Actual source code: invit.c

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

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

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

 14: struct HRtr
 15: {
 16:   PetscScalar *data;
 17:   PetscInt    m;
 18:   PetscInt    idx[2];
 19:   PetscInt    n[2];
 20:   PetscScalar tau[2];
 21:   PetscReal   alpha;
 22:   PetscReal   cs;
 23:   PetscReal   sn;
 24:   PetscInt    type;
 25: };

 27: /*
 28:   Generates a hyperbolic rotation
 29:     if x1*x1 - x2*x2 != 0
 30:       r = sqrt(|x1*x1 - x2*x2|)
 31:       c = x1/r  s = x2/r

 33:       | c -s||x1|   |d*r|
 34:       |-s  c||x2| = | 0 |
 35:       where d = 1 for type==1 and -1 for type==2
 36:   Returns the condition number of the reduction
 37: */
 38: static PetscErrorCode HRGen(PetscReal x1,PetscReal x2,PetscInt *type,PetscReal *c,PetscReal *s,PetscReal *r,PetscReal *cond)
 39: {
 40:   PetscReal t,n2,xa,xb;
 41:   PetscInt  type_;

 44:   if (x2==0.0) {
 45:     *r = PetscAbsReal(x1); *c = (x1>=0.0)?1.0:-1.0; *s = 0.0;
 46:     if (type) *type = 1;
 47:     return(0);
 48:   }
 49:   if (PetscAbsReal(x1) == PetscAbsReal(x2)) {
 50:     /* hyperbolic rotation doesn't exist */
 51:     *c = *s = *r = 0.0;
 52:     if (type) *type = 0;
 53:     *cond = PETSC_MAX_REAL;
 54:     return(0);
 55:   }

 57:   if (PetscAbsReal(x1)>PetscAbsReal(x2)) {
 58:     xa = x1; xb = x2; type_ = 1;
 59:   } else {
 60:     xa = x2; xb = x1; type_ = 2;
 61:   }
 62:   t = xb/xa;
 63:   n2 = PetscAbsReal(1 - t*t);
 64:   *r = PetscSqrtReal(n2)*PetscAbsReal(xa);
 65:   *c = x1/(*r);
 66:   *s = x2/(*r);
 67:   if (type_ == 2) *r *= -1;
 68:   if (type) *type = type_;
 69:   if (cond) *cond = (PetscAbsReal(*c) + PetscAbsReal(*s))/PetscAbsReal(PetscAbsReal(*c) - PetscAbsReal(*s));
 70:   return(0);
 71: }

 73: /*
 74:                                 |c  s|
 75:   Applies an hyperbolic rotator |s  c|
 76:            |c  s|
 77:     [x1 x2]|s  c|
 78: */
 79: static PetscErrorCode HRApply(PetscInt n,PetscScalar *x1,PetscInt inc1,PetscScalar *x2,PetscInt inc2,PetscReal c,PetscReal s)
 80: {
 81:   PetscInt    i;
 82:   PetscReal   t;
 83:   PetscScalar tmp;

 86:   if (PetscAbsReal(c)>PetscAbsReal(s)) { /* Type I */
 87:     t = s/c;
 88:     for (i=0;i<n;i++) {
 89:       x1[i*inc1] = c*x1[i*inc1] + s*x2[i*inc2];
 90:       x2[i*inc2] = t*x1[i*inc1] + x2[i*inc2]/c;
 91:     }
 92:   } else { /* Type II */
 93:     t = c/s;
 94:     for (i=0;i<n;i++) {
 95:       tmp = x1[i*inc1];
 96:       x1[i*inc1] = c*x1[i*inc1] + s*x2[i*inc2];
 97:       x2[i*inc2] = t*x1[i*inc1] + tmp/s;
 98:     }
 99:   }
100:   return(0);
101: }

103: /*
104:   Reduction to tridiagonal-diagonal form (see F. Tisseur, SIMAX 26(1), 2004).

106:   Input:
107:     A symmetric (only lower triangular part is referred)
108:     s vector +1 and -1 (signature matrix)
109:   Output:
110:     d,e
111:     s
112:     Q s-orthogonal matrix with Q^T*A*Q = T (symmetric tridiagonal matrix)
113: */
114: static PetscErrorCode TridiagDiag_HHR(PetscInt n,PetscScalar *A,PetscInt lda,PetscReal *s,PetscScalar* Q,PetscInt ldq,PetscBool flip,PetscReal *d,PetscReal *e,PetscInt *perm_,PetscScalar *work,PetscReal *rwork,PetscBLASInt *iwork)
115: {
117:   PetscInt       i,j,k,*ii,*jj,i0=0,ik=0,tmp,type;
118:   PetscInt       nwu=0;
119:   PetscReal      *ss,cond=1.0,cs,sn,r;
120:   PetscScalar    tau,t,*AA;
121:   PetscBLASInt   n0,n1,ni,inc=1,m,n_,lda_,ldq_,*perm;
122:   PetscBool      breakdown = PETSC_TRUE;

