Date: 2024-04-26T10:23:27+00:00

Actual source code: stsolve.c
slepc-3.21.1 2024-04-26
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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:    ST interface routines, callable by users
     12: */

     14: #include <slepc/private/stimpl.h>

     16: PetscErrorCode STApply_Generic(ST st,Vec x,Vec y)
     17: {
     18:   PetscFunctionBegin;
     19:   if (st->M && st->P) {
     20:     PetscCall(MatMult(st->M,x,st->work[0]));
     21:     PetscCall(STMatSolve(st,st->work[0],y));
     22:   } else if (st->M) PetscCall(MatMult(st->M,x,y));
     23:   else PetscCall(STMatSolve(st,x,y));
     24:   PetscFunctionReturn(PETSC_SUCCESS);
     25: }

     27: /*@
     28:    STApply - Applies the spectral transformation operator to a vector, for
     29:    instance (A - sB)^-1 B in the case of the shift-and-invert transformation
     30:    and generalized eigenproblem.

     32:    Collective

     34:    Input Parameters:
     35: +  st - the spectral transformation context
     36: -  x  - input vector

     38:    Output Parameter:
     39: .  y - output vector

     41:    Level: developer

     43: .seealso: STApplyTranspose(), STApplyHermitianTranspose()
     44: @*/
     45: PetscErrorCode STApply(ST st,Vec x,Vec y)
     46: {
     47:   Mat            Op;

     49:   PetscFunctionBegin;
     54:   STCheckMatrices(st,1);
     55:   PetscCheck(x!=y,PetscObjectComm((PetscObject)st),PETSC_ERR_ARG_IDN,"x and y must be different vectors");
     56:   PetscCall(VecSetErrorIfLocked(y,3));
     57:   PetscCall(STGetOperator_Private(st,&Op));
     58:   PetscCall(MatMult(Op,x,y));
     59:   PetscFunctionReturn(PETSC_SUCCESS);
     60: }

     62: PetscErrorCode STApplyMat_Generic(ST st,Mat B,Mat C)
     63: {
     64:   Mat            work;

     66:   PetscFunctionBegin;
     67:   if (st->M && st->P) {
     68:     PetscCall(MatMatMult(st->M,B,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&work));
     69:     PetscCall(STMatMatSolve(st,work,C));
     70:     PetscCall(MatDestroy(&work));
     71:   } else if (st->M) PetscCall(MatMatMult(st->M,B,MAT_REUSE_MATRIX,PETSC_DEFAULT,&C));
     72:   else PetscCall(STMatMatSolve(st,B,C));
     73:   PetscFunctionReturn(PETSC_SUCCESS);
     74: }

     76: /*@
     77:    STApplyMat - Applies the spectral transformation operator to a matrix, for
     78:    instance (A - sB)^-1 B in the case of the shift-and-invert transformation
     79:    and generalized eigenproblem.

     81:    Collective

     83:    Input Parameters:
     84: +  st - the spectral transformation context
     85: -  X  - input matrix

     87:    Output Parameter:
     88: .  Y - output matrix

     90:    Level: developer

     92: .seealso: STApply()
     93: @*/
     94: PetscErrorCode STApplyMat(ST st,Mat X,Mat Y)
     95: {
     96:   PetscFunctionBegin;
    101:   STCheckMatrices(st,1);
    102:   PetscCheck(X!=Y,PetscObjectComm((PetscObject)st),PETSC_ERR_ARG_IDN,"X and Y must be different matrices");
    103:   PetscUseTypeMethod(st,applymat,X,Y);
    104:   PetscFunctionReturn(PETSC_SUCCESS);
    105: }

    107: PetscErrorCode STApplyTranspose_Generic(ST st,Vec x,Vec y)
    108: {
    109:   PetscFunctionBegin;
    110:   if (st->M && st->P) {
    111:     PetscCall(STMatSolveTranspose(st,x,st->work[0]));
    112:     PetscCall(MatMultTranspose(st->M,st->work[0],y));
    113:   } else if (st->M) PetscCall(MatMultTranspose(st->M,x,y));
    114:   else PetscCall(STMatSolveTranspose(st,x,y));
    115:   PetscFunctionReturn(PETSC_SUCCESS);
    116: }

    118: /*@
    119:    STApplyTranspose - Applies the transpose of the operator to a vector, for
    120:    instance B^T(A - sB)^-T in the case of the shift-and-invert transformation
    121:    and generalized eigenproblem.

