Date: 2022-03-31T15:28:51+00:00

Actual source code: invit.c
slepc-3.17.0 2022-03-31
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-, Universitat Politecnica de Valencia, Spain

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

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

     14: 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_;

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

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

     72: /*
     73:                                 |c  s|
     74:   Applies an hyperbolic rotator |s  c|
     75:            |c  s|
     76:     [x1 x2]|s  c|
     77: */
     78: static PetscErrorCode HRApply(PetscInt n,PetscScalar *x1,PetscInt inc1,PetscScalar *x2,PetscInt inc2,PetscReal c,PetscReal s)
     79: {
     80:   PetscInt    i;
     81:   PetscReal   t;
     82:   PetscScalar tmp;

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

    101: /*
    102:   Reduction to tridiagonal-diagonal form (see F. Tisseur, SIMAX 26(1), 2004).

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

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

    237:         /* Update Q */
    238:         HRApply(n,Q+(ni-n0)*ldq,1,Q+(n-n1)*ldq,1,cs,-sn);
    239:         if (type==2) {
    240:           ss[ni-n0] = -ss[ni-n0]; ss[n-n1] = -ss[n-n1];
    241:           n0++;ni++;n1--;
    242:         }
    243:       }
    244:       if (n0>0) n0--;
    245:       else n1--;
    246:     }
    247:   }

    249:   /* flip matrices */
    250:   if (flip) {
    251:     for (i=0;i<n-1;i++) {
    252:       d[i] = PetscRealPart(A[n-i-1+(n-i-1)*lda]);
    253:       e[i] = PetscRealPart(A[n-i-1+(n-i-2)*lda]);
    254:       s[i] = ss[n-i-1];
    255:     }
    256:     s[n-1] = ss[0];
    257:     d[n-1] = PetscRealPart(A[0]);
    258:     for (i=0;i<n;i++) PetscArraycpy(work+i*n,Q+i*ldq,n);
    259:     for (i=0;i<n;i++)
    260:       for (j=0;j<n;j++)
    261:         Q[i+j*ldq] = work[i+(n-j-1)*n];
    262:   } else {
    263:     for (i=0;i<n-1;i++) {
    264:       d[i] = PetscRealPart(A[i+i*lda]);
    265:       e[i] = PetscRealPart(A[i+1+i*lda]);
    266:       s[i] = ss[i];
    267:     }
    268:     s[n-1] = ss[n-1];
    269:     d[n-1] = PetscRealPart(A[n-1 + (n-1)*lda]);
    270:   }
    271:   PetscFunctionReturn(0);
    272: }

    274: 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)
    275: {
    276:   PetscScalar    *x,*y;
    277:   PetscReal      ncond2=1.0;
    278:   PetscBLASInt   n0_,n1_,inc=1;

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

    343: /*
    344:   Auxiliary function to try perform one iteration of hr routine,
    345:   checking condition number. If it is < tolD, apply the
    346:   transformation to H and R, if not, ok=false and it do nothing
    347:   tolE, tolerance to exchange complex pairs to improve conditioning
    348: */
    349: 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)
    350: {
    351:   struct HRtr    *tr1,*tr2,tr1_t,tr2_t,tr1_te,tr2_te;
    352:   PetscScalar    *x,*y;
    353:   PetscReal      ncond,ncond_e;
    354:   PetscInt       nwu=0,i,d=1;
    355:   PetscBLASInt   n0_,n1_,inc=1,mh,mr,n_,ldr_,ldh_;
    356:   PetscReal      tolD = 1e+5;

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

    486: /*
    487:   compute V = HR whit H s-orthogonal and R upper triangular
    488: */
    489: static PetscErrorCode PseudoOrthog_HR(PetscInt *nv,PetscScalar *V,PetscInt ldv,PetscReal *s,PetscScalar *R,PetscInt ldr,PetscBLASInt *perm,PetscBLASInt *cmplxEig,PetscBool *breakdown,PetscScalar *work)
    490: {
    491:   PetscInt       i,j,n,n0,n1,np,idx0,idx1,sz=1,k=0,t1,t2,nwu=0;
    492:   PetscScalar    *col1,*col2;
    493:   PetscBool      exg=PETSC_FALSE,ok=PETSC_FALSE;

