Date: 2024-09-25T09:33:20+00:00

Actual source code: lyapii.c
slepc-3.21.2 2024-09-25
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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:    SLEPc eigensolver: "lyapii"

     13:    Method: Lyapunov inverse iteration

     15:    Algorithm:

     17:        Lyapunov inverse iteration using LME solvers

     19:    References:

     21:        [1] H.C. Elman and M. Wu, "Lyapunov inverse iteration for computing a
     22:            few rightmost eigenvalues of large generalized eigenvalue problems",
     23:            SIAM J. Matrix Anal. Appl. 34(4):1685-1707, 2013.

     25:        [2] K. Meerbergen and A. Spence, "Inverse iteration for purely imaginary
     26:            eigenvalues with application to the detection of Hopf bifurcations in
     27:            large-scale problems", SIAM J. Matrix Anal. Appl. 31:1982-1999, 2010.
     28: */

     30: #include <slepc/private/epsimpl.h>
     31: #include <slepcblaslapack.h>

     33: typedef struct {
     34:   LME      lme;      /* Lyapunov solver */
     35:   DS       ds;       /* used to compute the SVD for compression */
     36:   PetscInt rkl;      /* prescribed rank for the Lyapunov solver */
     37:   PetscInt rkc;      /* the compressed rank, cannot be larger than rkl */
     38: } EPS_LYAPII;

     40: typedef struct {
     41:   Mat      S;        /* the operator matrix, S=A^{-1}*B */
     42:   BV       Q;        /* orthogonal basis of converged eigenvectors */
     43: } EPS_LYAPII_MATSHELL;

     45: typedef struct {
     46:   Mat      S;        /* the matrix from which the implicit operator is built */
     47:   PetscInt n;        /* the size of matrix S, the operator is nxn */
     48:   LME      lme;      /* dummy LME object */
     49: #if defined(PETSC_USE_COMPLEX)
     50:   Mat      A,B,F;
     51:   Vec      w;
     52: #endif
     53: } EPS_EIG_MATSHELL;

     55: static PetscErrorCode EPSSetUp_LyapII(EPS eps)
     56: {
     57:   PetscRandom    rand;
     58:   EPS_LYAPII     *ctx = (EPS_LYAPII*)eps->data;

     60:   PetscFunctionBegin;
     61:   EPSCheckSinvert(eps);
     62:   if (eps->ncv!=PETSC_DEFAULT) {
     63:     PetscCheck(eps->ncv>=eps->nev+1,PetscObjectComm((PetscObject)eps),PETSC_ERR_USER_INPUT,"The value of ncv must be at least nev+1");
     64:   } else eps->ncv = eps->nev+1;
     65:   if (eps->mpd!=PETSC_DEFAULT) PetscCall(PetscInfo(eps,"Warning: parameter mpd ignored\n"));
     66:   if (eps->max_it==PETSC_DEFAULT) eps->max_it = PetscMax(1000*eps->nev,100*eps->n);
     67:   if (!eps->which) eps->which=EPS_LARGEST_REAL;
     68:   PetscCheck(eps->which==EPS_LARGEST_REAL,PetscObjectComm((PetscObject)eps),PETSC_ERR_SUP,"This solver supports only largest real eigenvalues");
     69:   EPSCheckUnsupported(eps,EPS_FEATURE_BALANCE | EPS_FEATURE_ARBITRARY | EPS_FEATURE_REGION | EPS_FEATURE_EXTRACTION | EPS_FEATURE_TWOSIDED);

     71:   if (!ctx->rkc) ctx->rkc = 10;
     72:   if (!ctx->rkl) ctx->rkl = 3*ctx->rkc;
     73:   if (!ctx->lme) PetscCall(EPSLyapIIGetLME(eps,&ctx->lme));
     74:   PetscCall(LMESetProblemType(ctx->lme,LME_LYAPUNOV));
     75:   PetscCall(LMESetErrorIfNotConverged(ctx->lme,PETSC_TRUE));

     77:   if (!ctx->ds) {
     78:     PetscCall(DSCreate(PetscObjectComm((PetscObject)eps),&ctx->ds));
     79:     PetscCall(DSSetType(ctx->ds,DSSVD));
     80:   }
     81:   PetscCall(DSAllocate(ctx->ds,ctx->rkl));

     83:   PetscCall(DSSetType(eps->ds,DSNHEP));
     84:   PetscCall(DSAllocate(eps->ds,eps->ncv));

     86:   PetscCall(EPSAllocateSolution(eps,0));
     87:   PetscCall(BVGetRandomContext(eps->V,&rand));  /* make sure the random context is available when duplicating */
     88:   PetscCall(EPSSetWorkVecs(eps,3));
     89:   PetscFunctionReturn(PETSC_SUCCESS);
     90: }

     92: static PetscErrorCode MatMult_EPSLyapIIOperator(Mat M,Vec x,Vec r)
     93: {
     94:   EPS_LYAPII_MATSHELL *matctx;

     96:   PetscFunctionBegin;
     97:   PetscCall(MatShellGetContext(M,&matctx));
     98:   PetscCall(MatMult(matctx->S,x,r));
     99:   PetscCall(BVOrthogonalizeVec(matctx->Q,r,NULL,NULL,NULL));
    100:   PetscFunctionReturn(PETSC_SUCCESS);
    101: }

    103: static PetscErrorCode MatDestroy_EPSLyapIIOperator(Mat M)
    104: {
    105:   EPS_LYAPII_MATSHELL *matctx;

