,NULL,NULL,NULL,work,&lwork,iwork,&liwork,&info));
    262: #else
    263:   PetscCallBLAS("LAPACKtgsen",LAPACKtgsen_(&zero_,&true_,&true_,selection,&n,S,&ld,T,&ld,wr,beta,Z,&ld,Q,&ld,&mout,NULL,NULL,NULL,work,&lwork,iwork,&liwork,&info));
    264: #endif
    265:   SlepcCheckLapackInfo("tgsen",info);
    266:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_A],&S));
    267:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_B],&T));
    268:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Q],&Q));
    269:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Z],&Z));
    270:   *k = mout;
    271:   for (i=0;i<n;i++) {
    272:     if (beta[i]==0.0) wr[i] = (PetscRealPart(wr[i])>0.0)? PETSC_MAX_REAL: PETSC_MIN_REAL;
    273:     else wr[i] /= beta[i];
    274: #if !defined(PETSC_USE_COMPLEX)
    275:     if (beta[i]==0.0) wi[i] = (wi[i]>0.0)? PETSC_MAX_REAL: PETSC_MIN_REAL;
    276:     else wi[i] /= beta[i];
    277: #endif
    278:   }
    279:   PetscFunctionReturn(PETSC_SUCCESS);
    280: }

    282: static PetscErrorCode DSSort_GNHEP_Total(DS ds,PetscScalar *wr,PetscScalar *wi)
    283: {
    284:   PetscScalar    re;
    285:   PetscInt       i,j,pos,result;
    286:   PetscBLASInt   ifst,ilst,info,n,ld,one=1;
    287:   PetscScalar    *S,*T,*Z,*Q;
    288: #if !defined(PETSC_USE_COMPLEX)
    289:   PetscBLASInt   lwork;
    290:   PetscScalar    *work,a,safmin,scale1,scale2,im;
    291: #endif