125:   if (n<3) {
126:     if (n==1) Q[0]=1;
127:     if (n==2) {
128:       Q[0] = Q[1+ldq] = 1;
129:       Q[1] = Q[ldq] = 0;
130:     }
131:     return(0);
132:   }
133:   PetscBLASIntCast(lda,&lda_);
134:   PetscBLASIntCast(n,&n_);
135:   PetscBLASIntCast(ldq,&ldq_);
136:   ss = rwork;
137:   perm = iwork;
138:   AA = work;
139:   for (i=0;i<n;i++) {
140:     PetscArraycpy(AA+i*n,A+i*lda,n);
141:   }
142:   nwu += n*n;
143:   k=0;
144:   while (breakdown && k<n) {
145:     breakdown = PETSC_FALSE;
146:     /* Classify (and flip) A and s according to sign */
147:     if (flip) {
148:       for (i=0;i<n;i++) {
149:         perm[i] = n-1-perm_[i];
150:         if (perm[i]==0) i0 = i;
151:         if (perm[i]==k) ik = i;
152:       }
153:     } else {
154:       for (i=0;i<n;i++) {
155:         perm[i] = perm_[i];
156:         if (perm[i]==0) i0 = i;
157:         if (perm[i]==k) ik = i;
158:       }
159:     }
160:     perm[ik] = 0;
161:     perm[i0] = k;
162:     i=1;
163:     while (i<n-1 && s[perm[i-1]]==s[perm[0]]) {
164:       if (s[perm[i]]!=s[perm[0]]) {
165:         j=i+1;
166:         while (j<n-1 && s[perm[j]]!=s[perm[0]])j++;
167:         tmp = perm[i]; perm[i] = perm[j]; perm[j] = tmp;
168:       }
169:       i++;
170:     }
171:     for (i=0;i<n;i++) {
172:       ss[i] = s[perm[i]];
173:     }
174:     if (flip) {
175:       ii = &j;
176:       jj = &i;
177:     } else {
178:       ii = &i;
179:       jj = &j;
180:     }
181:     for (i=0;i<n;i++)
182:       for (j=0;j<n;j++)
183:         A[i+j*lda] = AA[perm[*ii]+perm[*jj]*n];
184:     /* Initialize Q */
185:     for (i=0;i<n;i++) {
186:       PetscArrayzero(Q+i*ldq,n);
187:       Q[perm[i]+i*ldq] = 1.0;
188:     }
189:     for (ni=1;ni<n && ss[ni]==ss[0]; ni++);
190:     n0 = ni-1;
191:     n1 = n_-ni;
192:     for (j=0;j<n-2;j++) {
193:       PetscBLASIntCast(n-j-1,&m);
194:       /* Forming and applying reflectors */
195:       if (n0 > 1) {
196:         PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n0,A+ni-n0+j*lda,A+ni-n0+j*lda+1,&inc,&tau));
197:         /* Apply reflector */
198:         if (PetscAbsScalar(tau) != 0.0) {
199:           t=*(A+ni-n0+j*lda);  *(A+ni-n0+j*lda)=1.0;
200:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&m,&n0,A+ni-n0+j*lda,&inc,&tau,A+j+1+(j+1)*lda,&lda_,work+nwu));
201:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n0,&m,A+ni-n0+j*lda,&inc,&tau,A+j+1+(j+1)*lda,&lda_,work+nwu));
202:           /* Update Q */
203:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n0,A+ni-n0+j*lda,&inc,&tau,Q+(j+1)*ldq,&ldq_,work+nwu));
204:           *(A+ni-n0+j*lda) = t;
205:           for (i=1;i<n0;i++) {
206:             *(A+ni-n0+j*lda+i) = 0.0;  *(A+j+(ni-n0+i)*lda) = 0.0;
207:           }
208:           *(A+j+(ni-n0)*lda) = *(A+ni-n0+j*lda);
209:         }
210:       }
211:       if (n1 > 1) {
212:         PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n1,A+n-n1+j*lda,A+n-n1+j*lda+1,&inc,&tau));
213:         /* Apply reflector */
214:         if (PetscAbsScalar(tau) != 0.0) {
215:           t=*(A+n-n1+j*lda);  *(A+n-n1+j*lda)=1.0;
216:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&m,&n1,A+n-n1+j*lda,&inc,&tau,A+j+1+(n-n1)*lda,&lda_,work+nwu));
217:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n1,&m,A+n-n1+j*lda,&inc,&tau,A+n-n1+(j+1)*lda,&lda_,work+nwu));
218:           /* Update Q */
219:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n1,A+n-n1+j*lda,&inc,&tau,Q+(n-n1)*ldq,&ldq_,work+nwu));
220:           *(A+n-n1+j*lda) = t;
221:           for (i=1;i<n1;i++) {
222:             *(A+n-n1+i+j*lda) = 0.0;  *(A+j+(n-n1+i)*lda) = 0.0;
223:           }
224:           *(A+j+(n-n1)*lda) = *(A+n-n1+j*lda);
225:         }
226:       }
227:       /* Hyperbolic rotation */
228:       if (n0 > 0 && n1 > 0) {
229:         HRGen(PetscRealPart(A[ni-n0+j*lda]),PetscRealPart(A[n-n1+j*lda]),&type,&cs,&sn,&r,&cond);
230:         /* Check condition number */
231:         if (cond > 1.0/(10*PETSC_SQRT_MACHINE_EPSILON)) {
232:           breakdown = PETSC_TRUE;
233:           k++;
234:           if (k==n || flip) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Breakdown in construction of hyperbolic transformation");
235:           break;
236:         }
237:         A[ni-n0+j*lda] = r; A[n-n1+j*lda] = 0.0;
238:         A[j+(ni-n0)*lda] = r; A[j+(n-n1)*lda] = 0.0;
239:         /* Apply to A */
240:         HRApply(m,A+j+1+(ni-n0)*lda,1,A+j+1+(n-n1)*lda,1,cs,-sn);
241:         HRApply(m,A+ni-n0+(j+1)*lda,lda,A+n-n1+(j+1)*lda,lda,cs,-sn);

