Actual source code: invit.c
slepc-3.10.1 2018-10-23
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2018, 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);
     46:     *c = (x1>=0)?1.0:-1.0;
     47:     *s = 0.0;
     48:     if (type) *type = 1;
     49:     return(0);
     50:   }
     51:   if (PetscAbsReal(x1) == PetscAbsReal(x2)) {
     52:     /* hyperbolic rotation doesn't exist */
     53:     *c = 0.0;
     54:     *s = 0.0;
     55:     *r = 0.0;
     56:     if (type) *type = 0;
     57:     *cond = PETSC_MAX_REAL;
     58:     return(0);
     59:   }

     61:   if (PetscAbsReal(x1)>PetscAbsReal(x2)) {
     62:     xa = x1; xb = x2; type_ = 1;
     63:   } else {
     64:     xa = x2; xb = x1; type_ = 2;
     65:   }
     66:   t = xb/xa;
     67:   n2 = PetscAbsReal(1 - t*t);
     68:   *r = PetscSqrtReal(n2)*PetscAbsReal(xa);
     69:   *c = x1/(*r);
     70:   *s = x2/(*r);
     71:   if (type_ == 2) *r *= -1;
     72:   if (type) *type = type_;
     73:   if (cond) *cond = (PetscAbsReal(*c) + PetscAbsReal(*s))/PetscAbsReal(PetscAbsReal(*c) - PetscAbsReal(*s));
     74:   return(0);
     75: }

     77: /*
     78:                                 |c  s|
     79:   Applies an hyperbolic rotator |s  c|
     80:            |c  s|
     81:     [x1 x2]|s  c|
     82: */
     83: static PetscErrorCode HRApply(PetscInt n,PetscScalar *x1,PetscInt inc1,PetscScalar *x2,PetscInt inc2,PetscReal c,PetscReal s)
     84: {
     85:   PetscInt    i;
     86:   PetscReal   t;
     87:   PetscScalar tmp;

     90:   if (PetscAbsReal(c)>PetscAbsReal(s)) { /* Type I */
     91:     t = s/c;
     92:     for (i=0;i<n;i++) {
     93:       x1[i*inc1] = c*x1[i*inc1] + s*x2[i*inc2];
     94:       x2[i*inc2] = t*x1[i*inc1] + x2[i*inc2]/c;
     95:     }
     96:   } else { /* Type II */
     97:     t = c/s;
     98:     for (i=0;i<n;i++) {
     99:       tmp = x1[i*inc1];
    100:       x1[i*inc1] = c*x1[i*inc1] + s*x2[i*inc2];
    101:       x2[i*inc2] = t*x1[i*inc1] + tmp/s;
    102:     }
    103:   }
    104:   return(0);
    105: }

    107: /*
    108:   Reduction to tridiagonal-diagonal form (see F. Tisseur, SIMAX 26(1), 2004).

    110:   Input:
    111:     A symmetric (only lower triangular part is referred)
    112:     s vector +1 and -1 (signature matrix)
    113:   Output:
    114:     d,e
    115:     s
    116:     Q s-orthogonal matrix with Q^T*A*Q = T (symmetric tridiagonal matrix)
    117: */
    118: 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)
    119: {
    120: #if defined(SLEPC_MISSING_LAPACK_LARFG) || defined(SLEPC_MISSING_LAPACK_LARF)
    122:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LARFG/LARF - Lapack routines are unavailable");
    123: #else
    125:   PetscInt       i,j,k,*ii,*jj,i0=0,ik=0,tmp,type;
    126:   PetscInt       nwu=0;
    127:   PetscReal      *ss,cond=1.0,cs,sn,r;
    128:   PetscScalar    tau,t,*AA;
    129:   PetscBLASInt   n0,n1,ni,inc=1,m,n_,lda_,ldq_,*perm;
    130:   PetscBool      breakdown = PETSC_TRUE;

