Date: 2021-02-01T14:41:48+00:00

Actual source code: bvtensor.c
slepc-3.14.2 2021-02-01
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2020, 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:    Tensor BV that is represented in compact form as V = (I otimes U) S
     12: */

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

     17: typedef struct {
     18:   BV          U;        /* first factor */
     19:   Mat         S;        /* second factor */
     20:   PetscScalar *qB;      /* auxiliary matrix used in non-standard inner products */
     21:   PetscScalar *sw;      /* work space */
     22:   PetscInt    d;        /* degree of the tensor BV */
     23:   PetscInt    ld;       /* leading dimension of a single block in S */
     24:   PetscInt    puk;      /* copy of the k value */
     25:   Vec         u;        /* auxiliary work vector */
     26: } BV_TENSOR;

     28: PetscErrorCode BVMultInPlace_Tensor(BV V,Mat Q,PetscInt s,PetscInt e)
     29: {
     31:   BV_TENSOR      *ctx = (BV_TENSOR*)V->data;
     32:   PetscScalar    *pS,*q;
     33:   PetscInt       ldq,lds = ctx->ld*ctx->d;

     36:   MatGetSize(Q,&ldq,NULL);
     37:   MatDenseGetArray(ctx->S,&pS);
     38:   MatDenseGetArray(Q,&q);
     39:   BVMultInPlace_BLAS_Private(V,lds,V->k-V->l,ldq,s-V->l,e-V->l,pS+(V->nc+V->l)*lds,q+V->l*ldq+V->l,PETSC_FALSE);
     40:   MatDenseRestoreArray(Q,&q);
     41:   MatDenseRestoreArray(ctx->S,&pS);
     42:   return(0);
     43: }

     45: PetscErrorCode BVMultInPlaceTranspose_Tensor(BV V,Mat Q,PetscInt s,PetscInt e)
     46: {
     48:   BV_TENSOR      *ctx = (BV_TENSOR*)V->data;
     49:   PetscScalar    *pS,*q;
     50:   PetscInt       ldq,lds = ctx->ld*ctx->d;

     53:   MatGetSize(Q,&ldq,NULL);
     54:   MatDenseGetArray(ctx->S,&pS);
     55:   MatDenseGetArray(Q,&q);
     56:   BVMultInPlace_BLAS_Private(V,lds,V->k-V->l,ldq,s-V->l,e-V->l,pS+(V->nc+V->l)*lds,q+V->l*ldq+V->l,PETSC_TRUE);
     57:   MatDenseRestoreArray(Q,&q);
     58:   MatDenseRestoreArray(ctx->S,&pS);
     59:   return(0);
     60: }

     62: PetscErrorCode BVDot_Tensor(BV X,BV Y,Mat M)
     63: {
     65:   BV_TENSOR      *x = (BV_TENSOR*)X->data,*y = (BV_TENSOR*)Y->data;
     66:   PetscScalar    *m,*px,*py;
     67:   PetscInt       ldm,lds = x->ld*x->d;

     70:   if (x->U!=y->U) SETERRQ(PetscObjectComm((PetscObject)X),PETSC_ERR_SUP,"BVDot() in BVTENSOR requires that both operands have the same U factor");
     71:   if (lds!=y->ld*y->d) SETERRQ2(PetscObjectComm((PetscObject)X),PETSC_ERR_ARG_SIZ,"Mismatching dimensions ld*d %D %D",lds,y->ld*y->d);
     72:   MatGetSize(M,&ldm,NULL);
     73:   MatDenseGetArray(x->S,&px);
     74:   MatDenseGetArray(y->S,&py);
     75:   MatDenseGetArray(M,&m);
     76:   BVDot_BLAS_Private(X,Y->k-Y->l,X->k-X->l,lds,ldm,py+(Y->nc+Y->l)*lds,px+(X->nc+X->l)*lds,m+X->l*ldm+Y->l,PETSC_FALSE);
     77:   MatDenseRestoreArray(M,&m);
     78:   MatDenseRestoreArray(x->S,&px);
     79:   MatDenseRestoreArray(y->S,&py);
     80:   return(0);
     81: }

     83: PetscErrorCode BVScale_Tensor(BV bv,PetscInt j,PetscScalar alpha)
     84: {
     86:   BV_TENSOR      *ctx = (BV_TENSOR*)bv->data;
     87:   PetscScalar    *pS;
     88:   PetscInt       lds = ctx->ld*ctx->d;

     91:   MatDenseGetArray(ctx->S,&pS);
     92:   if (j<0) {
     93:     BVScale_BLAS_Private(bv,(bv->k-bv->l)*lds,pS+(bv->nc+bv->l)*lds,alpha);
     94:   } else {
     95:     BVScale_BLAS_Private(bv,lds,pS+(bv->nc+j)*lds,alpha);
     96:   }
     97:   MatDenseRestoreArray(ctx->S,&pS);
     98:   return(0);
     99: }

    101: PetscErrorCode BVNorm_Tensor(BV bv,PetscInt j,NormType type,PetscReal *val)
    102: {
    104:   BV_TENSOR      *ctx = (BV_TENSOR*)bv->data;
    105:   PetscScalar    *pS;
    106:   PetscInt       lds = ctx->ld*ctx->d;