    293:   PetscFunctionBegin;
    294:   if (!ds->sc) PetscFunctionReturn(PETSC_SUCCESS);
    295:   PetscCall(PetscBLASIntCast(ds->n,&n));
    296:   PetscCall(PetscBLASIntCast(ds->ld,&ld));
    297:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_A],&S));
    298:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_B],&T));
    299:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Q],&Q));
    300:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Z],&Z));
    301: #if !defined(PETSC_USE_COMPLEX)
    302:   lwork = -1;
    303:   PetscCallBLAS("LAPACKtgexc",LAPACKtgexc_(&one,&one,&ld,NULL,&ld,NULL,&ld,NULL,&ld,NULL,&ld,&one,&one,&a,&lwork,&info));
    304:   SlepcCheckLapackInfo("tgexc",info);
    305:   safmin = LAPACKlamch_("S");
    306:   PetscCall(PetscBLASIntCast((PetscInt)a,&lwork));
    307:   PetscCall(DSAllocateWork_Private(ds,lwork,0,0));
    308:   work = ds->work;
    309: #endif
    310:   /* selection sort */
    311:   for (i=ds->l;i<n-1;i++) {
    312:     re = wr[i];
    313: #if !defined(PETSC_USE_COMPLEX)
    314:     im = wi[i];
    315: #endif
    316:     pos = 0;
    317:     j = i+1; /* j points to the next eigenvalue */
    318: #if !defined(PETSC_USE_COMPLEX)
    319:     if (im != 0) j=i+2;
    320: #endif
    321:     /* find minimum eigenvalue */
    322:     for (;j<n;j++) {
    323: #if !defined(PETSC_USE_COMPLEX)
    324:       PetscCall(SlepcSCCompare(ds->sc,re,im,wr[j],wi[j],&result));
    325: #else
    326:       PetscCall(SlepcSCCompare(ds->sc,re,0.0,wr[j],0.0,&result));
    327: #endif
    328:       if (result > 0) {
    329:         re = wr[j];
    330: #if !defined(PETSC_USE_COMPLEX)
    331:         im = wi[j];
    332: #endif
    333:         pos = j;
    334:       }
    335: #if !defined(PETSC_USE_COMPLEX)
    336:       if (wi[j] != 0) j++;
    337: #endif
    338:     }
    339:     if (pos) {
    340:       /* interchange blocks */
    341:       PetscCall(PetscBLASIntCast(pos+1,&ifst));
    342:       PetscCall(PetscBLASIntCast(i+1,&ilst));
    343: #if !defined(PETSC_USE_COMPLEX)
    344:       PetscCallBLAS("LAPACKtgexc",LAPACKtgexc_(&one,&one,&n,S,&ld,T,&ld,Z,&ld,Q,&ld,&ifst,&ilst,work,&lwork,&info));
    345: #else
    346:       PetscCallBLAS("LAPACKtgexc",LAPACKtgexc_(&one,&one,&n,S,&ld,T,&ld,Z,&ld,Q,&ld,&ifst,&ilst,&info));
    347: #endif
    348:       SlepcCheckLapackInfo("tgexc",info);
    349:       /* recover original eigenvalues from T and S matrices */
    350:       for (j=i;j<n;j++) {
    351: #if !defined(PETSC_USE_COMPLEX)
    352:         if (j<n-1 && S[j*ld+j+1] != 0.0) {
    353:           /* complex conjugate eigenvalue */
    354:           PetscCallBLAS("LAPACKlag2",LAPACKlag2_(S+j*ld+j,&ld,T+j*ld+j,&ld,&safmin,&scale1,&scale2,&re,&a,&im));
