Date: 2020-01-13T11:59:06+00:00

Actual source code: dshep.c
slepc-3.12.2 2020-01-13
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2019, 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:       break;
    175:     default:
    176:       SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_ARG_OUTOFRANGE,"Invalid mat parameter");
    177:   }
    178:   return(0);
    179: }

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

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

    190:   to tridiagonal form

    192:                 [ d e 0 0 0 ]
    193:                 [ e d e 0 0 ]
    194:    T = Q'*A*Q = [ 0 e d e 0 ],
    195:                 [ 0 0 e d e ]
    196:                 [ 0 0 0 e d ]

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

    202:   Arguments
    203:   =========

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

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

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

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

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

    224:   Note
    225:   ====
    226:   Based on Fortran code contributed by Daniel Kressner
    227: */
    228: static PetscErrorCode ArrowTridiag(PetscBLASInt n,PetscReal *d,PetscReal *e,PetscScalar *Q,PetscBLASInt ld)
    229: {
    230: #if defined(SLEPC_MISSING_LAPACK_LARTG)
    232:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LARTG - Lapack routine is unavailable");
    233: #else
    234:   PetscBLASInt i,j,j2,one=1;
    235:   PetscReal    c,s,p,off,temp;

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

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

    242:     /* Eliminate entry e(j) by a rotation in the planes (j,j+1) */
    243:     temp = e[j+1];
    244:     PetscStackCallBLAS("LAPACKlartg",LAPACKREALlartg_(&temp,&e[j],&c,&s,&e[j+1]));
    245:     s = -s;

    247:     /* Apply rotation to diagonal elements */
    248:     temp   = d[j+1];
    249:     e[j]   = c*s*(temp-d[j]);
    250:     d[j+1] = s*s*d[j] + c*c*temp;
    251:     d[j]   = c*c*d[j] + s*s*temp;

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

    257:     /* Chase newly introduced off-diagonal entry to the top left corner */
    258:     for (i=j-1;i>=0;i--) {
    259:       off  = -s*e[i];
    260:       e[i] = c*e[i];
    261:       temp = e[i+1];
    262:       PetscStackCallBLAS("LAPACKlartg",LAPACKREALlartg_(&temp,&off,&c,&s,&e[i+1]));
    263:       s = -s;
    264:       temp = (d[i]-d[i+1])*s - 2.0*c*e[i];
    265:       p = s*temp;
    266:       d[i+1] += p;
    267:       d[i] -= p;
    268:       e[i] = -e[i] - c*temp;
    269:       j2 = j+2;
    270:       PetscStackCallBLAS("BLASrot",BLASMIXEDrot_(&j2,Q+i*ld,&one,Q+(i+1)*ld,&one,&c,&s));
    271:     }
    272:   }
    273:   return(0);
    274: #endif
    275: }

    277: /*
    278:    Reduce to tridiagonal form by means of ArrowTridiag.
    279: */
    280: static PetscErrorCode DSIntermediate_HEP(DS ds)
    281: {
    282: #if defined(SLEPC_MISSING_LAPACK_SYTRD) || defined(SLEPC_MISSING_LAPACK_ORGTR)
    284:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"SYTRD/ORGTR - Lapack routine is unavailable");
    285: #else
    287:   PetscInt       i;
    288:   PetscBLASInt   n1,n2,n3,lwork,info,l,n,ld,off;
    289:   PetscScalar    *A,*Q,*work,*tau;
    290:   PetscReal      *d,*e;