    109:   MatDenseGetArray(ctx->S,&pS);
    110:   if (j<0) {
    111:     BVNorm_LAPACK_Private(bv,lds,bv->k-bv->l,pS+(bv->nc+bv->l)*lds,type,val,PETSC_FALSE);
    112:   } else {
    113:     BVNorm_LAPACK_Private(bv,lds,1,pS+(bv->nc+j)*lds,type,val,PETSC_FALSE);
    114:   }
    115:   MatDenseRestoreArray(ctx->S,&pS);
    116:   return(0);
    117: }

    119: PetscErrorCode BVCopyColumn_Tensor(BV V,PetscInt j,PetscInt i)
    120: {
    122:   BV_TENSOR      *ctx = (BV_TENSOR*)V->data;
    123:   PetscScalar    *pS;
    124:   PetscInt       lds = ctx->ld*ctx->d;

    127:   MatDenseGetArray(ctx->S,&pS);
    128:   PetscArraycpy(pS+(V->nc+i)*lds,pS+(V->nc+j)*lds,lds);
    129:   MatDenseRestoreArray(ctx->S,&pS);
    130:   return(0);
    131: }

    133: static PetscErrorCode BVTensorNormColumn(BV bv,PetscInt j,PetscReal *norm)
    134: {
    136:   BV_TENSOR      *ctx = (BV_TENSOR*)bv->data;
    137:   PetscBLASInt   one=1,lds_;
    138:   PetscScalar    sone=1.0,szero=0.0,*S,*x,dot;
    139:   PetscReal      alpha=1.0,scale=0.0,aval;
    140:   PetscInt       i,lds,ld=ctx->ld;

    143:   lds = ld*ctx->d;
    144:   MatDenseGetArray(ctx->S,&S);
    145:   PetscBLASIntCast(lds,&lds_);
    146:   if (ctx->qB) {
    147:     x = ctx->sw;
    148:     PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&lds_,&lds_,&sone,ctx->qB,&lds_,S+j*lds,&one,&szero,x,&one));
    149:     dot = PetscRealPart(BLASdot_(&lds_,S+j*lds,&one,x,&one));
    150:     BV_SafeSqrt(bv,dot,norm);
    151:   } else {
    152:     /* Compute *norm = BLASnrm2_(&lds_,S+j*lds,&one); */
    153:     if (lds==1) *norm = PetscAbsScalar(S[j*lds]);
    154:     else {
    155:       for (i=0;i<lds;i++) {
    156:         aval = PetscAbsScalar(S[i+j*lds]);
    157:         if (aval!=0.0) {
    158:           if (scale<aval) {
    159:             alpha = 1.0 + alpha*PetscSqr(scale/aval);
    160:             scale = aval;
    161:           } else alpha += PetscSqr(aval/scale);
    162:         }
    163:       }
    164:       *norm = scale*PetscSqrtReal(alpha);
    165:     }
    166:   }
    167:   return(0);
    168: }

    170: PetscErrorCode BVOrthogonalizeGS1_Tensor(BV bv,PetscInt k,Vec v,PetscBool *which,PetscScalar *h,PetscScalar *c,PetscReal *onorm,PetscReal *norm)
    171: {
    172:   PetscErrorCode    ierr;
    173:   BV_TENSOR         *ctx = (BV_TENSOR*)bv->data;
    174:   PetscScalar       *pS,*cc,*x,dot,sonem=-1.0,sone=1.0,szero=0.0;
    175:   PetscInt          i,lds = ctx->ld*ctx->d;
    176:   PetscBLASInt      lds_,k_,one=1;
    177:   const PetscScalar *omega;

    180:   if (v) SETERRQ(PetscObjectComm((PetscObject)bv),PETSC_ERR_SUP,"Orthogonalization against an external vector is not allowed in BVTENSOR");
    181:   MatDenseGetArray(ctx->S,&pS);
    182:   if (!c) {
    183:     VecGetArray(bv->buffer,&cc);
    184:   } else cc = c;
    185:   PetscBLASIntCast(lds,&lds_);
    186:   PetscBLASIntCast(k,&k_);

    188:   if (onorm) { BVTensorNormColumn(bv,k,onorm); }

    190:   if (ctx->qB) x = ctx->sw;
    191:   else x = pS+k*lds;

    193:   if (bv->orthog_type==BV_ORTHOG_MGS) {  /* modified Gram-Schmidt */

    195:     if (bv->indef) { /* signature */
    196:       VecGetArrayRead(bv->omega,&omega);
    197:     }
    198:     for (i=-bv->nc;i<k;i++) {
    199:       if (which && i>=0 && !which[i]) continue;
    200:       if (ctx->qB) PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&lds_,&lds_,&sone,ctx->qB,&lds_,pS+k*lds,&one,&szero,x,&one));
    201:       /* c_i = (s_k, s_i) */
    202:       dot = PetscRealPart(BLASdot_(&lds_,pS+i*lds,&one,x,&one));
    203:       if (bv->indef) dot /= PetscRealPart(omega[i]);
    204:       BV_SetValue(bv,i,0,cc,dot);
    205:       /* s_k = s_k - c_i s_i */
    206:       dot = -dot;
    207:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&lds_,&dot,pS+i*lds,&one,pS+k*lds,&one));
    208:     }
    209:     if (bv->indef) {
    210:       VecRestoreArrayRead(bv->omega,&omega);
    211:     }

