Date: 2022-03-31T15:28:50+00:00

Actual source code: dshep.c
slepc-3.17.0 2022-03-31
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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: */

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

     14: PetscErrorCode DSAllocate_HEP(DS ds,PetscInt ld)
     15: {
     16:   DSAllocateMat_Private(ds,DS_MAT_A);
     17:   DSAllocateMat_Private(ds,DS_MAT_Q);
     18:   DSAllocateMatReal_Private(ds,DS_MAT_T);
     19:   PetscFree(ds->perm);
     20:   PetscMalloc1(ld,&ds->perm);
     21:   PetscLogObjectMemory((PetscObject)ds,ld*sizeof(PetscInt));
     22:   PetscFunctionReturn(0);
     23: }

     25: /*   0       l           k                 n-1
     26:     -----------------------------------------
     27:     |*       .           .                  |
     28:     |  *     .           .                  |
     29:     |    *   .           .                  |
     30:     |      * .           .                  |
     31:     |. . . . o           o                  |
     32:     |          o         o                  |
     33:     |            o       o                  |
     34:     |              o     o                  |
     35:     |                o   o                  |
     36:     |                  o o                  |
     37:     |. . . . o o o o o o o x                |
     38:     |                    x x x              |
     39:     |                      x x x            |
     40:     |                        x x x          |
     41:     |                          x x x        |
     42:     |                            x x x      |
     43:     |                              x x x    |
     44:     |                                x x x  |
     45:     |                                  x x x|
     46:     |                                    x x|
     47:     -----------------------------------------
     48: */

     50: static PetscErrorCode DSSwitchFormat_HEP(DS ds)
     51: {
     52:   PetscReal      *T = ds->rmat[DS_MAT_T];
     53:   PetscScalar    *A = ds->mat[DS_MAT_A];
     54:   PetscInt       i,n=ds->n,k=ds->k,ld=ds->ld;

     56:   /* switch from compact (arrow) to dense storage */
     57:   PetscArrayzero(A,ld*ld);
     58:   for (i=0;i<k;i++) {
     59:     A[i+i*ld] = T[i];
     60:     A[k+i*ld] = T[i+ld];
     61:     A[i+k*ld] = T[i+ld];
     62:   }
     63:   A[k+k*ld] = T[k];
     64:   for (i=k+1;i<n;i++) {
     65:     A[i+i*ld]     = T[i];
     66:     A[i-1+i*ld]   = T[i-1+ld];
     67:     A[i+(i-1)*ld] = T[i-1+ld];
     68:   }
     69:   if (ds->extrarow) A[n+(n-1)*ld] = T[n-1+ld];
     70:   PetscFunctionReturn(0);
     71: }

     73: PetscErrorCode DSView_HEP(DS ds,PetscViewer viewer)
     74: {
     75:   PetscViewerFormat format;
     76:   PetscInt          i,j,r,c,rows;
     77:   PetscReal         value;
     78:   const char        *methodname[] = {
     79:                      "Implicit QR method (_steqr)",
     80:                      "Relatively Robust Representations (_stevr)",
     81:                      "Divide and Conquer method (_stedc)",
     82:                      "Block Divide and Conquer method (dsbtdc)"
     83:   };
     84:   const int         nmeth=sizeof(methodname)/sizeof(methodname[0]);