    355:           wr[j] = re / scale1;
    356:           wi[j] = im / scale1;
    357:           wr[j+1] = a / scale2;
    358:           wi[j+1] = -wi[j];
    359:           j++;
    360:         } else
    361: #endif
    362:         {
    363:           if (T[j*ld+j] == 0.0) wr[j] = (PetscRealPart(S[j*ld+j])>0.0)? PETSC_MAX_REAL: PETSC_MIN_REAL;
    364:           else wr[j] = S[j*ld+j] / T[j*ld+j];
    365: #if !defined(PETSC_USE_COMPLEX)
    366:           wi[j] = 0.0;
    367: #endif
    368:         }
    369:       }
    370:     }
    371: #if !defined(PETSC_USE_COMPLEX)
    372:     if (wi[i] != 0.0) i++;
    373: #endif
    374:   }
    375:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_A],&S));
    376:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_B],&T));
    377:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Q],&Q));
    378:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Z],&Z));
    379:   PetscFunctionReturn(PETSC_SUCCESS);
    380: }

    382: static PetscErrorCode DSSort_GNHEP(DS ds,PetscScalar *wr,PetscScalar *wi,PetscScalar *rr,PetscScalar *ri,PetscInt *k)
    383: {
    384:   PetscFunctionBegin;
    385:   if (!rr || wr == rr) PetscCall(DSSort_GNHEP_Total(ds,wr,wi));
    386:   else PetscCall(DSSort_GNHEP_Arbitrary(ds,wr,wi,rr,ri,k));
    387:   PetscFunctionReturn(PETSC_SUCCESS);
    388: }

    390: static PetscErrorCode DSUpdateExtraRow_GNHEP(DS ds)
    391: {
    392:   PetscInt          i;
    393:   PetscBLASInt      n,ld,incx=1;
    394:   PetscScalar       *A,*B,*x,*y,one=1.0,zero=0.0;
    395:   const PetscScalar *Q;

    397:   PetscFunctionBegin;
    398:   PetscCall(PetscBLASIntCast(ds->n,&n));
    399:   PetscCall(PetscBLASIntCast(ds->ld,&ld));
    400:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_A],&A));
    401:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_B],&B));
    402:   PetscCall(MatDenseGetArrayRead(ds->omat[DS_MAT_Q],&Q));
    403:   PetscCall(DSAllocateWork_Private(ds,2*ld,0,0));
    404:   x = ds->work;
    405:   y = ds->work+ld;
    406:   for (i=0;i<n;i++) x[i] = PetscConj(A[n+i*ld]);
    407:   PetscCallBLAS("BLASgemv",BLASgemv_("C",&n,&n,&one,Q,&ld,x,&incx,&zero,y,&incx));
    408:   for (i=0;i<n;i++) A[n+i*ld] = PetscConj(y[i]);
    409:   for (i=0;i<n;i++) x[i] = PetscConj(B[n+i*ld]);
    410:   PetscCallBLAS("BLASgemv",BLASgemv_("C",&n,&n,&one,Q,&ld,x,&incx,&zero,y,&incx));
    411:   for (i=0;i<n;i++) B[n+i*ld] = PetscConj(y[i]);
    412:   ds->k = n;
    413:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_A],&A));
    414:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_B],&B));
    415:   PetscCall(MatDenseRestoreArrayRead(ds->omat[DS_MAT_Q],&Q));
    416:   PetscFunctionReturn(PETSC_SUCCESS);
    417: }