    213:   } else {  /* classical Gram-Schmidt */
    214:     if (ctx->qB) PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&lds_,&lds_,&sone,ctx->qB,&lds_,pS+k*lds,&one,&szero,x,&one));

    216:     /* cc = S_{0:k-1}^* s_k */
    217:     PetscStackCallBLAS("BLASgemv",BLASgemv_("C",&lds_,&k_,&sone,pS,&lds_,x,&one,&szero,cc,&one));

    219:     /* s_k = s_k - S_{0:k-1} cc */
    220:     if (bv->indef) { BV_ApplySignature(bv,k,cc,PETSC_TRUE); }
    221:      PetscStackCallBLAS("BLASgemv",BLASgemv_("N",&lds_,&k_,&sonem,pS,&lds_,cc,&one,&sone,pS+k*lds,&one));
    222:     if (bv->indef) { BV_ApplySignature(bv,k,cc,PETSC_FALSE); }
    223:   }

    225:   if (norm) { BVTensorNormColumn(bv,k,norm); }
    226:   BV_AddCoefficients(bv,k,h,cc);
    227:   MatDenseRestoreArray(ctx->S,&pS);
    228:   VecRestoreArray(bv->buffer,&cc);
    229:   return(0);
    230: }

    232: PetscErrorCode BVView_Tensor(BV bv,PetscViewer viewer)
    233: {
    234:   PetscErrorCode    ierr;
    235:   BV_TENSOR         *ctx = (BV_TENSOR*)bv->data;
    236:   PetscViewerFormat format;
    237:   PetscBool         isascii;
    238:   const char        *bvname,*uname,*sname;

    241:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);
    242:   if (isascii) {
    243:     PetscViewerGetFormat(viewer,&format);
    244:     if (format == PETSC_VIEWER_ASCII_INFO || format == PETSC_VIEWER_ASCII_INFO_DETAIL) {
    245:       PetscViewerASCIIPrintf(viewer,"number of tensor blocks (degree): %D\n",ctx->d);
    246:       PetscViewerASCIIPrintf(viewer,"number of columns of U factor: %D\n",ctx->ld);
    247:       return(0);
    248:     }
    249:     BVView(ctx->U,viewer);
    250:     MatView(ctx->S,viewer);
    251:     if (format == PETSC_VIEWER_ASCII_MATLAB) {
    252:       PetscObjectGetName((PetscObject)bv,&bvname);
    253:       PetscObjectGetName((PetscObject)ctx->U,&uname);
    254:       PetscObjectGetName((PetscObject)ctx->S,&sname);
    255:       PetscViewerASCIIPrintf(viewer,"%s=kron(eye(%D),%s)*%s(:,1:%D);\n",bvname,ctx->d,uname,sname,bv->k);
    256:     }
    257:   } else {
    258:     BVView(ctx->U,viewer);
    259:     MatView(ctx->S,viewer);
    260:   }
    261:   return(0);
    262: }

    264: static PetscErrorCode BVTensorUpdateMatrix(BV V,PetscInt ini,PetscInt end)
    265: {
    267:   BV_TENSOR      *ctx = (BV_TENSOR*)V->data;
    268:   PetscInt       i,j,r,c,l,k,ld=ctx->ld,lds=ctx->d*ctx->ld;
    269:   PetscScalar    *qB,*sqB;
    270:   Vec            u;
    271:   Mat            A;

    274:   if (!V->matrix) return(0);
    275:   l = ctx->U->l; k = ctx->U->k;
    276:   /* update inner product matrix */
    277:   if (!ctx->qB) {
    278:     PetscCalloc2(lds*lds,&ctx->qB,lds,&ctx->sw);
    279:     VecDuplicate(ctx->U->t,&ctx->u);
    280:   }
    281:   ctx->U->l = 0;
    282:   for (r=0;r<ctx->d;r++) {
    283:     for (c=0;c<=r;c++) {
    284:       MatNestGetSubMat(V->matrix,r,c,&A);
    285:       if (A) {
    286:         qB = ctx->qB+c*ld*lds+r*ld;
    287:         for (i=ini;i<end;i++) {
    288:           BVGetColumn(ctx->U,i,&u);
    289:           MatMult(A,u,ctx->u);
    290:           ctx->U->k = i+1;
    291:           BVDotVec(ctx->U,ctx->u,qB+i*lds);
    292:           BVRestoreColumn(ctx->U,i,&u);
    293:           for (j=0;j<i;j++) qB[i+j*lds] = PetscConj(qB[j+i*lds]);
    294:           qB[i*lds+i] = PetscRealPart(qB[i+i*lds]);
    295:         }
    296:         if (c!=r) {
    297:           sqB = ctx->qB+r*ld*lds+c*ld;
    298:           for (i=ini;i<end;i++) for (j=0;j<=i;j++) {
    299:             sqB[i+j*lds] = PetscConj(qB[j+i*lds]);
    300:             sqB[j+i*lds] = qB[j+i*lds];
    301:           }
    302:         }
    303:       }
    304:     }
    305:   }
    306:   ctx->U->l = l; ctx->U->k = k;
    307:   return(0);
    308: }