    495:   n = *nv;
    496:   col1 = work+nwu;
    497:   nwu += n;
    498:   col2 = work+nwu;
    499:   nwu += n;
    500:   /* Sort R and s according to sing(s) */
    501:   np = 0;
    502:   for (i=0;i<n;i++) if (s[i]>0) np++;
    503:   if (s[0]>0) n1 = np;
    504:   else n1 = n-np;
    505:   n0 = 0;
    506:   for (i=0;i<n;i++) {
    507:     if (s[i]==s[0]) {
    508:       s[n0] = s[0];
    509:       perm[n0++] = i;
    510:     } else perm[n1++] = i;
    511:   }
    512:   for (i=n0;i<n;i++) s[i] = -s[0];
    513:   n1 -= n0;
    514:   idx0 = 0;
    515:   idx1 = n0;
    516:   if (idx1==n) idx1=idx0;
    517:   for (i=0;i<n;i++) {
    518:     for (j=0;j<n;j++) R[j*ldr+i] = V[j*ldv+perm[i]];
    519:   }
    520:   /* Initialize H */
    521:   for (i=0;i<n;i++) {
    522:     PetscArrayzero(V+i*ldv,n);
    523:     V[perm[i]+i*ldv] = 1.0;
    524:   }
    525:   for (i=0;i<n;i++) perm[i] = i;
    526:   j = 0;
    527:   while (j<n-k) {
    528:     if (cmplxEig[j]==0) sz=1;
    529:     else sz=2;
    530:     TryHRIt(n,j,sz,V,ldv,R,ldr,s,&exg,&ok,&n0,&n1,&idx0,&idx1,NULL,work+nwu);
    531:     if (ok) {
    532:       if (exg) cmplxEig[j] = -cmplxEig[j];
    533:       j = j+sz;
    534:     } else { /* to be discarded */
    535:       k = k+1;
    536:       if (cmplxEig[j]==0) {
    537:         if (j<n) {
    538:           t1 = perm[j];
    539:           for (i=j;i<n-1;i++) perm[i] = perm[i+1];
    540:           perm[n-1] = t1;
    541:           t1 = cmplxEig[j];
    542:           for (i=j;i<n-1;i++) cmplxEig[i] = cmplxEig[i+1];
    543:           cmplxEig[n-1] = t1;
    544:           PetscArraycpy(col1,R+j*ldr,n);
    545:           for (i=j;i<n-1;i++) PetscArraycpy(R+i*ldr,R+(i+1)*ldr,n);
    546:           PetscArraycpy(R+(n-1)*ldr,col1,n);
    547:         }
    548:       } else {
    549:         k = k+1;
    550:         if (j<n-1) {
    551:           t1 = perm[j]; t2 = perm[j+1];
    552:           for (i=j;i<n-2;i++) perm[i] = perm[i+2];
    553:           perm[n-2] = t1; perm[n-1] = t2;
    554:           t1 = cmplxEig[j]; t2 = cmplxEig[j+1];
    555:           for (i=j;i<n-2;i++) cmplxEig[i] = cmplxEig[i+2];
    556:           cmplxEig[n-2] = t1; cmplxEig[n-1] = t2;
    557:           PetscArraycpy(col1,R+j*ldr,n);
    558:           PetscArraycpy(col2,R+(j+1)*ldr,n);
    559:           for (i=j;i<n-2;i++) PetscArraycpy(R+i*ldr,R+(i+2)*ldr,n);
    560:           PetscArraycpy(R+(n-2)*ldr,col1,n);
    561:           PetscArraycpy(R+(n-1)*ldr,col2,n);
    562:         }
    563:       }
    564:     }
    565:   }
    566:   if (k!=0) {
    567:     if (breakdown) *breakdown = PETSC_TRUE;
    568:     *nv = n-k;
    569:   }
    570:   PetscFunctionReturn(0);
    571: }