    107:   PetscFunctionBegin;
    108:   PetscCall(MatShellGetContext(M,&matctx));
    109:   PetscCall(MatDestroy(&matctx->S));
    110:   PetscCall(PetscFree(matctx));
    111:   PetscFunctionReturn(PETSC_SUCCESS);
    112: }

    114: static PetscErrorCode MatMult_EigOperator(Mat M,Vec x,Vec y)
    115: {
    116:   EPS_EIG_MATSHELL  *matctx;
    117: #if !defined(PETSC_USE_COMPLEX)
    118:   PetscInt          n,lds;
    119:   PetscScalar       *Y,*C,zero=0.0,done=1.0,dtwo=2.0;
    120:   const PetscScalar *S,*X;
    121:   PetscBLASInt      n_,lds_;
    122: #endif

    124:   PetscFunctionBegin;
    125:   PetscCall(MatShellGetContext(M,&matctx));

    127: #if defined(PETSC_USE_COMPLEX)
    128:   PetscCall(MatMult(matctx->B,x,matctx->w));
    129:   PetscCall(MatSolve(matctx->F,matctx->w,y));
    130: #else
    131:   PetscCall(VecGetArrayRead(x,&X));
    132:   PetscCall(VecGetArray(y,&Y));
    133:   PetscCall(MatDenseGetArrayRead(matctx->S,&S));
    134:   PetscCall(MatDenseGetLDA(matctx->S,&lds));

    136:   n = matctx->n;
    137:   PetscCall(PetscCalloc1(n*n,&C));
    138:   PetscCall(PetscBLASIntCast(n,&n_));
    139:   PetscCall(PetscBLASIntCast(lds,&lds_));

    141:   /* C = 2*S*X*S.' */
    142:   PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&dtwo,S,&lds_,X,&n_,&zero,Y,&n_));
    143:   PetscCallBLAS("BLASgemm",BLASgemm_("N","T",&n_,&n_,&n_,&done,Y,&n_,S,&lds_,&zero,C,&n_));

    145:   /* Solve S*Y + Y*S' = -C */
    146:   PetscCall(LMEDenseLyapunov(matctx->lme,n,(PetscScalar*)S,lds,C,n,Y,n));

    148:   PetscCall(PetscFree(C));
    149:   PetscCall(VecRestoreArrayRead(x,&X));
    150:   PetscCall(VecRestoreArray(y,&Y));
    151:   PetscCall(MatDenseRestoreArrayRead(matctx->S,&S));
    152: #endif
    153:   PetscFunctionReturn(PETSC_SUCCESS);
    154: }

    156: static PetscErrorCode MatDestroy_EigOperator(Mat M)
    157: {
    158:   EPS_EIG_MATSHELL *matctx;

    160:   PetscFunctionBegin;
    161:   PetscCall(MatShellGetContext(M,&matctx));
    162: #if defined(PETSC_USE_COMPLEX)
    163:   PetscCall(MatDestroy(&matctx->A));
    164:   PetscCall(MatDestroy(&matctx->B));
    165:   PetscCall(MatDestroy(&matctx->F));
    166:   PetscCall(VecDestroy(&matctx->w));
    167: #else
    168:   PetscCall(MatDestroy(&matctx->S));
    169: #endif
    170:   PetscCall(PetscFree(matctx));
    171:   PetscFunctionReturn(PETSC_SUCCESS);
    172: }

    174: /*
    175:    EV2x2: solve the eigenproblem for a 2x2 matrix M
    176:  */
    177: static PetscErrorCode EV2x2(PetscScalar *M,PetscInt ld,PetscScalar *wr,PetscScalar *wi,PetscScalar *vec)
    178: {
    179:   PetscBLASInt   lwork=10,ld_;
    180:   PetscScalar    work[10];
    181:   PetscBLASInt   two=2,info;
    182: #if defined(PETSC_USE_COMPLEX)
    183:   PetscReal      rwork[6];
    184: #endif

    186:   PetscFunctionBegin;
    187:   PetscCall(PetscBLASIntCast(ld,&ld_));
    188:   PetscCall(PetscFPTrapPush(PETSC_FP_TRAP_OFF));
    189: #if !defined(PETSC_USE_COMPLEX)
    190:   PetscCallBLAS("LAPACKgeev",LAPACKgeev_("N","V",&two,M,&ld_,wr,wi,NULL,&ld_,vec,&ld_,work,&lwork,&info));
    191: #else
    192:   PetscCallBLAS("LAPACKgeev",LAPACKgeev_("N","V",&two,M,&ld_,wr,NULL,&ld_,vec,&ld_,work,&lwork,rwork,&info));
    193: #endif
    194:   SlepcCheckLapackInfo("geev",info);
    195:   PetscCall(PetscFPTrapPop());
    196:   PetscFunctionReturn(PETSC_SUCCESS);
    197: }

    199: /*
    200:    LyapIIBuildRHS: prepare the right-hand side of the Lyapunov equation SY + YS' = -2*S*Z*S'
    201:    in factored form:
    202:       if (V)  U=sqrt(2)*S*V    (uses 1 work vector)
    203:       else    U=sqrt(2)*S*U    (uses 2 work vectors)
    204:    where U,V are assumed to have rk columns.
    205:  */
    206: static PetscErrorCode LyapIIBuildRHS(Mat S,PetscInt rk,Mat U,BV V,Vec *work)
    207: {
    208:   PetscScalar    *array,*uu;
    209:   PetscInt       i,nloc;
    210:   Vec            v,u=work[0];

