Date: 2021-02-01T14:41:43+00:00

Actual source code: dshep.c
slepc-3.14.2 2021-02-01
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2020, Universitat Politecnica de Valencia, Spain

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

     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,i;

    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:           PetscArrayzero(ds->mat[mat],ld*ld);
    166:           for (i=0;i<ds->n;i++) *(ds->mat[mat]+i+i*ld) = 1.0;
    167:         }
    168:       }
    169:       if (rnorm && j) *rnorm = PetscAbsScalar(Q[ds->n-1+(*j)*ld]);
    170:       break;
    171:     case DS_MAT_U:
    172:     case DS_MAT_VT:
    173:       SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"Not implemented yet");
    174:     default:
    175:       SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_OUTOFRANGE,"Invalid mat parameter");
    176:   }
    177:   return(0);
    178: }

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

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

    189:   to tridiagonal form

    191:                 [ d e 0 0 0 ]
    192:                 [ e d e 0 0 ]
    193:    T = Q'*A*Q = [ 0 e d e 0 ],
    194:                 [ 0 0 e d e ]
    195:                 [ 0 0 0 e d ]

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

    201:   Arguments
    202:   =========

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

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

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

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

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

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

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

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

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

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

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

    252:     /* Chase newly introduced off-diagonal entry to the top left corner */
    253:     for (i=j-1;i>=0;i--) {
    254:       off  = -s*e[i];
    255:       e[i] = c*e[i];
    256:       temp = e[i+1];
    257:       PetscStackCallBLAS("LAPACKlartg",LAPACKREALlartg_(&temp,&off,&c,&s,&e[i+1]));
    258:       s = -s;
    259:       temp = (d[i]-d[i+1])*s - 2.0*c*e[i];
    260:       p = s*temp;
    261:       d[i+1] += p;
    262:       d[i] -= p;
    263:       e[i] = -e[i] - c*temp;
    264:       j2 = j+2;
    265:       PetscStackCallBLAS("BLASrot",BLASMIXEDrot_(&j2,Q+i*ld,&one,Q+(i+1)*ld,&one,&c,&s));
    266:     }
    267:   }
    268:   return(0);
    269: }

    271: /*
    272:    Reduce to tridiagonal form by means of ArrowTridiag.
    273: */
    274: static PetscErrorCode DSIntermediate_HEP(DS ds)
    275: {
    277:   PetscInt       i;
    278:   PetscBLASInt   n1 = 0,n2,lwork,info,l = 0,n = 0,ld,off;
    279:   PetscScalar    *A,*Q,*work,*tau;
    280:   PetscReal      *d,*e;

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

    296:   if (ds->compact) {

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

    300:   } else {

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

    304:     if (ds->state<DS_STATE_INTERMEDIATE) {
    305:       DSCopyMatrix_Private(ds,DS_MAT_Q,DS_MAT_A);
    306:       DSAllocateWork_Private(ds,ld+ld*ld,0,0);
    307:       tau  = ds->work;
    308:       work = ds->work+ld;
    309:       lwork = ld*ld;
    310:       PetscStackCallBLAS("LAPACKsytrd",LAPACKsytrd_("L",&n2,Q+off,&ld,d+l,e+l,tau,work,&lwork,&info));
    311:       SlepcCheckLapackInfo("sytrd",info);
    312:       PetscStackCallBLAS("LAPACKorgtr",LAPACKorgtr_("L",&n2,Q+off,&ld,tau,work,&lwork,&info));
    313:       SlepcCheckLapackInfo("orgtr",info);
    314:     } else {
    315:       /* copy tridiagonal to d,e */
    316:       for (i=l;i<n;i++)   d[i] = PetscRealPart(A[i+i*ld]);
    317:       for (i=l;i<n-1;i++) e[i] = PetscRealPart(A[(i+1)+i*ld]);
    318:     }
    319:   }
    320:   return(0);
    321: }

    323: PetscErrorCode DSSort_HEP(DS ds,PetscScalar *wr,PetscScalar *wi,PetscScalar *rr,PetscScalar *ri,PetscInt *k)
    324: {
    326:   PetscInt       n,l,i,*perm,ld=ds->ld;
    327:   PetscScalar    *A;
    328:   PetscReal      *d;