    573: PetscErrorCode DSGHIEPOrthogEigenv(DS ds,DSMatType mat,PetscScalar *wr,PetscScalar *wi,PetscBool accum)
    574: {
    575:   PetscInt       lws,nwus=0,nwui=0,lwi;
    576:   PetscInt       off,n,nv,ld,i,ldr,l;
    577:   PetscScalar    *W,*X,*R,*ts,zeroS=0.0,oneS=1.0;
    578:   PetscReal      *s,vi,vr,tr,*d,*e;
    579:   PetscBLASInt   ld_,n_,nv_,*perm,*cmplxEig;

    581:   l = ds->l;
    582:   n = ds->n-l;
    583:   PetscBLASIntCast(n,&n_);
    584:   ld = ds->ld;
    585:   PetscBLASIntCast(ld,&ld_);
    586:   off = l*ld+l;
    587:   s = ds->rmat[DS_MAT_D];
    588:   if (!ds->compact) {
    589:     for (i=l;i<ds->n;i++) s[i] = PetscRealPart(*(ds->mat[DS_MAT_B]+i*ld+i));
    590:   }
    591:   lws = n*n+7*n;
    592:   lwi = 2*n;
    593:   DSAllocateWork_Private(ds,lws,0,lwi);
    594:   R = ds->work+nwus;
    595:   nwus += n*n;
    596:   ldr = n;
    597:   perm = ds->iwork + nwui;
    598:   nwui += n;
    599:   cmplxEig = ds->iwork+nwui;
    600:   X = ds->mat[mat];
    601:   for (i=0;i<n;i++) {
    602: #if defined(PETSC_USE_COMPLEX)
    603:     vi = PetscImaginaryPart(wr[l+i]);
    604: #else
    605:     vi = PetscRealPart(wi[l+i]);
    606: #endif
    607:     if (vi!=0) {
    608:       cmplxEig[i] = 1;
    609:       cmplxEig[i+1] = 2;
    610:       i++;
    611:     } else cmplxEig[i] = 0;
    612:   }
    613:   nv = n;