    293:   PetscBLASIntCast(ds->n,&n);
    294:   PetscBLASIntCast(ds->l,&l);
    295:   PetscBLASIntCast(ds->ld,&ld);
    296:   PetscBLASIntCast(ds->k-l+1,&n1); /* size of leading block, excl. locked */
    297:   PetscBLASIntCast(n-ds->k-1,&n2); /* size of trailing block */
    298:   n3 = n1+n2;
    299:   off = l+l*ld;
    300:   A  = ds->mat[DS_MAT_A];
    301:   Q  = ds->mat[DS_MAT_Q];
    302:   d  = ds->rmat[DS_MAT_T];
    303:   e  = ds->rmat[DS_MAT_T]+ld;
    304:   PetscArrayzero(Q,ld*ld);
    305:   for (i=0;i<n;i++) Q[i+i*ld] = 1.0;

    307:   if (ds->compact) {

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

    311:   } else {

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

    315:     if (ds->state<DS_STATE_INTERMEDIATE) {
    316:       DSCopyMatrix_Private(ds,DS_MAT_Q,DS_MAT_A);
    317:       DSAllocateWork_Private(ds,ld+ld*ld,0,0);
    318:       tau  = ds->work;
    319:       work = ds->work+ld;
    320:       lwork = ld*ld;
    321:       PetscStackCallBLAS("LAPACKsytrd",LAPACKsytrd_("L",&n3,Q+off,&ld,d+l,e+l,tau,work,&lwork,&info));
    322:       SlepcCheckLapackInfo("sytrd",info);
    323:       PetscStackCallBLAS("LAPACKorgtr",LAPACKorgtr_("L",&n3,Q+off,&ld,tau,work,&lwork,&info));
    324:       SlepcCheckLapackInfo("orgtr",info);
    325:     } else {
    326:       /* copy tridiagonal to d,e */
    327:       for (i=l;i<n;i++)   d[i] = PetscRealPart(A[i+i*ld]);
    328:       for (i=l;i<n-1;i++) e[i] = PetscRealPart(A[(i+1)+i*ld]);
    329:     }
    330:   }
    331:   return(0);
    332: #endif
    333: }

    335: PetscErrorCode DSSort_HEP(DS ds,PetscScalar *wr,PetscScalar *wi,PetscScalar *rr,PetscScalar *ri,PetscInt *k)
    336: {
    338:   PetscInt       n,l,i,*perm,ld=ds->ld;
    339:   PetscScalar    *A;
    340:   PetscReal      *d;

    343:   if (!ds->sc) return(0);
    344:   n = ds->n;
    345:   l = ds->l;
    346:   A = ds->mat[DS_MAT_A];
    347:   d = ds->rmat[DS_MAT_T];
    348:   perm = ds->perm;
    349:   if (!rr) {
    350:     DSSortEigenvaluesReal_Private(ds,d,perm);
    351:   } else {
    352:     DSSortEigenvalues_Private(ds,rr,ri,perm,PETSC_FALSE);
    353:   }
    354:   for (i=l;i<n;i++) wr[i] = d[perm[i]];
    355:   DSPermuteColumns_Private(ds,l,n,DS_MAT_Q,perm);
    356:   for (i=l;i<n;i++) d[i] = PetscRealPart(wr[i]);
    357:   if (!ds->compact) {
    358:     for (i=l;i<n;i++) A[i+i*ld] = wr[i];
    359:   }
    360:   return(0);
    361: }

    363: PetscErrorCode DSUpdateExtraRow_HEP(DS ds)
    364: {
    366:   PetscInt       i;
    367:   PetscBLASInt   n,ld,incx=1;
    368:   PetscScalar    *A,*Q,*x,*y,one=1.0,zero=0.0;
    369:   PetscReal      *e,beta;

    372:   PetscBLASIntCast(ds->n,&n);
    373:   PetscBLASIntCast(ds->ld,&ld);
    374:   A  = ds->mat[DS_MAT_A];
    375:   Q  = ds->mat[DS_MAT_Q];
    376:   e  = ds->rmat[DS_MAT_T]+ld;

