Date: 2021-09-20T09:43:40+00:00

Actual source code: dshep.c
slepc-3.15.2 2021-09-20
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2021, Universitat Politecnica de Valencia, Spain

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

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

     14: PetscErrorCode DSAllocate_HEP(DS ds,PetscInt ld)
     15: {

     19:   DSAllocateMat_Private(ds,DS_MAT_A);
     20:   DSAllocateMat_Private(ds,DS_MAT_Q);
     21:   DSAllocateMatReal_Private(ds,DS_MAT_T);
     22:   PetscFree(ds->perm);
     23:   PetscMalloc1(ld,&ds->perm);
     24:   PetscLogObjectMemory((PetscObject)ds,ld*sizeof(PetscInt));
     25:   return(0);
     26: }

     28: /*   0       l           k                 n-1
     29:     -----------------------------------------
     30:     |*       .           .                  |
     31:     |  *     .           .                  |
     32:     |    *   .           .                  |
     33:     |      * .           .                  |
     34:     |. . . . o           o                  |
     35:     |          o         o                  |
     36:     |            o       o                  |
     37:     |              o     o                  |
     38:     |                o   o                  |
     39:     |                  o o                  |
     40:     |. . . . o o o o o o o x                |
     41:     |                    x x x              |
     42:     |                      x x x            |
     43:     |                        x x x          |
     44:     |                          x x x        |
     45:     |                            x x x      |
     46:     |                              x x x    |
     47:     |                                x x x  |
     48:     |                                  x x x|
     49:     |                                    x x|
     50:     -----------------------------------------
     51: */

     53: static PetscErrorCode DSSwitchFormat_HEP(DS ds)
     54: {
     56:   PetscReal      *T = ds->rmat[DS_MAT_T];
     57:   PetscScalar    *A = ds->mat[DS_MAT_A];
     58:   PetscInt       i,n=ds->n,k=ds->k,ld=ds->ld;

     61:   /* switch from compact (arrow) to dense storage */
     62:   PetscArrayzero(A,ld*ld);
     63:   for (i=0;i<k;i++) {
     64:     A[i+i*ld] = T[i];
     65:     A[k+i*ld] = T[i+ld];
     66:     A[i+k*ld] = T[i+ld];
     67:   }
     68:   A[k+k*ld] = T[k];
     69:   for (i=k+1;i<n;i++) {
     70:     A[i+i*ld]     = T[i];
     71:     A[i-1+i*ld]   = T[i-1+ld];
     72:     A[i+(i-1)*ld] = T[i-1+ld];
     73:   }
     74:   if (ds->extrarow) A[n+(n-1)*ld] = T[n-1+ld];
     75:   return(0);
     76: }

     78: PetscErrorCode DSView_HEP(DS ds,PetscViewer viewer)
     79: {
     80:   PetscErrorCode    ierr;
     81:   PetscViewerFormat format;
     82:   PetscInt          i,j,r,c,rows;
     83:   PetscReal         value;
     84:   const char        *methodname[] = {
     85:                      "Implicit QR method (_steqr)",
     86:                      "Relatively Robust Representations (_stevr)",
     87:                      "Divide and Conquer method (_stedc)",
     88:                      "Block Divide and Conquer method (dsbtdc)"
     89:   };
     90:   const int         nmeth=sizeof(methodname)/sizeof(methodname[0]);

     93:   PetscViewerGetFormat(viewer,&format);
     94:   if (format == PETSC_VIEWER_ASCII_INFO || format == PETSC_VIEWER_ASCII_INFO_DETAIL) {
     95:     if (ds->bs>1) {
     96:       PetscViewerASCIIPrintf(viewer,"block size: %D\n",ds->bs);
     97:     }
     98:     if (ds->method<nmeth) {
     99:       PetscViewerASCIIPrintf(viewer,"solving the problem with: %s\n",methodname[ds->method]);
    100:     }
    101:     return(0);
    102:   }
    103:   if (ds->compact) {
    104:     PetscViewerASCIIUseTabs(viewer,PETSC_FALSE);
    105:     rows = ds->extrarow? ds->n+1: ds->n;
    106:     if (format == PETSC_VIEWER_ASCII_MATLAB) {
    107:       PetscViewerASCIIPrintf(viewer,"%% Size = %D %D\n",rows,ds->n);
    108:       PetscViewerASCIIPrintf(viewer,"zzz = zeros(%D,3);\n",3*ds->n);
    109:       PetscViewerASCIIPrintf(viewer,"zzz = [\n");
    110:       for (i=0;i<ds->n;i++) {
    111:         PetscViewerASCIIPrintf(viewer,"%D %D  %18.16e\n",i+1,i+1,(double)*(ds->rmat[DS_MAT_T]+i));
    112:       }
    113:       for (i=0;i<rows-1;i++) {
    114:         r = PetscMax(i+2,ds->k+1);
    115:         c = i+1;
    116:         PetscViewerASCIIPrintf(viewer,"%D %D  %18.16e\n",r,c,(double)*(ds->rmat[DS_MAT_T]+ds->ld+i));
    117:         if (i<ds->n-1 && ds->k<ds->n) { /* do not print vertical arrow when k=n */
    118:           PetscViewerASCIIPrintf(viewer,"%D %D  %18.16e\n",c,r,(double)*(ds->rmat[DS_MAT_T]+ds->ld+i));
    119:         }
    120:       }
    121:       PetscViewerASCIIPrintf(viewer,"];\n%s = spconvert(zzz);\n",DSMatName[DS_MAT_T]);
    122:     } else {
    123:       for (i=0;i<rows;i++) {
    124:         for (j=0;j<ds->n;j++) {
    125:           if (i==j) value = *(ds->rmat[DS_MAT_T]+i);
    126:           else if ((i<ds->k && j==ds->k) || (i==ds->k && j<ds->k)) value = *(ds->rmat[DS_MAT_T]+ds->ld+PetscMin(i,j));
    127:           else if (i==j+1 && i>ds->k) value = *(ds->rmat[DS_MAT_T]+ds->ld+i-1);
    128:           else if (i+1==j && j>ds->k) value = *(ds->rmat[DS_MAT_T]+ds->ld+j-1);
    129:           else value = 0.0;
    130:           PetscViewerASCIIPrintf(viewer," %18.16e ",(double)value);
    131:         }
    132:         PetscViewerASCIIPrintf(viewer,"\n");
    133:       }
    134:     }
    135:     PetscViewerASCIIUseTabs(viewer,PETSC_TRUE);
    136:     PetscViewerFlush(viewer);
    137:   } else {
    138:     DSViewMat(ds,viewer,DS_MAT_A);
    139:   }
    140:   if (ds->state>DS_STATE_INTERMEDIATE) { DSViewMat(ds,viewer,DS_MAT_Q); }
    141:   return(0);
    142: }