    123:    Collective

    125:    Input Parameters:
    126: +  st - the spectral transformation context
    127: -  x  - input vector

    129:    Output Parameter:
    130: .  y - output vector

    132:    Level: developer

    134: .seealso: STApply(), STApplyHermitianTranspose()
    135: @*/
    136: PetscErrorCode STApplyTranspose(ST st,Vec x,Vec y)
    137: {
    138:   Mat            Op;

    140:   PetscFunctionBegin;
    145:   STCheckMatrices(st,1);
    146:   PetscCheck(x!=y,PetscObjectComm((PetscObject)st),PETSC_ERR_ARG_IDN,"x and y must be different vectors");
    147:   PetscCall(VecSetErrorIfLocked(y,3));
    148:   PetscCall(STGetOperator_Private(st,&Op));
    149:   PetscCall(MatMultTranspose(Op,x,y));
    150:   PetscFunctionReturn(PETSC_SUCCESS);
    151: }

    153: PetscErrorCode STApplyHermitianTranspose_Generic(ST st,Vec x,Vec y)
    154: {
    155:   PetscFunctionBegin;
    156:   if (st->M && st->P) {
    157:     PetscCall(STMatSolveHermitianTranspose(st,x,st->work[0]));
    158:     PetscCall(MatMultHermitianTranspose(st->M,st->work[0],y));
    159:   } else if (st->M) PetscCall(MatMultHermitianTranspose(st->M,x,y));
    160:   else PetscCall(STMatSolveHermitianTranspose(st,x,y));
    161:   PetscFunctionReturn(PETSC_SUCCESS);
    162: }

    164: /*@
    165:    STApplyHermitianTranspose - Applies the hermitian-transpose of the operator
    166:    to a vector, for instance B^H(A - sB)^-H in the case of the shift-and-invert
    167:    transformation and generalized eigenproblem.

    169:    Collective

    171:    Input Parameters:
    172: +  st - the spectral transformation context
    173: -  x  - input vector

    175:    Output Parameter:
    176: .  y - output vector

    178:    Note:
    179:    Currently implemented via STApplyTranspose() with appropriate conjugation.

    181:    Level: developer

    183: .seealso: STApply(), STApplyTranspose()
    184: @*/
    185: PetscErrorCode STApplyHermitianTranspose(ST st,Vec x,Vec y)
    186: {
    187:   Mat            Op;

    189:   PetscFunctionBegin;
    194:   STCheckMatrices(st,1);
    195:   PetscCheck(x!=y,PetscObjectComm((PetscObject)st),PETSC_ERR_ARG_IDN,"x and y must be different vectors");
    196:   PetscCall(VecSetErrorIfLocked(y,3));
    197:   PetscCall(STGetOperator_Private(st,&Op));
    198:   PetscCall(MatMultHermitianTranspose(Op,x,y));
    199:   PetscFunctionReturn(PETSC_SUCCESS);
    200: }

    202: /*@
    203:    STGetBilinearForm - Returns the matrix used in the bilinear form with a
    204:    generalized problem with semi-definite B.

    206:    Logically Collective

    208:    Input Parameters:
    209: .  st - the spectral transformation context

    211:    Output Parameter:
    212: .  B - output matrix

    214:    Notes:
    215:    The output matrix B must be destroyed after use. It will be NULL in
    216:    case of standard eigenproblems.

    218:    Level: developer

    220: .seealso: BVSetMatrix()
    221: @*/
    222: PetscErrorCode STGetBilinearForm(ST st,Mat *B)
    223: {
    224:   PetscFunctionBegin;
    227:   PetscAssertPointer(B,2);
    228:   STCheckMatrices(st,1);
    229:   PetscUseTypeMethod(st,getbilinearform,B);
    230:   PetscFunctionReturn(PETSC_SUCCESS);
    231: }

    233: PetscErrorCode STGetBilinearForm_Default(ST st,Mat *B)
    234: {
    235:   PetscFunctionBegin;
    236:   if (st->nmat==1) *B = NULL;
    237:   else {
    238:     *B = st->A[1];
    239:     PetscCall(PetscObjectReference((PetscObject)*B));
    240:   }
    241:   PetscFunctionReturn(PETSC_SUCCESS);
    242: }

    244: static PetscErrorCode MatMult_STOperator(Mat Op,Vec x,Vec y)
    245: {
    246:   ST             st;