243:         /* Update Q */
244:         HRApply(n,Q+(ni-n0)*ldq,1,Q+(n-n1)*ldq,1,cs,-sn);
245:         if (type==2) {
246:           ss[ni-n0] = -ss[ni-n0]; ss[n-n1] = -ss[n-n1];
247:           n0++;ni++;n1--;
248:         }
249:       }
250:       if (n0>0) n0--;
251:       else n1--;
252:     }
253:   }

255:   /* flip matrices */
256:   if (flip) {
257:     for (i=0;i<n-1;i++) {
258:       d[i] = PetscRealPart(A[n-i-1+(n-i-1)*lda]);
259:       e[i] = PetscRealPart(A[n-i-1+(n-i-2)*lda]);
260:       s[i] = ss[n-i-1];
261:     }
262:     s[n-1] = ss[0];
263:     d[n-1] = PetscRealPart(A[0]);
264:     for (i=0;i<n;i++) {
265:       ierr=PetscArraycpy(work+i*n,Q+i*ldq,n);
266:     }
267:     for (i=0;i<n;i++)
268:       for (j=0;j<n;j++)
269:         Q[i+j*ldq] = work[i+(n-j-1)*n];
270:   } else {
271:     for (i=0;i<n-1;i++) {
272:       d[i] = PetscRealPart(A[i+i*lda]);
273:       e[i] = PetscRealPart(A[i+1+i*lda]);
274:       s[i] = ss[i];
275:     }
276:     s[n-1] = ss[n-1];
277:     d[n-1] = PetscRealPart(A[n-1 + (n-1)*lda]);
278:   }
279:   return(0);
280: }

282: static PetscErrorCode MadeHRtr(PetscInt sz,PetscInt n,PetscInt idx0,PetscInt n0,PetscInt idx1,PetscInt n1,struct HRtr *tr1,struct HRtr *tr2,PetscReal *ncond,PetscScalar *work)
283: {
285:   PetscScalar    *x,*y;
286:   PetscReal      ncond2=1.0;
287:   PetscBLASInt   n0_,n1_,inc=1;