    133:   if (n<3) {
    134:     if (n==1) Q[0]=1;
    135:     if (n==2) {
    136:       Q[0] = Q[1+ldq] = 1;
    137:       Q[1] = Q[ldq] = 0;
    138:     }
    139:     return(0);
    140:   }
    141:   PetscBLASIntCast(lda,&lda_);
    142:   PetscBLASIntCast(n,&n_);
    143:   PetscBLASIntCast(ldq,&ldq_);
    144:   ss = rwork;
    145:   perm = iwork;
    146:   AA = work;
    147:   for (i=0;i<n;i++) {
    148:     PetscMemcpy(AA+i*n,A+i*lda,n*sizeof(PetscScalar));
    149:   }
    150:   nwu += n*n;
    151:   k=0;
    152:   while (breakdown && k<n) {
    153:     breakdown = PETSC_FALSE;
    154:     /* Classify (and flip) A and s according to sign */
    155:     if (flip) {
    156:       for (i=0;i<n;i++) {
    157:         perm[i] = n-1-perm_[i];
    158:         if (perm[i]==0) i0 = i;
    159:         if (perm[i]==k) ik = i;
    160:       }
    161:     } else {
    162:       for (i=0;i<n;i++) {
    163:         perm[i] = perm_[i];
    164:         if (perm[i]==0) i0 = i;
    165:         if (perm[i]==k) ik = i;
    166:       }
    167:     }
    168:     perm[ik] = 0;
    169:     perm[i0] = k;
    170:     i=1;
    171:     while (i<n-1 && s[perm[i-1]]==s[perm[0]]) {
    172:       if (s[perm[i]]!=s[perm[0]]) {
    173:         j=i+1;
    174:         while (j<n-1 && s[perm[j]]!=s[perm[0]])j++;
    175:         tmp = perm[i]; perm[i] = perm[j]; perm[j] = tmp;
    176:       }
    177:       i++;
    178:     }
    179:     for (i=0;i<n;i++) {
    180:       ss[i] = s[perm[i]];
    181:     }
    182:     if (flip) {
    183:       ii = &j;
    184:       jj = &i;
    185:     } else {
    186:       ii = &i;
    187:       jj = &j;
    188:     }
    189:     for (i=0;i<n;i++)
    190:       for (j=0;j<n;j++)
    191:         A[i+j*lda] = AA[perm[*ii]+perm[*jj]*n];
    192:     /* Initialize Q */
    193:     for (i=0;i<n;i++) {
    194:       PetscMemzero(Q+i*ldq,n*sizeof(PetscScalar));
    195:       Q[perm[i]+i*ldq] = 1.0;
    196:     }
    197:     for (ni=1;ni<n && ss[ni]==ss[0]; ni++);
    198:     n0 = ni-1;
    199:     n1 = n_-ni;
    200:     for (j=0;j<n-2;j++) {
    201:       PetscBLASIntCast(n-j-1,&m);
    202:       /* Forming and applying reflectors */
    203:       if (n0 > 1) {
    204:         PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n0,A+ni-n0+j*lda,A+ni-n0+j*lda+1,&inc,&tau));
    205:         /* Apply reflector */
    206:         if (PetscAbsScalar(tau) != 0.0) {
    207:           t=*(A+ni-n0+j*lda);  *(A+ni-n0+j*lda)=1.0;
    208:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&m,&n0,A+ni-n0+j*lda,&inc,&tau,A+j+1+(j+1)*lda,&lda_,work+nwu));
    209:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n0,&m,A+ni-n0+j*lda,&inc,&tau,A+j+1+(j+1)*lda,&lda_,work+nwu));
    210:           /* Update Q */
    211:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n0,A+ni-n0+j*lda,&inc,&tau,Q+(j+1)*ldq,&ldq_,work+nwu));
    212:           *(A+ni-n0+j*lda) = t;
    213:           for (i=1;i<n0;i++) {
    214:             *(A+ni-n0+j*lda+i) = 0.0;  *(A+j+(ni-n0+i)*lda) = 0.0;
    215:           }
    216:           *(A+j+(ni-n0)*lda) = *(A+ni-n0+j*lda);
    217:         }
    218:       }
    219:       if (n1 > 1) {
    220:         PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n1,A+n-n1+j*lda,A+n-n1+j*lda+1,&inc,&tau));
    221:         /* Apply reflector */
    222:         if (PetscAbsScalar(tau) != 0.0) {
    223:           t=*(A+n-n1+j*lda);  *(A+n-n1+j*lda)=1.0;
    224:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&m,&n1,A+n-n1+j*lda,&inc,&tau,A+j+1+(n-n1)*lda,&lda_,work+nwu));
    225:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n1,&m,A+n-n1+j*lda,&inc,&tau,A+n-n1+(j+1)*lda,&lda_,work+nwu));
    226:           /* Update Q */
    227:           PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n1,A+n-n1+j*lda,&inc,&tau,Q+(n-n1)*ldq,&ldq_,work+nwu));
    228:           *(A+n-n1+j*lda) = t;
    229:           for (i=1;i<n1;i++) {
    230:             *(A+n-n1+i+j*lda) = 0.0;  *(A+j+(n-n1+i)*lda) = 0.0;
    231:           }
    232:           *(A+j+(n-n1)*lda) = *(A+n-n1+j*lda);
    233:         }
    234:       }
    235:       /* Hyperbolic rotation */
    236:       if (n0 > 0 && n1 > 0) {
    237:         HRGen(PetscRealPart(A[ni-n0+j*lda]),PetscRealPart(A[n-n1+j*lda]),&type,&cs,&sn,&r,&cond);
    238:         /* Check condition number */
    239:         if (cond > 1.0/(10*PETSC_SQRT_MACHINE_EPSILON)) {
    240:           breakdown = PETSC_TRUE;
    241:           k++;
    242:           if (k==n || flip) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Breakdown in construction of hyperbolic transformation");
    243:           break;
    244:         }
    245:         A[ni-n0+j*lda] = r; A[n-n1+j*lda] = 0.0;
    246:         A[j+(ni-n0)*lda] = r; A[j+(n-n1)*lda] = 0.0;
    247:         /* Apply to A */
    248:         HRApply(m,A+j+1+(ni-n0)*lda,1,A+j+1+(n-n1)*lda,1,cs,-sn);
    249:         HRApply(m,A+ni-n0+(j+1)*lda,lda,A+n-n1+(j+1)*lda,lda,cs,-sn);