    248:   PetscFunctionBegin;
    249:   PetscCall(MatShellGetContext(Op,&st));
    250:   PetscCall(STSetUp(st));
    251:   PetscCall(PetscLogEventBegin(ST_Apply,st,x,y,0));
    252:   if (st->D) { /* with balancing */
    253:     PetscCall(VecPointwiseDivide(st->wb,x,st->D));
    254:     PetscUseTypeMethod(st,apply,st->wb,y);
    255:     PetscCall(VecPointwiseMult(y,y,st->D));
    256:   } else PetscUseTypeMethod(st,apply,x,y);
    257:   PetscCall(PetscLogEventEnd(ST_Apply,st,x,y,0));
    258:   PetscFunctionReturn(PETSC_SUCCESS);
    259: }

    261: static PetscErrorCode MatMultTranspose_STOperator(Mat Op,Vec x,Vec y)
    262: {
    263:   ST             st;

    265:   PetscFunctionBegin;
    266:   PetscCall(MatShellGetContext(Op,&st));
    267:   PetscCall(STSetUp(st));
    268:   PetscCall(PetscLogEventBegin(ST_ApplyTranspose,st,x,y,0));
    269:   if (st->D) { /* with balancing */
    270:     PetscCall(VecPointwiseMult(st->wb,x,st->D));
    271:     PetscUseTypeMethod(st,applytrans,st->wb,y);
    272:     PetscCall(VecPointwiseDivide(y,y,st->D));
    273:   } else PetscUseTypeMethod(st,applytrans,x,y);
    274:   PetscCall(PetscLogEventEnd(ST_ApplyTranspose,st,x,y,0));
    275:   PetscFunctionReturn(PETSC_SUCCESS);
    276: }

    278: #if defined(PETSC_USE_COMPLEX)
    279: static PetscErrorCode MatMultHermitianTranspose_STOperator(Mat Op,Vec x,Vec y)
    280: {
    281:   ST             st;

    283:   PetscFunctionBegin;
    284:   PetscCall(MatShellGetContext(Op,&st));
    285:   PetscCall(STSetUp(st));
    286:   PetscCall(PetscLogEventBegin(ST_ApplyHermitianTranspose,st,x,y,0));
    287:   if (st->ops->applyhermtrans) {
    288:     if (st->D) { /* with balancing */
    289:       PetscCall(VecConjugate(st->D));
    290:       PetscCall(VecPointwiseMult(st->wb,x,st->D));
    291:       PetscUseTypeMethod(st,applyhermtrans,st->wb,y);
    292:       PetscCall(VecPointwiseDivide(y,y,st->D));
    293:       PetscCall(VecConjugate(st->D));
    294:     } else PetscUseTypeMethod(st,applyhermtrans,x,y);
    295:   } else {
    296:     if (!st->wht) PetscCall(MatCreateVecs(st->A[0],&st->wht,NULL));
    297:     PetscCall(VecCopy(x,st->wht));
    298:     PetscCall(VecConjugate(st->wht));
    299:     if (st->D) { /* with balancing */
    300:       PetscCall(VecPointwiseMult(st->wb,st->wht,st->D));
    301:       PetscUseTypeMethod(st,applytrans,st->wb,y);
    302:       PetscCall(VecPointwiseDivide(y,y,st->D));
    303:     } else PetscUseTypeMethod(st,applytrans,st->wht,y);
    304:     PetscCall(VecConjugate(y));
    305:   }
    306:   PetscCall(PetscLogEventEnd(ST_ApplyHermitianTranspose,st,x,y,0));
    307:   PetscFunctionReturn(PETSC_SUCCESS);
    308: }
    309: #endif

    311: static PetscErrorCode MatMatMult_STOperator(Mat Op,Mat B,Mat C,void *ctx)
    312: {
    313:   ST             st;

    315:   PetscFunctionBegin;
    316:   PetscCall(MatShellGetContext(Op,&st));
    317:   PetscCall(STSetUp(st));
    318:   PetscCall(PetscLogEventBegin(ST_Apply,st,B,C,0));
    319:   PetscCall(STApplyMat_Generic(st,B,C));
    320:   PetscCall(PetscLogEventEnd(ST_Apply,st,B,C,0));
    321:   PetscFunctionReturn(PETSC_SUCCESS);
    322: }

    324: PetscErrorCode STGetOperator_Private(ST st,Mat *Op)
    325: {
    326:   PetscInt       m,n,M,N;
    327:   Vec            v;
    328:   VecType        vtype;

