Date: 2020-01-13T11:59:14+00:00

Actual source code: invit.c
slepc-3.12.2 2020-01-13
Report Typos and Errors

      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2019, 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: {
    116: #if defined(SLEPC_MISSING_LAPACK_LARFG) || defined(SLEPC_MISSING_LAPACK_LARF)
    118:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LARFG/LARF - Lapack routines are unavailable");
    119: #else
    121:   PetscInt       i,j,k,*ii,*jj,i0=0,ik=0,tmp,type;
    122:   PetscInt       nwu=0;
    123:   PetscReal      *ss,cond=1.0,cs,sn,r;
    124:   PetscScalar    tau,t,*AA;
    125:   PetscBLASInt   n0,n1,ni,inc=1,m,n_,lda_,ldq_,*perm;
    126:   PetscBool      breakdown = PETSC_TRUE;

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

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

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

    287: 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)
    288: {
    289: #if defined(SLEPC_MISSING_LAPACK_LARFG) || defined(SLEPC_MISSING_LAPACK_LARF)
    291:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LARFG/LARF - Lapack routines are unavailable");
    292: #else
    294:   PetscScalar    *x,*y;
    295:   PetscReal      ncond2=1.0;
    296:   PetscBLASInt   n0_,n1_,inc=1;

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

    367: /*
    368:   Auxiliary function to try perform one iteration of hr routine,
    369:   checking condition number. If it is < tolD, apply the
    370:   transformation to H and R, if not, ok=false and it do nothing
    371:   tolE, tolerance to exchange complex pairs to improve conditioning
    372: */
    373: 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)
    374: {
    375: #if defined(SLEPC_MISSING_LAPACK_LARF)
    377:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"LARF - Lapack routine is unavailable");
    378: #else
    380:   struct HRtr    *tr1,*tr2,tr1_t,tr2_t,tr1_te,tr2_te;
    381:   PetscScalar    *x,*y;
    382:   PetscReal      ncond,ncond_e;
    383:   PetscInt       nwu=0,i,d=1;
    384:   PetscBLASInt   n0_,n1_,inc=1,mh,mr,n_,ldr_,ldh_;
    385:   PetscReal      tolD = 1e+5;

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

    519: /*
    520:   compute V = HR whit H s-orthogonal and R upper triangular
    521: */
    522: static PetscErrorCode PseudoOrthog_HR(PetscInt *nv,PetscScalar *V,PetscInt ldv,PetscReal *s,PetscScalar *R,PetscInt ldr,PetscBLASInt *perm,PetscBLASInt *cmplxEig,PetscBool *breakdown,PetscScalar *work)
    523: {
    525:   PetscInt       i,j,n,n0,n1,np,idx0,idx1,sz=1,k=0,t1,t2,nwu=0;
    526:   PetscScalar    *col1,*col2;
    527:   PetscBool      exg=PETSC_FALSE,ok=PETSC_FALSE;

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

    612: PetscErrorCode DSGHIEPOrthogEigenv(DS ds,DSMatType mat,PetscScalar *wr,PetscScalar *wi,PetscBool accum)
    613: {
    615:   PetscInt       lws,nwus=0,nwui=0,lwi;
    616:   PetscInt       off,n,nv,ld,i,ldr,l;
    617:   PetscScalar    *W,*X,*R,*ts,zeroS=0.0,oneS=1.0;
    618:   PetscReal      *s,vi,vr,tr,*d,*e;
    619:   PetscBLASInt   ld_,n_,nv_,*perm,*cmplxEig;

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

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

    708: /*
    709:    Reduce to tridiagonal-diagonal pair by means of TridiagDiag_HHR.
    710: */
    711: PetscErrorCode DSIntermediate_GHIEP(DS ds)
    712: {
    714:   PetscInt       i,ld,off;
    715:   PetscInt       nwall,nwallr,nwalli;
    716:   PetscScalar    *A,*B,*Q;
    717:   PetscReal      *d,*e,*s;

    720:   ld = ds->ld;
    721:   A = ds->mat[DS_MAT_A];
    722:   B = ds->mat[DS_MAT_B];
    723:   Q = ds->mat[DS_MAT_Q];
    724:   d = ds->rmat[DS_MAT_T];
    725:   e = ds->rmat[DS_MAT_T]+ld;
    726:   s = ds->rmat[DS_MAT_D];
    727:   off = ds->l+ds->l*ld;
    728:   PetscArrayzero(Q,ld*ld);
    729:   nwall = ld*ld+ld;
    730:   nwallr = ld;
    731:   nwalli = ld;
    732:   DSAllocateWork_Private(ds,nwall,nwallr,nwalli);
    733:   for (i=0;i<ds->n;i++) Q[i+i*ld]=1.0;
    734:   for (i=0;i<ds->n-ds->l;i++) *(ds->perm+i)=i;
    735:   if (ds->compact) {
    736:     if (ds->state < DS_STATE_INTERMEDIATE) {
    737:       DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
    738:       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);
    739:       ds->k = ds->l;
    740:       PetscArrayzero(d+2*ld+ds->l,ds->n-ds->l);
    741:     }
    742:   } else {
    743:     if (ds->state < DS_STATE_INTERMEDIATE) {
    744:       for (i=0;i<ds->n;i++) s[i] = PetscRealPart(B[i+i*ld]);
    745:       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);
    746:       PetscArrayzero(d+2*ld,ds->n);
    747:       ds->k = ds->l;
    748:       DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
    749:     } else {
    750:       DSSwitchFormat_GHIEP(ds,PETSC_TRUE);
    751:     }
    752:   }
    753:   return(0);
    754: }