    144: PetscErrorCode DSVectors_HEP(DS ds,DSMatType mat,PetscInt *j,PetscReal *rnorm)
    145: {
    146:   PetscScalar    *Q = ds->mat[DS_MAT_Q];
    147:   PetscInt       ld = ds->ld;

    151:   switch (mat) {
    152:     case DS_MAT_X:
    153:     case DS_MAT_Y:
    154:       if (j) {
    155:         if (ds->state>=DS_STATE_CONDENSED) {
    156:           PetscArraycpy(ds->mat[mat]+(*j)*ld,Q+(*j)*ld,ld);
    157:         } else {
    158:           PetscArrayzero(ds->mat[mat]+(*j)*ld,ld);
    159:           *(ds->mat[mat]+(*j)+(*j)*ld) = 1.0;
    160:         }
    161:       } else {
    162:         if (ds->state>=DS_STATE_CONDENSED) {
    163:           PetscArraycpy(ds->mat[mat],Q,ld*ld);
    164:         } else {
    165:           DSSetIdentity(ds,mat);
    166:         }
    167:       }
    168:       if (rnorm && j) *rnorm = PetscAbsScalar(Q[ds->n-1+(*j)*ld]);
    169:       break;
    170:     case DS_MAT_U:
    171:     case DS_MAT_VT:
    172:       SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"Not implemented yet");
    173:     default:
    174:       SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_OUTOFRANGE,"Invalid mat parameter");
    175:   }
    176:   return(0);
    177: }

    179: /*
    180:   ARROWTRIDIAG reduces a symmetric arrowhead matrix of the form

    182:                 [ d 0 0 0 e ]
    183:                 [ 0 d 0 0 e ]
    184:             A = [ 0 0 d 0 e ]
    185:                 [ 0 0 0 d e ]
    186:                 [ e e e e d ]

    188:   to tridiagonal form

    190:                 [ d e 0 0 0 ]
    191:                 [ e d e 0 0 ]
    192:    T = Q'*A*Q = [ 0 e d e 0 ],
    193:                 [ 0 0 e d e ]
    194:                 [ 0 0 0 e d ]

    196:   where Q is an orthogonal matrix. Rutishauser's algorithm is used to
    197:   perform the reduction, which requires O(n**2) flops. The accumulation
    198:   of the orthogonal factor Q, however, requires O(n**3) flops.

    200:   Arguments
    201:   =========

    203:   N       (input) INTEGER
    204:           The order of the matrix A.  N >= 0.

    206:   D       (input/output) DOUBLE PRECISION array, dimension (N)
    207:           On entry, the diagonal entries of the matrix A to be
    208:           reduced.
    209:           On exit, the diagonal entries of the reduced matrix T.

    211:   E       (input/output) DOUBLE PRECISION array, dimension (N-1)
    212:           On entry, the off-diagonal entries of the matrix A to be
    213:           reduced.
    214:           On exit, the subdiagonal entries of the reduced matrix T.

    216:   Q       (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
    217:           On exit, the orthogonal matrix Q.

    219:   LDQ     (input) INTEGER
    220:           The leading dimension of the array Q.