    419: /*
    420:    Write zeros from the column k to n in the lower triangular part of the
    421:    matrices S and T, and inside 2-by-2 diagonal blocks of T in order to
    422:    make (S,T) a valid Schur decompositon.
    423: */
    424: static PetscErrorCode CleanDenseSchur(PetscInt n,PetscInt k,PetscScalar *S,PetscInt ldS,PetscScalar *T,PetscInt ldT,PetscScalar *X,PetscInt ldX,PetscScalar *Y,PetscInt ldY)
    425: {
    426:   PetscInt       i;
    427: #if defined(PETSC_USE_COMPLEX)
    428:   PetscInt       j;
    429:   PetscScalar    s;
    430: #else
    431:   PetscBLASInt   ldS_,ldT_,n_i,n_i_2,one=1,n_,i_2,i_;
    432:   PetscScalar    b11,b22,sr,cr,sl,cl;
    433: #endif

    435:   PetscFunctionBegin;
    436: #if defined(PETSC_USE_COMPLEX)
    437:   for (i=k; i<n; i++) {
    438:     /* Some functions need the diagonal elements in T be real */
    439:     if (T && PetscImaginaryPart(T[ldT*i+i]) != 0.0) {
    440:       s = PetscConj(T[ldT*i+i])/PetscAbsScalar(T[ldT*i+i]);
    441:       for (j=0;j<=i;j++) {
    442:         T[ldT*i+j] *= s;
    443:         S[ldS*i+j] *= s;
    444:       }
    445:       T[ldT*i+i] = PetscRealPart(T[ldT*i+i]);
    446:       if (X) for (j=0;j<n;j++) X[ldX*i+j] *= s;
    447:     }
    448:     j = i+1;
    449:     if (j<n) {
    450:       S[ldS*i+j] = 0.0;
    451:       if (T) T[ldT*i+j] = 0.0;
    452:     }
    453:   }
    454: #else
    455:   PetscCall(PetscBLASIntCast(ldS,&ldS_));
    456:   PetscCall(PetscBLASIntCast(ldT,&ldT_));
    457:   PetscCall(PetscBLASIntCast(n,&n_));
    458:   for (i=k;i<n-1;i++) {
    459:     if (S[ldS*i+i+1] != 0.0) {
    460:       /* Check if T(i+1,i) and T(i,i+1) are zero */
    461:       if (T[ldT*(i+1)+i] != 0.0 || T[ldT*i+i+1] != 0.0) {
    462:         /* Check if T(i+1,i) and T(i,i+1) are negligible */
    463:         if (PetscAbs(T[ldT*(i+1)+i])+PetscAbs(T[ldT*i+i+1]) < (PetscAbs(T[ldT*i+i])+PetscAbs(T[ldT*(i+1)+i+1]))*PETSC_MACHINE_EPSILON) {
    464:           T[ldT*i+i+1] = 0.0;
    465:           T[ldT*(i+1)+i] = 0.0;
    466:         } else {
    467:           /* If one of T(i+1,i) or T(i,i+1) is negligible, we make zero the other element */
    468:           if (PetscAbs(T[ldT*i+i+1]) < (PetscAbs(T[ldT*i+i])+PetscAbs(T[ldT*(i+1)+i+1])+PetscAbs(T[ldT*(i+1)+i]))*PETSC_MACHINE_EPSILON) {
    469:             PetscCallBLAS("LAPACKlasv2",LAPACKlasv2_(&T[ldT*i+i],&T[ldT*(i+1)+i],&T[ldT*(i+1)+i+1],&b22,&b11,&sl,&cl,&sr,&cr));
    470:           } else if (PetscAbs(T[ldT*(i+1)+i]) < (PetscAbs(T[ldT*i+i])+PetscAbs(T[ldT*(i+1)+i+1])+PetscAbs(T[ldT*i+i+1]))*PETSC_MACHINE_EPSILON) {
    471:             PetscCallBLAS("LAPACKlasv2",LAPACKlasv2_(&T[ldT*i+i],&T[ldT*i+i+1],&T[ldT*(i+1)+i+1],&b22,&b11,&sr,&cr,&sl,&cl));
    472:           } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Unsupported format. Call DSSolve before this function");
    473:           PetscCall(PetscBLASIntCast(n-i,&n_i));
    474:           n_i_2 = n_i - 2;
    475:           PetscCall(PetscBLASIntCast(i+2,&i_2));
    476:           PetscCall(PetscBLASIntCast(i,&i_));
    477:           if (b11 < 0.0) {
    478:             cr = -cr; sr = -sr;
    479:             b11 = -b11; b22 = -b22;
    480:           }
    481:           PetscCallBLAS("BLASrot",BLASrot_(&n_i,&S[ldS*i+i],&ldS_,&S[ldS*i+i+1],&ldS_,&cl,&sl));
    482:           PetscCallBLAS("BLASrot",BLASrot_(&i_2,&S[ldS*i],&one,&S[ldS*(i+1)],&one,&cr,&sr));
    483:           PetscCallBLAS("BLASrot",BLASrot_(&n_i_2,&T[ldT*(i+2)+i],&ldT_,&T[ldT*(i+2)+i+1],&ldT_,&cl,&sl));
    484:           PetscCallBLAS("BLASrot",BLASrot_(&i_,&T[ldT*i],&one,&T[ldT*(i+1)],&one,&cr,&sr));
    485:           if (X) PetscCallBLAS("BLASrot",BLASrot_(&n_,&X[ldX*i],&one,&X[ldX*(i+1)],&one,&cr,&sr));
    486:           if (Y) PetscCallBLAS("BLASrot",BLASrot_(&n_,&Y[ldY*i],&one,&Y[ldY*(i+1)],&one,&cl,&sl));
    487:           T[ldT*i+i] = b11; T[ldT*i+i+1] = 0.0;
    488:           T[ldT*(i+1)+i] = 0.0; T[ldT*(i+1)+i+1] = b22;
    489:         }
    490:       }
    491:       i++;
    492:     }
    493:   }
    494: #endif
    495:   PetscFunctionReturn(PETSC_SUCCESS);
    496: }

    498: static PetscErrorCode DSSolve_GNHEP(DS ds,PetscScalar *wr,PetscScalar *wi)
    499: {
    500:   PetscScalar    *work,*beta,a;
    501:   PetscInt       i;
    502:   PetscBLASInt   lwork,info,n,ld,iaux;
    503:   PetscScalar    *A,*B,*Z,*Q;
    504:   PetscBool      usegges3=(ds->method==1)?PETSC_TRUE:PETSC_FALSE;