    331:   if (!ds->sc) return(0);
    332:   n = ds->n;
    333:   l = ds->l;
    334:   A = ds->mat[DS_MAT_A];
    335:   d = ds->rmat[DS_MAT_T];
    336:   perm = ds->perm;
    337:   if (!rr) {
    338:     DSSortEigenvaluesReal_Private(ds,d,perm);
    339:   } else {
    340:     DSSortEigenvalues_Private(ds,rr,ri,perm,PETSC_FALSE);
    341:   }
    342:   for (i=l;i<n;i++) wr[i] = d[perm[i]];
    343:   DSPermuteColumns_Private(ds,l,n,DS_MAT_Q,perm);
    344:   for (i=l;i<n;i++) d[i] = PetscRealPart(wr[i]);
    345:   if (!ds->compact) {
    346:     for (i=l;i<n;i++) A[i+i*ld] = wr[i];
    347:   }
    348:   return(0);
    349: }

    351: PetscErrorCode DSUpdateExtraRow_HEP(DS ds)
    352: {
    354:   PetscInt       i;
    355:   PetscBLASInt   n,ld,incx=1;
    356:   PetscScalar    *A,*Q,*x,*y,one=1.0,zero=0.0;
    357:   PetscReal      *e,beta;

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

    366:   if (ds->compact) {
    367:     beta = e[n-1];   /* in compact, we assume all entries are zero except the last one */
    368:     for (i=0;i<n;i++) e[i] = PetscRealPart(beta*Q[n-1+i*ld]);
    369:     ds->k = n;
    370:   } else {
    371:     DSAllocateWork_Private(ds,2*ld,0,0);
    372:     x = ds->work;
    373:     y = ds->work+ld;
    374:     for (i=0;i<n;i++) x[i] = PetscConj(A[n+i*ld]);
    375:     PetscStackCallBLAS("BLASgemv",BLASgemv_("C",&n,&n,&one,Q,&ld,x,&incx,&zero,y,&incx));
    376:     for (i=0;i<n;i++) A[n+i*ld] = PetscConj(y[i]);
    377:     ds->k = n;
    378:   }
    379:   return(0);
    380: }

    382: PetscErrorCode DSSolve_HEP_QR(DS ds,PetscScalar *wr,PetscScalar *wi)
    383: {
    385:   PetscInt       i;
    386:   PetscBLASInt   n1,info,l = 0,n = 0,ld,off;
    387:   PetscScalar    *Q,*A;
    388:   PetscReal      *d,*e;

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

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

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

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

    413:   /* Create diagonal matrix as a result */
    414:   if (ds->compact) {
    415:     PetscArrayzero(e,n-1);
    416:   } else {
    417:     for (i=l;i<n;i++) {
    418:       PetscArrayzero(A+l+i*ld,n-l);
    419:     }
    420:     for (i=l;i<n;i++) A[i+i*ld] = d[i];
    421:   }

    423:   /* Set zero wi */
    424:   if (wi) for (i=l;i<n;i++) wi[i] = 0.0;
    425:   return(0);
    426: }

    428: PetscErrorCode DSSolve_HEP_MRRR(DS ds,PetscScalar *wr,PetscScalar *wi)
    429: {
    431:   PetscInt       i;
    432:   PetscBLASInt   n1 = 0,n2 = 0,n3,lwork,liwork,info,l = 0,n = 0,m = 0,ld,off,il,iu,*isuppz;
    433:   PetscScalar    *A,*Q,*W=NULL,one=1.0,zero=0.0;
    434:   PetscReal      *d,*e,abstol=0.0,vl,vu;
    435: #if defined(PETSC_USE_COMPLEX)
    436:   PetscInt       j;
    437:   PetscReal      *ritz;
    438: #endif

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

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

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

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

    491:   /* Create diagonal matrix as a result */
    492:   if (ds->compact) {
    493:     PetscArrayzero(e,n-1);
    494:   } else {
    495:     for (i=l;i<n;i++) {
    496:       PetscArrayzero(A+l+i*ld,n-l);
    497:     }
    498:     for (i=l;i<n;i++) A[i+i*ld] = d[i];
    499:   }

    501:   /* Set zero wi */
    502:   if (wi) for (i=l;i<n;i++) wi[i] = 0.0;
    503:   return(0);
    504: }

    506: PetscErrorCode DSSolve_HEP_DC(DS ds,PetscScalar *wr,PetscScalar *wi)
    507: {
    509:   PetscInt       i;
    510:   PetscBLASInt   n1,info,l = 0,ld,off,lrwork,liwork;
    511:   PetscScalar    *Q,*A;
    512:   PetscReal      *d,*e;
    513: #if defined(PETSC_USE_COMPLEX)
    514:   PetscBLASInt   lwork;
    515:   PetscInt       j;
    516: #endif

