Actual source code: dshep.c

slepc-3.16.0 2021-09-30
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2021, Universitat Politecnica de Valencia, Spain

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

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

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

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

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

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

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

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

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

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

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

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

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

188:   to tridiagonal form

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

295:   if (ds->compact) {

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

299:   } else {

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

806: /*MC
807:    DSHEP - Dense Hermitian Eigenvalue Problem.

809:    Level: beginner

811:    Notes:
812:    The problem is expressed as A*X = X*Lambda, where A is real symmetric
813:    (or complex Hermitian). Lambda is a diagonal matrix whose diagonal
814:    elements are the arguments of DSSolve(). After solve, A is overwritten
815:    with Lambda.

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

820:    Used DS matrices:
821: +  DS_MAT_A - problem matrix
822: .  DS_MAT_T - symmetric tridiagonal matrix
823: -  DS_MAT_Q - orthogonal/unitary transformation that reduces to tridiagonal form
824:    (intermediate step) or matrix of orthogonal eigenvectors, which is equal to X

826:    Implemented methods:
827: +  0 - Implicit QR (_steqr)
828: .  1 - Multiple Relatively Robust Representations (_stevr)
829: .  2 - Divide and Conquer (_stedc)
830: -  3 - Block Divide and Conquer (real scalars only)

832: .seealso: DSCreate(), DSSetType(), DSType
833: M*/
834: SLEPC_EXTERN PetscErrorCode DSCreate_HEP(DS ds)
835: {
837:   ds->ops->allocate      = DSAllocate_HEP;
838:   ds->ops->view          = DSView_HEP;
839:   ds->ops->vectors       = DSVectors_HEP;
840:   ds->ops->solve[0]      = DSSolve_HEP_QR;
841:   ds->ops->solve[1]      = DSSolve_HEP_MRRR;
842:   ds->ops->solve[2]      = DSSolve_HEP_DC;
843: #if !defined(PETSC_USE_COMPLEX)
844:   ds->ops->solve[3]      = DSSolve_HEP_BDC;
845: #endif
846:   ds->ops->sort          = DSSort_HEP;
847:   ds->ops->synchronize   = DSSynchronize_HEP;
848:   ds->ops->truncate      = DSTruncate_HEP;
849:   ds->ops->update        = DSUpdateExtraRow_HEP;
850:   ds->ops->cond          = DSCond_HEP;
851:   ds->ops->transrks      = DSTranslateRKS_HEP;
852:   ds->ops->hermitian     = DSHermitian_HEP;
853:   return(0);
854: }