    506:   PetscFunctionBegin;
    507: #if !defined(PETSC_USE_COMPLEX)
    508:   PetscAssertPointer(wi,3);
    509: #endif
    510:   PetscCall(PetscBLASIntCast(ds->n,&n));
    511:   PetscCall(PetscBLASIntCast(ds->ld,&ld));
    512:   lwork = -1;
    513:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_A],&A));
    514:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_B],&B));
    515:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Q],&Q));
    516:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Z],&Z));
    517: #if !defined(PETSC_USE_COMPLEX)
    518:   if (usegges3) PetscCallBLAS("LAPACKgges3",LAPACKgges3_("V","V","N",NULL,&n,A,&ld,B,&ld,&iaux,wr,wi,NULL,Z,&ld,Q,&ld,&a,&lwork,NULL,&info));
    519:   else PetscCallBLAS("LAPACKgges",LAPACKgges_("V","V","N",NULL,&n,A,&ld,B,&ld,&iaux,wr,wi,NULL,Z,&ld,Q,&ld,&a,&lwork,NULL,&info));
    520:   PetscCall(PetscBLASIntCast((PetscInt)a,&lwork));
    521:   PetscCall(DSAllocateWork_Private(ds,lwork+ld,0,0));
    522:   beta = ds->work;
    523:   work = beta+ds->n;
    524:   PetscCall(PetscBLASIntCast(ds->lwork-ds->n,&lwork));
    525:   if (usegges3) PetscCallBLAS("LAPACKgges3",LAPACKgges3_("V","V","N",NULL,&n,A,&ld,B,&ld,&iaux,wr,wi,beta,Z,&ld,Q,&ld,work,&lwork,NULL,&info));
    526:   else PetscCallBLAS("LAPACKgges",LAPACKgges_("V","V","N",NULL,&n,A,&ld,B,&ld,&iaux,wr,wi,beta,Z,&ld,Q,&ld,work,&lwork,NULL,&info));
    527: #else
    528:   if (usegges3) PetscCallBLAS("LAPACKgges3",LAPACKgges3_("V","V","N",NULL,&n,A,&ld,B,&ld,&iaux,wr,NULL,Z,&ld,Q,&ld,&a,&lwork,NULL,NULL,&info));
    529:   else PetscCallBLAS("LAPACKgges",LAPACKgges_("V","V","N",NULL,&n,A,&ld,B,&ld,&iaux,wr,NULL,Z,&ld,Q,&ld,&a,&lwork,NULL,NULL,&info));
    530:   PetscCall(PetscBLASIntCast((PetscInt)PetscRealPart(a),&lwork));
    531:   PetscCall(DSAllocateWork_Private(ds,lwork+ld,8*ld,0));
    532:   beta = ds->work;
    533:   work = beta+ds->n;
    534:   PetscCall(PetscBLASIntCast(ds->lwork-ds->n,&lwork));
    535:   if (usegges3) PetscCallBLAS("LAPACKgges3",LAPACKgges3_("V","V","N",NULL,&n,A,&ld,B,&ld,&iaux,wr,beta,Z,&ld,Q,&ld,work,&lwork,ds->rwork,NULL,&info));
    536:   else PetscCallBLAS("LAPACKgges",LAPACKgges_("V","V","N",NULL,&n,A,&ld,B,&ld,&iaux,wr,beta,Z,&ld,Q,&ld,work,&lwork,ds->rwork,NULL,&info));
    537: #endif
    538:   SlepcCheckLapackInfo(usegges3?"gges3":"gges",info);
    539:   for (i=0;i<n;i++) {
    540:     if (beta[i]==0.0) wr[i] = (PetscRealPart(wr[i])>0.0)? PETSC_MAX_REAL: PETSC_MIN_REAL;
    541:     else wr[i] /= beta[i];
    542: #if !defined(PETSC_USE_COMPLEX)
    543:     if (beta[i]==0.0) wi[i] = (wi[i]>0.0)? PETSC_MAX_REAL: PETSC_MIN_REAL;
    544:     else wi[i] /= beta[i];
    545: #else
    546:     if (wi) wi[i] = 0.0;
    547: #endif
    548:   }
    549:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_A],&A));
    550:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_B],&B));
    551:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Q],&Q));
    552:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Z],&Z));
    553:   PetscFunctionReturn(PETSC_SUCCESS);
    554: }

    556: #if !defined(PETSC_HAVE_MPIUNI)
    557: static PetscErrorCode DSSynchronize_GNHEP(DS ds,PetscScalar eigr[],PetscScalar eigi[])
    558: {
    559:   PetscInt       ld=ds->ld,l=ds->l,k;
    560:   PetscMPIInt    n,rank,off=0,size,ldn;
    561:   PetscScalar    *A,*B,*Q,*Z;