    222:   Note
    223:   ====
    224:   Based on Fortran code contributed by Daniel Kressner
    225: */
    226: static PetscErrorCode ArrowTridiag(PetscBLASInt n,PetscReal *d,PetscReal *e,PetscScalar *Q,PetscBLASInt ld)
    227: {
    228:   PetscBLASInt i,j,j2,one=1;
    229:   PetscReal    c,s,p,off,temp;

    232:   if (n<=2) return(0);

    234:   for (j=0;j<n-2;j++) {

    236:     /* Eliminate entry e(j) by a rotation in the planes (j,j+1) */
    237:     temp = e[j+1];
    238:     PetscStackCallBLAS("LAPACKlartg",LAPACKREALlartg_(&temp,&e[j],&c,&s,&e[j+1]));
    239:     s = -s;

    241:     /* Apply rotation to diagonal elements */
    242:     temp   = d[j+1];
    243:     e[j]   = c*s*(temp-d[j]);
    244:     d[j+1] = s*s*d[j] + c*c*temp;
    245:     d[j]   = c*c*d[j] + s*s*temp;

    247:     /* Apply rotation to Q */
    248:     j2 = j+2;
    249:     PetscStackCallBLAS("BLASrot",BLASMIXEDrot_(&j2,Q+j*ld,&one,Q+(j+1)*ld,&one,&c,&s));

    251:     /* Chase newly introduced off-diagonal entry to the top left corner */
    252:     for (i=j-1;i>=0;i--) {
    253:       off  = -s*e[i];
    254:       e[i] = c*e[i];
    255:       temp = e[i+1];
    256:       PetscStackCallBLAS("LAPACKlartg",LAPACKREALlartg_(&temp,&off,&c,&s,&e[i+1]));
    257:       s = -s;
    258:       temp = (d[i]-d[i+1])*s - 2.0*c*e[i];
    259:       p = s*temp;
    260:       d[i+1] += p;
    261:       d[i] -= p;
    262:       e[i] = -e[i] - c*temp;
    263:       j2 = j+2;
    264:       PetscStackCallBLAS("BLASrot",BLASMIXEDrot_(&j2,Q+i*ld,&one,Q+(i+1)*ld,&one,&c,&s));
    265:     }
    266:   }
    267:   return(0);
    268: }

    270: /*
    271:    Reduce to tridiagonal form by means of ArrowTridiag.
    272: */
    273: static PetscErrorCode DSIntermediate_HEP(DS ds)
    274: {
    276:   PetscInt       i;
    277:   PetscBLASInt   n1 = 0,n2,lwork,info,l = 0,n = 0,ld,off;
    278:   PetscScalar    *A,*Q,*work,*tau;
    279:   PetscReal      *d,*e;

    282:   PetscBLASIntCast(ds->n,&n);
    283:   PetscBLASIntCast(ds->l,&l);
    284:   PetscBLASIntCast(ds->ld,&ld);
    285:   PetscBLASIntCast(PetscMax(0,ds->k-l+1),&n1); /* size of leading block, excl. locked */
    286:   n2 = n-l;     /* n2 = n1 + size of trailing block */
    287:   off = l+l*ld;
    288:   A  = ds->mat[DS_MAT_A];
    289:   Q  = ds->mat[DS_MAT_Q];
    290:   d  = ds->rmat[DS_MAT_T];
    291:   e  = ds->rmat[DS_MAT_T]+ld;
    292:   PetscArrayzero(Q,ld*ld);
    293:   for (i=0;i<n;i++) Q[i+i*ld] = 1.0;

    295:   if (ds->compact) {

    297:     if (ds->state<DS_STATE_INTERMEDIATE) ArrowTridiag(n1,d+l,e+l,Q+off,ld);

    299:   } else {

    301:     for (i=0;i<l;i++) { d[i] = PetscRealPart(A[i+i*ld]); e[i] = 0.0; }

    303:     if (ds->state<DS_STATE_INTERMEDIATE) {
    304:       DSCopyMatrix_Private(ds,DS_MAT_Q,DS_MAT_A);
    305:       DSAllocateWork_Private(ds,ld+ld*ld,0,0);
    306:       tau  = ds->work;
    307:       work = ds->work+ld;
    308:       lwork = ld*ld;
    309:       PetscStackCallBLAS("LAPACKsytrd",LAPACKsytrd_("L",&n2,Q+off,&ld,d+l,e+l,tau,work,&lwork,&info));
    310:       SlepcCheckLapackInfo("sytrd",info);
    311:       PetscStackCallBLAS("LAPACKorgtr",LAPACKorgtr_("L",&n2,Q+off,&ld,tau,work,&lwork,&info));
    312:       SlepcCheckLapackInfo("orgtr",info);
    313:     } else {
    314:       /* copy tridiagonal to d,e */
    315:       for (i=l;i<n;i++)   d[i] = PetscRealPart(A[i+i*ld]);
    316:       for (i=l;i<n-1;i++) e[i] = PetscRealPart(A[(i+1)+i*ld]);
    317:     }
    318:   }
    319:   return(0);
    320: }

    322: PetscErrorCode DSSort_HEP(DS ds,PetscScalar *wr,PetscScalar *wi,PetscScalar *rr,PetscScalar *ri,PetscInt *k)
    323: {
    325:   PetscInt       n,l,i,*perm,ld=ds->ld;
    326:   PetscScalar    *A;
    327:   PetscReal      *d;