    310: static PetscErrorCode BVTensorBuildFirstColumn_Tensor(BV V,PetscInt k)
    311: {
    313:   BV_TENSOR      *ctx = (BV_TENSOR*)V->data;
    314:   PetscInt       i,nq=0;
    315:   PetscScalar    *pS,*omega;
    316:   PetscReal      norm;
    317:   PetscBool      breakdown=PETSC_FALSE;

    320:   MatDenseGetArray(ctx->S,&pS);
    321:   for (i=0;i<ctx->d;i++) {
    322:     if (i>=k) {
    323:       BVSetRandomColumn(ctx->U,nq);
    324:     } else {
    325:       BVCopyColumn(ctx->U,i,nq);
    326:     }
    327:     BVOrthogonalizeColumn(ctx->U,nq,pS+i*ctx->ld,&norm,&breakdown);
    328:     if (!breakdown) {
    329:       BVScaleColumn(ctx->U,nq,1.0/norm);
    330:       pS[nq+i*ctx->ld] = norm;
    331:       nq++;
    332:     }
    333:   }
    334:   MatDenseRestoreArray(ctx->S,&pS);
    335:   if (!nq) SETERRQ1(PetscObjectComm((PetscObject)V),1,"Cannot build first column of tensor BV; U should contain k=%D nonzero columns",k);
    336:   BVTensorUpdateMatrix(V,0,nq);
    337:   BVTensorNormColumn(V,0,&norm);
    338:   BVScale_Tensor(V,0,1.0/norm);
    339:   if (V->indef) {
    340:     BV_AllocateSignature(V);
    341:     VecGetArray(V->omega,&omega);
    342:     omega[0] = (norm<0.0)? -1.0: 1.0;
    343:     VecRestoreArray(V->omega,&omega);
    344:   }
    345:   /* set active columns */
    346:   ctx->U->l = 0;
    347:   ctx->U->k = nq;
    348:   return(0);
    349: }

    351: /*@
    352:    BVTensorBuildFirstColumn - Builds the first column of the tensor basis vectors
    353:    V from the data contained in the first k columns of U.

    355:    Collective on V

    357:    Input Parameters:
    358: +  V - the basis vectors context
    359: -  k - the number of columns of U with relevant information

    361:    Notes:
    362:    At most d columns are considered, where d is the degree of the tensor BV.
    363:    Given V = (I otimes U) S, this function computes the first column of V, that
    364:    is, it computes the coefficients of the first column of S by orthogonalizing
    365:    the first d columns of U. If k is less than d (or linearly dependent columns
    366:    are found) then additional random columns are used.

    368:    The computed column has unit norm.

    370:    Level: advanced

    372: .seealso: BVCreateTensor()
    373: @*/
    374: PetscErrorCode BVTensorBuildFirstColumn(BV V,PetscInt k)
    375: {

    381:   PetscUseMethod(V,"BVTensorBuildFirstColumn_C",(BV,PetscInt),(V,k));
    382:   return(0);
    383: }

    385: static PetscErrorCode BVTensorCompress_Tensor(BV V,PetscInt newc)
    386: {
    388:   BV_TENSOR      *ctx = (BV_TENSOR*)V->data;
    389:   PetscInt       nwu=0,nnc,nrow,lwa,r,c;
    390:   PetscInt       i,j,k,n,lds=ctx->ld*ctx->d,deg=ctx->d,lock,cs1=V->k,rs1=ctx->U->k,rk=0,offu;
    391:   PetscScalar    *S,*M,*Z,*pQ,*SS,*SS2,t,sone=1.0,zero=0.0,mone=-1.0,*p,*tau,*work,*qB,*sqB;
    392:   PetscReal      *sg,tol,*rwork;
    393:   PetscBLASInt   ld_,cs1_,rs1_,cs1tdeg,n_,info,lw_,newc_,newctdeg,nnc_,nrow_,nnctdeg,lds_,rk_;
    394:   Mat            Q,A;

    397:   if (!cs1) return(0);
    398:   lwa = 6*ctx->ld*lds+2*cs1;
    399:   n = PetscMin(rs1,deg*cs1);
    400:   lock = ctx->U->l;
    401:   nnc = cs1-lock-newc;
    402:   nrow = rs1-lock;
    403:   PetscCalloc6(deg*newc*nnc,&SS,newc*nnc,&SS2,(rs1+lock+newc)*n,&pQ,deg*rs1,&tau,lwa,&work,6*n,&rwork);
    404:   offu = lock*(rs1+1);
    405:   M = work+nwu;
    406:   nwu += rs1*cs1*deg;
    407:   sg = rwork;
    408:   Z = work+nwu;
    409:   nwu += deg*cs1*n;
    410:   PetscBLASIntCast(n,&n_);
    411:   PetscBLASIntCast(nnc,&nnc_);
    412:   PetscBLASIntCast(cs1,&cs1_);
    413:   PetscBLASIntCast(rs1,&rs1_);
    414:   PetscBLASIntCast(newc,&newc_);
    415:   PetscBLASIntCast(newc*deg,&newctdeg);
    416:   PetscBLASIntCast(nnc*deg,&nnctdeg);
    417:   PetscBLASIntCast(cs1*deg,&cs1tdeg);
    418:   PetscBLASIntCast(lwa-nwu,&lw_);
    419:   PetscBLASIntCast(nrow,&nrow_);
    420:   PetscBLASIntCast(lds,&lds_);
    421:   MatDenseGetArray(ctx->S,&S);