    563:   PetscFunctionBegin;
    564:   k = 2*(ds->n-l)*ld;
    565:   if (ds->state>DS_STATE_RAW) k += 2*(ds->n-l)*ld;
    566:   if (eigr) k += (ds->n-l);
    567:   if (eigi) k += (ds->n-l);
    568:   PetscCall(DSAllocateWork_Private(ds,k,0,0));
    569:   PetscCall(PetscMPIIntCast(k*sizeof(PetscScalar),&size));
    570:   PetscCall(PetscMPIIntCast(ds->n-l,&n));
    571:   PetscCall(PetscMPIIntCast(ld*(ds->n-l),&ldn));
    572:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_A],&A));
    573:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_B],&B));
    574:   if (ds->state>DS_STATE_RAW) {
    575:     PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Q],&Q));
    576:     PetscCall(MatDenseGetArray(ds->omat[DS_MAT_Z],&Z));
    577:   }
    578:   PetscCallMPI(MPI_Comm_rank(PetscObjectComm((PetscObject)ds),&rank));
    579:   if (!rank) {
    580:     PetscCallMPI(MPI_Pack(A+l*ld,ldn,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds)));
    581:     PetscCallMPI(MPI_Pack(B+l*ld,ldn,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds)));
    582:     if (ds->state>DS_STATE_RAW) {
    583:       PetscCallMPI(MPI_Pack(Q+l*ld,ldn,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds)));
    584:       PetscCallMPI(MPI_Pack(Z+l*ld,ldn,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds)));
    585:     }
    586:     if (eigr) PetscCallMPI(MPI_Pack(eigr+l,n,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds)));
    587: #if !defined(PETSC_USE_COMPLEX)
    588:     if (eigi) PetscCallMPI(MPI_Pack(eigi+l,n,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds)));
    589: #endif
    590:   }
    591:   PetscCallMPI(MPI_Bcast(ds->work,size,MPI_BYTE,0,PetscObjectComm((PetscObject)ds)));
    592:   if (rank) {
    593:     PetscCallMPI(MPI_Unpack(ds->work,size,&off,A+l*ld,ldn,MPIU_SCALAR,PetscObjectComm((PetscObject)ds)));
    594:     PetscCallMPI(MPI_Unpack(ds->work,size,&off,B+l*ld,ldn,MPIU_SCALAR,PetscObjectComm((PetscObject)ds)));
    595:     if (ds->state>DS_STATE_RAW) {
    596:       PetscCallMPI(MPI_Unpack(ds->work,size,&off,Q+l*ld,ldn,MPIU_SCALAR,PetscObjectComm((PetscObject)ds)));
    597:       PetscCallMPI(MPI_Unpack(ds->work,size,&off,Z+l*ld,ldn,MPIU_SCALAR,PetscObjectComm((PetscObject)ds)));
    598:     }
    599:     if (eigr) PetscCallMPI(MPI_Unpack(ds->work,size,&off,eigr+l,n,MPIU_SCALAR,PetscObjectComm((PetscObject)ds)));
    600: #if !defined(PETSC_USE_COMPLEX)
    601:     if (eigi) PetscCallMPI(MPI_Unpack(ds->work,size,&off,eigi+l,n,MPIU_SCALAR,PetscObjectComm((PetscObject)ds)));
    602: #endif
    603:   }
    604:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_A],&A));
    605:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_B],&B));
    606:   if (ds->state>DS_STATE_RAW) {
    607:     PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Q],&Q));
    608:     PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_Z],&Z));
    609:   }
    610:   PetscFunctionReturn(PETSC_SUCCESS);
    611: }
    612: #endif

    614: static PetscErrorCode DSTruncate_GNHEP(DS ds,PetscInt n,PetscBool trim)
    615: {
    616:   PetscInt    i,ld=ds->ld,l=ds->l;
    617:   PetscScalar *A,*B;