    330:   if (!ds->sc) return(0);
    331:   n = ds->n;
    332:   l = ds->l;
    333:   A = ds->mat[DS_MAT_A];
    334:   d = ds->rmat[DS_MAT_T];
    335:   perm = ds->perm;
    336:   if (!rr) {
    337:     DSSortEigenvaluesReal_Private(ds,d,perm);
    338:   } else {
    339:     DSSortEigenvalues_Private(ds,rr,ri,perm,PETSC_FALSE);
    340:   }
    341:   for (i=l;i<n;i++) wr[i] = d[perm[i]];
    342:   DSPermuteColumns_Private(ds,l,n,DS_MAT_Q,perm);
    343:   for (i=l;i<n;i++) d[i] = PetscRealPart(wr[i]);
    344:   if (!ds->compact) {
    345:     for (i=l;i<n;i++) A[i+i*ld] = wr[i];
    346:   }
    347:   return(0);
    348: }

    350: PetscErrorCode DSUpdateExtraRow_HEP(DS ds)
    351: {
    353:   PetscInt       i;
    354:   PetscBLASInt   n,ld,incx=1;
    355:   PetscScalar    *A,*Q,*x,*y,one=1.0,zero=0.0;
    356:   PetscReal      *e,beta;

    359:   PetscBLASIntCast(ds->n,&n);
    360:   PetscBLASIntCast(ds->ld,&ld);
    361:   A  = ds->mat[DS_MAT_A];
    362:   Q  = ds->mat[DS_MAT_Q];
    363:   e  = ds->rmat[DS_MAT_T]+ld;

    365:   if (ds->compact) {
    366:     beta = e[n-1];   /* in compact, we assume all entries are zero except the last one */
    367:     for (i=0;i<n;i++) e[i] = PetscRealPart(beta*Q[n-1+i*ld]);
    368:     ds->k = n;
    369:   } else {
    370:     DSAllocateWork_Private(ds,2*ld,0,0);
    371:     x = ds->work;
    372:     y = ds->work+ld;
    373:     for (i=0;i<n;i++) x[i] = PetscConj(A[n+i*ld]);
    374:     PetscStackCallBLAS("BLASgemv",BLASgemv_("C",&n,&n,&one,Q,&ld,x,&incx,&zero,y,&incx));
    375:     for (i=0;i<n;i++) A[n+i*ld] = PetscConj(y[i]);
    376:     ds->k = n;
    377:   }
    378:   return(0);
    379: }

    381: PetscErrorCode DSSolve_HEP_QR(DS ds,PetscScalar *wr,PetscScalar *wi)
    382: {
    384:   PetscInt       i;
    385:   PetscBLASInt   n1,info,l = 0,n = 0,ld,off;
    386:   PetscScalar    *Q,*A;
    387:   PetscReal      *d,*e;

    390:   if (ds->bs>1) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"This method is not prepared for bs>1");
    391:   PetscBLASIntCast(ds->n,&n);
    392:   PetscBLASIntCast(ds->l,&l);
    393:   PetscBLASIntCast(ds->ld,&ld);
    394:   n1 = n-l;     /* n1 = size of leading block, excl. locked + size of trailing block */
    395:   off = l+l*ld;
    396:   Q  = ds->mat[DS_MAT_Q];
    397:   A  = ds->mat[DS_MAT_A];
    398:   d  = ds->rmat[DS_MAT_T];
    399:   e  = ds->rmat[DS_MAT_T]+ld;

    401:   /* Reduce to tridiagonal form */
    402:   DSIntermediate_HEP(ds);

    404:   /* Solve the tridiagonal eigenproblem */
    405:   for (i=0;i<l;i++) wr[i] = d[i];

    407:   DSAllocateWork_Private(ds,0,2*ld,0);
    408:   PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("V",&n1,d+l,e+l,Q+off,&ld,ds->rwork,&info));
    409:   SlepcCheckLapackInfo("steqr",info);
    410:   for (i=l;i<n;i++) wr[i] = d[i];

    412:   /* Create diagonal matrix as a result */
    413:   if (ds->compact) {
    414:     PetscArrayzero(e,n-1);
    415:   } else {
    416:     for (i=l;i<n;i++) {
    417:       PetscArrayzero(A+l+i*ld,n-l);
    418:     }
    419:     for (i=l;i<n;i++) A[i+i*ld] = d[i];
    420:   }

    422:   /* Set zero wi */
    423:   if (wi) for (i=l;i<n;i++) wi[i] = 0.0;
    424:   return(0);
    425: }

    427: PetscErrorCode DSSolve_HEP_MRRR(DS ds,PetscScalar *wr,PetscScalar *wi)
    428: {
    430:   PetscInt       i;
    431:   PetscBLASInt   n1 = 0,n2 = 0,n3,lwork,liwork,info,l = 0,n = 0,m = 0,ld,off,il,iu,*isuppz;
    432:   PetscScalar    *A,*Q,*W=NULL,one=1.0,zero=0.0;
    433:   PetscReal      *d,*e,abstol=0.0,vl,vu;
    434: #if defined(PETSC_USE_COMPLEX)
    435:   PetscInt       j;
    436:   PetscReal      *ritz;
    437: #endif