290:   /* Hyperbolic transformation to make zeros in x */
291:   x = tr1->data;
292:   tr1->n[0] = n0;
293:   tr1->n[1] = n1;
294:   tr1->idx[0] = idx0;
295:   tr1->idx[1] = idx1;
296:   PetscBLASIntCast(tr1->n[0],&n0_);
297:   PetscBLASIntCast(tr1->n[1],&n1_);
298:   if (tr1->n[0] > 1) {
299:     PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n0_,x+tr1->idx[0],x+tr1->idx[0]+1,&inc,tr1->tau));
300:   }
301:   if (tr1->n[1]> 1) {
302:     PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n1_,x+tr1->idx[1],x+tr1->idx[1]+1,&inc,tr1->tau+1));
303:   }
304:   if (tr1->idx[0]<tr1->idx[1]) {
305:     HRGen(PetscRealPart(x[tr1->idx[0]]),PetscRealPart(x[tr1->idx[1]]),&(tr1->type),&(tr1->cs),&(tr1->sn),&(tr1->alpha),ncond);
306:   } else {
307:     tr1->alpha = PetscRealPart(x[tr1->idx[0]]);
308:     *ncond = 1.0;
309:   }
310:   if (sz==2) {
311:     y = tr2->data;
312:     /* Apply first transformation to second column */
313:     if (tr1->n[0] > 1 && PetscAbsScalar(tr1->tau[0])!=0.0) {
314:       x[tr1->idx[0]] = 1.0;
315:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n0_,&inc,x+tr1->idx[0],&inc,tr1->tau,y+tr1->idx[0],&n0_,work));
316:     }
317:     if (tr1->n[1] > 1 && PetscAbsScalar(tr1->tau[1])!=0.0) {
318:       x[tr1->idx[1]] = 1.0;
319:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n1_,&inc,x+tr1->idx[1],&inc,tr1->tau+1,y+tr1->idx[1],&n1_,work));
320:     }
321:     if (tr1->idx[0]<tr1->idx[1]) {
322:       HRApply(1,y+tr1->idx[0],1,y+tr1->idx[1],1,tr1->cs,-tr1->sn);
323:     }
324:     tr2->n[0] = tr1->n[0];
325:     tr2->n[1] = tr1->n[1];
326:     tr2->idx[0] = tr1->idx[0];
327:     tr2->idx[1] = tr1->idx[1];
328:     if (tr1->idx[0]<tr1->idx[1] && tr1->type==2) {
329:       tr2->idx[1]++; tr2->n[1]--; tr2->n[0]++;
330:     }
331:     if (tr2->n[0]>0) {
332:       tr2->n[0]--; tr2->idx[0]++;
333:       if (tr2->n[1]==0) tr2->idx[1] = tr2->idx[0];
334:     } else {
335:       tr2->n[1]--; tr2->idx[1]++; tr2->idx[0] = tr2->idx[1];
336:     }
337:     /* Hyperbolic transformation to make zeros in y */
338:     PetscBLASIntCast(tr2->n[0],&n0_);
339:     PetscBLASIntCast(tr2->n[1],&n1_);
340:     if (tr2->n[0] > 1) {
341:       PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n0_,y+tr2->idx[0],y+tr2->idx[0]+1,&inc,tr2->tau));
342:     }
343:     if (tr2->n[1]> 1) {
344:       PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n1_,y+tr2->idx[1],y+tr2->idx[1]+1,&inc,tr2->tau+1));
345:     }
346:     if (tr2->idx[0]<tr2->idx[1]) {
347:       HRGen(PetscRealPart(y[tr2->idx[0]]),PetscRealPart(y[tr2->idx[1]]),&(tr2->type),&(tr2->cs),&(tr2->sn),&(tr2->alpha),&ncond2);
348:     } else {
349:       tr2->alpha = PetscRealPart(y[tr2->idx[0]]);
350:       ncond2 = 1.0;
351:     }
352:     if (ncond2>*ncond) *ncond = ncond2;
353:   }
354:   return(0);
355: }

357: /*
358:   Auxiliary function to try perform one iteration of hr routine,
359:   checking condition number. If it is < tolD, apply the
360:   transformation to H and R, if not, ok=false and it do nothing
361:   tolE, tolerance to exchange complex pairs to improve conditioning
362: */
363: static PetscErrorCode TryHRIt(PetscInt n,PetscInt j,PetscInt sz,PetscScalar *H,PetscInt ldh,PetscScalar *R,PetscInt ldr,PetscReal *s,PetscBool *exg,PetscBool *ok,PetscInt *n0,PetscInt *n1,PetscInt *idx0,PetscInt *idx1,PetscReal *cond,PetscScalar *work)
364: {
366:   struct HRtr    *tr1,*tr2,tr1_t,tr2_t,tr1_te,tr2_te;
367:   PetscScalar    *x,*y;
368:   PetscReal      ncond,ncond_e;
369:   PetscInt       nwu=0,i,d=1;
370:   PetscBLASInt   n0_,n1_,inc=1,mh,mr,n_,ldr_,ldh_;
371:   PetscReal      tolD = 1e+5;