     86:   PetscViewerGetFormat(viewer,&format);
     87:   if (format == PETSC_VIEWER_ASCII_INFO || format == PETSC_VIEWER_ASCII_INFO_DETAIL) {
     88:     if (ds->bs>1) PetscViewerASCIIPrintf(viewer,"block size: %" PetscInt_FMT "\n",ds->bs);
     89:     if (ds->method<nmeth) PetscViewerASCIIPrintf(viewer,"solving the problem with: %s\n",methodname[ds->method]);
     90:     PetscFunctionReturn(0);
     91:   }
     92:   if (ds->compact) {
     93:     PetscViewerASCIIUseTabs(viewer,PETSC_FALSE);
     94:     rows = ds->extrarow? ds->n+1: ds->n;
     95:     if (format == PETSC_VIEWER_ASCII_MATLAB) {
     96:       PetscViewerASCIIPrintf(viewer,"%% Size = %" PetscInt_FMT " %" PetscInt_FMT "\n",rows,ds->n);
     97:       PetscViewerASCIIPrintf(viewer,"zzz = zeros(%" PetscInt_FMT ",3);\n",3*ds->n);
     98:       PetscViewerASCIIPrintf(viewer,"zzz = [\n");
     99:       for (i=0;i<ds->n;i++) PetscViewerASCIIPrintf(viewer,"%" PetscInt_FMT " %" PetscInt_FMT "  %18.16e\n",i+1,i+1,(double)*(ds->rmat[DS_MAT_T]+i));
    100:       for (i=0;i<rows-1;i++) {
    101:         r = PetscMax(i+2,ds->k+1);
    102:         c = i+1;
    103:         PetscViewerASCIIPrintf(viewer,"%" PetscInt_FMT " %" PetscInt_FMT "  %18.16e\n",r,c,(double)*(ds->rmat[DS_MAT_T]+ds->ld+i));
    104:         if (i<ds->n-1 && ds->k<ds->n) { /* do not print vertical arrow when k=n */
    105:           PetscViewerASCIIPrintf(viewer,"%" PetscInt_FMT " %" PetscInt_FMT "  %18.16e\n",c,r,(double)*(ds->rmat[DS_MAT_T]+ds->ld+i));
    106:         }
    107:       }
    108:       PetscViewerASCIIPrintf(viewer,"];\n%s = spconvert(zzz);\n",DSMatName[DS_MAT_T]);
    109:     } else {
    110:       for (i=0;i<rows;i++) {
    111:         for (j=0;j<ds->n;j++) {
    112:           if (i==j) value = *(ds->rmat[DS_MAT_T]+i);
    113:           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));
    114:           else if (i==j+1 && i>ds->k) value = *(ds->rmat[DS_MAT_T]+ds->ld+i-1);
    115:           else if (i+1==j && j>ds->k) value = *(ds->rmat[DS_MAT_T]+ds->ld+j-1);
    116:           else value = 0.0;
    117:           PetscViewerASCIIPrintf(viewer," %18.16e ",(double)value);
    118:         }
    119:         PetscViewerASCIIPrintf(viewer,"\n");
    120:       }
    121:     }
    122:     PetscViewerASCIIUseTabs(viewer,PETSC_TRUE);
    123:     PetscViewerFlush(viewer);
    124:   } else DSViewMat(ds,viewer,DS_MAT_A);
    125:   if (ds->state>DS_STATE_INTERMEDIATE) DSViewMat(ds,viewer,DS_MAT_Q);
    126:   PetscFunctionReturn(0);
    127: }

    129: PetscErrorCode DSVectors_HEP(DS ds,DSMatType mat,PetscInt *j,PetscReal *rnorm)
    130: {
    131:   PetscScalar    *Q = ds->mat[DS_MAT_Q];
    132:   PetscInt       ld = ds->ld;

    134:   switch (mat) {
    135:     case DS_MAT_X:
    136:     case DS_MAT_Y:
    137:       if (j) {
    138:         if (ds->state>=DS_STATE_CONDENSED) PetscArraycpy(ds->mat[mat]+(*j)*ld,Q+(*j)*ld,ld);
    139:         else {
    140:           PetscArrayzero(ds->mat[mat]+(*j)*ld,ld);
    141:           *(ds->mat[mat]+(*j)+(*j)*ld) = 1.0;
    142:         }
    143:       } else {
    144:         if (ds->state>=DS_STATE_CONDENSED) PetscArraycpy(ds->mat[mat],Q,ld*ld);
    145:         else DSSetIdentity(ds,mat);
    146:       }
    147:       if (rnorm && j) *rnorm = PetscAbsScalar(Q[ds->n-1+(*j)*ld]);
    148:       break;
    149:     case DS_MAT_U:
    150:     case DS_MAT_V:
    151:       SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"Not implemented yet");
    152:     default:
    153:       SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_OUTOFRANGE,"Invalid mat parameter");
    154:   }
    155:   PetscFunctionReturn(0);
    156: }

    158: /*
    159:   ARROWTRIDIAG reduces a symmetric arrowhead matrix of the form

    161:                 [ d 0 0 0 e ]
    162:                 [ 0 d 0 0 e ]
    163:             A = [ 0 0 d 0 e ]
    164:                 [ 0 0 0 d e ]
    165:                 [ e e e e d ]

    167:   to tridiagonal form

    169:                 [ d e 0 0 0 ]
    170:                 [ e d e 0 0 ]
    171:    T = Q'*A*Q = [ 0 e d e 0 ],
    172:                 [ 0 0 e d e ]
    173:                 [ 0 0 0 e d ]

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

    179:   Arguments
    180:   =========

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

    185:   D       (input/output) DOUBLE PRECISION array, dimension (N)
    186:           On entry, the diagonal entries of the matrix A to be
    187:           reduced.
    188:           On exit, the diagonal entries of the reduced matrix T.

    190:   E       (input/output) DOUBLE PRECISION array, dimension (N-1)
    191:           On entry, the off-diagonal entries of the matrix A to be
    192:           reduced.
    193:           On exit, the subdiagonal entries of the reduced matrix T.

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

    198:   LDQ     (input) INTEGER
    199:           The leading dimension of the array Q.