    212:   PetscFunctionBegin;
    213:   PetscCall(MatGetLocalSize(U,&nloc,NULL));
    214:   for (i=0;i<rk;i++) {
    215:     PetscCall(MatDenseGetColumn(U,i,&array));
    216:     if (V) PetscCall(BVGetColumn(V,i,&v));
    217:     else {
    218:       v = work[1];
    219:       PetscCall(VecPlaceArray(v,array));
    220:     }
    221:     PetscCall(MatMult(S,v,u));
    222:     if (V) PetscCall(BVRestoreColumn(V,i,&v));
    223:     else PetscCall(VecResetArray(v));
    224:     PetscCall(VecScale(u,PETSC_SQRT2));
    225:     PetscCall(VecGetArray(u,&uu));
    226:     PetscCall(PetscArraycpy(array,uu,nloc));
    227:     PetscCall(VecRestoreArray(u,&uu));
    228:     PetscCall(MatDenseRestoreColumn(U,&array));
    229:   }
    230:   PetscFunctionReturn(PETSC_SUCCESS);
    231: }

    233: /*
    234:    LyapIIBuildEigenMat: create shell matrix Op=A\B with A = kron(I,S)+kron(S,I), B = -2*kron(S,S)
    235:    where S is a sequential square dense matrix of order n.
    236:    v0 is the initial vector, should have the form v0 = w*w' (for instance 1*1')
    237:  */
    238: static PetscErrorCode LyapIIBuildEigenMat(LME lme,Mat S,Mat *Op,Vec *v0)
    239: {
    240:   PetscInt          n,m;
    241:   PetscBool         create=PETSC_FALSE;
    242:   EPS_EIG_MATSHELL  *matctx;
    243: #if defined(PETSC_USE_COMPLEX)
    244:   PetscScalar       theta,*aa,*bb;
    245:   const PetscScalar *ss;
    246:   PetscInt          i,j,f,c,off,ld,lds;
    247:   IS                perm;
    248: #endif

    250:   PetscFunctionBegin;
    251:   PetscCall(MatGetSize(S,&n,NULL));
    252:   if (!*Op) create=PETSC_TRUE;
    253:   else {
    254:     PetscCall(MatGetSize(*Op,&m,NULL));
    255:     if (m!=n*n) create=PETSC_TRUE;
    256:   }
    257:   if (create) {
    258:     PetscCall(MatDestroy(Op));
    259:     PetscCall(VecDestroy(v0));
    260:     PetscCall(PetscNew(&matctx));
    261: #if defined(PETSC_USE_COMPLEX)
    262:     PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,n*n,n*n,NULL,&matctx->A));
    263:     PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,n*n,n*n,NULL,&matctx->B));
    264:     PetscCall(MatCreateVecs(matctx->A,NULL,&matctx->w));
    265: #else
    266:     PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,n,n,NULL,&matctx->S));
    267: #endif
    268:     PetscCall(MatCreateShell(PETSC_COMM_SELF,n*n,n*n,PETSC_DETERMINE,PETSC_DETERMINE,matctx,Op));
    269:     PetscCall(MatShellSetOperation(*Op,MATOP_MULT,(void(*)(void))MatMult_EigOperator));
    270:     PetscCall(MatShellSetOperation(*Op,MATOP_DESTROY,(void(*)(void))MatDestroy_EigOperator));
    271:     PetscCall(MatCreateVecs(*Op,NULL,v0));
    272:   } else {
    273:     PetscCall(MatShellGetContext(*Op,&matctx));
    274: #if defined(PETSC_USE_COMPLEX)
    275:     PetscCall(MatZeroEntries(matctx->A));
    276: #endif
    277:   }
    278: #if defined(PETSC_USE_COMPLEX)
    279:   PetscCall(MatDenseGetArray(matctx->A,&aa));
    280:   PetscCall(MatDenseGetArray(matctx->B,&bb));
    281:   PetscCall(MatDenseGetArrayRead(S,&ss));
    282:   PetscCall(MatDenseGetLDA(S,&lds));
    283:   ld = n*n;
    284:   for (f=0;f<n;f++) {
    285:     off = f*n+f*n*ld;
    286:     for (i=0;i<n;i++) for (j=0;j<n;j++) aa[off+i+j*ld] = ss[i+j*lds];
    287:     for (c=0;c<n;c++) {
    288:       off = f*n+c*n*ld;
    289:       theta = ss[f+c*lds];
    290:       for (i=0;i<n;i++) aa[off+i+i*ld] += theta;
    291:       for (i=0;i<n;i++) for (j=0;j<n;j++) bb[off+i+j*ld] = -2*theta*ss[i+j*lds];
    292:     }
    293:   }
    294:   PetscCall(MatDenseRestoreArray(matctx->A,&aa));
    295:   PetscCall(MatDenseRestoreArray(matctx->B,&bb));
    296:   PetscCall(MatDenseRestoreArrayRead(S,&ss));
    297:   PetscCall(ISCreateStride(PETSC_COMM_SELF,n*n,0,1,&perm));
    298:   PetscCall(MatDestroy(&matctx->F));
    299:   PetscCall(MatDuplicate(matctx->A,MAT_COPY_VALUES,&matctx->F));
    300:   PetscCall(MatLUFactor(matctx->F,perm,perm,NULL));
    301:   PetscCall(ISDestroy(&perm));
    302: #else
    303:   PetscCall(MatCopy(S,matctx->S,SAME_NONZERO_PATTERN));
    304: #endif
    305:   matctx->lme = lme;
    306:   matctx->n = n;
    307:   PetscCall(VecSet(*v0,1.0));
    308:   PetscFunctionReturn(PETSC_SUCCESS);
    309: }