    378:   if (ds->compact) {
    379:     beta = e[n-1];   /* in compact, we assume all entries are zero except the last one */
    380:     for (i=0;i<n;i++) e[i] = PetscRealPart(beta*Q[n-1+i*ld]);
    381:     ds->k = n;
    382:   } else {
    383:     DSAllocateWork_Private(ds,2*ld,0,0);
    384:     x = ds->work;
    385:     y = ds->work+ld;
    386:     for (i=0;i<n;i++) x[i] = PetscConj(A[n+i*ld]);
    387:     PetscStackCallBLAS("BLASgemv",BLASgemv_("C",&n,&n,&one,Q,&ld,x,&incx,&zero,y,&incx));
    388:     for (i=0;i<n;i++) A[n+i*ld] = PetscConj(y[i]);
    389:     ds->k = n;
    390:   }
    391:   return(0);
    392: }

    394: PetscErrorCode DSSolve_HEP_QR(DS ds,PetscScalar *wr,PetscScalar *wi)
    395: {
    396: #if defined(PETSC_MISSING_LAPACK_STEQR)
    398:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"STEQR - Lapack routine is unavailable");
    399: #else
    401:   PetscInt       i;
    402:   PetscBLASInt   n1,n2,n3,info,l,n,ld,off;
    403:   PetscScalar    *Q,*A;
    404:   PetscReal      *d,*e;

    407:   if (ds->bs>1) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"This method is not prepared for bs>1");
    408:   PetscBLASIntCast(ds->n,&n);
    409:   PetscBLASIntCast(ds->l,&l);
    410:   PetscBLASIntCast(ds->ld,&ld);
    411:   PetscBLASIntCast(ds->k-l+1,&n1); /* size of leading block, excl. locked */
    412:   PetscBLASIntCast(n-ds->k-1,&n2); /* size of trailing block */
    413:   n3 = n1+n2;
    414:   off = l+l*ld;
    415:   Q  = ds->mat[DS_MAT_Q];
    416:   A  = ds->mat[DS_MAT_A];
    417:   d  = ds->rmat[DS_MAT_T];
    418:   e  = ds->rmat[DS_MAT_T]+ld;

    420:   /* Reduce to tridiagonal form */
    421:   DSIntermediate_HEP(ds);

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

    426:   DSAllocateWork_Private(ds,0,2*ld,0);
    427:   PetscStackCallBLAS("LAPACKsteqr",LAPACKsteqr_("V",&n3,d+l,e+l,Q+off,&ld,ds->rwork,&info));
    428:   SlepcCheckLapackInfo("steqr",info);
    429:   for (i=l;i<n;i++) wr[i] = d[i];

    431:   /* Create diagonal matrix as a result */
    432:   if (ds->compact) {
    433:     PetscArrayzero(e,n-1);
    434:   } else {
    435:     for (i=l;i<n;i++) {
    436:       PetscArrayzero(A+l+i*ld,n-l);
    437:     }
    438:     for (i=l;i<n;i++) A[i+i*ld] = d[i];
    439:   }

    441:   /* Set zero wi */
    442:   if (wi) for (i=l;i<n;i++) wi[i] = 0.0;
    443:   return(0);
    444: #endif
    445: }

    447: PetscErrorCode DSSolve_HEP_MRRR(DS ds,PetscScalar *wr,PetscScalar *wi)
    448: {
    449: #if defined(SLEPC_MISSING_LAPACK_STEVR)
    451:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"STEVR - Lapack routine is unavailable");
    452: #else
    454:   PetscInt       i;
    455:   PetscBLASInt   n1,n2,n3,lwork,liwork,info,l,n,m,ld,off,il,iu,*isuppz;
    456:   PetscScalar    *A,*Q,*W=NULL,one=1.0,zero=0.0;
    457:   PetscReal      *d,*e,abstol=0.0,vl,vu;
    458: #if defined(PETSC_USE_COMPLEX)
    459:   PetscInt       j;
    460:   PetscReal      *ritz;
    461: #endif