374:   if (cond) *cond = 1.0;
375:   PetscBLASIntCast(n,&n_);
376:   PetscBLASIntCast(ldr,&ldr_);
377:   PetscBLASIntCast(ldh,&ldh_);
378:   x = work+nwu;
379:   nwu += n;
380:   PetscArraycpy(x,R+j*ldr,n);
381:   *exg = PETSC_FALSE;
382:   *ok = PETSC_TRUE;
383:   tr1_t.data = x;
384:   if (sz==1) {
385:     /* Hyperbolic transformation to make zeros in x */
386:     MadeHRtr(sz,n,*idx0,*n0,*idx1,*n1,&tr1_t,NULL,&ncond,work+nwu);
387:     /* Check condition number to single column*/
388:     if (ncond>tolD) *ok = PETSC_FALSE;
389:     tr1 = &tr1_t;
390:     tr2 = &tr2_t;
391:   } else {
392:     y = work+nwu;
393:     nwu += n;
394:     PetscArraycpy(y,R+(j+1)*ldr,n);
395:     tr2_t.data = y;
396:     MadeHRtr(sz,n,*idx0,*n0,*idx1,*n1,&tr1_t,&tr2_t,&ncond,work+nwu);
397:     /* Computing hyperbolic transformations also for exchanged vectors */
398:     tr1_te.data = work+nwu;
399:     nwu += n;
400:     PetscArraycpy(tr1_te.data,R+(j+1)*ldr,n);
401:     tr2_te.data = work+nwu;
402:     nwu += n;
403:     PetscArraycpy(tr2_te.data,R+j*ldr,n);
404:     MadeHRtr(sz,n,*idx0,*n0,*idx1,*n1,&tr1_te,&tr2_te,&ncond_e,work+nwu);
405:     if (ncond > d*ncond_e) {
406:       *exg = PETSC_TRUE;
407:       tr1 = &tr1_te;
408:       tr2 = &tr2_te;
409:       ncond = ncond_e;
410:     } else {
411:       tr1 = &tr1_t;
412:       tr2 = &tr2_t;
413:     }
414:     if (ncond>tolD) *ok = PETSC_FALSE;
415:   }
416:   if (*ok) {
417:     /* Everything is OK, apply transformations to R and H */
418:     /* First column */
419:     if (cond && *cond<ncond) *cond = ncond;
420:     x = tr1->data;
421:     PetscBLASIntCast(tr1->n[0],&n0_);
422:     PetscBLASIntCast(tr1->n[1],&n1_);
423:     PetscBLASIntCast(n-j-sz,&mr);
424:     if (tr1->n[0] > 1 && PetscAbsScalar(tr1->tau[0])!=0.0) {
425:       x[tr1->idx[0]] = 1.0;
426:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n0_,&mr,x+tr1->idx[0],&inc,tr1->tau,R+(j+sz)*ldr+tr1->idx[0],&ldr_,work+nwu));
427:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n0_,x+tr1->idx[0],&inc,tr1->tau,H+(tr1->idx[0])*ldh,&ldh_,work+nwu));
428:     }
429:     if (tr1->n[1] > 1 && PetscAbsScalar(tr1->tau[1])!=0.0) {
430:       x[tr1->idx[1]] = 1.0;
431:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n1_,&mr,x+tr1->idx[1],&inc,tr1->tau+1,R+(j+sz)*ldr+tr1->idx[1],&ldr_,work+nwu));
432:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n1_,x+tr1->idx[1],&inc,tr1->tau+1,H+(tr1->idx[1])*ldh,&ldh_,work+nwu));
433:     }
434:     if (tr1->idx[0]<tr1->idx[1]) {
435:       HRApply(mr,R+(j+sz)*ldr+tr1->idx[0],ldr,R+(j+sz)*ldr+tr1->idx[1],ldr,tr1->cs,-tr1->sn);
436:       if (tr1->type==1) {
437:         HRApply(n,H+(tr1->idx[0])*ldh,1,H+(tr1->idx[1])*ldh,1,tr1->cs,tr1->sn);
438:       } else {
439:         HRApply(n,H+(tr1->idx[0])*ldh,1,H+(tr1->idx[1])*ldh,1,-tr1->cs,-tr1->sn);