    251:         /* Update Q */
    252:         HRApply(n,Q+(ni-n0)*ldq,1,Q+(n-n1)*ldq,1,cs,-sn);
    253:         if (type==2) {
    254:           ss[ni-n0] = -ss[ni-n0]; ss[n-n1] = -ss[n-n1];
    255:           n0++;ni++;n1--;
    256:         }
    257:       }
    258:       if (n0>0) n0--;
    259:       else n1--;
    260:     }
    261:   }

    263:   /* flip matrices */
    264:   if (flip) {
    265:     for (i=0;i<n-1;i++) {
    266:       d[i] = PetscRealPart(A[n-i-1+(n-i-1)*lda]);
    267:       e[i] = PetscRealPart(A[n-i-1+(n-i-2)*lda]);
    268:       s[i] = ss[n-i-1];
    269:     }
    270:     s[n-1] = ss[0];
    271:     d[n-1] = PetscRealPart(A[0]);
    272:     for (i=0;i<n;i++) {
    273:       ierr=PetscMemcpy(work+i*n,Q+i*ldq,n*sizeof(PetscScalar));
    274:     }
    275:     for (i=0;i<n;i++)
    276:       for (j=0;j<n;j++)
    277:         Q[i+j*ldq] = work[i+(n-j-1)*n];
    278:   } else {
    279:     for (i=0;i<n-1;i++) {
    280:       d[i] = PetscRealPart(A[i+i*lda]);
    281:       e[i] = PetscRealPart(A[i+1+i*lda]);
    282:       s[i] = ss[i];
    283:     }
    284:     s[n-1] = ss[n-1];
    285:     d[n-1] = PetscRealPart(A[n-1 + (n-1)*lda]);
    286:   }
    287:   return(0);
    288: #endif
    289: }

    291: 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)
    292: {
    293: #if defined(SLEPC_MISSING_LAPACK_LARFG) || defined(SLEPC_MISSING_LAPACK_LARF)
    295:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LARFG/LARF - Lapack routines are unavailable");
    296: #else
    298:   PetscScalar    *x,*y;
    299:   PetscReal       ncond2;
    300:   PetscBLASInt   n0_,n1_,inc=1;

    303:   /* Hyperbolic transformation to make zeros in x */
    304:   x = tr1->data;
    305:   tr1->n[0] = n0;
    306:   tr1->n[1] = n1;
    307:   tr1->idx[0] = idx0;
    308:   tr1->idx[1] = idx1;
    309:   PetscBLASIntCast(tr1->n[0],&n0_);
    310:   PetscBLASIntCast(tr1->n[1],&n1_);
    311:   if (tr1->n[0] > 1) {
    312:     PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n0_,x+tr1->idx[0],x+tr1->idx[0]+1,&inc,tr1->tau));
    313:   }
    314:   if (tr1->n[1]> 1) {
    315:     PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n1_,x+tr1->idx[1],x+tr1->idx[1]+1,&inc,tr1->tau+1));
    316:   }
    317:   if (tr1->idx[0]<tr1->idx[1]) {
    318:     HRGen(PetscRealPart(x[tr1->idx[0]]),PetscRealPart(x[tr1->idx[1]]),&(tr1->type),&(tr1->cs),&(tr1->sn),&(tr1->alpha),ncond);
    319:   } else {
    320:     tr1->alpha = PetscRealPart(x[tr1->idx[0]]);
    321:     *ncond = 1.0;
    322:   }
    323:   if (sz==2) {
    324:     y = tr2->data;
    325:     /* Apply first transformation to second column */
    326:     if (tr1->n[0] > 1 && PetscAbsScalar(tr1->tau[0])!=0.0) {
    327:       x[tr1->idx[0]] = 1.0;
    328:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n0_,&inc,x+tr1->idx[0],&inc,tr1->tau,y+tr1->idx[0],&n0_,work));
    329:     }
    330:     if (tr1->n[1] > 1 && PetscAbsScalar(tr1->tau[1])!=0.0) {
    331:       x[tr1->idx[1]] = 1.0;
    332:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n1_,&inc,x+tr1->idx[1],&inc,tr1->tau+1,y+tr1->idx[1],&n1_,work));
    333:     }
    334:     if (tr1->idx[0]<tr1->idx[1]) {
    335:       HRApply(1,y+tr1->idx[0],1,y+tr1->idx[1],1,tr1->cs,-tr1->sn);
    336:     }
    337:     tr2->n[0] = tr1->n[0];
    338:     tr2->n[1] = tr1->n[1];
    339:     tr2->idx[0] = tr1->idx[0];
    340:     tr2->idx[1] = tr1->idx[1];
    341:     if (tr1->idx[0]<tr1->idx[1] && tr1->type==2) {
    342:       tr2->idx[1]++; tr2->n[1]--; tr2->n[0]++;
    343:     }
    344:     if (tr2->n[0]>0) {
    345:       tr2->n[0]--; tr2->idx[0]++;
    346:       if (tr2->n[1]==0) tr2->idx[1] = tr2->idx[0];
    347:     } else {
    348:       tr2->n[1]--; tr2->idx[1]++; tr2->idx[0] = tr2->idx[1];
    349:     }
    350:     /* Hyperbolic transformation to make zeros in y */
    351:     PetscBLASIntCast(tr2->n[0],&n0_);
    352:     PetscBLASIntCast(tr2->n[1],&n1_);
    353:     if (tr2->n[0] > 1) {
    354:       PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n0_,y+tr2->idx[0],y+tr2->idx[0]+1,&inc,tr2->tau));
    355:     }
    356:     if (tr2->n[1]> 1) {
    357:       PetscStackCallBLAS("LAPACKlarfg",LAPACKlarfg_(&n1_,y+tr2->idx[1],y+tr2->idx[1]+1,&inc,tr2->tau+1));
    358:     }
    359:     if (tr2->idx[0]<tr2->idx[1]) {
    360:       HRGen(PetscRealPart(y[tr2->idx[0]]),PetscRealPart(y[tr2->idx[1]]),&(tr2->type),&(tr2->cs),&(tr2->sn),&(tr2->alpha),&ncond2);
    361:     } else {
    362:     tr2->alpha = PetscRealPart(y[tr2->idx[0]]);
    363:     ncond2 = 1.0;
    364:     }
    365:     if (ncond2>*ncond) *ncond = ncond2;
    366:   }
    367:   return(0);
    368: #endif
    369: }