    201:   Note
    202:   ====
    203:   Based on Fortran code contributed by Daniel Kressner
    204: */
    205: static PetscErrorCode ArrowTridiag(PetscBLASInt n,PetscReal *d,PetscReal *e,PetscScalar *Q,PetscBLASInt ld)
    206: {
    207:   PetscBLASInt i,j,j2,one=1;
    208:   PetscReal    c,s,p,off,temp;

    210:   if (n<=2) PetscFunctionReturn(0);

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

    214:     /* Eliminate entry e(j) by a rotation in the planes (j,j+1) */
    215:     temp = e[j+1];
    216:     PetscStackCallBLAS("LAPACKlartg",LAPACKREALlartg_(&temp,&e[j],&c,&s,&e[j+1]));
    217:     s = -s;

    219:     /* Apply rotation to diagonal elements */
    220:     temp   = d[j+1];
    221:     e[j]   = c*s*(temp-d[j]);
    222:     d[j+1] = s*s*d[j] + c*c*temp;
    223:     d[j]   = c*c*d[j] + s*s*temp;

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

    229:     /* Chase newly introduced off-diagonal entry to the top left corner */
    230:     for (i=j-1;i>=0;i--) {
    231:       off  = -s*e[i];
    232:       e[i] = c*e[i];
    233:       temp = e[i+1];
    234:       PetscStackCallBLAS("LAPACKlartg",LAPACKREALlartg_(&temp,&off,&c,&s,&e[i+1]));
    235:       s = -s;
    236:       temp = (d[i]-d[i+1])*s - 2.0*c*e[i];
    237:       p = s*temp;
    238:       d[i+1] += p;
    239:       d[i] -= p;
    240:       e[i] = -e[i] - c*temp;
    241:       j2 = j+2;
    242:       PetscStackCallBLAS("BLASrot",BLASMIXEDrot_(&j2,Q+i*ld,&one,Q+(i+1)*ld,&one,&c,&s));
    243:     }
    244:   }
    245:   PetscFunctionReturn(0);
    246: }

    248: /*
    249:    Reduce to tridiagonal form by means of ArrowTridiag.
    250: */
    251: static PetscErrorCode DSIntermediate_HEP(DS ds)
    252: {
    253:   PetscInt       i;
    254:   PetscBLASInt   n1 = 0,n2,lwork,info,l = 0,n = 0,ld,off;
    255:   PetscScalar    *A,*Q,*work,*tau;
    256:   PetscReal      *d,*e;

    258:   PetscBLASIntCast(ds->n,&n);
    259:   PetscBLASIntCast(ds->l,&l);
    260:   PetscBLASIntCast(ds->ld,&ld);
    261:   PetscBLASIntCast(PetscMax(0,ds->k-l+1),&n1); /* size of leading block, excl. locked */
    262:   n2 = n-l;     /* n2 = n1 + size of trailing block */
    263:   off = l+l*ld;
    264:   A  = ds->mat[DS_MAT_A];
    265:   Q  = ds->mat[DS_MAT_Q];
    266:   d  = ds->rmat[DS_MAT_T];
    267:   e  = ds->rmat[DS_MAT_T]+ld;
    268:   PetscArrayzero(Q,ld*ld);
    269:   for (i=0;i<n;i++) Q[i+i*ld] = 1.0;

    271:   if (ds->compact) {

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

    275:   } else {

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

    279:     if (ds->state<DS_STATE_INTERMEDIATE) {
    280:       DSCopyMatrix_Private(ds,DS_MAT_Q,DS_MAT_A);
    281:       DSAllocateWork_Private(ds,ld+ld*ld,0,0);
    282:       tau  = ds->work;
    283:       work = ds->work+ld;
    284:       lwork = ld*ld;
    285:       PetscStackCallBLAS("LAPACKsytrd",LAPACKsytrd_("L",&n2,Q+off,&ld,d+l,e+l,tau,work,&lwork,&info));
    286:       SlepcCheckLapackInfo("sytrd",info);
    287:       PetscStackCallBLAS("LAPACKorgtr",LAPACKorgtr_("L",&n2,Q+off,&ld,tau,work,&lwork,&info));
    288:       SlepcCheckLapackInfo("orgtr",info);
    289:     } else {
    290:       /* copy tridiagonal to d,e */
    291:       for (i=l;i<n;i++)   d[i] = PetscRealPart(A[i+i*ld]);
    292:       for (i=l;i<n-1;i++) e[i] = PetscRealPart(A[(i+1)+i*ld]);
    293:     }
    294:   }
    295:   PetscFunctionReturn(0);
    296: }

    298: PetscErrorCode DSSort_HEP(DS ds,PetscScalar *wr,PetscScalar *wi,PetscScalar *rr,PetscScalar *ri,PetscInt *k)
    299: {
    300:   PetscInt       n,l,i,*perm,ld=ds->ld;
    301:   PetscScalar    *A;
    302:   PetscReal      *d;