    464:   if (ds->bs>1) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"This method is not prepared for bs>1");
    465:   PetscBLASIntCast(ds->n,&n);
    466:   PetscBLASIntCast(ds->l,&l);
    467:   PetscBLASIntCast(ds->ld,&ld);
    468:   PetscBLASIntCast(ds->k-l+1,&n1); /* size of leading block, excl. locked */
    469:   PetscBLASIntCast(n-ds->k-1,&n2); /* size of trailing block */
    470:   n3 = n1+n2;
    471:   off = l+l*ld;
    472:   A  = ds->mat[DS_MAT_A];
    473:   Q  = ds->mat[DS_MAT_Q];
    474:   d  = ds->rmat[DS_MAT_T];
    475:   e  = ds->rmat[DS_MAT_T]+ld;

    477:   /* Reduce to tridiagonal form */
    478:   DSIntermediate_HEP(ds);

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

    483:   if (ds->state<DS_STATE_INTERMEDIATE) {  /* Q contains useful info */
    484:     DSAllocateMat_Private(ds,DS_MAT_W);
    485:     DSCopyMatrix_Private(ds,DS_MAT_W,DS_MAT_Q);
    486:     W = ds->mat[DS_MAT_W];
    487:   }
    488: #if defined(PETSC_USE_COMPLEX)
    489:   DSAllocateMatReal_Private(ds,DS_MAT_Q);
    490: #endif
    491:   lwork = 20*ld;
    492:   liwork = 10*ld;
    493:   DSAllocateWork_Private(ds,0,lwork+ld,liwork+2*ld);
    494:   isuppz = ds->iwork+liwork;
    495: #if defined(PETSC_USE_COMPLEX)
    496:   ritz = ds->rwork+lwork;
    497:   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));
    498:   for (i=l;i<n;i++) wr[i] = ritz[i];
    499: #else
    500:   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));
    501: #endif
    502:   SlepcCheckLapackInfo("stevr",info);
    503: #if defined(PETSC_USE_COMPLEX)
    504:   for (i=l;i<n;i++)
    505:     for (j=l;j<n;j++)
    506:       Q[i+j*ld] = (ds->rmat[DS_MAT_Q])[i+j*ld];
    507: #endif
    508:   if (ds->state<DS_STATE_INTERMEDIATE) {  /* accumulate previous Q */
    509:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n3,&n3,&n3,&one,W+off,&ld,Q+off,&ld,&zero,A+off,&ld));
    510:     DSCopyMatrix_Private(ds,DS_MAT_Q,DS_MAT_A);
    511:   }
    512:   for (i=l;i<n;i++) d[i] = PetscRealPart(wr[i]);

    514:   /* Create diagonal matrix as a result */
    515:   if (ds->compact) {
    516:     PetscArrayzero(e,n-1);
    517:   } else {
    518:     for (i=l;i<n;i++) {
    519:       PetscArrayzero(A+l+i*ld,n-l);
    520:     }
    521:     for (i=l;i<n;i++) A[i+i*ld] = d[i];
    522:   }

    524:   /* Set zero wi */
    525:   if (wi) for (i=l;i<n;i++) wi[i] = 0.0;
    526:   return(0);
    527: #endif
    528: }

    530: PetscErrorCode DSSolve_HEP_DC(DS ds,PetscScalar *wr,PetscScalar *wi)
    531: {
    532: #if defined(SLEPC_MISSING_LAPACK_STEDC)
    534:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"STEDC - Lapack routine is unavailable");
    535: #else
    537:   PetscInt       i;
    538:   PetscBLASInt   n1,info,l,ld,off,lrwork,liwork;
    539:   PetscScalar    *Q,*A;
    540:   PetscReal      *d,*e;
    541: #if defined(PETSC_USE_COMPLEX)
    542:   PetscBLASInt   lwork;
    543:   PetscInt       j;
    544: #endif