    371: /*
    372:   Auxiliary function to try perform one iteration of hr routine,
    373:   checking condition number. If it is < tolD, apply the
    374:   transformation to H and R, if not, ok=false and it do nothing
    375:   tolE, tolerance to exchange complex pairs to improve conditioning
    376: */
    377: 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)
    378: {
    379: #if defined(SLEPC_MISSING_LAPACK_LARF)
    381:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LARF - Lapack routine is unavailable");
    382: #else
    384:   struct HRtr    *tr1,*tr2,tr1_t,tr2_t,tr1_te,tr2_te;
    385:   PetscScalar    *x,*y;
    386:   PetscReal      ncond,ncond_e;
    387:   PetscInt       nwu=0,i,d=1;
    388:   PetscBLASInt   n0_,n1_,inc=1,mh,mr,n_,ldr_,ldh_;
    389:   PetscReal      tolD = 1e+5;

    392:   if (cond) *cond = 1.0;
    393:   PetscBLASIntCast(n,&n_);
    394:   PetscBLASIntCast(ldr,&ldr_);
    395:   PetscBLASIntCast(ldh,&ldh_);
    396:   x = work+nwu;
    397:   nwu += n;
    398:   PetscMemcpy(x,R+j*ldr,n*sizeof(PetscScalar));
    399:   *exg = PETSC_FALSE;
    400:   *ok = PETSC_TRUE;
    401:   tr1_t.data = x;
    402:   if (sz==1) {
    403:     /* Hyperbolic transformation to make zeros in x */
    404:     MadeHRtr(sz,n,*idx0,*n0,*idx1,*n1,&tr1_t,NULL,&ncond,work+nwu);
    405:     /* Check condition number to single column*/
    406:     if (ncond>tolD) *ok = PETSC_FALSE;
    407:     tr1 = &tr1_t;
    408:     tr2 = &tr2_t;
    409:   } else {
    410:     y = work+nwu;
    411:     nwu += n;
    412:     PetscMemcpy(y,R+(j+1)*ldr,n*sizeof(PetscScalar));
    413:     tr2_t.data = y;
    414:     MadeHRtr(sz,n,*idx0,*n0,*idx1,*n1,&tr1_t,&tr2_t,&ncond,work+nwu);
    415:     /* Computing hyperbolic transformations also for exchanged vectors */
    416:     tr1_te.data = work+nwu;
    417:     nwu += n;
    418:     PetscMemcpy(tr1_te.data,R+(j+1)*ldr,n*sizeof(PetscScalar));
    419:     tr2_te.data = work+nwu;
    420:     nwu += n;
    421:     PetscMemcpy(tr2_te.data,R+j*ldr,n*sizeof(PetscScalar));
    422:     MadeHRtr(sz,n,*idx0,*n0,*idx1,*n1,&tr1_te,&tr2_te,&ncond_e,work+nwu);
    423:     if (ncond > d*ncond_e) {
    424:       *exg = PETSC_TRUE;
    425:       tr1 = &tr1_te;
    426:       tr2 = &tr2_te;
    427:       ncond = ncond_e;
    428:     } else {
    429:       tr1 = &tr1_t;
    430:       tr2 = &tr2_t;
    431:     }
    432:     if (ncond>tolD) *ok = PETSC_FALSE;
    433:   }
    434:   if (*ok) {
    435:     /* Everything is OK, apply transformations to R and H */
    436:     /* First column */
    437:     if (cond && *cond<ncond) *cond = ncond;
    438:     x = tr1->data;
    439:     PetscBLASIntCast(tr1->n[0],&n0_);
    440:     PetscBLASIntCast(tr1->n[1],&n1_);
    441:     PetscBLASIntCast(n-j-sz,&mr);
    442:     if (tr1->n[0] > 1 && PetscAbsScalar(tr1->tau[0])!=0.0) {
    443:       x[tr1->idx[0]] = 1.0;
    444:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n0_,&mr,x+tr1->idx[0],&inc,tr1->tau,R+(j+sz)*ldr+tr1->idx[0],&ldr_,work+nwu));
    445:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n0_,x+tr1->idx[0],&inc,tr1->tau,H+(tr1->idx[0])*ldh,&ldh_,work+nwu));
    446:     }
    447:     if (tr1->n[1] > 1 && PetscAbsScalar(tr1->tau[1])!=0.0) {
    448:       x[tr1->idx[1]] = 1.0;
    449:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n1_,&mr,x+tr1->idx[1],&inc,tr1->tau+1,R+(j+sz)*ldr+tr1->idx[1],&ldr_,work+nwu));