    330:   PetscFunctionBegin;
    331:   if (!st->Op) {
    332:     if (Op) *Op = NULL;
    333:     /* create the shell matrix */
    334:     PetscCall(MatGetLocalSize(st->A[0],&m,&n));
    335:     PetscCall(MatGetSize(st->A[0],&M,&N));
    336:     PetscCall(MatCreateShell(PetscObjectComm((PetscObject)st),m,n,M,N,st,&st->Op));
    337:     PetscCall(MatShellSetOperation(st->Op,MATOP_MULT,(void(*)(void))MatMult_STOperator));
    338:     PetscCall(MatShellSetOperation(st->Op,MATOP_MULT_TRANSPOSE,(void(*)(void))MatMultTranspose_STOperator));
    339: #if defined(PETSC_USE_COMPLEX)
    340:     PetscCall(MatShellSetOperation(st->Op,MATOP_MULT_HERMITIAN_TRANSPOSE,(void(*)(void))MatMultHermitianTranspose_STOperator));
    341: #else
    342:     PetscCall(MatShellSetOperation(st->Op,MATOP_MULT_HERMITIAN_TRANSPOSE,(void(*)(void))MatMultTranspose_STOperator));
    343: #endif
    344:     if (!st->D && st->ops->apply==STApply_Generic) {
    345:       PetscCall(MatShellSetMatProductOperation(st->Op,MATPRODUCT_AB,NULL,MatMatMult_STOperator,NULL,MATDENSE,MATDENSE));
    346:       PetscCall(MatShellSetMatProductOperation(st->Op,MATPRODUCT_AB,NULL,MatMatMult_STOperator,NULL,MATDENSECUDA,MATDENSECUDA));
    347:     }
    348:     /* make sure the shell matrix generates a vector of the same type as the problem matrices */
    349:     PetscCall(MatCreateVecs(st->A[0],&v,NULL));
    350:     PetscCall(VecGetType(v,&vtype));
    351:     PetscCall(MatShellSetVecType(st->Op,vtype));
    352:     PetscCall(VecDestroy(&v));
    353:     /* build the operator matrices */
    354:     PetscCall(STComputeOperator(st));
    355:   }
    356:   if (Op) *Op = st->Op;
    357:   PetscFunctionReturn(PETSC_SUCCESS);
    358: }

    360: /*@
    361:    STGetOperator - Returns a shell matrix that represents the operator of the
    362:    spectral transformation.

    364:    Collective

    366:    Input Parameter:
    367: .  st - the spectral transformation context

    369:    Output Parameter:
    370: .  Op - operator matrix

    372:    Notes:
    373:    The operator is defined in linear eigenproblems only, not in polynomial ones,
    374:    so the call will fail if more than 2 matrices were passed in STSetMatrices().

    376:    The returned shell matrix is essentially a wrapper to the STApply() and
    377:    STApplyTranspose() operations. The operator can often be expressed as

    379: $     Op = D*inv(K)*M*inv(D)

    381:    where D is the balancing matrix, and M and K are two matrices corresponding
    382:    to the numerator and denominator for spectral transformations that represent
    383:    a rational matrix function. In the case of STSHELL, the inner part inv(K)*M
    384:    is replaced by the user-provided operation from STShellSetApply().

    386:    The preconditioner matrix K typically depends on the value of the shift, and
    387:    its inverse is handled via an internal KSP object. Normal usage does not
    388:    require explicitly calling STGetOperator(), but it can be used to force the
    389:    creation of K and M, and then K is passed to the KSP. This is useful for
    390:    setting options associated with the PCFactor (to set MUMPS options, for instance).

    392:    The returned matrix must NOT be destroyed by the user. Instead, when no
    393:    longer needed it must be returned with STRestoreOperator(). In particular,
    394:    this is required before modifying the ST matrices or the shift.

    396:    A NULL pointer can be passed in Op in case the matrix is not required but we
    397:    want to force its creation. In this case, STRestoreOperator() should not be
    398:    called.

    400:    Level: advanced

    402: .seealso: STApply(), STApplyTranspose(), STSetBalanceMatrix(), STShellSetApply(),
    403:           STGetKSP(), STSetShift(), STRestoreOperator(), STSetMatrices()
    404: @*/
    405: PetscErrorCode STGetOperator(ST st,Mat *Op)
    406: {
    407:   PetscFunctionBegin;
    410:   STCheckMatrices(st,1);
    411:   STCheckNotSeized(st,1);
    412:   PetscCheck(st->nmat<=2,PetscObjectComm((PetscObject)st),PETSC_ERR_ARG_WRONGSTATE,"The operator is not defined in polynomial eigenproblems");
    413:   PetscCall(STGetOperator_Private(st,Op));
    414:   if (Op) st->opseized = PETSC_TRUE;
    415:   PetscFunctionReturn(PETSC_SUCCESS);
    416: }

    418: /*@
    419:    STRestoreOperator - Restore the previously seized operator matrix.