    547:   if (ds->bs>1) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"This method is not prepared for bs>1");
    548:   PetscBLASIntCast(ds->l,&l);
    549:   PetscBLASIntCast(ds->ld,&ld);
    550:   PetscBLASIntCast(ds->n-ds->l,&n1);
    551:   off = l+l*ld;
    552:   Q  = ds->mat[DS_MAT_Q];
    553:   A  = ds->mat[DS_MAT_A];
    554:   d  = ds->rmat[DS_MAT_T];
    555:   e  = ds->rmat[DS_MAT_T]+ld;

    557:   /* Reduce to tridiagonal form */
    558:   DSIntermediate_HEP(ds);

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

    563:   lrwork = 5*n1*n1+3*n1+1;
    564:   liwork = 5*n1*n1+6*n1+6;
    565: #if !defined(PETSC_USE_COMPLEX)
    566:   DSAllocateWork_Private(ds,0,lrwork,liwork);
    567:   PetscStackCallBLAS("LAPACKstedc",LAPACKstedc_("V",&n1,d+l,e+l,Q+off,&ld,ds->rwork,&lrwork,ds->iwork,&liwork,&info));
    568: #else
    569:   lwork = ld*ld;
    570:   DSAllocateWork_Private(ds,lwork,lrwork,liwork);
    571:   PetscStackCallBLAS("LAPACKstedc",LAPACKstedc_("V",&n1,d+l,e+l,Q+off,&ld,ds->work,&lwork,ds->rwork,&lrwork,ds->iwork,&liwork,&info));
    572:   /* Fixing Lapack bug*/
    573:   for (j=ds->l;j<ds->n;j++)
    574:     for (i=0;i<ds->l;i++) Q[i+j*ld] = 0.0;
    575: #endif
    576:   SlepcCheckLapackInfo("stedc",info);
    577:   for (i=l;i<ds->n;i++) wr[i] = d[i];

    579:   /* Create diagonal matrix as a result */
    580:   if (ds->compact) {
    581:     PetscArrayzero(e,ds->n-1);
    582:   } else {
    583:     for (i=l;i<ds->n;i++) {
    584:       PetscArrayzero(A+l+i*ld,ds->n-l);
    585:     }
    586:     for (i=l;i<ds->n;i++) A[i+i*ld] = d[i];
    587:   }

    589:   /* Set zero wi */
    590:   if (wi) for (i=l;i<ds->n;i++) wi[i] = 0.0;
    591:   return(0);
    592: #endif
    593: }

    595: #if !defined(PETSC_USE_COMPLEX)
    596: PetscErrorCode DSSolve_HEP_BDC(DS ds,PetscScalar *wr,PetscScalar *wi)
    597: {
    599:   PetscBLASInt   i,j,k,m,n,info,nblks,bs,ld,lde,lrwork,liwork,*ksizes,*iwork,mingapi;
    600:   PetscScalar    *Q,*A;
    601:   PetscReal      *D,*E,*d,*e,tol=PETSC_MACHINE_EPSILON/2,tau1=1e-16,tau2=1e-18,*rwork,mingap;

    604:   if (ds->l>0) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"This method is not prepared for l>1");
    605:   if (ds->compact) SETERRQ(PetscObjectComm((PetscObject)ds),PETSC_ERR_SUP,"Not implemented for compact storage");
    606:   PetscBLASIntCast(ds->ld,&ld);
    607:   PetscBLASIntCast(ds->bs,&bs);
    608:   PetscBLASIntCast(ds->n,&n);
    609:   nblks = n/bs;
    610:   Q  = ds->mat[DS_MAT_Q];
    611:   A  = ds->mat[DS_MAT_A];
    612:   d  = ds->rmat[DS_MAT_T];
    613:   e  = ds->rmat[DS_MAT_T]+ld;
    614:   lrwork = 4*n*n+60*n+1;
    615:   liwork = 5*n+5*nblks-1;
    616:   lde = 2*bs+1;
    617:   DSAllocateWork_Private(ds,bs*n+lde*lde*(nblks-1),lrwork,nblks+liwork);
    618:   D      = ds->work;
    619:   E      = ds->work+bs*n;
    620:   rwork  = ds->rwork;
    621:   ksizes = ds->iwork;
    622:   iwork  = ds->iwork+nblks;
    623:   PetscArrayzero(iwork,liwork);