    311: static PetscErrorCode EPSSolve_LyapII(EPS eps)
    312: {
    313:   EPS_LYAPII          *ctx = (EPS_LYAPII*)eps->data;
    314:   PetscInt            i,ldds,rk,nloc,mloc,nv,idx,k;
    315:   Vec                 v,w,z=eps->work[0],v0=NULL;
    316:   Mat                 S,C,Ux[2],Y,Y1,R,U,W,X,Op=NULL;
    317:   BV                  V;
    318:   BVOrthogType        type;
    319:   BVOrthogRefineType  refine;
    320:   PetscScalar         eigr[2],eigi[2],*array,er,ei,*uu,*s,*xx,*aa,pM[4],vec[4];
    321:   PetscReal           eta;
    322:   EPS                 epsrr;
    323:   PetscReal           norm;
    324:   EPS_LYAPII_MATSHELL *matctx;

    326:   PetscFunctionBegin;
    327:   PetscCall(DSGetLeadingDimension(ctx->ds,&ldds));

    329:   /* Operator for the Lyapunov equation */
    330:   PetscCall(PetscNew(&matctx));
    331:   PetscCall(STGetOperator(eps->st,&matctx->S));
    332:   PetscCall(MatGetLocalSize(matctx->S,&mloc,&nloc));
    333:   PetscCall(MatCreateShell(PetscObjectComm((PetscObject)eps),mloc,nloc,PETSC_DETERMINE,PETSC_DETERMINE,matctx,&S));
    334:   matctx->Q = eps->V;
    335:   PetscCall(MatShellSetOperation(S,MATOP_MULT,(void(*)(void))MatMult_EPSLyapIIOperator));
    336:   PetscCall(MatShellSetOperation(S,MATOP_DESTROY,(void(*)(void))MatDestroy_EPSLyapIIOperator));
    337:   PetscCall(LMESetCoefficients(ctx->lme,S,NULL,NULL,NULL));

    339:   /* Right-hand side */
    340:   PetscCall(BVDuplicateResize(eps->V,ctx->rkl,&V));
    341:   PetscCall(BVGetOrthogonalization(V,&type,&refine,&eta,NULL));
    342:   PetscCall(BVSetOrthogonalization(V,type,refine,eta,BV_ORTHOG_BLOCK_TSQR));
    343:   PetscCall(MatCreateDense(PetscObjectComm((PetscObject)eps),eps->nloc,PETSC_DECIDE,PETSC_DECIDE,1,NULL,&Ux[0]));
    344:   PetscCall(MatCreateDense(PetscObjectComm((PetscObject)eps),eps->nloc,PETSC_DECIDE,PETSC_DECIDE,2,NULL,&Ux[1]));
    345:   nv = ctx->rkl;
    346:   PetscCall(PetscMalloc1(nv,&s));

    348:   /* Initialize first column */
    349:   PetscCall(EPSGetStartVector(eps,0,NULL));
    350:   PetscCall(BVGetColumn(eps->V,0,&v));
    351:   PetscCall(BVInsertVec(V,0,v));
    352:   PetscCall(BVRestoreColumn(eps->V,0,&v));
    353:   PetscCall(BVSetActiveColumns(eps->V,0,0));  /* no deflation at the beginning */
    354:   PetscCall(LyapIIBuildRHS(S,1,Ux[0],V,eps->work));
    355:   idx = 0;

    357:   /* EPS for rank reduction */
    358:   PetscCall(EPSCreate(PETSC_COMM_SELF,&epsrr));
    359:   PetscCall(EPSSetOptionsPrefix(epsrr,((PetscObject)eps)->prefix));
    360:   PetscCall(EPSAppendOptionsPrefix(epsrr,"eps_lyapii_"));
    361:   PetscCall(EPSSetDimensions(epsrr,1,PETSC_DEFAULT,PETSC_DEFAULT));
    362:   PetscCall(EPSSetTolerances(epsrr,PETSC_MACHINE_EPSILON*100,PETSC_DEFAULT));

    364:   while (eps->reason == EPS_CONVERGED_ITERATING) {
    365:     eps->its++;

    367:     /* Matrix for placing the solution of the Lyapunov equation (an alias of V) */
    368:     PetscCall(BVSetActiveColumns(V,0,nv));
    369:     PetscCall(BVGetMat(V,&Y1));
    370:     PetscCall(MatZeroEntries(Y1));
    371:     PetscCall(MatCreateLRC(NULL,Y1,NULL,NULL,&Y));
    372:     PetscCall(LMESetSolution(ctx->lme,Y));

    374:     /* Solve the Lyapunov equation SY + YS' = -2*S*Z*S' */
    375:     PetscCall(MatCreateLRC(NULL,Ux[idx],NULL,NULL,&C));
    376:     PetscCall(LMESetRHS(ctx->lme,C));
    377:     PetscCall(MatDestroy(&C));
    378:     PetscCall(LMESolve(ctx->lme));
    379:     PetscCall(BVRestoreMat(V,&Y1));
    380:     PetscCall(MatDestroy(&Y));

    382:     /* SVD of the solution: [Q,R]=qr(V); [U,Sigma,~]=svd(R) */
    383:     PetscCall(DSSetDimensions(ctx->ds,nv,0,0));
    384:     PetscCall(DSSVDSetDimensions(ctx->ds,nv));
    385:     PetscCall(DSGetMat(ctx->ds,DS_MAT_A,&R));
    386:     PetscCall(BVOrthogonalize(V,R));
    387:     PetscCall(DSRestoreMat(ctx->ds,DS_MAT_A,&R));
    388:     PetscCall(DSSetState(ctx->ds,DS_STATE_RAW));
    389:     PetscCall(DSSolve(ctx->ds,s,NULL));