    304:   if (!ds->sc) PetscFunctionReturn(0);
    305:   n = ds->n;
    306:   l = ds->l;
    307:   A = ds->mat[DS_MAT_A];
    308:   d = ds->rmat[DS_MAT_T];
    309:   perm = ds->perm;
    310:   if (!rr) DSSortEigenvaluesReal_Private(ds,d,perm);
    311:   else DSSortEigenvalues_Private(ds,rr,ri,perm,PETSC_FALSE);
    312:   for (i=l;i<n;i++) wr[i] = d[perm[i]];
    313:   DSPermuteColumns_Private(ds,l,n,n,DS_MAT_Q,perm);
    314:   for (i=l;i<n;i++) d[i] = PetscRealPart(wr[i]);
    315:   if (!ds->compact) {
    316:     for (i=l;i<n;i++) A[i+i*ld] = wr[i];
    317:   }
    318:   PetscFunctionReturn(0);
    319: }

    321: PetscErrorCode DSUpdateExtraRow_HEP(DS ds)
    322: {
    323:   PetscInt       i;
    324:   PetscBLASInt   n,ld,incx=1;
    325:   PetscScalar    *A,*Q,*x,*y,one=1.0,zero=0.0;
    326:   PetscReal      *e,beta;

    328:   PetscBLASIntCast(ds->n,&n);
    329:   PetscBLASIntCast(ds->ld,&ld);
    330:   A  = ds->mat[DS_MAT_A];
    331:   Q  = ds->mat[DS_MAT_Q];
    332:   e  = ds->rmat[DS_MAT_T]+ld;

    334:   if (ds->compact) {
    335:     beta = e[n-1];   /* in compact, we assume all entries are zero except the last one */
    336:     for (i=0;i<n;i++) e[i] = PetscRealPart(beta*Q[n-1+i*ld]);
    337:     ds->k = n;
    338:   } else {
    339:     DSAllocateWork_Private(ds,2*ld,0,0);
    340:     x = ds->work;
    341:     y = ds->work+ld;
    342:     for (i=0;i<n;i++) x[i] = PetscConj(A[n+i*ld]);
    343:     PetscStackCallBLAS("BLASgemv",BLASgemv_("C",&n,&n,&one,Q,&ld,x,&incx,&zero,y,&incx));
    344:     for (i=0;i<n;i++) A[n+i*ld] = PetscConj(y[i]);
    345:     ds->k = n;
    346:   }
    347:   PetscFunctionReturn(0);
    348: }

    350: PetscErrorCode DSSolve_HEP_QR(DS ds,PetscScalar *wr,PetscScalar *wi)
    351: {
    352:   PetscInt       i;
    353:   PetscBLASInt   n1,info,l = 0,n = 0,ld,off;
    354:   PetscScalar    *Q,*A;
    355:   PetscReal      *d,*e;

    358:   PetscBLASIntCast(ds->n,&n);
    359:   PetscBLASIntCast(ds->l,&l);
    360:   PetscBLASIntCast(ds->ld,&ld);
    361:   n1 = n-l;     /* n1 = size of leading block, excl. locked + size of trailing block */
    362:   off = l+l*ld;
    363:   Q  = ds->mat[DS_MAT_Q];
    364:   A  = ds->mat[DS_MAT_A];
    365:   d  = ds->rmat[DS_MAT_T];
    366:   e  = ds->rmat[DS_MAT_T]+ld;

    368:   /* Reduce to tridiagonal form */
    369:   DSIntermediate_HEP(ds);

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

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

    379:   /* Create diagonal matrix as a result */
    380:   if (ds->compact) PetscArrayzero(e,n-1);
    381:   else {
    382:     for (i=l;i<n;i++) PetscArrayzero(A+l+i*ld,n-l);
    383:     for (i=l;i<n;i++) A[i+i*ld] = d[i];
    384:   }

    386:   /* Set zero wi */
    387:   if (wi) for (i=l;i<n;i++) wi[i] = 0.0;
    388:   PetscFunctionReturn(0);
    389: }

    391: PetscErrorCode DSSolve_HEP_MRRR(DS ds,PetscScalar *wr,PetscScalar *wi)
    392: {
    393:   PetscInt       i;
    394:   PetscBLASInt   n1 = 0,n2 = 0,n3,lwork,liwork,info,l = 0,n = 0,m = 0,ld,off,il,iu,*isuppz;
    395:   PetscScalar    *A,*Q,*W=NULL,one=1.0,zero=0.0;
    396:   PetscReal      *d,*e,abstol=0.0,vl,vu;
    397: #if defined(PETSC_USE_COMPLEX)
    398:   PetscInt       j;
    399:   PetscReal      *ritz;
    400: #endif