    423:   if (newc>0) {
    424:     /* truncate columns associated with new converged eigenpairs */
    425:     for (j=0;j<deg;j++) {
    426:       for (i=lock;i<lock+newc;i++) {
    427:         PetscArraycpy(M+(i-lock+j*newc)*nrow,S+i*lds+j*ctx->ld+lock,nrow);
    428:       }
    429:     }
    430: #if !defined (PETSC_USE_COMPLEX)
    431:     PetscStackCallBLAS("LAPACKgesvd",LAPACKgesvd_("S","S",&nrow_,&newctdeg,M,&nrow_,sg,pQ+offu,&rs1_,Z,&n_,work+nwu,&lw_,&info));
    432: #else
    433:     PetscStackCallBLAS("LAPACKgesvd",LAPACKgesvd_("S","S",&nrow_,&newctdeg,M,&nrow_,sg,pQ+offu,&rs1_,Z,&n_,work+nwu,&lw_,rwork+n,&info));
    434: #endif
    435:     SlepcCheckLapackInfo("gesvd",info);
    436:     /* SVD has rank min(newc,nrow) */
    437:     rk = PetscMin(newc,nrow);
    438:     for (i=0;i<rk;i++) {
    439:       t = sg[i];
    440:       PetscStackCallBLAS("BLASscal",BLASscal_(&newctdeg,&t,Z+i,&n_));
    441:     }
    442:     for (i=0;i<deg;i++) {
    443:       for (j=lock;j<lock+newc;j++) {
    444:         PetscArraycpy(S+j*lds+i*ctx->ld+lock,Z+(newc*i+j-lock)*n,rk);
    445:         PetscArrayzero(S+j*lds+i*ctx->ld+lock+rk,(ctx->ld-lock-rk));
    446:       }
    447:     }
    448:     /*
    449:       update columns associated with non-converged vectors, orthogonalize
    450:       against pQ so that next M has rank nnc+d-1 insted of nrow+d-1
    451:     */
    452:     for (i=0;i<deg;i++) {
    453:       PetscStackCallBLAS("BLASgemm",BLASgemm_("C","N",&newc_,&nnc_,&nrow_,&sone,pQ+offu,&rs1_,S+(lock+newc)*lds+i*ctx->ld+lock,&lds_,&zero,SS+i*newc*nnc,&newc_));
    454:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&nrow_,&nnc_,&newc_,&mone,pQ+offu,&rs1_,SS+i*newc*nnc,&newc_,&sone,S+(lock+newc)*lds+i*ctx->ld+lock,&lds_));
    455:       /* repeat orthogonalization step */
    456:       PetscStackCallBLAS("BLASgemm",BLASgemm_("C","N",&newc_,&nnc_,&nrow_,&sone,pQ+offu,&rs1_,S+(lock+newc)*lds+i*ctx->ld+lock,&lds_,&zero,SS2,&newc_));
    457:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&nrow_,&nnc_,&newc_,&mone,pQ+offu,&rs1_,SS2,&newc_,&sone,S+(lock+newc)*lds+i*ctx->ld+lock,&lds_));
    458:       for (j=0;j<newc*nnc;j++) *(SS+i*newc*nnc+j) += SS2[j];
    459:     }
    460:   }

    462:   /* truncate columns associated with non-converged eigenpairs */
    463:   for (j=0;j<deg;j++) {
    464:     for (i=lock+newc;i<cs1;i++) {
    465:       PetscArraycpy(M+(i-lock-newc+j*nnc)*nrow,S+i*lds+j*ctx->ld+lock,nrow);
    466:     }
    467:   }
    468: #if !defined (PETSC_USE_COMPLEX)
    469:   PetscStackCallBLAS("LAPACKgesvd",LAPACKgesvd_("S","S",&nrow_,&nnctdeg,M,&nrow_,sg,pQ+offu+newc*rs1,&rs1_,Z,&n_,work+nwu,&lw_,&info));
    470: #else
    471:   PetscStackCallBLAS("LAPACKgesvd",LAPACKgesvd_("S","S",&nrow_,&nnctdeg,M,&nrow_,sg,pQ+offu+newc*rs1,&rs1_,Z,&n_,work+nwu,&lw_,rwork+n,&info));
    472: #endif
    473:   SlepcCheckLapackInfo("gesvd",info);
    474:   tol = PetscMax(rs1,deg*cs1)*PETSC_MACHINE_EPSILON*sg[0];
    475:   rk = 0;
    476:   for (i=0;i<PetscMin(nrow,nnctdeg);i++) if (sg[i]>tol) rk++;
    477:   rk = PetscMin(nnc+deg-1,rk);
    478:   /* the SVD has rank (at most) nnc+deg-1 */
    479:   for (i=0;i<rk;i++) {
    480:     t = sg[i];
    481:     PetscStackCallBLAS("BLASscal",BLASscal_(&nnctdeg,&t,Z+i,&n_));
    482:   }
    483:   /* update S */
    484:   PetscArrayzero(S+cs1*lds,(V->m-cs1)*lds);
    485:   k = ctx->ld-lock-newc-rk;
    486:   for (i=0;i<deg;i++) {
    487:     for (j=lock+newc;j<cs1;j++) {
    488:       PetscArraycpy(S+j*lds+i*ctx->ld+lock+newc,Z+(nnc*i+j-lock-newc)*n,rk);
    489:       PetscArrayzero(S+j*lds+i*ctx->ld+lock+newc+rk,k);
    490:     }
    491:   }
    492:   if (newc>0) {
    493:     for (i=0;i<deg;i++) {
    494:       p = SS+nnc*newc*i;
    495:       for (j=lock+newc;j<cs1;j++) {
    496:         for (k=0;k<newc;k++) S[j*lds+i*ctx->ld+lock+k] = *(p++);
    497:       }
    498:     }
    499:   }