    625:   /* Copy matrix to block tridiagonal format */
    626:   j=0;
    627:   for (i=0;i<nblks;i++) {
    628:     ksizes[i]=bs;
    629:     for (k=0;k<bs;k++)
    630:       for (m=0;m<bs;m++)
    631:         D[k+m*bs+i*bs*bs] = PetscRealPart(A[j+k+(j+m)*n]);
    632:     j = j + bs;
    633:   }
    634:   j=0;
    635:   for (i=0;i<nblks-1;i++) {
    636:     for (k=0;k<bs;k++)
    637:       for (m=0;m<bs;m++)
    638:         E[k+m*lde+i*lde*lde] = PetscRealPart(A[j+bs+k+(j+m)*n]);
    639:     j = j + bs;
    640:   }

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

    647:   /* Create diagonal matrix as a result */
    648:   if (ds->compact) {
    649:     PetscArrayzero(e,ds->n-1);
    650:   } else {
    651:     for (i=0;i<ds->n;i++) {
    652:       PetscArrayzero(A+i*ld,ds->n);
    653:     }
    654:     for (i=0;i<ds->n;i++) A[i+i*ld] = wr[i];
    655:   }

    657:   /* Set zero wi */
    658:   if (wi) for (i=0;i<ds->n;i++) wi[i] = 0.0;
    659:   return(0);
    660: }
    661: #endif

    663: PetscErrorCode DSTruncate_HEP(DS ds,PetscInt n)
    664: {
    665:   PetscInt    i,ld=ds->ld,l=ds->l;
    666:   PetscScalar *A;

    669:   if (ds->state==DS_STATE_CONDENSED) ds->t = ds->n;
    670:   A = ds->mat[DS_MAT_A];
    671:   if (!ds->compact && ds->extrarow && ds->k==ds->n) {
    672:     for (i=l;i<n;i++) A[n+i*ld] = A[ds->n+i*ld];
    673:   }
    674:   if (ds->extrarow) ds->k = n;
    675:   else ds->k = 0;
    676:   ds->n = n;
    677:   return(0);
    678: }

    680: PetscErrorCode DSSynchronize_HEP(DS ds,PetscScalar eigr[],PetscScalar eigi[])
    681: {
    683:   PetscInt       ld=ds->ld,l=ds->l,k=0,kr=0;
    684:   PetscMPIInt    n,rank,off=0,size,ldn,ld3;