440:         s[tr1->idx[0]] = -s[tr1->idx[0]];
441:         s[tr1->idx[1]] = -s[tr1->idx[1]];
442:       }
443:     }
444:     for (i=0;i<tr1->idx[0];i++) *(R+j*ldr+i) = x[i];
445:     for (i=tr1->idx[0]+1;i<n;i++) *(R+j*ldr+i) = 0.0;
446:     *(R+j*ldr+tr1->idx[0]) = tr1->alpha;
447:     if (sz==2) {
448:       y = tr2->data;
449:       /* Second column */
450:       PetscBLASIntCast(tr2->n[0],&n0_);
451:       PetscBLASIntCast(tr2->n[1],&n1_);
452:       PetscBLASIntCast(n-j-sz,&mr);
453:       PetscBLASIntCast(n-tr2->idx[0],&mh);
454:       if (tr2->n[0] > 1 && PetscAbsScalar(tr2->tau[0])!=0.0) {
455:         y[tr2->idx[0]] = 1.0;
456:         PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n0_,&mr,y+tr2->idx[0],&inc,tr2->tau,R+(j+2)*ldr+tr2->idx[0],&ldr_,work+nwu));
457:         PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n0_,y+tr2->idx[0],&inc,tr2->tau,H+(tr2->idx[0])*ldh,&ldh_,work+nwu));
458:       }
459:       if (tr2->n[1] > 1 && PetscAbsScalar(tr2->tau[1])!=0.0) {
460:         y[tr2->idx[1]] = 1.0;
461:         PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n1_,&mr,y+tr2->idx[1],&inc,tr2->tau+1,R+(j+2)*ldr+tr2->idx[1],&ldr_,work+nwu));
462:         PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n1_,y+tr2->idx[1],&inc,tr2->tau+1,H+(tr2->idx[1])*ldh,&ldh_,work+nwu));
463:       }
464:       if (tr2->idx[0]<tr2->idx[1]) {
465:         HRApply(mr,R+(j+2)*ldr+tr2->idx[0],ldr,R+(j+2)*ldr+tr2->idx[1],ldr,tr2->cs,-tr2->sn);
466:         if (tr2->type==1) {
467:           HRApply(n,H+(tr2->idx[0])*ldh,1,H+(tr2->idx[1])*ldh,1,tr2->cs,tr2->sn);
468:         } else {
469:           HRApply(n,H+(tr2->idx[0])*ldh,1,H+(tr2->idx[1])*ldh,1,-tr2->cs,-tr2->sn);
470:           s[tr2->idx[0]] = -s[tr2->idx[0]];
471:           s[tr2->idx[1]] = -s[tr2->idx[1]];
472:         }
473:       }
474:       for (i=0;i<tr2->idx[0]-1;i++) *(R+(j+1)*ldr+i) = y[i];
475:       *(R+(j+1)*ldr+tr2->idx[0]-1) = y[tr2->idx[0]-1];
476:       for (i=tr2->idx[0]+1;i<n;i++) *(R+(j+1)*ldr+i) = 0.0;
477:       *(R+(j+1)*ldr+tr2->idx[0]) = tr2->alpha;
478:       *n0 = tr2->n[0];
479:       *n1 = tr2->n[1];
480:       *idx0 = tr2->idx[0];
481:       *idx1 = tr2->idx[1];
482:       if (tr2->idx[0]<tr2->idx[1] && tr2->type==2) {
483:         (*idx1)++; (*n1)--; (*n0)++;
484:       }
485:     } else {
486:       *n0 = tr1->n[0];
487:       *n1 = tr1->n[1];
488:       *idx0 = tr1->idx[0];
489:       *idx1 = tr1->idx[1];
490:       if (tr1->idx[0]<tr1->idx[1] && tr1->type==2) {
491:         (*idx1)++; (*n1)--; (*n0)++;
492:       }
493:     }
494:     if (*n0>0) {
495:       (*n0)--; (*idx0)++;
496:       if (*n1==0) *idx1 = *idx0;
497:     } else {
498:       (*n1)--; (*idx1)++; *idx0 = *idx1;
499:     }
500:   }
501:   return(0);
502: }