    403:   PetscBLASIntCast(ds->n,&n);
    404:   PetscBLASIntCast(ds->l,&l);
    405:   PetscBLASIntCast(ds->ld,&ld);
    406:   PetscBLASIntCast(ds->k-l+1,&n1); /* size of leading block, excl. locked */
    407:   PetscBLASIntCast(n-ds->k-1,&n2); /* size of trailing block */
    408:   n3 = n1+n2;
    409:   off = l+l*ld;
    410:   A  = ds->mat[DS_MAT_A];
    411:   Q  = ds->mat[DS_MAT_Q];
    412:   d  = ds->rmat[DS_MAT_T];
    413:   e  = ds->rmat[DS_MAT_T]+ld;

    415:   /* Reduce to tridiagonal form */
    416:   DSIntermediate_HEP(ds);

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

    421:   if (ds->state<DS_STATE_INTERMEDIATE) {  /* Q contains useful info */
    422:     DSAllocateMat_Private(ds,DS_MAT_W);
    423:     DSCopyMatrix_Private(ds,DS_MAT_W,DS_MAT_Q);
    424:     W = ds->mat[DS_MAT_W];
    425:   }
    426: #if defined(PETSC_USE_COMPLEX)
    427:   DSAllocateMatReal_Private(ds,DS_MAT_Q);
    428: #endif
    429:   lwork = 20*ld;
    430:   liwork = 10*ld;
    431:   DSAllocateWork_Private(ds,0,lwork+ld,liwork+2*ld);
    432:   isuppz = ds->iwork+liwork;
    433: #if defined(PETSC_USE_COMPLEX)
    434:   ritz = ds->rwork+lwork;
    435:   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));
    436:   for (i=l;i<n;i++) wr[i] = ritz[i];
    437: #else
    438:   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));
    439: #endif
    440:   SlepcCheckLapackInfo("stevr",info);
    441: #if defined(PETSC_USE_COMPLEX)
    442:   for (i=l;i<n;i++)
    443:     for (j=l;j<n;j++)
    444:       Q[i+j*ld] = (ds->rmat[DS_MAT_Q])[i+j*ld];
    445: #endif
    446:   if (ds->state<DS_STATE_INTERMEDIATE) {  /* accumulate previous Q */
    447:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n3,&n3,&n3,&one,W+off,&ld,Q+off,&ld,&zero,A+off,&ld));
    448:     DSCopyMatrix_Private(ds,DS_MAT_Q,DS_MAT_A);
    449:   }
    450:   for (i=l;i<n;i++) d[i] = PetscRealPart(wr[i]);

    452:   /* Create diagonal matrix as a result */
    453:   if (ds->compact) PetscArrayzero(e,n-1);
    454:   else {
    455:     for (i=l;i<n;i++) PetscArrayzero(A+l+i*ld,n-l);
    456:     for (i=l;i<n;i++) A[i+i*ld] = d[i];
    457:   }

    459:   /* Set zero wi */
    460:   if (wi) for (i=l;i<n;i++) wi[i] = 0.0;
    461:   PetscFunctionReturn(0);
    462: }

    464: PetscErrorCode DSSolve_HEP_DC(DS ds,PetscScalar *wr,PetscScalar *wi)
    465: {
    466:   PetscInt       i;
    467:   PetscBLASInt   n1,info,l = 0,ld,off,lrwork,liwork;
    468:   PetscScalar    *Q,*A;
    469:   PetscReal      *d,*e;
    470: #if defined(PETSC_USE_COMPLEX)
    471:   PetscBLASInt   lwork;
    472:   PetscInt       j;
    473: #endif

    476:   PetscBLASIntCast(ds->l,&l);
    477:   PetscBLASIntCast(ds->ld,&ld);
    478:   PetscBLASIntCast(ds->n-ds->l,&n1);
    479:   off = l+l*ld;
    480:   Q  = ds->mat[DS_MAT_Q];
    481:   A  = ds->mat[DS_MAT_A];
    482:   d  = ds->rmat[DS_MAT_T];
    483:   e  = ds->rmat[DS_MAT_T]+ld;

    485:   /* Reduce to tridiagonal form */
    486:   DSIntermediate_HEP(ds);

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

    491:   lrwork = 5*n1*n1+3*n1+1;
    492:   liwork = 5*n1*n1+6*n1+6;
    493: #if !defined(PETSC_USE_COMPLEX)
    494:   DSAllocateWork_Private(ds,0,lrwork,liwork);
    495:   PetscStackCallBLAS("LAPACKstedc",LAPACKstedc_("V",&n1,d+l,e+l,Q+off,&ld,ds->rwork,&lrwork,ds->iwork,&liwork,&info));
    496: #else
    497:   lwork = ld*ld;
    498:   DSAllocateWork_Private(ds,lwork,lrwork,liwork);
    499:   PetscStackCallBLAS("LAPACKstedc",LAPACKstedc_("V",&n1,d+l,e+l,Q+off,&ld,ds->work,&lwork,ds->rwork,&lrwork,ds->iwork,&liwork,&info));
    500:   /* Fixing Lapack bug*/
    501:   for (j=ds->l;j<ds->n;j++)
    502:     for (i=0;i<ds->l;i++) Q[i+j*ld] = 0.0;
    503: #endif
    504:   SlepcCheckLapackInfo("stedc",info);
    505:   for (i=l;i<ds->n;i++) wr[i] = d[i];