    501:   /* orthogonalize pQ */
    502:   rk = rk+newc;
    503:   PetscBLASIntCast(rk,&rk_);
    504:   PetscBLASIntCast(cs1-lock,&nnc_);
    505:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&nrow_,&rk_,pQ+offu,&rs1_,tau,work+nwu,&lw_,&info));
    506:   SlepcCheckLapackInfo("geqrf",info);
    507:   for (i=0;i<deg;i++) {
    508:     PetscStackCallBLAS("BLAStrmm",BLAStrmm_("L","U","N","N",&rk_,&nnc_,&sone,pQ+offu,&rs1_,S+lock*lds+lock+i*ctx->ld,&lds_));
    509:   }
    510:   PetscStackCallBLAS("LAPACKorgqr",LAPACKorgqr_(&nrow_,&rk_,&rk_,pQ+offu,&rs1_,tau,work+nwu,&lw_,&info));
    511:   SlepcCheckLapackInfo("orgqr",info);

    513:   /* update vectors U(:,idx) = U*Q(:,idx) */
    514:   rk = rk+lock;
    515:   for (i=0;i<lock;i++) pQ[i*(1+rs1)] = 1.0;
    516:   MatCreateSeqDense(PETSC_COMM_SELF,rs1,rk,pQ,&Q);
    517:   ctx->U->k = rs1;
    518:   BVMultInPlace(ctx->U,Q,lock,rk);
    519:   MatDestroy(&Q);

    521:   if (ctx->qB) {
    522:    /* update matrix qB */
    523:     PetscBLASIntCast(ctx->ld,&ld_);
    524:     PetscBLASIntCast(rk,&rk_);
    525:     for (r=0;r<ctx->d;r++) {
    526:       for (c=0;c<=r;c++) {
    527:         MatNestGetSubMat(V->matrix,r,c,&A);
    528:         if (A) {
    529:           qB = ctx->qB+r*ctx->ld+c*ctx->ld*lds;
    530:           PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&rs1_,&rk_,&rs1_,&sone,qB,&lds_,pQ,&rs1_,&zero,work+nwu,&rs1_));
    531:           PetscStackCallBLAS("BLASgemm",BLASgemm_("C","N",&rk_,&rk_,&rs1_,&sone,pQ,&rs1_,work+nwu,&rs1_,&zero,qB,&lds_));
    532:           for (i=0;i<rk;i++) {
    533:             for (j=0;j<i;j++) qB[i+j*lds] = PetscConj(qB[j+i*lds]);
    534:             qB[i+i*lds] = PetscRealPart(qB[i+i*lds]);
    535:           }
    536:           for (i=rk;i<ctx->ld;i++) {
    537:             PetscArrayzero(qB+i*lds,ctx->ld);
    538:           }
    539:           for (i=0;i<rk;i++) {
    540:             PetscArrayzero(qB+i*lds+rk,(ctx->ld-rk));
    541:           }
    542:           if (c!=r) {
    543:             sqB = ctx->qB+r*ctx->ld*lds+c*ctx->ld;
    544:             for (i=0;i<ctx->ld;i++) for (j=0;j<ctx->ld;j++) sqB[i+j*lds] = PetscConj(qB[j+i*lds]);
    545:           }
    546:         }
    547:       }
    548:     }
    549:   }

    551:   /* free work space */
    552:   PetscFree6(SS,SS2,pQ,tau,work,rwork);
    553:   MatDenseRestoreArray(ctx->S,&S);

    555:   /* set active columns */
    556:   if (newc) ctx->U->l += newc;
    557:   ctx->U->k = rk;
    558:   return(0);
    559: }

    561: /*@
    562:    BVTensorCompress - Updates the U and S factors of the tensor basis vectors
    563:    object V by means of an SVD, removing redundant information.

    565:    Collective on V

    567:    Input Parameters:
    568: +  V - the tensor basis vectors context
    569: -  newc - additional columns to be locked

    571:    Notes:
    572:    This function is typically used when restarting Krylov solvers. Truncating a
    573:    tensor BV V = (I otimes U) S to its leading columns amounts to keeping the
    574:    leading columns of S. However, to effectively reduce the size of the
    575:    decomposition, it is necessary to compress it in a way that fewer columns of
    576:    U are employed. This can be achieved by means of an update that involves the
    577:    SVD of the low-rank matrix [S_0 S_1 ... S_{d-1}], where S_i are the pieces of S.

    579:    If newc is nonzero, then newc columns are added to the leading columns of V.
    580:    This means that the corresponding columns of the U and S factors will remain
    581:    invariant in subsequent operations.