    421:    Logically Collective

    423:    Input Parameters:
    424: +  st - the spectral transformation context
    425: -  Op - operator matrix

    427:    Notes:
    428:    The arguments must match the corresponding call to STGetOperator().

    430:    Level: advanced

    432: .seealso: STGetOperator()
    433: @*/
    434: PetscErrorCode STRestoreOperator(ST st,Mat *Op)
    435: {
    436:   PetscFunctionBegin;
    438:   PetscAssertPointer(Op,2);
    440:   PetscCheck(st->opseized,PetscObjectComm((PetscObject)st),PETSC_ERR_ARG_WRONGSTATE,"Must be called after STGetOperator()");
    441:   *Op = NULL;
    442:   st->opseized = PETSC_FALSE;
    443:   PetscFunctionReturn(PETSC_SUCCESS);
    444: }

    446: /*
    447:    STComputeOperator - Computes the matrices that constitute the operator

    449:       Op = D*inv(K)*M*inv(D).

    451:    K and M are computed here (D is user-provided) from the system matrices
    452:    and the shift sigma (whenever these are changed, this function recomputes
    453:    K and M). This is used only in linear eigenproblems (nmat<3).

    455:    K is the "preconditioner matrix": it is the denominator in rational operators,
    456:    e.g. (A-sigma*B) in shift-and-invert. In non-rational transformations such
    457:    as STFILTER, K=NULL which means identity. After computing K, it is passed to
    458:    the internal KSP object via KSPSetOperators.

    460:    M is the numerator in rational operators. If unused it is set to NULL (e.g.
    461:    in STPRECOND).

    463:    STSHELL does not compute anything here, but sets the flag as if it was ready.
    464: */
    465: PetscErrorCode STComputeOperator(ST st)
    466: {
    467:   PC             pc;

    469:   PetscFunctionBegin;
    472:   if (!st->opready && st->ops->computeoperator) {
    473:     PetscCall(PetscInfo(st,"Building the operator matrices\n"));
    474:     STCheckMatrices(st,1);
    475:     if (!st->T) PetscCall(PetscCalloc1(PetscMax(2,st->nmat),&st->T));
    476:     PetscCall(PetscLogEventBegin(ST_ComputeOperator,st,0,0,0));
    477:     PetscUseTypeMethod(st,computeoperator);
    478:     PetscCall(PetscLogEventEnd(ST_ComputeOperator,st,0,0,0));
    479:     if (st->usesksp) {
    480:       if (!st->ksp) PetscCall(STGetKSP(st,&st->ksp));
    481:       if (st->P) {
    482:         PetscCall(STSetDefaultKSP(st));
    483:         PetscCall(ST_KSPSetOperators(st,st->P,st->Pmat?st->Pmat:st->P));
    484:       } else {
    485:         /* STPRECOND defaults to PCNONE if st->P is empty */
    486:         PetscCall(KSPGetPC(st->ksp,&pc));
    487:         PetscCall(PCSetType(pc,PCNONE));
    488:       }
    489:     }
    490:   }
    491:   st->opready = PETSC_TRUE;
    492:   PetscFunctionReturn(PETSC_SUCCESS);
    493: }

    495: /*@
    496:    STSetUp - Prepares for the use of a spectral transformation.

    498:    Collective

    500:    Input Parameter:
    501: .  st - the spectral transformation context

    503:    Level: advanced

    505: .seealso: STCreate(), STApply(), STDestroy()
    506: @*/
    507: PetscErrorCode STSetUp(ST st)
    508: {
    509:   PetscInt       i,n,k;