    440:   if (ds->bs>1) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"This method is not prepared for bs>1");
    441:   PetscBLASIntCast(ds->n,&n);
    442:   PetscBLASIntCast(ds->l,&l);
    443:   PetscBLASIntCast(ds->ld,&ld);
    444:   PetscBLASIntCast(ds->k-l+1,&n1); /* size of leading block, excl. locked */
    445:   PetscBLASIntCast(n-ds->k-1,&n2); /* size of trailing block */
    446:   n3 = n1+n2;
    447:   off = l+l*ld;
    448:   A  = ds->mat[DS_MAT_A];
    449:   Q  = ds->mat[DS_MAT_Q];
    450:   d  = ds->rmat[DS_MAT_T];
    451:   e  = ds->rmat[DS_MAT_T]+ld;

    453:   /* Reduce to tridiagonal form */
    454:   DSIntermediate_HEP(ds);

    456:   /* Solve the tridiagonal eigenproblem */
    457:   for (i=0;i<l;i++) wr[i] = d[i];

    459:   if (ds->state<DS_STATE_INTERMEDIATE) {  /* Q contains useful info */
    460:     DSAllocateMat_Private(ds,DS_MAT_W);
    461:     DSCopyMatrix_Private(ds,DS_MAT_W,DS_MAT_Q);
    462:     W = ds->mat[DS_MAT_W];
    463:   }
    464: #if defined(PETSC_USE_COMPLEX)
    465:   DSAllocateMatReal_Private(ds,DS_MAT_Q);
    466: #endif
    467:   lwork = 20*ld;
    468:   liwork = 10*ld;
    469:   DSAllocateWork_Private(ds,0,lwork+ld,liwork+2*ld);
    470:   isuppz = ds->iwork+liwork;
    471: #if defined(PETSC_USE_COMPLEX)
    472:   ritz = ds->rwork+lwork;
    473:   PetscStackCallBLAS("LAPACKstevr",LAPACKstevr_("V","A",&n3,d+l,e+l,&vl,&vu,&il,&iu,&abstol,&m,ritz+l,ds->rmat[DS_MAT_Q]+off,&ld,isuppz,ds->rwork,&lwork,ds->iwork,&liwork,&info));
    474:   for (i=l;i<n;i++) wr[i] = ritz[i];
    475: #else
    476:   PetscStackCallBLAS("LAPACKstevr",LAPACKstevr_("V","A",&n3,d+l,e+l,&vl,&vu,&il,&iu,&abstol,&m,wr+l,Q+off,&ld,isuppz,ds->rwork,&lwork,ds->iwork,&liwork,&info));
    477: #endif
    478:   SlepcCheckLapackInfo("stevr",info);
    479: #if defined(PETSC_USE_COMPLEX)
    480:   for (i=l;i<n;i++)
    481:     for (j=l;j<n;j++)
    482:       Q[i+j*ld] = (ds->rmat[DS_MAT_Q])[i+j*ld];
    483: #endif
    484:   if (ds->state<DS_STATE_INTERMEDIATE) {  /* accumulate previous Q */
    485:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n3,&n3,&n3,&one,W+off,&ld,Q+off,&ld,&zero,A+off,&ld));
    486:     DSCopyMatrix_Private(ds,DS_MAT_Q,DS_MAT_A);
    487:   }
    488:   for (i=l;i<n;i++) d[i] = PetscRealPart(wr[i]);

    490:   /* Create diagonal matrix as a result */
    491:   if (ds->compact) {
    492:     PetscArrayzero(e,n-1);
    493:   } else {
    494:     for (i=l;i<n;i++) {
    495:       PetscArrayzero(A+l+i*ld,n-l);
    496:     }
    497:     for (i=l;i<n;i++) A[i+i*ld] = d[i];
    498:   }

    500:   /* Set zero wi */
    501:   if (wi) for (i=l;i<n;i++) wi[i] = 0.0;
    502:   return(0);
    503: }

    505: PetscErrorCode DSSolve_HEP_DC(DS ds,PetscScalar *wr,PetscScalar *wi)
    506: {
    508:   PetscInt       i;
    509:   PetscBLASInt   n1,info,l = 0,ld,off,lrwork,liwork;
    510:   PetscScalar    *Q,*A;
    511:   PetscReal      *d,*e;
    512: #if defined(PETSC_USE_COMPLEX)
    513:   PetscBLASInt   lwork;
    514:   PetscInt       j;
    515: #endif

    518:   if (ds->bs>1) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"This method is not prepared for bs>1");
    519:   PetscBLASIntCast(ds->l,&l);
    520:   PetscBLASIntCast(ds->ld,&ld);
    521:   PetscBLASIntCast(ds->n-ds->l,&n1);
    522:   off = l+l*ld;
    523:   Q  = ds->mat[DS_MAT_Q];
    524:   A  = ds->mat[DS_MAT_A];
    525:   d  = ds->rmat[DS_MAT_T];
    526:   e  = ds->rmat[DS_MAT_T]+ld;

    528:   /* Reduce to tridiagonal form */
    529:   DSIntermediate_HEP(ds);

    531:   /* Solve the tridiagonal eigenproblem */
    532:   for (i=0;i<l;i++) wr[i] = d[i];