    583:    Level: advanced

    585: .seealso: BVCreateTensor(), BVSetActiveColumns()
    586: @*/
    587: PetscErrorCode BVTensorCompress(BV V,PetscInt newc)
    588: {

    594:   PetscUseMethod(V,"BVTensorCompress_C",(BV,PetscInt),(V,newc));
    595:   return(0);
    596: }

    598: static PetscErrorCode BVTensorGetDegree_Tensor(BV bv,PetscInt *d)
    599: {
    600:   BV_TENSOR *ctx = (BV_TENSOR*)bv->data;

    603:   *d = ctx->d;
    604:   return(0);
    605: }

    607: /*@
    608:    BVTensorGetDegree - Returns the number of blocks (degree) of the tensor BV.

    610:    Not collective

    612:    Input Parameter:
    613: .  bv - the basis vectors context

    615:    Output Parameter:
    616: .  d - the degree

    618:    Level: advanced

    620: .seealso: BVCreateTensor()
    621: @*/
    622: PetscErrorCode BVTensorGetDegree(BV bv,PetscInt *d)
    623: {

    629:   PetscUseMethod(bv,"BVTensorGetDegree_C",(BV,PetscInt*),(bv,d));
    630:   return(0);
    631: }

    633: static PetscErrorCode BVTensorGetFactors_Tensor(BV V,BV *U,Mat *S)
    634: {
    635:   BV_TENSOR *ctx = (BV_TENSOR*)V->data;

    638:   if (ctx->puk>-1) SETERRQ(PetscObjectComm((PetscObject)V),PETSC_ERR_ORDER,"Previous call to BVTensonGetFactors without a BVTensorRestoreFactors call");
    639:   ctx->puk = ctx->U->k;
    640:   if (U) *U = ctx->U;
    641:   if (S) *S = ctx->S;
    642:   return(0);
    643: }

    645: /*@C
    646:    BVTensorGetFactors - Returns the two factors involved in the definition of the
    647:    tensor basis vectors object, V = (I otimes U) S.

    649:    Logically Collective on V

    651:    Input Parameter:
    652: .  V - the basis vectors context

    654:    Output Parameters:
    655: +  U - the BV factor
    656: -  S - the Mat factor

    658:    Notes:
    659:    The returned factors are references (not copies) of the internal factors,
    660:    so modifying them will change the tensor BV as well. Some operations of the
    661:    tensor BV assume that U has orthonormal columns, so if the user modifies U
    662:    this restriction must be taken into account.

    664:    The returned factors must not be destroyed. BVTensorRestoreFactors() must
    665:    be called when they are no longer needed.

    667:    Pass a NULL vector for any of the arguments that is not needed.

    669:    Level: advanced

    671: .seealso: BVTensorRestoreFactors()
    672: @*/
    673: PetscErrorCode BVTensorGetFactors(BV V,BV *U,Mat *S)
    674: {

    679:   PetscUseMethod(V,"BVTensorGetFactors_C",(BV,BV*,Mat*),(V,U,S));
    680:   return(0);
    681: }

    683: static PetscErrorCode BVTensorRestoreFactors_Tensor(BV V,BV *U,Mat *S)
    684: {
    686:   BV_TENSOR      *ctx = (BV_TENSOR*)V->data;

    689:   PetscObjectStateIncrease((PetscObject)V);
    690:   if (U) *U = NULL;
    691:   if (S) *S = NULL;
    692:   BVTensorUpdateMatrix(V,ctx->puk,ctx->U->k);
    693:   ctx->puk = -1;
    694:   return(0);
    695: }

    697: /*@C
    698:    BVTensorRestoreFactors - Restore the two factors that were obtained with
    699:    BVTensorGetFactors().

    701:    Logically Collective on V

    703:    Input Parameters:
    704: +  V - the basis vectors context
    705: .  U - the BV factor (or NULL)
    706: -  S - the Mat factor (or NULL)

    708:    Nots:
    709:    The arguments must match the corresponding call to BVTensorGetFactors().

    711:    Level: advanced

    713: .seealso: BVTensorGetFactors()
    714: @*/
    715: PetscErrorCode BVTensorRestoreFactors(BV V,BV *U,Mat *S)
    716: {

    723:   PetscUseMethod(V,"BVTensorRestoreFactors_C",(BV,BV*,Mat*),(V,U,S));
    724:   return(0);
    725: }

    727: PetscErrorCode BVDestroy_Tensor(BV bv)
    728: {
    730:   BV_TENSOR      *ctx = (BV_TENSOR*)bv->data;

    733:   BVDestroy(&ctx->U);
    734:   MatDestroy(&ctx->S);
    735:   if (ctx->u) {
    736:     PetscFree2(ctx->qB,ctx->sw);
    737:     VecDestroy(&ctx->u);
    738:   }
    739:   PetscFree(bv->data);
    740:   PetscObjectComposeFunction((PetscObject)bv,"BVTensorBuildFirstColumn_C",NULL);
    741:   PetscObjectComposeFunction((PetscObject)bv,"BVTensorCompress_C",NULL);
    742:   PetscObjectComposeFunction((PetscObject)bv,"BVTensorGetDegree_C",NULL);
    743:   PetscObjectComposeFunction((PetscObject)bv,"BVTensorGetFactors_C",NULL);
    744:   PetscObjectComposeFunction((PetscObject)bv,"BVTensorRestoreFactors_C",NULL);
    745:   return(0);
    746: }