    391:     /* Determine rank */
    392:     rk = nv;
    393:     for (i=1;i<nv;i++) if (PetscAbsScalar(s[i]/s[0])<PETSC_SQRT_MACHINE_EPSILON) {rk=i; break;}
    394:     PetscCall(PetscInfo(eps,"The computed solution of the Lyapunov equation has rank %" PetscInt_FMT "\n",rk));
    395:     rk = PetscMin(rk,ctx->rkc);
    396:     PetscCall(DSGetMat(ctx->ds,DS_MAT_U,&U));
    397:     PetscCall(BVMultInPlace(V,U,0,rk));
    398:     PetscCall(DSRestoreMat(ctx->ds,DS_MAT_U,&U));
    399:     PetscCall(BVSetActiveColumns(V,0,rk));

    401:     /* Rank reduction */
    402:     PetscCall(DSSetDimensions(ctx->ds,rk,0,0));
    403:     PetscCall(DSSVDSetDimensions(ctx->ds,rk));
    404:     PetscCall(DSGetMat(ctx->ds,DS_MAT_A,&W));
    405:     PetscCall(BVMatProject(V,S,V,W));
    406:     PetscCall(LyapIIBuildEigenMat(ctx->lme,W,&Op,&v0)); /* Op=A\B, A=kron(I,S)+kron(S,I), B=-2*kron(S,S) */
    407:     PetscCall(DSRestoreMat(ctx->ds,DS_MAT_A,&W));
    408:     PetscCall(EPSSetOperators(epsrr,Op,NULL));
    409:     PetscCall(EPSSetInitialSpace(epsrr,1,&v0));
    410:     PetscCall(EPSSolve(epsrr));
    411:     PetscCall(EPSComputeVectors(epsrr));
    412:     /* Copy first eigenvector, vec(A)=x */
    413:     PetscCall(BVGetArray(epsrr->V,&xx));
    414:     PetscCall(DSGetArray(ctx->ds,DS_MAT_A,&aa));
    415:     for (i=0;i<rk;i++) PetscCall(PetscArraycpy(aa+i*ldds,xx+i*rk,rk));
    416:     PetscCall(DSRestoreArray(ctx->ds,DS_MAT_A,&aa));
    417:     PetscCall(BVRestoreArray(epsrr->V,&xx));
    418:     PetscCall(DSSetState(ctx->ds,DS_STATE_RAW));
    419:     /* Compute [U,Sigma,~] = svd(A), its rank should be 1 or 2 */
    420:     PetscCall(DSSolve(ctx->ds,s,NULL));
    421:     if (PetscAbsScalar(s[1]/s[0])<PETSC_SQRT_MACHINE_EPSILON) rk=1;
    422:     else rk = 2;
    423:     PetscCall(PetscInfo(eps,"The eigenvector has rank %" PetscInt_FMT "\n",rk));
    424:     PetscCall(DSGetMat(ctx->ds,DS_MAT_U,&U));
    425:     PetscCall(BVMultInPlace(V,U,0,rk));
    426:     PetscCall(DSRestoreMat(ctx->ds,DS_MAT_U,&U));

    428:     /* Save V in Ux */
    429:     idx = (rk==2)?1:0;
    430:     for (i=0;i<rk;i++) {
    431:       PetscCall(BVGetColumn(V,i,&v));
    432:       PetscCall(VecGetArray(v,&uu));
    433:       PetscCall(MatDenseGetColumn(Ux[idx],i,&array));
    434:       PetscCall(PetscArraycpy(array,uu,eps->nloc));
    435:       PetscCall(MatDenseRestoreColumn(Ux[idx],&array));
    436:       PetscCall(VecRestoreArray(v,&uu));
    437:       PetscCall(BVRestoreColumn(V,i,&v));
    438:     }