    534:   lrwork = 5*n1*n1+3*n1+1;
    535:   liwork = 5*n1*n1+6*n1+6;
    536: #if !defined(PETSC_USE_COMPLEX)
    537:   DSAllocateWork_Private(ds,0,lrwork,liwork);
    538:   PetscStackCallBLAS("LAPACKstedc",LAPACKstedc_("V",&n1,d+l,e+l,Q+off,&ld,ds->rwork,&lrwork,ds->iwork,&liwork,&info));
    539: #else
    540:   lwork = ld*ld;
    541:   DSAllocateWork_Private(ds,lwork,lrwork,liwork);
    542:   PetscStackCallBLAS("LAPACKstedc",LAPACKstedc_("V",&n1,d+l,e+l,Q+off,&ld,ds->work,&lwork,ds->rwork,&lrwork,ds->iwork,&liwork,&info));
    543:   /* Fixing Lapack bug*/
    544:   for (j=ds->l;j<ds->n;j++)
    545:     for (i=0;i<ds->l;i++) Q[i+j*ld] = 0.0;
    546: #endif
    547:   SlepcCheckLapackInfo("stedc",info);
    548:   for (i=l;i<ds->n;i++) wr[i] = d[i];

    550:   /* Create diagonal matrix as a result */
    551:   if (ds->compact) {
    552:     PetscArrayzero(e,ds->n-1);
    553:   } else {
    554:     for (i=l;i<ds->n;i++) {
    555:       PetscArrayzero(A+l+i*ld,ds->n-l);
    556:     }
    557:     for (i=l;i<ds->n;i++) A[i+i*ld] = d[i];
    558:   }

    560:   /* Set zero wi */
    561:   if (wi) for (i=l;i<ds->n;i++) wi[i] = 0.0;
    562:   return(0);
    563: }

    565: #if !defined(PETSC_USE_COMPLEX)
    566: PetscErrorCode DSSolve_HEP_BDC(DS ds,PetscScalar *wr,PetscScalar *wi)
    567: {
    569:   PetscBLASInt   i,j,k,m,n = 0,info,nblks,bs = 0,ld = 0,lde,lrwork,liwork,*ksizes,*iwork,mingapi;
    570:   PetscScalar    *Q,*A;
    571:   PetscReal      *D,*E,*d,*e,tol=PETSC_MACHINE_EPSILON/2,tau1=1e-16,tau2=1e-18,*rwork,mingap;

    574:   if (ds->l>0) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"This method is not prepared for l>1");
    575:   if (ds->compact) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"Not implemented for compact storage");
    576:   PetscBLASIntCast(ds->ld,&ld);
    577:   PetscBLASIntCast(ds->bs,&bs);
    578:   PetscBLASIntCast(ds->n,&n);
    579:   nblks = n/bs;
    580:   Q  = ds->mat[DS_MAT_Q];
    581:   A  = ds->mat[DS_MAT_A];
    582:   d  = ds->rmat[DS_MAT_T];
    583:   e  = ds->rmat[DS_MAT_T]+ld;
    584:   lrwork = 4*n*n+60*n+1;
    585:   liwork = 5*n+5*nblks-1;
    586:   lde = 2*bs+1;
    587:   DSAllocateWork_Private(ds,bs*n+lde*lde*(nblks-1),lrwork,nblks+liwork);
    588:   D      = ds->work;
    589:   E      = ds->work+bs*n;
    590:   rwork  = ds->rwork;
    591:   ksizes = ds->iwork;
    592:   iwork  = ds->iwork+nblks;
    593:   PetscArrayzero(iwork,liwork);

    595:   /* Copy matrix to block tridiagonal format */
    596:   j=0;
    597:   for (i=0;i<nblks;i++) {
    598:     ksizes[i]=bs;
    599:     for (k=0;k<bs;k++)
    600:       for (m=0;m<bs;m++)
    601:         D[k+m*bs+i*bs*bs] = PetscRealPart(A[j+k+(j+m)*n]);
    602:     j = j + bs;
    603:   }
    604:   j=0;
    605:   for (i=0;i<nblks-1;i++) {
    606:     for (k=0;k<bs;k++)
    607:       for (m=0;m<bs;m++)
    608:         E[k+m*lde+i*lde*lde] = PetscRealPart(A[j+bs+k+(j+m)*n]);
    609:     j = j + bs;
    610:   }

    612:   /* Solve the block tridiagonal eigenproblem */
    613:   BDC_dsbtdc_("D","A",n,nblks,ksizes,D,bs,bs,E,lde,lde,tol,tau1,tau2,d,
    614:            Q,n,rwork,lrwork,iwork,liwork,&mingap,&mingapi,&info,1,1);
    615:   for (i=0;i<ds->n;i++) wr[i] = d[i];

    617:   /* Create diagonal matrix as a result */
    618:   if (ds->compact) {
    619:     PetscArrayzero(e,ds->n-1);
    620:   } else {
    621:     for (i=0;i<ds->n;i++) {
    622:       PetscArrayzero(A+i*ld,ds->n);
    623:     }
    624:     for (i=0;i<ds->n;i++) A[i+i*ld] = wr[i];
    625:   }

    627:   /* Set zero wi */
    628:   if (wi) for (i=0;i<ds->n;i++) wi[i] = 0.0;
    629:   return(0);
    630: }
    631: #endif