    450:       PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n1_,x+tr1->idx[1],&inc,tr1->tau+1,H+(tr1->idx[1])*ldh,&ldh_,work+nwu));
    451:     }
    452:     if (tr1->idx[0]<tr1->idx[1]) {
    453:       HRApply(mr,R+(j+sz)*ldr+tr1->idx[0],ldr,R+(j+sz)*ldr+tr1->idx[1],ldr,tr1->cs,-tr1->sn);
    454:       if (tr1->type==1) {
    455:         HRApply(n,H+(tr1->idx[0])*ldh,1,H+(tr1->idx[1])*ldh,1,tr1->cs,tr1->sn);
    456:       } else {
    457:         HRApply(n,H+(tr1->idx[0])*ldh,1,H+(tr1->idx[1])*ldh,1,-tr1->cs,-tr1->sn);
    458:         s[tr1->idx[0]] = -s[tr1->idx[0]];
    459:         s[tr1->idx[1]] = -s[tr1->idx[1]];
    460:       }
    461:     }
    462:     for (i=0;i<tr1->idx[0];i++) *(R+j*ldr+i) = x[i];
    463:     for (i=tr1->idx[0]+1;i<n;i++) *(R+j*ldr+i) = 0.0;
    464:     *(R+j*ldr+tr1->idx[0]) = tr1->alpha;
    465:     if (sz==2) {
    466:       y = tr2->data;
    467:       /* Second column */
    468:       PetscBLASIntCast(tr2->n[0],&n0_);
    469:       PetscBLASIntCast(tr2->n[1],&n1_);
    470:       PetscBLASIntCast(n-j-sz,&mr);
    471:       PetscBLASIntCast(n-tr2->idx[0],&mh);
    472:       if (tr2->n[0] > 1 && PetscAbsScalar(tr2->tau[0])!=0.0) {
    473:         y[tr2->idx[0]] = 1.0;
    474:         PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n0_,&mr,y+tr2->idx[0],&inc,tr2->tau,R+(j+2)*ldr+tr2->idx[0],&ldr_,work+nwu));
    475:         PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n0_,y+tr2->idx[0],&inc,tr2->tau,H+(tr2->idx[0])*ldh,&ldh_,work+nwu));
    476:       }
    477:       if (tr2->n[1] > 1 && PetscAbsScalar(tr2->tau[1])!=0.0) {
    478:         y[tr2->idx[1]] = 1.0;
    479:         PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("L",&n1_,&mr,y+tr2->idx[1],&inc,tr2->tau+1,R+(j+2)*ldr+tr2->idx[1],&ldr_,work+nwu));
    480:         PetscStackCallBLAS("LAPACKlarf",LAPACKlarf_("R",&n_,&n1_,y+tr2->idx[1],&inc,tr2->tau+1,H+(tr2->idx[1])*ldh,&ldh_,work+nwu));
    481:       }
    482:       if (tr2->idx[0]<tr2->idx[1]) {
    483:         HRApply(mr,R+(j+2)*ldr+tr2->idx[0],ldr,R+(j+2)*ldr+tr2->idx[1],ldr,tr2->cs,-tr2->sn);
    484:         if (tr2->type==1) {
    485:           HRApply(n,H+(tr2->idx[0])*ldh,1,H+(tr2->idx[1])*ldh,1,tr2->cs,tr2->sn);
    486:         } else {
    487:           HRApply(n,H+(tr2->idx[0])*ldh,1,H+(tr2->idx[1])*ldh,1,-tr2->cs,-tr2->sn);
    488:           s[tr2->idx[0]] = -s[tr2->idx[0]];
    489:           s[tr2->idx[1]] = -s[tr2->idx[1]];
    490:         }
    491:       }
    492:       for (i=0;i<tr2->idx[0]-1;i++) *(R+(j+1)*ldr+i) = y[i];
    493:       *(R+(j+1)*ldr+tr2->idx[0]-1) = y[tr2->idx[0]-1];
    494:       for (i=tr2->idx[0]+1;i<n;i++) *(R+(j+1)*ldr+i) = 0.0;
    495:       *(R+(j+1)*ldr+tr2->idx[0]) = tr2->alpha;
    496:       *n0 = tr2->n[0];
    497:       *n1 = tr2->n[1];
    498:       *idx0 = tr2->idx[0];
    499:       *idx1 = tr2->idx[1];
    500:       if (tr2->idx[0]<tr2->idx[1] && tr2->type==2) {
    501:         (*idx1)++; (*n1)--; (*n0)++;
    502:       }
    503:     } else {
    504:       *n0 = tr1->n[0];
    505:       *n1 = tr1->n[1];
    506:       *idx0 = tr1->idx[0];
    507:       *idx1 = tr1->idx[1];
    508:       if (tr1->idx[0]<tr1->idx[1] && tr1->type==2) {
    509:         (*idx1)++; (*n1)--; (*n0)++;
    510:       }
    511:     }
    512:     if (*n0>0) {
    513:       (*n0)--; (*idx0)++;
    514:       if (*n1==0) *idx1 = *idx0;
    515:     } else {
    516:       (*n1)--; (*idx1)++; *idx0 = *idx1;
    517:     }
    518:   }
    519:   return(0);
    520: #endif
    521: }