    440:     /* Eigenpair approximation */
    441:     PetscCall(BVGetColumn(V,0,&v));
    442:     PetscCall(MatMult(S,v,z));
    443:     PetscCall(VecDot(z,v,pM));
    444:     PetscCall(BVRestoreColumn(V,0,&v));
    445:     if (rk>1) {
    446:       PetscCall(BVGetColumn(V,1,&w));
    447:       PetscCall(VecDot(z,w,pM+1));
    448:       PetscCall(MatMult(S,w,z));
    449:       PetscCall(VecDot(z,w,pM+3));
    450:       PetscCall(BVGetColumn(V,0,&v));
    451:       PetscCall(VecDot(z,v,pM+2));
    452:       PetscCall(BVRestoreColumn(V,0,&v));
    453:       PetscCall(BVRestoreColumn(V,1,&w));
    454:       PetscCall(EV2x2(pM,2,eigr,eigi,vec));
    455:       PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,2,2,vec,&X));
    456:       PetscCall(BVSetActiveColumns(V,0,rk));
    457:       PetscCall(BVMultInPlace(V,X,0,rk));
    458:       PetscCall(MatDestroy(&X));
    459: #if !defined(PETSC_USE_COMPLEX)
    460:       norm = eigr[0]*eigr[0]+eigi[0]*eigi[0];
    461:       er = eigr[0]/norm; ei = -eigi[0]/norm;
    462: #else
    463:       er =1.0/eigr[0]; ei = 0.0;
    464: #endif
    465:     } else {
    466:       eigr[0] = pM[0]; eigi[0] = 0.0;
    467:       er = 1.0/eigr[0]; ei = 0.0;
    468:     }
    469:     PetscCall(BVGetColumn(V,0,&v));
    470:     if (eigi[0]!=0.0) PetscCall(BVGetColumn(V,1,&w));
    471:     else w = NULL;
    472:     eps->eigr[eps->nconv] = eigr[0]; eps->eigi[eps->nconv] = eigi[0];
    473:     PetscCall(EPSComputeResidualNorm_Private(eps,PETSC_FALSE,er,ei,v,w,eps->work,&norm));
    474:     PetscCall(BVRestoreColumn(V,0,&v));
    475:     if (w) PetscCall(BVRestoreColumn(V,1,&w));
    476:     PetscCall((*eps->converged)(eps,er,ei,norm,&eps->errest[eps->nconv],eps->convergedctx));
    477:     k = 0;
    478:     if (eps->errest[eps->nconv]<eps->tol) {
    479:       k++;
    480:       if (rk==2) {
    481: #if !defined (PETSC_USE_COMPLEX)
    482:         eps->eigr[eps->nconv+k] = eigr[0]; eps->eigi[eps->nconv+k] = -eigi[0];
    483: #else
    484:         eps->eigr[eps->nconv+k] = PetscConj(eps->eigr[eps->nconv]);
    485: #endif
    486:         k++;
    487:       }
    488:       /* Store converged eigenpairs and vectors for deflation */
    489:       for (i=0;i<k;i++) {
    490:         PetscCall(BVGetColumn(V,i,&v));
    491:         PetscCall(BVInsertVec(eps->V,eps->nconv+i,v));
    492:         PetscCall(BVRestoreColumn(V,i,&v));
    493:       }
    494:       eps->nconv += k;
    495:       PetscCall(BVSetActiveColumns(eps->V,eps->nconv-rk,eps->nconv));
    496:       PetscCall(BVOrthogonalize(eps->V,NULL));
    497:       PetscCall(DSSetDimensions(eps->ds,eps->nconv,0,0));
    498:       PetscCall(DSGetMat(eps->ds,DS_MAT_A,&W));
    499:       PetscCall(BVMatProject(eps->V,matctx->S,eps->V,W));
    500:       PetscCall(DSRestoreMat(eps->ds,DS_MAT_A,&W));
    501:       if (eps->nconv<eps->nev) {
    502:         idx = 0;
    503:         PetscCall(BVSetRandomColumn(V,0));
    504:         PetscCall(BVNormColumn(V,0,NORM_2,&norm));
    505:         PetscCall(BVScaleColumn(V,0,1.0/norm));
    506:         PetscCall(LyapIIBuildRHS(S,1,Ux[idx],V,eps->work));
    507:       }
    508:     } else {
    509:       /* Prepare right-hand side */
    510:       PetscCall(LyapIIBuildRHS(S,rk,Ux[idx],NULL,eps->work));
    511:     }
    512:     PetscCall((*eps->stopping)(eps,eps->its,eps->max_it,eps->nconv,eps->nev,&eps->reason,eps->stoppingctx));
    513:     PetscCall(EPSMonitor(eps,eps->its,eps->nconv,eps->eigr,eps->eigi,eps->errest,eps->nconv+1));
    514:   }
    515:   PetscCall(STRestoreOperator(eps->st,&matctx->S));
    516:   PetscCall(MatDestroy(&S));
    517:   PetscCall(MatDestroy(&Ux[0]));
    518:   PetscCall(MatDestroy(&Ux[1]));
    519:   PetscCall(MatDestroy(&Op));
    520:   PetscCall(VecDestroy(&v0));
    521:   PetscCall(BVDestroy(&V));
    522:   PetscCall(EPSDestroy(&epsrr));
    523:   PetscCall(PetscFree(s));
    524:   PetscFunctionReturn(PETSC_SUCCESS);
    525: }

    527: static PetscErrorCode EPSSetFromOptions_LyapII(EPS eps,PetscOptionItems *PetscOptionsObject)
    528: {
    529:   EPS_LYAPII     *ctx = (EPS_LYAPII*)eps->data;
    530:   PetscInt       k,array[2]={PETSC_DEFAULT,PETSC_DEFAULT};
    531:   PetscBool      flg;

    533:   PetscFunctionBegin;
    534:   PetscOptionsHeadBegin(PetscOptionsObject,"EPS Lyapunov Inverse Iteration Options");

    536:     k = 2;
    537:     PetscCall(PetscOptionsIntArray("-eps_lyapii_ranks","Ranks for Lyapunov equation (one or two comma-separated integers)","EPSLyapIISetRanks",array,&k,&flg));
    538:     if (flg) PetscCall(EPSLyapIISetRanks(eps,array[0],array[1]));

    540:   PetscOptionsHeadEnd();

    542:   if (!ctx->lme) PetscCall(EPSLyapIIGetLME(eps,&ctx->lme));
    543:   PetscCall(LMESetFromOptions(ctx->lme));
    544:   PetscFunctionReturn(PETSC_SUCCESS);
    545: }

    547: static PetscErrorCode EPSLyapIISetRanks_LyapII(EPS eps,PetscInt rkc,PetscInt rkl)
    548: {
    549:   EPS_LYAPII *ctx = (EPS_LYAPII*)eps->data;

    551:   PetscFunctionBegin;
    552:   if (rkc==PETSC_DEFAULT) rkc = 10;
    553:   PetscCheck(rkc>1,PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_OUTOFRANGE,"The compressed rank %" PetscInt_FMT " must be larger than 1",rkc);
    554:   if (rkl==PETSC_DEFAULT) rkl = 3*rkc;
    555:   PetscCheck(rkl>=rkc,PetscObjectComm((PetscObject)eps),PETSC_ERR_ARG_OUTOFRANGE,"The Lyapunov rank %" PetscInt_FMT " cannot be smaller than the compressed rank %" PetscInt_FMT,rkl,rkc);
    556:   if (rkc != ctx->rkc) {
    557:     ctx->rkc   = rkc;
    558:     eps->state = EPS_STATE_INITIAL;
    559:   }
    560:   if (rkl != ctx->rkl) {
    561:     ctx->rkl   = rkl;
    562:     eps->state = EPS_STATE_INITIAL;
    563:   }
    564:   PetscFunctionReturn(PETSC_SUCCESS);
    565: }

    567: /*@
    568:    EPSLyapIISetRanks - Set the ranks used in the solution of the Lyapunov equation.