    507:   /* Create diagonal matrix as a result */
    508:   if (ds->compact) PetscArrayzero(e,ds->n-1);
    509:   else {
    510:     for (i=l;i<ds->n;i++) PetscArrayzero(A+l+i*ld,ds->n-l);
    511:     for (i=l;i<ds->n;i++) A[i+i*ld] = d[i];
    512:   }

    514:   /* Set zero wi */
    515:   if (wi) for (i=l;i<ds->n;i++) wi[i] = 0.0;
    516:   PetscFunctionReturn(0);
    517: }

    519: #if !defined(PETSC_USE_COMPLEX)
    520: PetscErrorCode DSSolve_HEP_BDC(DS ds,PetscScalar *wr,PetscScalar *wi)
    521: {
    522:   PetscBLASInt   i,j,k,m,n = 0,info,nblks,bs = 0,ld = 0,lde,lrwork,liwork,*ksizes,*iwork,mingapi;
    523:   PetscScalar    *Q,*A;
    524:   PetscReal      *D,*E,*d,*e,tol=PETSC_MACHINE_EPSILON/2,tau1=1e-16,tau2=1e-18,*rwork,mingap;

    528:   PetscBLASIntCast(ds->ld,&ld);
    529:   PetscBLASIntCast(ds->bs,&bs);
    530:   PetscBLASIntCast(ds->n,&n);
    531:   nblks = n/bs;
    532:   Q  = ds->mat[DS_MAT_Q];
    533:   A  = ds->mat[DS_MAT_A];
    534:   d  = ds->rmat[DS_MAT_T];
    535:   e  = ds->rmat[DS_MAT_T]+ld;
    536:   lrwork = 4*n*n+60*n+1;
    537:   liwork = 5*n+5*nblks-1;
    538:   lde = 2*bs+1;
    539:   DSAllocateWork_Private(ds,bs*n+lde*lde*(nblks-1),lrwork,nblks+liwork);
    540:   D      = ds->work;
    541:   E      = ds->work+bs*n;
    542:   rwork  = ds->rwork;
    543:   ksizes = ds->iwork;
    544:   iwork  = ds->iwork+nblks;
    545:   PetscArrayzero(iwork,liwork);

    547:   /* Copy matrix to block tridiagonal format */
    548:   j=0;
    549:   for (i=0;i<nblks;i++) {
    550:     ksizes[i]=bs;
    551:     for (k=0;k<bs;k++)
    552:       for (m=0;m<bs;m++)
    553:         D[k+m*bs+i*bs*bs] = PetscRealPart(A[j+k+(j+m)*n]);
    554:     j = j + bs;
    555:   }
    556:   j=0;
    557:   for (i=0;i<nblks-1;i++) {
    558:     for (k=0;k<bs;k++)
    559:       for (m=0;m<bs;m++)
    560:         E[k+m*lde+i*lde*lde] = PetscRealPart(A[j+bs+k+(j+m)*n]);
    561:     j = j + bs;
    562:   }

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

    569:   /* Create diagonal matrix as a result */
    570:   if (ds->compact) PetscArrayzero(e,ds->n-1);
    571:   else {
    572:     for (i=0;i<ds->n;i++) PetscArrayzero(A+i*ld,ds->n);
    573:     for (i=0;i<ds->n;i++) A[i+i*ld] = wr[i];
    574:   }

    576:   /* Set zero wi */
    577:   if (wi) for (i=0;i<ds->n;i++) wi[i] = 0.0;
    578:   PetscFunctionReturn(0);
    579: }
    580: #endif

    582: PetscErrorCode DSTruncate_HEP(DS ds,PetscInt n,PetscBool trim)
    583: {
    584:   PetscInt    i,ld=ds->ld,l=ds->l;
    585:   PetscScalar *A = ds->mat[DS_MAT_A];

    587:   if (trim) {
    588:     if (!ds->compact && ds->extrarow) {   /* clean extra row */
    589:       for (i=l;i<ds->n;i++) A[ds->n+i*ld] = 0.0;
    590:     }
    591:     ds->l = 0;
    592:     ds->k = 0;
    593:     ds->n = n;
    594:     ds->t = ds->n;   /* truncated length equal to the new dimension */
    595:   } else {
    596:     if (!ds->compact && ds->extrarow && ds->k==ds->n) {
    597:       /* copy entries of extra row to the new position, then clean last row */
    598:       for (i=l;i<n;i++) A[n+i*ld] = A[ds->n+i*ld];
    599:       for (i=l;i<ds->n;i++) A[ds->n+i*ld] = 0.0;
    600:     }
    601:     ds->k = (ds->extrarow)? n: 0;
    602:     ds->t = ds->n;   /* truncated length equal to previous dimension */
    603:     ds->n = n;
    604:   }
    605:   PetscFunctionReturn(0);
    606: }