504: /*
505:   compute V = HR whit H s-orthogonal and R upper triangular
506: */
507: static PetscErrorCode PseudoOrthog_HR(PetscInt *nv,PetscScalar *V,PetscInt ldv,PetscReal *s,PetscScalar *R,PetscInt ldr,PetscBLASInt *perm,PetscBLASInt *cmplxEig,PetscBool *breakdown,PetscScalar *work)
508: {
510:   PetscInt       i,j,n,n0,n1,np,idx0,idx1,sz=1,k=0,t1,t2,nwu=0;
511:   PetscScalar    *col1,*col2;
512:   PetscBool      exg=PETSC_FALSE,ok=PETSC_FALSE;

515:   n = *nv;
516:   col1 = work+nwu;
517:   nwu += n;
518:   col2 = work+nwu;
519:   nwu += n;
520:   /* Sort R and s according to sing(s) */
521:   np = 0;
522:   for (i=0;i<n;i++) if (s[i]>0) np++;
523:   if (s[0]>0) n1 = np;
524:   else n1 = n-np;
525:   n0 = 0;
526:   for (i=0;i<n;i++) {
527:     if (s[i]==s[0]) {
528:       s[n0] = s[0];
529:       perm[n0++] = i;
530:     } else perm[n1++] = i;
531:   }
532:   for (i=n0;i<n;i++) s[i] = -s[0];
533:   n1 -= n0;
534:   idx0 = 0;
535:   idx1 = n0;
536:   if (idx1==n) idx1=idx0;
537:   for (i=0;i<n;i++) {
538:     for (j=0;j<n;j++) R[j*ldr+i] = V[j*ldv+perm[i]];
539:   }
540:   /* Initialize H */
541:   for (i=0;i<n;i++) {
542:     PetscArrayzero(V+i*ldv,n);
543:     V[perm[i]+i*ldv] = 1.0;
544:   }
545:   for (i=0;i<n;i++) perm[i] = i;
546:   j = 0;
547:   while (j<n-k) {
548:     if (cmplxEig[j]==0) sz=1;
549:     else sz=2;
550:     TryHRIt(n,j,sz,V,ldv,R,ldr,s,&exg,&ok,&n0,&n1,&idx0,&idx1,NULL,work+nwu);
551:     if (ok) {
552:       if (exg) cmplxEig[j] = -cmplxEig[j];
553:       j = j+sz;
554:     } else { /* to be discarded */
555:       k = k+1;
556:       if (cmplxEig[j]==0) {
557:         if (j<n) {
558:           t1 = perm[j];
559:           for (i=j;i<n-1;i++) perm[i] = perm[i+1];
560:           perm[n-1] = t1;
561:           t1 = cmplxEig[j];
562:           for (i=j;i<n-1;i++) cmplxEig[i] = cmplxEig[i+1];
563:           cmplxEig[n-1] = t1;
564:           PetscArraycpy(col1,R+j*ldr,n*sizeof(PetscScalar));
565:           for (i=j;i<n-1;i++) {
566:             PetscArraycpy(R+i*ldr,R+(i+1)*ldr,n*sizeof(PetscScalar));
567:           }
568:           PetscArraycpy(R+(n-1)*ldr,col1,n*sizeof(PetscScalar));
569:         }
570:       } else {
571:         k = k+1;
572:         if (j<n-1) {
573:           t1 = perm[j]; t2 = perm[j+1];
574:           for (i=j;i<n-2;i++) perm[i] = perm[i+2];
575:           perm[n-2] = t1; perm[n-1] = t2;
576:           t1 = cmplxEig[j]; t2 = cmplxEig[j+1];
577:           for (i=j;i<n-2;i++) cmplxEig[i] = cmplxEig[i+2];
578:           cmplxEig[n-2] = t1; cmplxEig[n-1] = t2;
579:           PetscArraycpy(col1,R+j*ldr,n);
580:           PetscArraycpy(col2,R+(j+1)*ldr,n);
581:           for (i=j;i<n-2;i++) {
582:             PetscArraycpy(R+i*ldr,R+(i+2)*ldr,n);
583:           }
584:           PetscArraycpy(R+(n-2)*ldr,col1,n);
585:           PetscArraycpy(R+(n-1)*ldr,col2,n);
586:         }
587:       }
588:     }
589:   }
590:   if (k!=0) {
591:     if (breakdown) *breakdown = PETSC_TRUE;
592:     *nv = n-k;
593:   }
594:   return(0);
595: }

597: PetscErrorCode DSGHIEPOrthogEigenv(DS ds,DSMatType mat,PetscScalar *wr,PetscScalar *wi,PetscBool accum)
598: {
600:   PetscInt       lws,nwus=0,nwui=0,lwi;
601:   PetscInt       off,n,nv,ld,i,ldr,l;
602:   PetscScalar    *W,*X,*R,*ts,zeroS=0.0,oneS=1.0;
603:   PetscReal      *s,vi,vr,tr,*d,*e;
604:   PetscBLASInt   ld_,n_,nv_,*perm,*cmplxEig;

607:   l = ds->l;
608:   n = ds->n-l;
609:   PetscBLASIntCast(n,&n_);
610:   ld = ds->ld;
611:   PetscBLASIntCast(ld,&ld_);
612:   off = l*ld+l;
613:   s = ds->rmat[DS_MAT_D];
614:   if (!ds->compact) {
615:     for (i=l;i<ds->n;i++) s[i] = PetscRealPart(*(ds->mat[DS_MAT_B]+i*ld+i));
616:   }
617:   lws = n*n+7*n;
618:   lwi = 2*n;
619:   DSAllocateWork_Private(ds,lws,0,lwi);
620:   R = ds->work+nwus;
621:   nwus += n*n;
622:   ldr = n;
623:   perm = ds->iwork + nwui;
624:   nwui += n;
625:   cmplxEig = ds->iwork+nwui;
626:   X = ds->mat[mat];
627:   for (i=0;i<n;i++) {
628: #if defined(PETSC_USE_COMPLEX)
629:     vi = PetscImaginaryPart(wr[l+i]);
630: #else
631:     vi = PetscRealPart(wi[l+i]);
632: #endif
633:     if (vi!=0) {
634:       cmplxEig[i] = 1;
635:       cmplxEig[i+1] = 2;
636:       i++;
637:     } else cmplxEig[i] = 0;
638:   }
639:   nv = n;