    570:    Logically Collective

    572:    Input Parameters:
    573: +  eps - the eigenproblem solver context
    574: .  rkc - the compressed rank
    575: -  rkl - the Lyapunov rank

    577:    Options Database Key:
    578: .  -eps_lyapii_ranks <rkc,rkl> - Sets the rank parameters

    580:    Notes:
    581:    Lyapunov inverse iteration needs to solve a large-scale Lyapunov equation
    582:    at each iteration of the eigensolver. For this, an iterative solver (LME)
    583:    is used, which requires to prescribe the rank of the solution matrix X. This
    584:    is the meaning of parameter rkl. Later, this matrix is compressed into
    585:    another matrix of rank rkc. If not provided, rkl is a small multiple of rkc.

    587:    Level: intermediate

    589: .seealso: EPSLyapIIGetRanks()
    590: @*/
    591: PetscErrorCode EPSLyapIISetRanks(EPS eps,PetscInt rkc,PetscInt rkl)
    592: {
    593:   PetscFunctionBegin;
    597:   PetscTryMethod(eps,"EPSLyapIISetRanks_C",(EPS,PetscInt,PetscInt),(eps,rkc,rkl));
    598:   PetscFunctionReturn(PETSC_SUCCESS);
    599: }

    601: static PetscErrorCode EPSLyapIIGetRanks_LyapII(EPS eps,PetscInt *rkc,PetscInt *rkl)
    602: {
    603:   EPS_LYAPII *ctx = (EPS_LYAPII*)eps->data;

    605:   PetscFunctionBegin;
    606:   if (rkc) *rkc = ctx->rkc;
    607:   if (rkl) *rkl = ctx->rkl;
    608:   PetscFunctionReturn(PETSC_SUCCESS);
    609: }

    611: /*@
    612:    EPSLyapIIGetRanks - Return the rank values used for the Lyapunov step.

    614:    Not Collective

    616:    Input Parameter:
    617: .  eps - the eigenproblem solver context

    619:    Output Parameters:
    620: +  rkc - the compressed rank
    621: -  rkl - the Lyapunov rank

    623:    Level: intermediate

    625: .seealso: EPSLyapIISetRanks()
    626: @*/
    627: PetscErrorCode EPSLyapIIGetRanks(EPS eps,PetscInt *rkc,PetscInt *rkl)
    628: {
    629:   PetscFunctionBegin;
    631:   PetscUseMethod(eps,"EPSLyapIIGetRanks_C",(EPS,PetscInt*,PetscInt*),(eps,rkc,rkl));
    632:   PetscFunctionReturn(PETSC_SUCCESS);
    633: }

    635: static PetscErrorCode EPSLyapIISetLME_LyapII(EPS eps,LME lme)
    636: {
    637:   EPS_LYAPII     *ctx = (EPS_LYAPII*)eps->data;

    639:   PetscFunctionBegin;
    640:   PetscCall(PetscObjectReference((PetscObject)lme));
    641:   PetscCall(LMEDestroy(&ctx->lme));
    642:   ctx->lme = lme;
    643:   eps->state = EPS_STATE_INITIAL;
    644:   PetscFunctionReturn(PETSC_SUCCESS);
    645: }

    647: /*@
    648:    EPSLyapIISetLME - Associate a linear matrix equation solver object (LME) to the
    649:    eigenvalue solver.

    651:    Collective

    653:    Input Parameters:
    654: +  eps - the eigenproblem solver context
    655: -  lme - the linear matrix equation solver object

    657:    Level: advanced

    659: .seealso: EPSLyapIIGetLME()
    660: @*/
    661: PetscErrorCode EPSLyapIISetLME(EPS eps,LME lme)
    662: {
    663:   PetscFunctionBegin;
    666:   PetscCheckSameComm(eps,1,lme,2);
    667:   PetscTryMethod(eps,"EPSLyapIISetLME_C",(EPS,LME),(eps,lme));
    668:   PetscFunctionReturn(PETSC_SUCCESS);
    669: }

    671: static PetscErrorCode EPSLyapIIGetLME_LyapII(EPS eps,LME *lme)
    672: {
    673:   EPS_LYAPII     *ctx = (EPS_LYAPII*)eps->data;

    675:   PetscFunctionBegin;
    676:   if (!ctx->lme) {
    677:     PetscCall(LMECreate(PetscObjectComm((PetscObject)eps),&ctx->lme));
    678:     PetscCall(LMESetOptionsPrefix(ctx->lme,((PetscObject)eps)->prefix));
    679:     PetscCall(LMEAppendOptionsPrefix(ctx->lme,"eps_lyapii_"));
    680:     PetscCall(PetscObjectIncrementTabLevel((PetscObject)ctx->lme,(PetscObject)eps,1));
    681:   }
    682:   *lme = ctx->lme;
    683:   PetscFunctionReturn(PETSC_SUCCESS);
    684: }

    686: /*@
    687:    EPSLyapIIGetLME - Retrieve the linear matrix equation solver object (LME)
    688:    associated with the eigenvalue solver.