    511:   PetscFunctionBegin;
    514:   STCheckMatrices(st,1);
    515:   switch (st->state) {
    516:     case ST_STATE_INITIAL:
    517:       PetscCall(PetscInfo(st,"Setting up new ST\n"));
    518:       if (!((PetscObject)st)->type_name) PetscCall(STSetType(st,STSHIFT));
    519:       break;
    520:     case ST_STATE_SETUP:
    521:       PetscFunctionReturn(PETSC_SUCCESS);
    522:     case ST_STATE_UPDATED:
    523:       PetscCall(PetscInfo(st,"Setting up updated ST\n"));
    524:       break;
    525:   }
    526:   PetscCall(PetscLogEventBegin(ST_SetUp,st,0,0,0));
    527:   if (st->state!=ST_STATE_UPDATED) {
    528:     if (!(st->nmat<3 && st->opready)) {
    529:       if (st->T) {
    530:         for (i=0;i<PetscMax(2,st->nmat);i++) PetscCall(MatDestroy(&st->T[i]));
    531:       }
    532:       PetscCall(MatDestroy(&st->P));
    533:     }
    534:   }
    535:   if (st->D) {
    536:     PetscCall(MatGetLocalSize(st->A[0],NULL,&n));
    537:     PetscCall(VecGetLocalSize(st->D,&k));
    538:     PetscCheck(n==k,PetscObjectComm((PetscObject)st),PETSC_ERR_ARG_SIZ,"Balance matrix has wrong dimension %" PetscInt_FMT " (should be %" PetscInt_FMT ")",k,n);
    539:     if (!st->wb) PetscCall(VecDuplicate(st->D,&st->wb));
    540:   }
    541:   if (st->nmat<3 && st->transform) PetscCall(STComputeOperator(st));
    542:   else {
    543:     if (!st->T) PetscCall(PetscCalloc1(PetscMax(2,st->nmat),&st->T));
    544:   }
    545:   PetscTryTypeMethod(st,setup);
    546:   st->state = ST_STATE_SETUP;
    547:   PetscCall(PetscLogEventEnd(ST_SetUp,st,0,0,0));
    548:   PetscFunctionReturn(PETSC_SUCCESS);
    549: }

    551: /*
    552:    Computes coefficients for the transformed polynomial,
    553:    and stores the result in argument S.

    555:    alpha - value of the parameter of the transformed polynomial
    556:    beta - value of the previous shift (only used in inplace mode)
    557:    k - index of first matrix included in the computation
    558:    coeffs - coefficients of the expansion
    559:    initial - true if this is the first time
    560:    precond - whether the preconditioner matrix must be computed
    561: */
    562: PetscErrorCode STMatMAXPY_Private(ST st,PetscScalar alpha,PetscScalar beta,PetscInt k,PetscScalar *coeffs,PetscBool initial,PetscBool precond,Mat *S)
    563: {
    564:   PetscInt       *matIdx=NULL,nmat,i,ini=-1;
    565:   PetscScalar    t=1.0,ta,gamma;
    566:   PetscBool      nz=PETSC_FALSE;
    567:   Mat            *A=precond?st->Psplit:st->A;
    568:   MatStructure   str=precond?st->strp:st->str;