    619:   PetscFunctionBegin;
    620:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_A],&A));
    621:   PetscCall(MatDenseGetArray(ds->omat[DS_MAT_B],&B));
    622: #if defined(PETSC_USE_DEBUG)
    623:   /* make sure diagonal 2x2 block is not broken */
    624:   PetscCheck(ds->state<DS_STATE_CONDENSED || n==0 || n==ds->n || (A[n+(n-1)*ld]==0.0 && B[n+(n-1)*ld]==0.0),PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"The given size would break a 2x2 block, call DSGetTruncateSize() first");
    625: #endif
    626:   if (trim) {
    627:     if (ds->extrarow) {   /* clean extra row */
    628:       for (i=l;i<ds->n;i++) A[ds->n+i*ld] = 0.0;
    629:       for (i=l;i<ds->n;i++) B[ds->n+i*ld] = 0.0;
    630:     }
    631:     ds->l = 0;
    632:     ds->k = 0;
    633:     ds->n = n;
    634:     ds->t = ds->n;   /* truncated length equal to the new dimension */
    635:   } else {
    636:     if (ds->extrarow && ds->k==ds->n) {
    637:       /* copy entries of extra row to the new position, then clean last row */
    638:       for (i=l;i<n;i++) A[n+i*ld] = A[ds->n+i*ld];
    639:       for (i=l;i<ds->n;i++) A[ds->n+i*ld] = 0.0;
    640:       for (i=l;i<n;i++) B[n+i*ld] = B[ds->n+i*ld];
    641:       for (i=l;i<ds->n;i++) B[ds->n+i*ld] = 0.0;
    642:     }
    643:     ds->k = (ds->extrarow)? n: 0;
    644:     ds->t = ds->n;   /* truncated length equal to previous dimension */
    645:     ds->n = n;
    646:   }
    647:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_A],&A));
    648:   PetscCall(MatDenseRestoreArray(ds->omat[DS_MAT_B],&B));
    649:   PetscFunctionReturn(PETSC_SUCCESS);
    650: }

    652: /*MC
    653:    DSGNHEP - Dense Generalized Non-Hermitian Eigenvalue Problem.

    655:    Notes:
    656:    The problem is expressed as $AX = BX\Lambda$, where $(A,B)$ is the input
    657:    matrix pencil. $\Lambda$ is a diagonal matrix whose diagonal elements are the
    658:    arguments of `DSSolve()`. After solve, $(A,B)$ is overwritten with the
    659:    generalized (real) Schur form $(S,T) = (Z^*AQ,Z^*BQ)$, with the first
    660:    matrix being upper quasi-triangular and the second one triangular.

    662:    Used DS matrices:
    663: +  `DS_MAT_A` - first problem matrix
    664: .  `DS_MAT_B` - second problem matrix
    665: .  `DS_MAT_Q` - first orthogonal/unitary transformation that reduces to
    666:    generalized (real) Schur form
    667: -  `DS_MAT_Z` - second orthogonal/unitary transformation that reduces to
    668:    generalized (real) Schur form

    670:    Implemented methods:
    671: +  0 - QZ iteration (`_gges`)
    672: -  1 - blocked QZ iteration (`_gges3`, if available)

    674:    Level: beginner

    676: .seealso: [](sec:ds), `DSCreate()`, `DSSetType()`, `DSType`
    677: M*/
    678: SLEPC_EXTERN PetscErrorCode DSCreate_GNHEP(DS ds)
    679: {
    680:   PetscFunctionBegin;
    681:   ds->ops->allocate        = DSAllocate_GNHEP;
    682:   ds->ops->view            = DSView_GNHEP;
    683:   ds->ops->vectors         = DSVectors_GNHEP;
    684:   ds->ops->solve[0]        = DSSolve_GNHEP;
    685: #if !defined(SLEPC_MISSING_LAPACK_GGES3)
    686:   ds->ops->solve[1]        = DSSolve_GNHEP;
    687: #endif
    688:   ds->ops->sort            = DSSort_GNHEP;
    689: #if !defined(PETSC_HAVE_MPIUNI)
    690:   ds->ops->synchronize     = DSSynchronize_GNHEP;
    691: #endif
    692:   ds->ops->gettruncatesize = DSGetTruncateSize_Default;
    693:   ds->ops->truncate        = DSTruncate_GNHEP;
    694:   ds->ops->update          = DSUpdateExtraRow_GNHEP;
    695:   PetscFunctionReturn(PETSC_SUCCESS);
    696: }