    687:   if (ds->compact) kr = 3*ld;
    688:   else k = (ds->n-l)*ld;
    689:   if (ds->state>DS_STATE_RAW) k += (ds->n-l)*ld;
    690:   if (eigr) k += (ds->n-l);
    691:   DSAllocateWork_Private(ds,k+kr,0,0);
    692:   PetscMPIIntCast(k*sizeof(PetscScalar)+kr*sizeof(PetscReal),&size);
    693:   PetscMPIIntCast(ds->n-l,&n);
    694:   PetscMPIIntCast(ld*(ds->n-l),&ldn);
    695:   PetscMPIIntCast(ld*3,&ld3);
    696:   MPI_Comm_rank(PetscObjectComm((PetscObject)ds),&rank);
    697:   if (!rank) {
    698:     if (ds->compact) {
    699:       MPI_Pack(ds->rmat[DS_MAT_T],ld3,MPIU_REAL,ds->work,size,&off,PetscObjectComm((PetscObject)ds));
    700:     } else {
    701:       MPI_Pack(ds->mat[DS_MAT_A]+l*ld,ldn,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds));
    702:     }
    703:     if (ds->state>DS_STATE_RAW) {
    704:       MPI_Pack(ds->mat[DS_MAT_Q]+l*ld,ldn,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds));
    705:     }
    706:     if (eigr) {
    707:       MPI_Pack(eigr+l,n,MPIU_SCALAR,ds->work,size,&off,PetscObjectComm((PetscObject)ds));
    708:     }
    709:   }
    710:   MPI_Bcast(ds->work,size,MPI_BYTE,0,PetscObjectComm((PetscObject)ds));
    711:   if (rank) {
    712:     if (ds->compact) {
    713:       MPI_Unpack(ds->work,size,&off,ds->rmat[DS_MAT_T],ld3,MPIU_REAL,PetscObjectComm((PetscObject)ds));
    714:     } else {
    715:       MPI_Unpack(ds->work,size,&off,ds->mat[DS_MAT_A]+l*ld,ldn,MPIU_SCALAR,PetscObjectComm((PetscObject)ds));
    716:     }
    717:     if (ds->state>DS_STATE_RAW) {
    718:       MPI_Unpack(ds->work,size,&off,ds->mat[DS_MAT_Q]+l*ld,ldn,MPIU_SCALAR,PetscObjectComm((PetscObject)ds));
    719:     }
    720:     if (eigr) {
    721:       MPI_Unpack(ds->work,size,&off,eigr+l,n,MPIU_SCALAR,PetscObjectComm((PetscObject)ds));
    722:     }
    723:   }
    724:   return(0);
    725: }

    727: PetscErrorCode DSCond_HEP(DS ds,PetscReal *cond)
    728: {
    729: #if defined(PETSC_MISSING_LAPACK_GETRF) || defined(PETSC_MISSING_LAPACK_GETRI) || defined(SLEPC_MISSING_LAPACK_LANGE)
    731:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"GETRF/GETRI/LANGE - Lapack routines are unavailable");
    732: #else
    734:   PetscScalar    *work;
    735:   PetscReal      *rwork;
    736:   PetscBLASInt   *ipiv;
    737:   PetscBLASInt   lwork,info,n,ld;
    738:   PetscReal      hn,hin;
    739:   PetscScalar    *A;

    742:   PetscBLASIntCast(ds->n,&n);
    743:   PetscBLASIntCast(ds->ld,&ld);
    744:   lwork = 8*ld;
    745:   DSAllocateWork_Private(ds,lwork,ld,ld);
    746:   work  = ds->work;
    747:   rwork = ds->rwork;
    748:   ipiv  = ds->iwork;
    749:   DSSwitchFormat_HEP(ds);

    751:   /* use workspace matrix W to avoid overwriting A */
    752:   DSAllocateMat_Private(ds,DS_MAT_W);
    753:   A = ds->mat[DS_MAT_W];
    754:   PetscArraycpy(A,ds->mat[DS_MAT_A],ds->ld*ds->ld);

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

    759:   /* norm of inv(A) */
    760:   PetscStackCallBLAS("LAPACKgetrf",LAPACKgetrf_(&n,&n,A,&ld,ipiv,&info));
    761:   SlepcCheckLapackInfo("getrf",info);
    762:   PetscStackCallBLAS("LAPACKgetri",LAPACKgetri_(&n,A,&ld,ipiv,work,&lwork,&info));
    763:   SlepcCheckLapackInfo("getri",info);
    764:   hin = LAPACKlange_("I",&n,&n,A,&ld,rwork);

    766:   *cond = hn*hin;
    767:   return(0);
    768: #endif
    769: }

    771: PetscErrorCode DSTranslateRKS_HEP(DS ds,PetscScalar alpha)
    772: {
    773: #if defined(PETSC_MISSING_LAPACK_GEQRF) || defined(PETSC_MISSING_LAPACK_ORGQR)
    775:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"GEQRF/ORGQR - Lapack routines are unavailable");
    776: #else
    778:   PetscInt       i,j,k=ds->k;
    779:   PetscScalar    *Q,*A,*R,*tau,*work;
    780:   PetscBLASInt   ld,n1,n0,lwork,info;