    633: PetscErrorCode DSTruncate_HEP(DS ds,PetscInt n,PetscBool trim)
    634: {
    635:   PetscInt    i,ld=ds->ld,l=ds->l;
    636:   PetscScalar *A = ds->mat[DS_MAT_A];

    639:   if (trim) {
    640:     if (!ds->compact && ds->extrarow) {   /* clean extra row */
    641:       for (i=l;i<ds->n;i++) A[ds->n+i*ld] = 0.0;
    642:     }
    643:     ds->l = 0;
    644:     ds->k = 0;
    645:     ds->n = n;
    646:     ds->t = ds->n;   /* truncated length equal to the new dimension */
    647:   } else {
    648:     if (!ds->compact && ds->extrarow && ds->k==ds->n) {
    649:       /* copy entries of extra row to the new position, then clean last row */
    650:       for (i=l;i<n;i++) A[n+i*ld] = A[ds->n+i*ld];
    651:       for (i=l;i<ds->n;i++) A[ds->n+i*ld] = 0.0;
    652:     }
    653:     ds->k = (ds->extrarow)? n: 0;
    654:     ds->t = ds->n;   /* truncated length equal to previous dimension */
    655:     ds->n = n;
    656:   }
    657:   return(0);
    658: }

    660: PetscErrorCode DSSynchronize_HEP(DS ds,PetscScalar eigr[],PetscScalar eigi[])
    661: {
    663:   PetscInt       ld=ds->ld,l=ds->l,k=0,kr=0;
    664:   PetscMPIInt    n,rank,off=0,size,ldn,ld3;

    667:   if (ds->compact) kr = 3*ld;
    668:   else k = (ds->n-l)*ld;
    669:   if (ds->state>DS_STATE_RAW) k += (ds->n-l)*ld;
    670:   if (eigr) k += (ds->n-l);
    671:   DSAllocateWork_Private(ds,k+kr,0,0);
    672:   PetscMPIIntCast(k*sizeof(PetscScalar)+kr*sizeof(PetscReal),&size);
    673:   PetscMPIIntCast(ds->n-l,&n);
    674:   PetscMPIIntCast(ld*(ds->n-l),&ldn);
    675:   PetscMPIIntCast(ld*3,&ld3);
    676:   MPI_Comm_rank(PetscObjectComm((PetscObject)ds),&rank);CHKERRMPI(ierr);
    677:   if (!rank) {
    678:     if (ds->compact) {
    679:       MPI_Pack(ds->rmat[DS_MAT_T],ld3,MPIU_REAL,ds->work,size,&off,PetscObjectComm((PetscObject)ds));CHKERRMPI(ierr);
    680:     } else {
    681:       MPI_Pack(ds->mat[DS_MAT_A]+l*ld,ldn,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds));CHKERRMPI(ierr);
    682:     }
    683:     if (ds->state>DS_STATE_RAW) {
    684:       MPI_Pack(ds->mat[DS_MAT_Q]+l*ld,ldn,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds));CHKERRMPI(ierr);
    685:     }
    686:     if (eigr) {
    687:       MPI_Pack(eigr+l,n,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds));CHKERRMPI(ierr);
    688:     }
    689:   }
    690:   MPI_Bcast(ds->work,size,MPI_BYTE,0,PetscObjectComm((PetscObject)ds));CHKERRMPI(ierr);
    691:   if (rank) {
    692:     if (ds->compact) {
    693:       MPI_Unpack(ds->work,size,&off,ds->rmat[DS_MAT_T],ld3,MPIU_REAL,PetscObjectComm((PetscObject)ds));CHKERRMPI(ierr);
    694:     } else {
    695:       MPI_Unpack(ds->work,size,&off,ds->mat[DS_MAT_A]+l*ld,ldn,MPIU_SCALAR,PetscObjectComm((PetscObject)ds));CHKERRMPI(ierr);
    696:     }
    697:     if (ds->state>DS_STATE_RAW) {
    698:       MPI_Unpack(ds->work,size,&off,ds->mat[DS_MAT_Q]+l*ld,ldn,MPIU_SCALAR,PetscObjectComm((PetscObject)ds));CHKERRMPI(ierr);
    699:     }
    700:     if (eigr) {
    701:       MPI_Unpack(ds->work,size,&off,eigr+l,n,MPIU_SCALAR,PetscObjectComm((PetscObject)ds));CHKERRMPI(ierr);
    702:     }
    703:   }
    704:   return(0);
    705: }

    707: PetscErrorCode DSCond_HEP(DS ds,PetscReal *cond)
    708: {
    710:   PetscScalar    *work;
    711:   PetscReal      *rwork;
    712:   PetscBLASInt   *ipiv;
    713:   PetscBLASInt   lwork,info,n,ld;
    714:   PetscReal      hn,hin;
    715:   PetscScalar    *A;

    718:   PetscBLASIntCast(ds->n,&n);
    719:   PetscBLASIntCast(ds->ld,&ld);
    720:   lwork = 8*ld;
    721:   DSAllocateWork_Private(ds,lwork,ld,ld);
    722:   work  = ds->work;
    723:   rwork = ds->rwork;
    724:   ipiv  = ds->iwork;
    725:   DSSwitchFormat_HEP(ds);