    748: SLEPC_EXTERN PetscErrorCode BVCreate_Tensor(BV bv)
    749: {
    751:   BV_TENSOR      *ctx;

    754:   PetscNewLog(bv,&ctx);
    755:   bv->data = (void*)ctx;
    756:   ctx->puk = -1;

    758:   bv->ops->multinplace      = BVMultInPlace_Tensor;
    759:   bv->ops->multinplacetrans = BVMultInPlaceTranspose_Tensor;
    760:   bv->ops->dot              = BVDot_Tensor;
    761:   bv->ops->scale            = BVScale_Tensor;
    762:   bv->ops->norm             = BVNorm_Tensor;
    763:   bv->ops->copycolumn       = BVCopyColumn_Tensor;
    764:   bv->ops->gramschmidt      = BVOrthogonalizeGS1_Tensor;
    765:   bv->ops->destroy          = BVDestroy_Tensor;
    766:   bv->ops->view             = BVView_Tensor;

    768:   PetscObjectComposeFunction((PetscObject)bv,"BVTensorBuildFirstColumn_C",BVTensorBuildFirstColumn_Tensor);
    769:   PetscObjectComposeFunction((PetscObject)bv,"BVTensorCompress_C",BVTensorCompress_Tensor);
    770:   PetscObjectComposeFunction((PetscObject)bv,"BVTensorGetDegree_C",BVTensorGetDegree_Tensor);
    771:   PetscObjectComposeFunction((PetscObject)bv,"BVTensorGetFactors_C",BVTensorGetFactors_Tensor);
    772:   PetscObjectComposeFunction((PetscObject)bv,"BVTensorRestoreFactors_C",BVTensorRestoreFactors_Tensor);
    773:   return(0);
    774: }

    776: /*@
    777:    BVCreateTensor - Creates a tensor BV that is represented in compact form
    778:    as V = (I otimes U) S, where U has orthonormal columns.

    780:    Collective on U

    782:    Input Parameters:
    783: +  U - a basis vectors object
    784: -  d - the number of blocks (degree) of the tensor BV

    786:    Output Parameter:
    787: .  V - the new basis vectors context

    789:    Notes:
    790:    The new basis vectors object is V = (I otimes U) S, where otimes denotes
    791:    the Kronecker product, I is the identity matrix of order d, and S is a
    792:    sequential matrix allocated internally. This compact representation is
    793:    used e.g. to represent the Krylov basis generated with the linearization
    794:    of a matrix polynomial of degree d.

    796:    The size of V (number of rows) is equal to d times n, where n is the size
    797:    of U. The dimensions of S are d times m rows and m-d+1 columns, where m is
    798:    the number of columns of U, so m should be at least d.

    800:    The communicator of V will be the same as U.

    802:    On input, the content of U is irrelevant. Alternatively, it may contain
    803:    some nonzero columns that will be used by BVTensorBuildFirstColumn().

    805:    Level: advanced

    807: .seealso: BVTensorGetDegree(), BVTensorGetFactors(), BVTensorBuildFirstColumn()
    808: @*/
    809: PetscErrorCode BVCreateTensor(BV U,PetscInt d,BV *V)
    810: {
    812:   PetscBool      match;
    813:   PetscInt       n,N,m;
    814:   BV_TENSOR      *ctx;

    819:   PetscObjectTypeCompare((PetscObject)U,BVTENSOR,&match);
    820:   if (match) SETERRQ(PetscObjectComm((PetscObject)U),PETSC_ERR_SUP,"U cannot be of type tensor");

    822:   BVCreate(PetscObjectComm((PetscObject)U),V);
    823:   BVGetSizes(U,&n,&N,&m);
    824:   if (m<d) SETERRQ2(PetscObjectComm((PetscObject)U),PETSC_ERR_ARG_SIZ,"U has %D columns, it should have at least d=%D",m,d);
    825:   BVSetSizes(*V,d*n,d*N,m-d+1);
    826:   PetscObjectChangeTypeName((PetscObject)*V,BVTENSOR);
    827:   PetscLogEventBegin(BV_Create,*V,0,0,0);
    828:   BVCreate_Tensor(*V);
    829:   PetscLogEventEnd(BV_Create,*V,0,0,0);

    831:   ctx = (BV_TENSOR*)(*V)->data;
    832:   ctx->U  = U;
    833:   ctx->d  = d;
    834:   ctx->ld = m;
    835:   PetscObjectReference((PetscObject)U);
    836:   PetscLogObjectParent((PetscObject)*V,(PetscObject)U);
    837:   MatCreateSeqDense(PETSC_COMM_SELF,d*m,m-d+1,NULL,&ctx->S);
    838:   PetscLogObjectParent((PetscObject)*V,(PetscObject)ctx->S);
    839:   PetscObjectSetName((PetscObject)ctx->S,"S");

    841:   /* Copy user-provided attributes of U */
    842:   (*V)->orthog_type  = U->orthog_type;
    843:   (*V)->orthog_ref   = U->orthog_ref;
    844:   (*V)->orthog_eta   = U->orthog_eta;
    845:   (*V)->orthog_block = U->orthog_block;
    846:   (*V)->vmm          = U->vmm;
    847:   (*V)->rrandom      = U->rrandom;
    848:   return(0);
    849: }