    783:   PetscBLASIntCast(ds->ld,&ld);
    784:   DSAllocateWork_Private(ds,ld*ld,0,0);
    785:   tau = ds->work;
    786:   work = ds->work+ld;
    787:   PetscBLASIntCast(ld*(ld-1),&lwork);
    788:   DSAllocateMat_Private(ds,DS_MAT_W);
    789:   A  = ds->mat[DS_MAT_A];
    790:   Q  = ds->mat[DS_MAT_Q];
    791:   R  = ds->mat[DS_MAT_W];

    793:   /* copy I+alpha*A */
    794:   PetscArrayzero(Q,ld*ld);
    795:   PetscArrayzero(R,ld*ld);
    796:   for (i=0;i<k;i++) {
    797:     Q[i+i*ld] = 1.0 + alpha*A[i+i*ld];
    798:     Q[k+i*ld] = alpha*A[k+i*ld];
    799:   }

    801:   /* compute qr */
    802:   PetscBLASIntCast(k+1,&n1);
    803:   PetscBLASIntCast(k,&n0);
    804:   PetscStackCallBLAS("LAPACKgeqrf",LAPACKgeqrf_(&n1,&n0,Q,&ld,tau,work,&lwork,&info));
    805:   SlepcCheckLapackInfo("geqrf",info);

    807:   /* copy R from Q */
    808:   for (j=0;j<k;j++)
    809:     for (i=0;i<=j;i++)
    810:       R[i+j*ld] = Q[i+j*ld];

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

    816:   /* compute the updated matrix of projected problem */
    817:   for (j=0;j<k;j++)
    818:     for (i=0;i<k+1;i++)
    819:       A[j*ld+i] = Q[i*ld+j];
    820:   alpha = -1.0/alpha;
    821:   PetscStackCallBLAS("BLAStrsm",BLAStrsm_("R","U","N","N",&n1,&n0,&alpha,R,&ld,A,&ld));
    822:   for (i=0;i<k;i++)
    823:     A[ld*i+i] -= alpha;
    824:   return(0);
    825: #endif
    826: }

    828: PetscErrorCode DSHermitian_HEP(DS ds,DSMatType m,PetscBool *flg)
    829: {
    831:   if (m==DS_MAT_A && !ds->extrarow) *flg = PETSC_TRUE;
    832:   else *flg = PETSC_FALSE;
    833:   return(0);
    834: }

    836: SLEPC_EXTERN PetscErrorCode DSCreate_HEP(DS ds)
    837: {
    839:   ds->ops->allocate      = DSAllocate_HEP;
    840:   ds->ops->view          = DSView_HEP;
    841:   ds->ops->vectors       = DSVectors_HEP;
    842:   ds->ops->solve[0]      = DSSolve_HEP_QR;
    843:   ds->ops->solve[1]      = DSSolve_HEP_MRRR;
    844:   ds->ops->solve[2]      = DSSolve_HEP_DC;
    845: #if !defined(PETSC_USE_COMPLEX)
    846:   ds->ops->solve[3]      = DSSolve_HEP_BDC;
    847: #endif
    848:   ds->ops->sort          = DSSort_HEP;
    849:   ds->ops->synchronize   = DSSynchronize_HEP;
    850:   ds->ops->truncate      = DSTruncate_HEP;
    851:   ds->ops->update        = DSUpdateExtraRow_HEP;
    852:   ds->ops->cond          = DSCond_HEP;
    853:   ds->ops->transrks      = DSTranslateRKS_HEP;
    854:   ds->ops->hermitian     = DSHermitian_HEP;
    855:   return(0);
    856: }