    523: /*
    524:   compute V = HR whit H s-orthogonal and R upper triangular
    525: */
    526: static PetscErrorCode PseudoOrthog_HR(PetscInt *nv,PetscScalar *V,PetscInt ldv,PetscReal *s,PetscScalar *R,PetscInt ldr,PetscBLASInt *perm,PetscBLASInt *cmplxEig,PetscBool *breakdown,PetscScalar *work)
    527: {
    529:   PetscInt       i,j,n,n0,n1,np,idx0,idx1,sz=1,k=0,t1,t2,nwu=0;
    530:   PetscScalar    *col1,*col2;
    531:   PetscBool      exg=PETSC_FALSE,ok=PETSC_FALSE;

    534:   n = *nv;
    535:   col1 = work+nwu;
    536:   nwu += n;
    537:   col2 = work+nwu;
    538:   nwu += n;
    539:   /* Sort R and s according to sing(s) */
    540:   np = 0;
    541:   for (i=0;i<n;i++) if (s[i]>0) np++;
    542:   if (s[0]>0) n1 = np;
    543:   else n1 = n-np;
    544:   n0 = 0;
    545:   for (i=0;i<n;i++) {
    546:     if (s[i]==s[0]) {
    547:       s[n0] = s[0];
    548:       perm[n0++] = i;
    549:     } else perm[n1++] = i;
    550:   }
    551:   for (i=n0;i<n;i++) s[i] = -s[0];
    552:   n1 -= n0;
    553:   idx0 = 0;
    554:   idx1 = n0;
    555:   if (idx1==n) idx1=idx0;
    556:   for (i=0;i<n;i++) {
    557:     for (j=0;j<n;j++) R[j*ldr+i] = V[j*ldv+perm[i]];
    558:   }
    559:   /* Initialize H */
    560:   for (i=0;i<n;i++) {
    561:     PetscMemzero(V+i*ldv,n*sizeof(PetscScalar));
    562:     V[perm[i]+i*ldv] = 1.0;
    563:   }
    564:   for (i=0;i<n;i++) perm[i] = i;
    565:   j = 0;
    566:   while (j<n-k) {
    567:     if (cmplxEig[j]==0) sz=1;
    568:     else sz=2;
    569:     TryHRIt(n,j,sz,V,ldv,R,ldr,s,&exg,&ok,&n0,&n1,&idx0,&idx1,NULL,work+nwu);
    570:     if (ok) {
    571:       if (exg) cmplxEig[j] = -cmplxEig[j];
    572:       j = j+sz;
    573:     } else { /* to be discarded */
    574:       k = k+1;
    575:       if (cmplxEig[j]==0) {
    576:         if (j<n) {
    577:           t1 = perm[j];
    578:           for (i=j;i<n-1;i++) perm[i] = perm[i+1];
    579:           perm[n-1] = t1;
    580:           t1 = cmplxEig[j];
    581:           for (i=j;i<n-1;i++) cmplxEig[i] = cmplxEig[i+1];
    582:           cmplxEig[n-1] = t1;
    583:           PetscMemcpy(col1,R+j*ldr,n*sizeof(PetscScalar));
    584:           for (i=j;i<n-1;i++) {
    585:             PetscMemcpy(R+i*ldr,R+(i+1)*ldr,n*sizeof(PetscScalar));
    586:           }
    587:           PetscMemcpy(R+(n-1)*ldr,col1,n*sizeof(PetscScalar));
    588:         }
    589:       } else {
    590:         k = k+1;