641:   /* Perform HR decomposition */
642:   /* Hyperbolic rotators */
643:   PseudoOrthog_HR(&nv,X+off,ld,s+l,R,ldr,perm,cmplxEig,NULL,ds->work+nwus);
644:   /* Sort wr,wi perm */
645:   ts = ds->work+nwus;
646:   PetscArraycpy(ts,wr+l,n);
647:   for (i=0;i<n;i++) wr[i+l] = ts[perm[i]];
648: #if !defined(PETSC_USE_COMPLEX)
649:   PetscArraycpy(ts,wi+l,n);
650:   for (i=0;i<n;i++) wi[i+l] = ts[perm[i]];
651: #endif
652:   /* Projected Matrix */
653:   PetscArrayzero(ds->rmat[DS_MAT_T]+2*ld,ld);
654:   d = ds->rmat[DS_MAT_T];
655:   e = d+ld;
656:   for (i=0;i<nv;i++) {
657:     if (cmplxEig[i]==0) { /* Real */
658:       d[l+i] = PetscRealPart(wr[l+i]*s[l+i]);
659:       e[l+i] = 0.0;
660:     } else {
661:       vr = PetscRealPart(wr[l+i]);
662: #if defined(PETSC_USE_COMPLEX)
663:       vi = PetscImaginaryPart(wr[l+i]);
664: #else
665:       vi = PetscRealPart(wi[l+i]);
666: #endif
667:       if (cmplxEig[i]==-1) vi = -vi;
668:       tr = PetscRealPart((R[i+(i+1)*ldr]/R[i+i*ldr]))*vi;
669:       d[l+i] = (vr-tr)*s[l+i];
670:       d[l+i+1] = (vr+tr)*s[l+i+1];
671:       e[l+i] = PetscRealPart(s[l+i]*(R[(i+1)+(i+1)*ldr]/R[i+i*ldr])*vi);
672:       e[l+i+1] = 0.0;
673:       i++;
674:     }
675:   }
676:   /* accumulate previous Q */
677:   if (accum) {
678:     PetscBLASIntCast(nv,&nv_);
679:     DSAllocateMat_Private(ds,DS_MAT_W);
680:     W = ds->mat[DS_MAT_W];
681:     DSCopyMatrix_Private(ds,DS_MAT_W,DS_MAT_Q);
682:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&nv_,&n_,&oneS,W+off,&ld_,X+off,&ld_,&zeroS,ds->mat[DS_MAT_Q]+off,&ld_));
683:   } else {
684:     PetscArrayzero(ds->mat[DS_MAT_Q],ld*ld);
685:     for (i=0;i<ds->l;i++) *(ds->mat[DS_MAT_Q]+i+i*ld) = 1.0;
686:     for (i=0;i<n;i++) { PetscArraycpy(ds->mat[DS_MAT_Q]+off+i*ld,X+off+i*ld,n); }
687:   }
688:   ds->t = nv+l;
689:   if (!ds->compact) { DSSwitchFormat_GHIEP(ds,PETSC_FALSE); }
690:   return(0);
691: }

693: /*
694:    Reduce to tridiagonal-diagonal pair by means of TridiagDiag_HHR.
695: */
696: PetscErrorCode DSIntermediate_GHIEP(DS ds)
697: {
699:   PetscInt       i,ld,off;
700:   PetscInt       nwall,nwallr,nwalli;
701:   PetscScalar    *A,*B,*Q;
702:   PetscReal      *d,*e,*s;

705:   ld = ds->ld;
706:   A = ds->mat[DS_MAT_A];
707:   B = ds->mat[DS_MAT_B];
708:   Q = ds->mat[DS_MAT_Q];
709:   d = ds->rmat[DS_MAT_T];
710:   e = ds->rmat[DS_MAT_T]+ld;
711:   s = ds->rmat[DS_MAT_D];
712:   off = ds->l+ds->l*ld;
713:   PetscArrayzero(Q,ld*ld);
714:   nwall = ld*ld+ld;
715:   nwallr = ld;
716:   nwalli = ld;
717:   DSAllocateWork_Private(ds,nwall,nwallr,nwalli);
718:   for (i=0;i<ds->n;i++) Q[i+i*ld]=1.0;
719:   for (i=0;i<ds->n-ds->l;i++) *(ds->perm+i)=i;
720:   if (ds->compact) {
721:     if (ds->state < DS_STATE_INTERMEDIATE) {
722:       DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
723:       TridiagDiag_HHR(ds->k-ds->l+1,A+off,ld,s+ds->l,Q+off,ld,PETSC_TRUE,d+ds->l,e+ds->l,ds->perm,ds->work,ds->rwork,ds->iwork);
724:       ds->k = ds->l;
725:       PetscArrayzero(d+2*ld+ds->l,ds->n-ds->l);
726:     }
727:   } else {
728:     if (ds->state < DS_STATE_INTERMEDIATE) {
729:       for (i=0;i<ds->n;i++) s[i] = PetscRealPart(B[i+i*ld]);
730:       TridiagDiag_HHR(ds->n-ds->l,A+off,ld,s+ds->l,Q+off,ld,PETSC_FALSE,d+ds->l,e+ds->l,ds->perm,ds->work,ds->rwork,ds->iwork);
731:       PetscArrayzero(d+2*ld,ds->n);
732:       ds->k = ds->l;
733:       DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
734:     } else {
735:       DSSwitchFormat_GHIEP(ds,PETSC_TRUE);
736:     }
737:   }
738:   return(0);
739: }