    608: PetscErrorCode DSSynchronize_HEP(DS ds,PetscScalar eigr[],PetscScalar eigi[])
    609: {
    610:   PetscInt       ld=ds->ld,l=ds->l,k=0,kr=0;
    611:   PetscMPIInt    n,rank,off=0,size,ldn,ld3;

    613:   if (ds->compact) kr = 3*ld;
    614:   else k = (ds->n-l)*ld;
    615:   if (ds->state>DS_STATE_RAW) k += (ds->n-l)*ld;
    616:   if (eigr) k += (ds->n-l);
    617:   DSAllocateWork_Private(ds,k+kr,0,0);
    618:   PetscMPIIntCast(k*sizeof(PetscScalar)+kr*sizeof(PetscReal),&size);
    619:   PetscMPIIntCast(ds->n-l,&n);
    620:   PetscMPIIntCast(ld*(ds->n-l),&ldn);
    621:   PetscMPIIntCast(ld*3,&ld3);
    622:   MPI_Comm_rank(PetscObjectComm((PetscObject)ds),&rank);
    623:   if (!rank) {
    624:     if (ds->compact) MPI_Pack(ds->rmat[DS_MAT_T],ld3,MPIU_REAL,ds->work,size,&off,PetscObjectComm((PetscObject)ds));
    625:     else MPI_Pack(ds->mat[DS_MAT_A]+l*ld,ldn,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds));
    626:     if (ds->state>DS_STATE_RAW) MPI_Pack(ds->mat[DS_MAT_Q]+l*ld,ldn,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds));
    627:     if (eigr) MPI_Pack(eigr+l,n,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds));
    628:   }
    629:   MPI_Bcast(ds->work,size,MPI_BYTE,0,PetscObjectComm((PetscObject)ds));
    630:   if (rank) {
    631:     if (ds->compact) MPI_Unpack(ds->work,size,&off,ds->rmat[DS_MAT_T],ld3,MPIU_REAL,PetscObjectComm((PetscObject)ds));
    632:     else MPI_Unpack(ds->work,size,&off,ds->mat[DS_MAT_A]+l*ld,ldn,MPIU_SCALAR,PetscObjectComm((PetscObject)ds));
    633:     if (ds->state>DS_STATE_RAW) MPI_Unpack(ds->work,size,&off,ds->mat[DS_MAT_Q]+l*ld,ldn,MPIU_SCALAR,PetscObjectComm((PetscObject)ds));
    634:     if (eigr) MPI_Unpack(ds->work,size,&off,eigr+l,n,MPIU_SCALAR,PetscObjectComm((PetscObject)ds));
    635:   }
    636:   PetscFunctionReturn(0);
    637: }

    639: PetscErrorCode DSCond_HEP(DS ds,PetscReal *cond)
    640: {
    641:   PetscScalar    *work;
    642:   PetscReal      *rwork;
    643:   PetscBLASInt   *ipiv;
    644:   PetscBLASInt   lwork,info,n,ld;
    645:   PetscReal      hn,hin;
    646:   PetscScalar    *A;

    648:   PetscBLASIntCast(ds->n,&n);
    649:   PetscBLASIntCast(ds->ld,&ld);
    650:   lwork = 8*ld;
    651:   DSAllocateWork_Private(ds,lwork,ld,ld);
    652:   work  = ds->work;
    653:   rwork = ds->rwork;
    654:   ipiv  = ds->iwork;
    655:   DSSwitchFormat_HEP(ds);

    657:   /* use workspace matrix W to avoid overwriting A */
    658:   DSAllocateMat_Private(ds,DS_MAT_W);
    659:   A = ds->mat[DS_MAT_W];
    660:   PetscArraycpy(A,ds->mat[DS_MAT_A],ds->ld*ds->ld);

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

    665:   /* norm of inv(A) */
    666:   PetscStackCallBLAS("LAPACKgetrf",LAPACKgetrf_(&n,&n,A,&ld,ipiv,&info));
    667:   SlepcCheckLapackInfo("getrf",info);
    668:   PetscStackCallBLAS("LAPACKgetri",LAPACKgetri_(&n,A,&ld,ipiv,work,&lwork,&info));
    669:   SlepcCheckLapackInfo("getri",info);
    670:   hin = LAPACKlange_("I",&n,&n,A,&ld,rwork);

    672:   *cond = hn*hin;
    673:   PetscFunctionReturn(0);
    674: }

    676: PetscErrorCode DSTranslateRKS_HEP(DS ds,PetscScalar alpha)
    677: {
    678:   PetscInt       i,j,k=ds->k;
    679:   PetscScalar    *Q,*A,*R,*tau,*work;
    680:   PetscBLASInt   ld,n1,n0,lwork,info;