    615:   /* Perform HR decomposition */
    616:   /* Hyperbolic rotators */
    617:   PseudoOrthog_HR(&nv,X+off,ld,s+l,R,ldr,perm,cmplxEig,NULL,ds->work+nwus);
    618:   /* Sort wr,wi perm */
    619:   ts = ds->work+nwus;
    620:   PetscArraycpy(ts,wr+l,n);
    621:   for (i=0;i<n;i++) wr[i+l] = ts[perm[i]];
    622: #if !defined(PETSC_USE_COMPLEX)
    623:   PetscArraycpy(ts,wi+l,n);
    624:   for (i=0;i<n;i++) wi[i+l] = ts[perm[i]];
    625: #endif
    626:   /* Projected Matrix */
    627:   PetscArrayzero(ds->rmat[DS_MAT_T]+2*ld,ld);
    628:   d = ds->rmat[DS_MAT_T];
    629:   e = d+ld;
    630:   d[l+nv-1] = PetscRealPart(wr[l+nv-1]*s[l+nv-1]);
    631:   for (i=0;i<nv-1;i++) {
    632:     if (cmplxEig[i]==0) { /* Real */
    633:       d[l+i] = PetscRealPart(wr[l+i]*s[l+i]);
    634:       e[l+i] = 0.0;
    635:     } else {
    636:       vr = PetscRealPart(wr[l+i]);
    637: #if defined(PETSC_USE_COMPLEX)
    638:       vi = PetscImaginaryPart(wr[l+i]);
    639: #else
    640:       vi = PetscRealPart(wi[l+i]);
    641: #endif
    642:       if (cmplxEig[i]==-1) vi = -vi;
    643:       tr = PetscRealPart((R[i+(i+1)*ldr]/R[i+i*ldr]))*vi;
    644:       d[l+i] = (vr-tr)*s[l+i];
    645:       d[l+i+1] = (vr+tr)*s[l+i+1];
    646:       e[l+i] = PetscRealPart(s[l+i]*(R[(i+1)+(i+1)*ldr]/R[i+i*ldr])*vi);
    647:       if (i<nv-2) e[l+i+1] = 0.0;
    648:       i++;
    649:     }
    650:   }
    651:   /* accumulate previous Q */
    652:   if (accum) {
    653:     PetscBLASIntCast(nv,&nv_);
    654:     DSAllocateMat_Private(ds,DS_MAT_W);
    655:     W = ds->mat[DS_MAT_W];
    656:     DSCopyMatrix_Private(ds,DS_MAT_W,DS_MAT_Q);
    657:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&nv_,&n_,&oneS,W+off,&ld_,X+off,&ld_,&zeroS,ds->mat[DS_MAT_Q]+off,&ld_));
    658:   } else {
    659:     PetscArrayzero(ds->mat[DS_MAT_Q],ld*ld);
    660:     for (i=0;i<ds->l;i++) *(ds->mat[DS_MAT_Q]+i+i*ld) = 1.0;
    661:     for (i=0;i<n;i++) PetscArraycpy(ds->mat[DS_MAT_Q]+off+i*ld,X+off+i*ld,n);
    662:   }
    663:   ds->t = nv+l;
    664:   if (!ds->compact) DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
    665:   PetscFunctionReturn(0);
    666: }

    668: /*
    669:    Reduce to tridiagonal-diagonal pair by means of TridiagDiag_HHR.
    670: */
    671: PetscErrorCode DSIntermediate_GHIEP(DS ds)
    672: {
    673:   PetscInt       i,ld,off;
    674:   PetscInt       nwall,nwallr,nwalli;
    675:   PetscScalar    *A,*B,*Q;
    676:   PetscReal      *d,*e,*s;

    678:   ld = ds->ld;
    679:   A = ds->mat[DS_MAT_A];
    680:   B = ds->mat[DS_MAT_B];
    681:   Q = ds->mat[DS_MAT_Q];
    682:   d = ds->rmat[DS_MAT_T];
    683:   e = ds->rmat[DS_MAT_T]+ld;
    684:   s = ds->rmat[DS_MAT_D];
    685:   off = ds->l+ds->l*ld;
    686:   PetscArrayzero(Q,ld*ld);
    687:   nwall = ld*ld+ld;
    688:   nwallr = ld;
    689:   nwalli = ld;
    690:   DSAllocateWork_Private(ds,nwall,nwallr,nwalli);
    691:   for (i=0;i<ds->n;i++) Q[i+i*ld]=1.0;
    692:   for (i=0;i<ds->n-ds->l;i++) *(ds->perm+i)=i;
    693:   if (ds->compact) {
    694:     if (ds->state < DS_STATE_INTERMEDIATE) {
    695:       DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
    696:       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);
    697:       ds->k = ds->l;
    698:       PetscArrayzero(d+2*ld+ds->l,ds->n-ds->l);
    699:     }
    700:   } else {
    701:     if (ds->state < DS_STATE_INTERMEDIATE) {
    702:       for (i=0;i<ds->n;i++) s[i] = PetscRealPart(B[i+i*ld]);
    703:       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);
    704:       PetscArrayzero(d+2*ld,ds->n);
    705:       ds->k = ds->l;
    706:       DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
    707:     } else DSSwitchFormat_GHIEP(ds,PETSC_TRUE);
    708:   }
    709:   PetscFunctionReturn(0);
    710: }