    570:   PetscFunctionBegin;
    571:   nmat = st->nmat-k;
    572:   switch (st->matmode) {
    573:   case ST_MATMODE_INPLACE:
    574:     PetscCheck(st->nmat<=2,PetscObjectComm((PetscObject)st),PETSC_ERR_SUP,"ST_MATMODE_INPLACE not supported for polynomial eigenproblems");
    575:     PetscCheck(!precond,PetscObjectComm((PetscObject)st),PETSC_ERR_SUP,"ST_MATMODE_INPLACE not supported for split preconditioner");
    576:     if (initial) {
    577:       PetscCall(PetscObjectReference((PetscObject)A[0]));
    578:       *S = A[0];
    579:       gamma = alpha;
    580:     } else gamma = alpha-beta;
    581:     if (gamma != 0.0) {
    582:       if (st->nmat>1) PetscCall(MatAXPY(*S,gamma,A[1],str));
    583:       else PetscCall(MatShift(*S,gamma));
    584:     }
    585:     break;
    586:   case ST_MATMODE_SHELL:
    587:     PetscCheck(!precond,PetscObjectComm((PetscObject)st),PETSC_ERR_SUP,"ST_MATMODE_SHELL not supported for split preconditioner");
    588:     if (initial) {
    589:       if (st->nmat>2) {
    590:         PetscCall(PetscMalloc1(nmat,&matIdx));
    591:         for (i=0;i<nmat;i++) matIdx[i] = k+i;
    592:       }
    593:       PetscCall(STMatShellCreate(st,alpha,nmat,matIdx,coeffs,S));
    594:       if (st->nmat>2) PetscCall(PetscFree(matIdx));
    595:     } else PetscCall(STMatShellShift(*S,alpha));
    596:     break;
    597:   case ST_MATMODE_COPY:
    598:     if (coeffs) {
    599:       for (i=0;i<nmat && ini==-1;i++) {
    600:         if (coeffs[i]!=0.0) ini = i;
    601:         else t *= alpha;
    602:       }
    603:       if (coeffs[ini] != 1.0) nz = PETSC_TRUE;
    604:       for (i=ini+1;i<nmat&&!nz;i++) if (coeffs[i]!=0.0) nz = PETSC_TRUE;
    605:     } else { nz = PETSC_TRUE; ini = 0; }
    606:     if ((alpha == 0.0 || !nz) && t==1.0) {
    607:       PetscCall(PetscObjectReference((PetscObject)A[k+ini]));
    608:       PetscCall(MatDestroy(S));
    609:       *S = A[k+ini];
    610:     } else {
    611:       if (*S && *S!=A[k+ini]) {
    612:         PetscCall(MatSetOption(*S,MAT_NEW_NONZERO_ALLOCATION_ERR,PETSC_FALSE));
    613:         PetscCall(MatCopy(A[k+ini],*S,DIFFERENT_NONZERO_PATTERN));
    614:       } else {
    615:         PetscCall(MatDestroy(S));
    616:         PetscCall(MatDuplicate(A[k+ini],MAT_COPY_VALUES,S));
    617:         PetscCall(MatSetOption(*S,MAT_NEW_NONZERO_ALLOCATION_ERR,PETSC_FALSE));
    618:       }
    619:       if (coeffs && coeffs[ini]!=1.0) PetscCall(MatScale(*S,coeffs[ini]));
    620:       for (i=ini+k+1;i<PetscMax(2,st->nmat);i++) {
    621:         t *= alpha;
    622:         ta = t;
    623:         if (coeffs) ta *= coeffs[i-k];
    624:         if (ta!=0.0) {
    625:           if (st->nmat>1) PetscCall(MatAXPY(*S,ta,A[i],str));
    626:           else PetscCall(MatShift(*S,ta));
    627:         }
    628:       }
    629:     }
    630:   }
    631:   PetscCall(MatSetOption(*S,MAT_SYMMETRIC,st->asymm));
    632:   PetscCall(MatSetOption(*S,MAT_HERMITIAN,(PetscImaginaryPart(st->sigma)==0.0)?st->aherm:PETSC_FALSE));
    633:   PetscFunctionReturn(PETSC_SUCCESS);
    634: }

    636: /*
    637:    Computes the values of the coefficients required by STMatMAXPY_Private
    638:    for the case of monomial basis.
    639: */
    640: PetscErrorCode STCoeffs_Monomial(ST st, PetscScalar *coeffs)
    641: {
    642:   PetscInt  k,i,ini,inip;

    644:   PetscFunctionBegin;
    645:   /* Compute binomial coefficients */
    646:   ini = (st->nmat*(st->nmat-1))/2;
    647:   for (i=0;i<st->nmat;i++) coeffs[ini+i]=1.0;
    648:   for (k=st->nmat-1;k>=1;k--) {
    649:     inip = ini+1;
    650:     ini = (k*(k-1))/2;
    651:     coeffs[ini] = 1.0;
    652:     for (i=1;i<k;i++) coeffs[ini+i] = coeffs[ini+i-1]+coeffs[inip+i-1];
    653:   }
    654:   PetscFunctionReturn(PETSC_SUCCESS);
    655: }

    657: /*@
    658:    STPostSolve - Optional post-solve phase, intended for any actions that must
    659:    be performed on the ST object after the eigensolver has finished.

    661:    Collective

    663:    Input Parameters:
    664: .  st  - the spectral transformation context

    666:    Level: developer

    668: .seealso: EPSSolve()
    669: @*/
    670: PetscErrorCode STPostSolve(ST st)
    671: {
    672:   PetscFunctionBegin;
    675:   PetscTryTypeMethod(st,postsolve);
    676:   PetscFunctionReturn(PETSC_SUCCESS);
    677: }

    679: /*@
    680:    STBackTransform - Back-transformation phase, intended for
    681:    spectral transformations which require to transform the computed
    682:    eigenvalues back to the original eigenvalue problem.

