Date: 2021-02-01T14:41:44+00:00

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: }