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

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

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

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

    551:   /* Create diagonal matrix as a result */
    552:   if (ds->compact) {
    553:     PetscArrayzero(e,ds->n-1);
    554:   } else {
    555:     for (i=l;i<ds->n;i++) {
    556:       PetscArrayzero(A+l+i*ld,ds->n-l);
    557:     }
    558:     for (i=l;i<ds->n;i++) A[i+i*ld] = d[i];
    559:   }

    561:   /* Set zero wi */
    562:   if (wi) for (i=l;i<ds->n;i++) wi[i] = 0.0;
    563:   return(0);
    564: }

    566: #if !defined(PETSC_USE_COMPLEX)
    567: PetscErrorCode DSSolve_HEP_BDC(DS ds,PetscScalar *wr,PetscScalar *wi)
    568: {
    570:   PetscBLASInt   i,j,k,m,n = 0,info,nblks,bs = 0,ld = 0,lde,lrwork,liwork,*ksizes,*iwork,mingapi;
    571:   PetscScalar    *Q,*A;
    572:   PetscReal      *D,*E,*d,*e,tol=PETSC_MACHINE_EPSILON/2,tau1=1e-16,tau2=1e-18,*rwork,mingap;

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

    596:   /* Copy matrix to block tridiagonal format */
    597:   j=0;
    598:   for (i=0;i<nblks;i++) {
    599:     ksizes[i]=bs;
    600:     for (k=0;k<bs;k++)
    601:       for (m=0;m<bs;m++)
    602:         D[k+m*bs+i*bs*bs] = PetscRealPart(A[j+k+(j+m)*n]);
    603:     j = j + bs;
    604:   }
    605:   j=0;
    606:   for (i=0;i<nblks-1;i++) {
    607:     for (k=0;k<bs;k++)
    608:       for (m=0;m<bs;m++)
    609:         E[k+m*lde+i*lde*lde] = PetscRealPart(A[j+bs+k+(j+m)*n]);
    610:     j = j + bs;
    611:   }

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

    618:   /* Create diagonal matrix as a result */
    619:   if (ds->compact) {
    620:     PetscArrayzero(e,ds->n-1);
    621:   } else {
    622:     for (i=0;i<ds->n;i++) {
    623:       PetscArrayzero(A+i*ld,ds->n);
    624:     }
    625:     for (i=0;i<ds->n;i++) A[i+i*ld] = wr[i];
    626:   }

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

    634: PetscErrorCode DSTruncate_HEP(DS ds,PetscInt n)
    635: {
    637:   PetscInt       i,ld=ds->ld,l=ds->l;
    638:   PetscScalar    *A;

    641:   if (ds->state==DS_STATE_CONDENSED) ds->t = ds->n;
    642:   A = ds->mat[DS_MAT_A];
    643:   if (!ds->compact && ds->extrarow && ds->k==ds->n) {
    644:     for (i=l;i<n;i++) A[n+i*ld] = A[ds->n+i*ld];
    645:   }
    646:   if (ds->extrarow) ds->k = n;
    647:   else ds->k = 0;
    648:   ds->n = n;
    649:   PetscInfo1(ds,"Decomposition truncated to size n=%D\n",n);
    650:   return(0);
    651: }

    653: PetscErrorCode DSSynchronize_HEP(DS ds,PetscScalar eigr[],PetscScalar eigi[])
    654: {
    656:   PetscInt       ld=ds->ld,l=ds->l,k=0,kr=0;
    657:   PetscMPIInt    n,rank,off=0,size,ldn,ld3;

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

    700: PetscErrorCode DSCond_HEP(DS ds,PetscReal *cond)
    701: {
    703:   PetscScalar    *work;
    704:   PetscReal      *rwork;
    705:   PetscBLASInt   *ipiv;
    706:   PetscBLASInt   lwork,info,n,ld;
    707:   PetscReal      hn,hin;
    708:   PetscScalar    *A;

    711:   PetscBLASIntCast(ds->n,&n);
    712:   PetscBLASIntCast(ds->ld,&ld);
    713:   lwork = 8*ld;
    714:   DSAllocateWork_Private(ds,lwork,ld,ld);
    715:   work  = ds->work;
    716:   rwork = ds->rwork;
    717:   ipiv  = ds->iwork;
    718:   DSSwitchFormat_HEP(ds);

    720:   /* use workspace matrix W to avoid overwriting A */
    721:   DSAllocateMat_Private(ds,DS_MAT_W);
    722:   A = ds->mat[DS_MAT_W];
    723:   PetscArraycpy(A,ds->mat[DS_MAT_A],ds->ld*ds->ld);