    727:   /* use workspace matrix W to avoid overwriting A */
    728:   DSAllocateMat_Private(ds,DS_MAT_W);
    729:   A = ds->mat[DS_MAT_W];
    730:   PetscArraycpy(A,ds->mat[DS_MAT_A],ds->ld*ds->ld);

    732:   /* norm of A */
    733:   hn = LAPACKlange_("I",&n,&n,A,&ld,rwork);

    735:   /* norm of inv(A) */
    736:   PetscStackCallBLAS("LAPACKgetrf",LAPACKgetrf_(&n,&n,A,&ld,ipiv,&info));
    737:   SlepcCheckLapackInfo("getrf",info);
    738:   PetscStackCallBLAS("LAPACKgetri",LAPACKgetri_(&n,A,&ld,ipiv,work,&lwork,&info));
    739:   SlepcCheckLapackInfo("getri",info);
    740:   hin = LAPACKlange_("I",&n,&n,A,&ld,rwork);

    742:   *cond = hn*hin;
    743:   return(0);
    744: }

    746: PetscErrorCode DSTranslateRKS_HEP(DS ds,PetscScalar alpha)
    747: {
    749:   PetscInt       i,j,k=ds->k;
    750:   PetscScalar    *Q,*A,*R,*tau,*work;
    751:   PetscBLASInt   ld,n1,n0,lwork,info;

    754:   PetscBLASIntCast(ds->ld,&ld);
    755:   DSAllocateWork_Private(ds,ld*ld,0,0);
    756:   tau = ds->work;
    757:   work = ds->work+ld;
    758:   PetscBLASIntCast(ld*(ld-1),&lwork);
    759:   DSAllocateMat_Private(ds,DS_MAT_W);
    760:   A  = ds->mat[DS_MAT_A];
    761:   Q  = ds->mat[DS_MAT_Q];
    762:   R  = ds->mat[DS_MAT_W];

    764:   /* copy I+alpha*A */
    765:   PetscArrayzero(Q,ld*ld);
    766:   PetscArrayzero(R,ld*ld);
    767:   for (i=0;i<k;i++) {
    768:     Q[i+i*ld] = 1.0 + alpha*A[i+i*ld];
    769:     Q[k+i*ld] = alpha*A[k+i*ld];
    770:   }

    772:   /* compute qr */
    773:   PetscBLASIntCast(k+1,&n1);
    774:   PetscBLASIntCast(k,&n0);
    775:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&n1,&n0,Q,&ld,tau,work,&lwork,&info));
    776:   SlepcCheckLapackInfo("geqrf",info);

    778:   /* copy R from Q */
    779:   for (j=0;j<k;j++)
    780:     for (i=0;i<=j;i++)
    781:       R[i+j*ld] = Q[i+j*ld];

    783:   /* compute orthogonal matrix in Q */
    784:   PetscStackCallBLAS("LAPACKorgqr",LAPACKorgqr_(&n1,&n1,&n0,Q,&ld,tau,work,&lwork,&info));
    785:   SlepcCheckLapackInfo("orgqr",info);

    787:   /* compute the updated matrix of projected problem */
    788:   for (j=0;j<k;j++)
    789:     for (i=0;i<k+1;i++)
    790:       A[j*ld+i] = Q[i*ld+j];
    791:   alpha = -1.0/alpha;
    792:   PetscStackCallBLAS("BLAStrsm",BLAStrsm_("R","U","N","N",&n1,&n0,&alpha,R,&ld,A,&ld));
    793:   for (i=0;i<k;i++)
    794:     A[ld*i+i] -= alpha;
    795:   return(0);
    796: }

    798: PetscErrorCode DSHermitian_HEP(DS ds,DSMatType m,PetscBool *flg)
    799: {
    801:   if (m==DS_MAT_A && !ds->extrarow) *flg = PETSC_TRUE;
    802:   else *flg = PETSC_FALSE;
    803:   return(0);
    804: }

    806: SLEPC_EXTERN PetscErrorCode DSCreate_HEP(DS ds)
    807: {
    809:   ds->ops->allocate      = DSAllocate_HEP;
    810:   ds->ops->view          = DSView_HEP;
    811:   ds->ops->vectors       = DSVectors_HEP;
    812:   ds->ops->solve[0]      = DSSolve_HEP_QR;
    813:   ds->ops->solve[1]      = DSSolve_HEP_MRRR;
    814:   ds->ops->solve[2]      = DSSolve_HEP_DC;
    815: #if !defined(PETSC_USE_COMPLEX)
    816:   ds->ops->solve[3]      = DSSolve_HEP_BDC;
    817: #endif
    818:   ds->ops->sort          = DSSort_HEP;
    819:   ds->ops->synchronize   = DSSynchronize_HEP;
    820:   ds->ops->truncate      = DSTruncate_HEP;
    821:   ds->ops->update        = DSUpdateExtraRow_HEP;
    822:   ds->ops->cond          = DSCond_HEP;
    823:   ds->ops->transrks      = DSTranslateRKS_HEP;
    824:   ds->ops->hermitian     = DSHermitian_HEP;
    825:   return(0);
    826: }