    684:    Not Collective

    686:    Input Parameters:
    687: +  st   - the spectral transformation context
    688: .  n    - number of eigenvalues
    689: .  eigr - real part of a computed eigenvalues
    690: -  eigi - imaginary part of a computed eigenvalues

    692:    Level: developer

    694: .seealso: STIsInjective()
    695: @*/
    696: PetscErrorCode STBackTransform(ST st,PetscInt n,PetscScalar* eigr,PetscScalar* eigi)
    697: {
    698:   PetscFunctionBegin;
    701:   PetscTryTypeMethod(st,backtransform,n,eigr,eigi);
    702:   PetscFunctionReturn(PETSC_SUCCESS);
    703: }

    705: /*@
    706:    STIsInjective - Ask if this spectral transformation is injective or not
    707:    (that is, if it corresponds to a one-to-one mapping). If not, then it
    708:    does not make sense to call STBackTransform().

    710:    Not Collective

    712:    Input Parameter:
    713: .  st   - the spectral transformation context

    715:    Output Parameter:
    716: .  is - the answer

    718:    Level: developer

    720: .seealso: STBackTransform()
    721: @*/
    722: PetscErrorCode STIsInjective(ST st,PetscBool* is)
    723: {
    724:   PetscBool      shell;

    726:   PetscFunctionBegin;
    729:   PetscAssertPointer(is,2);

    731:   PetscCall(PetscObjectTypeCompare((PetscObject)st,STSHELL,&shell));
    732:   if (shell) PetscCall(STIsInjective_Shell(st,is));
    733:   else *is = st->ops->backtransform? PETSC_TRUE: PETSC_FALSE;
    734:   PetscFunctionReturn(PETSC_SUCCESS);
    735: }

    737: /*@
    738:    STMatSetUp - Build the preconditioner matrix used in STMatSolve().

    740:    Collective

    742:    Input Parameters:
    743: +  st     - the spectral transformation context
    744: .  sigma  - the shift
    745: -  coeffs - the coefficients (may be NULL)

    747:    Note:
    748:    This function is not intended to be called by end users, but by SLEPc
    749:    solvers that use ST. It builds matrix st->P as follows, then calls KSPSetUp().
    750: .vb
    751:     If (coeffs)  st->P = Sum_{i=0..nmat-1} coeffs[i]*sigma^i*A_i
    752:     else         st->P = Sum_{i=0..nmat-1} sigma^i*A_i
    753: .ve

    755:    Level: developer

    757: .seealso: STMatSolve()
    758: @*/
    759: PetscErrorCode STMatSetUp(ST st,PetscScalar sigma,PetscScalar *coeffs)
    760: {
    761:   PetscFunctionBegin;
    764:   STCheckMatrices(st,1);

    766:   PetscCall(PetscLogEventBegin(ST_MatSetUp,st,0,0,0));
    767:   PetscCall(STMatMAXPY_Private(st,sigma,0.0,0,coeffs,PETSC_TRUE,PETSC_FALSE,&st->P));
    768:   if (st->Psplit) PetscCall(STMatMAXPY_Private(st,sigma,0.0,0,coeffs,PETSC_TRUE,PETSC_TRUE,&st->Pmat));
    769:   PetscCall(ST_KSPSetOperators(st,st->P,st->Pmat?st->Pmat:st->P));
    770:   PetscCall(KSPSetUp(st->ksp));
    771:   PetscCall(PetscLogEventEnd(ST_MatSetUp,st,0,0,0));
    772:   PetscFunctionReturn(PETSC_SUCCESS);
    773: }

    775: /*@
    776:    STSetWorkVecs - Sets a number of work vectors into the ST object.

    778:    Collective

    780:    Input Parameters:
    781: +  st - the spectral transformation context
    782: -  nw - number of work vectors to allocate

    784:    Developer Notes:
    785:    This is SLEPC_EXTERN because it may be required by shell STs.

    787:    Level: developer

    789: .seealso: STMatCreateVecs()
    790: @*/
    791: PetscErrorCode STSetWorkVecs(ST st,PetscInt nw)
    792: {
    793:   PetscInt       i;

    795:   PetscFunctionBegin;
    798:   PetscCheck(nw>0,PetscObjectComm((PetscObject)st),PETSC_ERR_ARG_OUTOFRANGE,"nw must be > 0: nw = %" PetscInt_FMT,nw);
    799:   if (st->nwork < nw) {
    800:     PetscCall(VecDestroyVecs(st->nwork,&st->work));
    801:     st->nwork = nw;
    802:     PetscCall(PetscMalloc1(nw,&st->work));
    803:     for (i=0;i<nw;i++) PetscCall(STMatCreateVecs(st,&st->work[i],NULL));
    804:   }
    805:   PetscFunctionReturn(PETSC_SUCCESS);
    806: }