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

    728:   /* norm of inv(A) */
    729:   PetscStackCallBLAS("LAPACKgetrf",LAPACKgetrf_(&n,&n,A,&ld,ipiv,&info));
    730:   SlepcCheckLapackInfo("getrf",info);
    731:   PetscStackCallBLAS("LAPACKgetri",LAPACKgetri_(&n,A,&ld,ipiv,work,&lwork,&info));
    732:   SlepcCheckLapackInfo("getri",info);
    733:   hin = LAPACKlange_("I",&n,&n,A,&ld,rwork);

    735:   *cond = hn*hin;
    736:   return(0);
    737: }

    739: PetscErrorCode DSTranslateRKS_HEP(DS ds,PetscScalar alpha)
    740: {
    742:   PetscInt       i,j,k=ds->k;
    743:   PetscScalar    *Q,*A,*R,*tau,*work;
    744:   PetscBLASInt   ld,n1,n0,lwork,info;

    747:   PetscBLASIntCast(ds->ld,&ld);
    748:   DSAllocateWork_Private(ds,ld*ld,0,0);
    749:   tau = ds->work;
    750:   work = ds->work+ld;
    751:   PetscBLASIntCast(ld*(ld-1),&lwork);
    752:   DSAllocateMat_Private(ds,DS_MAT_W);
    753:   A  = ds->mat[DS_MAT_A];
    754:   Q  = ds->mat[DS_MAT_Q];
    755:   R  = ds->mat[DS_MAT_W];

    757:   /* copy I+alpha*A */
    758:   PetscArrayzero(Q,ld*ld);
    759:   PetscArrayzero(R,ld*ld);
    760:   for (i=0;i<k;i++) {
    761:     Q[i+i*ld] = 1.0 + alpha*A[i+i*ld];
    762:     Q[k+i*ld] = alpha*A[k+i*ld];
    763:   }

    765:   /* compute qr */
    766:   PetscBLASIntCast(k+1,&n1);
    767:   PetscBLASIntCast(k,&n0);
    768:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&n1,&n0,Q,&ld,tau,work,&lwork,&info));
    769:   SlepcCheckLapackInfo("geqrf",info);

    771:   /* copy R from Q */
    772:   for (j=0;j<k;j++)
    773:     for (i=0;i<=j;i++)
    774:       R[i+j*ld] = Q[i+j*ld];

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

    780:   /* compute the updated matrix of projected problem */
    781:   for (j=0;j<k;j++)
    782:     for (i=0;i<k+1;i++)
    783:       A[j*ld+i] = Q[i*ld+j];
    784:   alpha = -1.0/alpha;
    785:   PetscStackCallBLAS("BLAStrsm",BLAStrsm_("R","U","N","N",&n1,&n0,&alpha,R,&ld,A,&ld));
    786:   for (i=0;i<k;i++)
    787:     A[ld*i+i] -= alpha;
    788:   return(0);
    789: }

    791: PetscErrorCode DSHermitian_HEP(DS ds,DSMatType m,PetscBool *flg)
    792: {
    794:   if (m==DS_MAT_A && !ds->extrarow) *flg = PETSC_TRUE;
    795:   else *flg = PETSC_FALSE;
    796:   return(0);
    797: }

    799: SLEPC_EXTERN PetscErrorCode DSCreate_HEP(DS ds)
    800: {
    802:   ds->ops->allocate      = DSAllocate_HEP;
    803:   ds->ops->view          = DSView_HEP;
    804:   ds->ops->vectors       = DSVectors_HEP;
    805:   ds->ops->solve[0]      = DSSolve_HEP_QR;
    806:   ds->ops->solve[1]      = DSSolve_HEP_MRRR;
    807:   ds->ops->solve[2]      = DSSolve_HEP_DC;
    808: #if !defined(PETSC_USE_COMPLEX)
    809:   ds->ops->solve[3]      = DSSolve_HEP_BDC;
    810: #endif
    811:   ds->ops->sort          = DSSort_HEP;
    812:   ds->ops->synchronize   = DSSynchronize_HEP;
    813:   ds->ops->truncate      = DSTruncate_HEP;
    814:   ds->ops->update        = DSUpdateExtraRow_HEP;
    815:   ds->ops->cond          = DSCond_HEP;
    816:   ds->ops->transrks      = DSTranslateRKS_HEP;
    817:   ds->ops->hermitian     = DSHermitian_HEP;
    818:   return(0);
    819: }