    690:    Not Collective

    692:    Input Parameter:
    693: .  eps - the eigenproblem solver context

    695:    Output Parameter:
    696: .  lme - the linear matrix equation solver object

    698:    Level: advanced

    700: .seealso: EPSLyapIISetLME()
    701: @*/
    702: PetscErrorCode EPSLyapIIGetLME(EPS eps,LME *lme)
    703: {
    704:   PetscFunctionBegin;
    706:   PetscAssertPointer(lme,2);
    707:   PetscUseMethod(eps,"EPSLyapIIGetLME_C",(EPS,LME*),(eps,lme));
    708:   PetscFunctionReturn(PETSC_SUCCESS);
    709: }

    711: static PetscErrorCode EPSView_LyapII(EPS eps,PetscViewer viewer)
    712: {
    713:   EPS_LYAPII     *ctx = (EPS_LYAPII*)eps->data;
    714:   PetscBool      isascii;

    716:   PetscFunctionBegin;
    717:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii));
    718:   if (isascii) {
    719:     PetscCall(PetscViewerASCIIPrintf(viewer,"  ranks: for Lyapunov solver=%" PetscInt_FMT ", after compression=%" PetscInt_FMT "\n",ctx->rkl,ctx->rkc));
    720:     if (!ctx->lme) PetscCall(EPSLyapIIGetLME(eps,&ctx->lme));
    721:     PetscCall(PetscViewerASCIIPushTab(viewer));
    722:     PetscCall(LMEView(ctx->lme,viewer));
    723:     PetscCall(PetscViewerASCIIPopTab(viewer));
    724:   }
    725:   PetscFunctionReturn(PETSC_SUCCESS);
    726: }

    728: static PetscErrorCode EPSReset_LyapII(EPS eps)
    729: {
    730:   EPS_LYAPII     *ctx = (EPS_LYAPII*)eps->data;

    732:   PetscFunctionBegin;
    733:   if (!ctx->lme) PetscCall(LMEReset(ctx->lme));
    734:   PetscFunctionReturn(PETSC_SUCCESS);
    735: }

    737: static PetscErrorCode EPSDestroy_LyapII(EPS eps)
    738: {
    739:   EPS_LYAPII     *ctx = (EPS_LYAPII*)eps->data;

    741:   PetscFunctionBegin;
    742:   PetscCall(LMEDestroy(&ctx->lme));
    743:   PetscCall(DSDestroy(&ctx->ds));
    744:   PetscCall(PetscFree(eps->data));
    745:   PetscCall(PetscObjectComposeFunction((PetscObject)eps,"EPSLyapIISetLME_C",NULL));
    746:   PetscCall(PetscObjectComposeFunction((PetscObject)eps,"EPSLyapIIGetLME_C",NULL));
    747:   PetscCall(PetscObjectComposeFunction((PetscObject)eps,"EPSLyapIISetRanks_C",NULL));
    748:   PetscCall(PetscObjectComposeFunction((PetscObject)eps,"EPSLyapIIGetRanks_C",NULL));
    749:   PetscFunctionReturn(PETSC_SUCCESS);
    750: }

    752: static PetscErrorCode EPSSetDefaultST_LyapII(EPS eps)
    753: {
    754:   PetscFunctionBegin;
    755:   if (!((PetscObject)eps->st)->type_name) PetscCall(STSetType(eps->st,STSINVERT));
    756:   PetscFunctionReturn(PETSC_SUCCESS);
    757: }

    759: SLEPC_EXTERN PetscErrorCode EPSCreate_LyapII(EPS eps)
    760: {
    761:   EPS_LYAPII     *ctx;

    763:   PetscFunctionBegin;
    764:   PetscCall(PetscNew(&ctx));
    765:   eps->data = (void*)ctx;

    767:   eps->useds = PETSC_TRUE;

    769:   eps->ops->solve          = EPSSolve_LyapII;
    770:   eps->ops->setup          = EPSSetUp_LyapII;
    771:   eps->ops->setupsort      = EPSSetUpSort_Default;
    772:   eps->ops->setfromoptions = EPSSetFromOptions_LyapII;
    773:   eps->ops->reset          = EPSReset_LyapII;
    774:   eps->ops->destroy        = EPSDestroy_LyapII;
    775:   eps->ops->view           = EPSView_LyapII;
    776:   eps->ops->setdefaultst   = EPSSetDefaultST_LyapII;
    777:   eps->ops->backtransform  = EPSBackTransform_Default;
    778:   eps->ops->computevectors = EPSComputeVectors_Schur;

    780:   PetscCall(PetscObjectComposeFunction((PetscObject)eps,"EPSLyapIISetLME_C",EPSLyapIISetLME_LyapII));
    781:   PetscCall(PetscObjectComposeFunction((PetscObject)eps,"EPSLyapIIGetLME_C",EPSLyapIIGetLME_LyapII));
    782:   PetscCall(PetscObjectComposeFunction((PetscObject)eps,"EPSLyapIISetRanks_C",EPSLyapIISetRanks_LyapII));
    783:   PetscCall(PetscObjectComposeFunction((PetscObject)eps,"EPSLyapIIGetRanks_C",EPSLyapIIGetRanks_LyapII));
    784:   PetscFunctionReturn(PETSC_SUCCESS);
    785: }