    591:         if (j<n-1) {
    592:           t1 = perm[j];
    593:           t2 = perm[j+1];
    594:           for (i=j;i<n-2;i++) perm[i] = perm[i+2];
    595:           perm[n-2] = t1;
    596:           perm[n-1] = t2;
    597:           t1 = cmplxEig[j];
    598:           t2 = cmplxEig[j+1];
    599:           for (i=j;i<n-2;i++) cmplxEig[i] = cmplxEig[i+2];
    600:           cmplxEig[n-2] = t1;
    601:           cmplxEig[n-1] = t2;
    602:           PetscMemcpy(col1,R+j*ldr,n*sizeof(PetscScalar));
    603:           PetscMemcpy(col2,R+(j+1)*ldr,n*sizeof(PetscScalar));
    604:           for (i=j;i<n-2;i++) {
    605:             PetscMemcpy(R+i*ldr,R+(i+2)*ldr,n*sizeof(PetscScalar));
    606:           }
    607:           PetscMemcpy(R+(n-2)*ldr,col1,n*sizeof(PetscScalar));
    608:           PetscMemcpy(R+(n-1)*ldr,col2,n*sizeof(PetscScalar));
    609:         }
    610:       }
    611:     }
    612:   }
    613:   if (k!=0) {
    614:     if (breakdown) *breakdown = PETSC_TRUE;
    615:     *nv = n-k;
    616:   }
    617:   return(0);
    618: }

    620: PetscErrorCode DSGHIEPOrthogEigenv(DS ds,DSMatType mat,PetscScalar *wr,PetscScalar *wi,PetscBool accum)
    621: {
    623:   PetscInt       lws,nwus=0,nwui=0,lwi;
    624:   PetscInt       off,n,nv,ld,i,ldr,l;
    625:   PetscScalar    *W,*X,*R,*ts,zeroS=0.0,oneS=1.0;
    626:   PetscReal      *s,vi,vr,tr,*d,*e;
    627:   PetscBLASInt   ld_,n_,nv_,*perm,*cmplxEig;

    630:   l = ds->l;
    631:   n = ds->n-l;
    632:   PetscBLASIntCast(n,&n_);
    633:   ld = ds->ld;
    634:   PetscBLASIntCast(ld,&ld_);
    635:   off = l*ld+l;
    636:   s = ds->rmat[DS_MAT_D];
    637:   if (!ds->compact) {
    638:     for (i=l;i<ds->n;i++) s[i] = PetscRealPart(*(ds->mat[DS_MAT_B]+i*ld+i));
    639:   }
    640:   lws = n*n+7*n;
    641:   lwi = 2*n;
    642:   DSAllocateWork_Private(ds,lws,0,lwi);
    643:   R = ds->work+nwus;
    644:   nwus += n*n;
    645:   ldr = n;
    646:   perm = ds->iwork + nwui;
    647:   nwui += n;
    648:   cmplxEig = ds->iwork+nwui;
    649:   X = ds->mat[mat];
    650:   for (i=0;i<n;i++) {
    651: #if defined(PETSC_USE_COMPLEX)
    652:     vi = PetscImaginaryPart(wr[l+i]);
    653: #else
    654:     vi = PetscRealPart(wi[l+i]);
    655: #endif
    656:     if (vi!=0) {
    657:       cmplxEig[i] = 1;
    658:       cmplxEig[i+1] = 2;
    659:       i++;
    660:     } else cmplxEig[i] = 0;
    661:   }
    662:   nv = n;