    682:   PetscBLASIntCast(ds->ld,&ld);
    683:   DSAllocateWork_Private(ds,ld*ld,0,0);
    684:   tau = ds->work;
    685:   work = ds->work+ld;
    686:   PetscBLASIntCast(ld*(ld-1),&lwork);
    687:   DSAllocateMat_Private(ds,DS_MAT_W);
    688:   A  = ds->mat[DS_MAT_A];
    689:   Q  = ds->mat[DS_MAT_Q];
    690:   R  = ds->mat[DS_MAT_W];

    692:   /* copy I+alpha*A */
    693:   PetscArrayzero(Q,ld*ld);
    694:   PetscArrayzero(R,ld*ld);
    695:   for (i=0;i<k;i++) {
    696:     Q[i+i*ld] = 1.0 + alpha*A[i+i*ld];
    697:     Q[k+i*ld] = alpha*A[k+i*ld];
    698:   }

    700:   /* compute qr */
    701:   PetscBLASIntCast(k+1,&n1);
    702:   PetscBLASIntCast(k,&n0);
    703:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&n1,&n0,Q,&ld,tau,work,&lwork,&info));
    704:   SlepcCheckLapackInfo("geqrf",info);

    706:   /* copy R from Q */
    707:   for (j=0;j<k;j++)
    708:     for (i=0;i<=j;i++)
    709:       R[i+j*ld] = Q[i+j*ld];

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

    715:   /* compute the updated matrix of projected problem */
    716:   for (j=0;j<k;j++)
    717:     for (i=0;i<k+1;i++)
    718:       A[j*ld+i] = Q[i*ld+j];
    719:   alpha = -1.0/alpha;
    720:   PetscStackCallBLAS("BLAStrsm",BLAStrsm_("R","U","N","N",&n1,&n0,&alpha,R,&ld,A,&ld));
    721:   for (i=0;i<k;i++)
    722:     A[ld*i+i] -= alpha;
    723:   PetscFunctionReturn(0);
    724: }

    726: PetscErrorCode DSHermitian_HEP(DS ds,DSMatType m,PetscBool *flg)
    727: {
    728:   if (m==DS_MAT_A && !ds->extrarow) *flg = PETSC_TRUE;
    729:   else *flg = PETSC_FALSE;
    730:   PetscFunctionReturn(0);
    731: }

    733: /*MC
    734:    DSHEP - Dense Hermitian Eigenvalue Problem.

    736:    Level: beginner

    738:    Notes:
    739:    The problem is expressed as A*X = X*Lambda, where A is real symmetric
    740:    (or complex Hermitian). Lambda is a diagonal matrix whose diagonal
    741:    elements are the arguments of DSSolve(). After solve, A is overwritten
    742:    with Lambda.

    744:    In the intermediate state A is reduced to tridiagonal form. In compact
    745:    storage format, the symmetric tridiagonal matrix is stored in T.

    747:    Used DS matrices:
    748: +  DS_MAT_A - problem matrix
    749: .  DS_MAT_T - symmetric tridiagonal matrix
    750: -  DS_MAT_Q - orthogonal/unitary transformation that reduces to tridiagonal form
    751:    (intermediate step) or matrix of orthogonal eigenvectors, which is equal to X

    753:    Implemented methods:
    754: +  0 - Implicit QR (_steqr)
    755: .  1 - Multiple Relatively Robust Representations (_stevr)
    756: .  2 - Divide and Conquer (_stedc)
    757: -  3 - Block Divide and Conquer (real scalars only)

    759: .seealso: DSCreate(), DSSetType(), DSType
    760: M*/
    761: SLEPC_EXTERN PetscErrorCode DSCreate_HEP(DS ds)
    762: {
    763:   ds->ops->allocate      = DSAllocate_HEP;
    764:   ds->ops->view          = DSView_HEP;
    765:   ds->ops->vectors       = DSVectors_HEP;
    766:   ds->ops->solve[0]      = DSSolve_HEP_QR;
    767:   ds->ops->solve[1]      = DSSolve_HEP_MRRR;
    768:   ds->ops->solve[2]      = DSSolve_HEP_DC;
    769: #if !defined(PETSC_USE_COMPLEX)
    770:   ds->ops->solve[3]      = DSSolve_HEP_BDC;
    771: #endif
    772:   ds->ops->sort          = DSSort_HEP;
    773:   ds->ops->synchronize   = DSSynchronize_HEP;
    774:   ds->ops->truncate      = DSTruncate_HEP;
    775:   ds->ops->update        = DSUpdateExtraRow_HEP;
    776:   ds->ops->cond          = DSCond_HEP;
    777:   ds->ops->transrks      = DSTranslateRKS_HEP;
    778:   ds->ops->hermitian     = DSHermitian_HEP;
    779:   PetscFunctionReturn(0);
    780: }