    664:   /* Perform HR decomposition */
    665:   /* Hyperbolic rotators */
    666:   PseudoOrthog_HR(&nv,X+off,ld,s+l,R,ldr,perm,cmplxEig,NULL,ds->work+nwus);
    667:   /* Sort wr,wi perm */
    668:   ts = ds->work+nwus;
    669:   PetscMemcpy(ts,wr+l,n*sizeof(PetscScalar));
    670:   for (i=0;i<n;i++) wr[i+l] = ts[perm[i]];
    671: #if !defined(PETSC_USE_COMPLEX)
    672:   PetscMemcpy(ts,wi+l,n*sizeof(PetscScalar));
    673:   for (i=0;i<n;i++) wi[i+l] = ts[perm[i]];
    674: #endif
    675:   /* Projected Matrix */
    676:   PetscMemzero(ds->rmat[DS_MAT_T]+2*ld,ld*sizeof(PetscReal));
    677:   d = ds->rmat[DS_MAT_T];
    678:   e = d+ld;
    679:   for (i=0;i<nv;i++) {
    680:     if (cmplxEig[i]==0) { /* Real */
    681:       d[l+i] = PetscRealPart(wr[l+i]*s[l+i]);
    682:       e[l+i] = 0.0;
    683:     } else {
    684:       vr = PetscRealPart(wr[l+i]);
    685: #if defined(PETSC_USE_COMPLEX)
    686:       vi = PetscImaginaryPart(wr[l+i]);
    687: #else
    688:       vi = PetscRealPart(wi[l+i]);
    689: #endif
    690:       if (cmplxEig[i]==-1) vi = -vi;
    691:       tr = PetscRealPart((R[i+(i+1)*ldr]/R[i+i*ldr]))*vi;
    692:       d[l+i] = (vr-tr)*s[l+i];
    693:       d[l+i+1] = (vr+tr)*s[l+i+1];
    694:       e[l+i] = PetscRealPart(s[l+i]*(R[(i+1)+(i+1)*ldr]/R[i+i*ldr])*vi);
    695:       e[l+i+1] = 0.0;
    696:       i++;
    697:     }
    698:   }
    699:   /* accumulate previous Q */
    700:   if (accum) {
    701:     PetscBLASIntCast(nv,&nv_);
    702:     DSAllocateMat_Private(ds,DS_MAT_W);
    703:     W = ds->mat[DS_MAT_W];
    704:     DSCopyMatrix_Private(ds,DS_MAT_W,DS_MAT_Q);
    705:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&nv_,&n_,&oneS,W+off,&ld_,X+off,&ld_,&zeroS,ds->mat[DS_MAT_Q]+off,&ld_));
    706:   } else {
    707:     PetscMemzero(ds->mat[DS_MAT_Q],ld*ld*sizeof(PetscScalar));
    708:     for (i=0;i<ds->l;i++) *(ds->mat[DS_MAT_Q]+i+i*ld) = 1.0;
    709:     for (i=0;i<n;i++) { PetscMemcpy(ds->mat[DS_MAT_Q]+off+i*ld,X+off+i*ld,n*sizeof(PetscScalar)); }
    710:   }
    711:   ds->t = nv+l;
    712:   if (!ds->compact) { DSSwitchFormat_GHIEP(ds,PETSC_FALSE); }
    713:   return(0);
    714: }

    716: /*
    717:    Reduce to tridiagonal-diagonal pair by means of TridiagDiag_HHR.
    718: */
    719: PetscErrorCode DSIntermediate_GHIEP(DS ds)
    720: {
    722:   PetscInt       i,ld,off;
    723:   PetscInt       nwall,nwallr,nwalli;
    724:   PetscScalar    *A,*B,*Q;
    725:   PetscReal      *d,*e,*s;

    728:   ld = ds->ld;
    729:   A = ds->mat[DS_MAT_A];
    730:   B = ds->mat[DS_MAT_B];
    731:   Q = ds->mat[DS_MAT_Q];
    732:   d = ds->rmat[DS_MAT_T];
    733:   e = ds->rmat[DS_MAT_T]+ld;
    734:   s = ds->rmat[DS_MAT_D];
    735:   off = ds->l+ds->l*ld;
    736:   PetscMemzero(Q,ld*ld*sizeof(PetscScalar));
    737:   nwall = ld*ld+ld;
    738:   nwallr = ld;
    739:   nwalli = ld;
    740:   DSAllocateWork_Private(ds,nwall,nwallr,nwalli);
    741:   for (i=0;i<ds->n;i++) Q[i+i*ld]=1.0;
    742:   for (i=0;i<ds->n-ds->l;i++) *(ds->perm+i)=i;
    743:   if (ds->compact) {
    744:     if (ds->state < DS_STATE_INTERMEDIATE) {
    745:       DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
    746:       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);
    747:       ds->k = ds->l;
    748:       PetscMemzero(d+2*ld+ds->l,(ds->n-ds->l)*sizeof(PetscReal));
    749:     }
    750:   } else {
    751:     if (ds->state < DS_STATE_INTERMEDIATE) {
    752:       for (i=0;i<ds->n;i++) s[i] = PetscRealPart(B[i+i*ld]);
    753:       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);
    754:       PetscMemzero(d+2*ld,(ds->n)*sizeof(PetscReal));
    755:       ds->k = ds->l;
    756:       DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
    757:     } else {
    758:       DSSwitchFormat_GHIEP(ds,PETSC_TRUE);
    759:     }
    760:   }
    761:   return(0);
    762: }