Actual source code: hz.c

slepc-3.7.4 2017-05-17
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  1: /*
  2:    HZ iteration for generalized symmetric-indefinite eigenproblem.
  3:    Based on Matlab code from David Watkins.

  5:    References:

  7:        [1] D.S. Watkins, The Matrix Eigenvalue Problem: GR and Krylov Subspace
  8:            Methods, SIAM, 2007.

 10:        [2] M.A. Brebner, J. Grad, "Eigenvalues of Ax = lambda Bx for real
 11:            symmetric matrices A and B computed by reduction to pseudosymmetric
 12:            form and the HR process", Linear Alg. Appl. 43:99-118, 1982.

 14:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 15:    SLEPc - Scalable Library for Eigenvalue Problem Computations
 16:    Copyright (c) 2002-2016, Universitat Politecnica de Valencia, Spain

 18:    This file is part of SLEPc.

 20:    SLEPc is free software: you can redistribute it and/or modify it under  the
 21:    terms of version 3 of the GNU Lesser General Public License as published by
 22:    the Free Software Foundation.

 24:    SLEPc  is  distributed in the hope that it will be useful, but WITHOUT  ANY
 25:    WARRANTY;  without even the implied warranty of MERCHANTABILITY or  FITNESS
 26:    FOR  A  PARTICULAR PURPOSE. See the GNU Lesser General Public  License  for
 27:    more details.

 29:    You  should have received a copy of the GNU Lesser General  Public  License
 30:    along with SLEPc. If not, see <http://www.gnu.org/licenses/>.
 31:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 32: */
 33: #include <slepc/private/dsimpl.h>
 34: #include <slepcblaslapack.h>

 38: /*
 39:    Sets up a 2-by-2 matrix to eliminate y in the vector [x y]'.
 40:    Transformation is rotator if sygn = 1 and hyperbolic if sygn = -1.
 41: */
 42: static PetscErrorCode UnifiedRotation(PetscReal x,PetscReal y,PetscReal sygn,PetscReal *rot,PetscReal *rcond,PetscBool *swap)
 43: {
 44:   PetscReal nrm,c,s;

 47:   *swap = PETSC_FALSE;
 48:   if (y == 0) {
 49:     rot[0] = 1.0; rot[1] = 0.0; rot[2] = 0.0; rot[3] = 1.0;
 50:     *rcond = 1.0;
 51:   } else {
 52:     nrm = PetscMax(PetscAbs(x),PetscAbs(y));
 53:     c = x/nrm; s = y/nrm;
 54:     if (sygn == 1.0) {  /* set up a rotator */
 55:       nrm = PetscSqrtReal(c*c+s*s);
 56:       c = c/nrm; s = s/nrm;
 57:       /* rot = [c s; -s c]; */
 58:       rot[0] = c; rot[1] = -s; rot[2] = s; rot[3] = c;
 59:       *rcond = 1.0;
 60:     } else if (sygn == -1) {  /* set up a hyperbolic transformation */
 61:       nrm = c*c-s*s;
 62:       if (nrm > 0) nrm = PetscSqrtReal(nrm);
 63:       else if (nrm < 0) {
 64:         nrm = PetscSqrtReal(-nrm);
 65:         *swap = PETSC_TRUE;
 66:       } else   /* breakdown */
 67:         SETERRQ(PETSC_COMM_SELF,1,"Breakdown in construction of hyperbolic transformation");
 68:       c = c/nrm; s = s/nrm;
 69:       /* rot = [c -s; -s c]; */
 70:       rot[0] = c; rot[1] = -s; rot[2] = -s; rot[3] = c;
 71:       *rcond = PetscAbs(PetscAbs(s)-PetscAbs(c))/(PetscAbs(s)+PetscAbs(c));
 72:     } else SETERRQ(PETSC_COMM_SELF,1,"Value of sygn sent to transetup must be 1 or -1");
 73:   }
 74:   return(0);
 75: }

 79: static PetscErrorCode HZStep(PetscBLASInt ntop,PetscBLASInt nn,PetscReal tr,PetscReal dt,PetscReal *aa,PetscReal *bb,PetscReal *dd,PetscScalar *uu,PetscInt n,PetscInt ld,PetscBool *flag)
 80: {
 82:   PetscBLASInt   one=1;
 83:   PetscInt       k,jj,ii;
 84:   PetscBLASInt   n_;
 85:   PetscReal      bulge10,bulge20,bulge30,bulge31,bulge41,bulge42;
 86:   PetscReal      sygn,rcond=1.0,worstcond,rot[4],buf[2],t;
 87:   PetscScalar    rtmp;
 88:   PetscBool      swap;

 91:   worstcond = 1.0;
 92:   PetscBLASIntCast(n,&n_);

 94:   /* Build initial bulge that sets step in motion */
 95:   bulge10 = dd[ntop+1]*(aa[ntop]*(aa[ntop] - dd[ntop]*tr) + dt*dd[ntop]*dd[ntop]) + dd[ntop]*bb[ntop]*bb[ntop];
 96:   bulge20 = bb[ntop]*(dd[ntop+1]*aa[ntop] + dd[ntop]*aa[ntop+1] - dd[ntop]*dd[ntop+1]*tr);
 97:   bulge30 = bb[ntop]*bb[ntop+1]*dd[ntop];
 98:   bulge31 = 0.0;
 99:   bulge41 = 0.0;
100:   bulge42 = 0.0;

102:   /* Chase the bulge */
103:   for (jj=ntop;jj<nn-1;jj++) {

105:     /* Check for trivial bulge */
106:     if (jj>ntop && PetscMax(PetscMax(PetscAbs(bulge10),PetscAbs(bulge20)),PetscAbs(bulge30))<PETSC_MACHINE_EPSILON*(PetscAbs(aa[jj]) + PetscAbs(aa[jj+1]))) {
107:       bb[jj-1] = 0.0;  /* deflate and move on */

109:     } else { /* carry out the step */

111:       /* Annihilate tip entry bulge30 */
112:       if (bulge30 != 0.0) {

114:         /* Make an interchange if necessary to ensure that the
115:            first transformation is othogonal, not hyperbolic.  */
116:         if (dd[jj+1] != dd[jj+2]) { /* make an interchange */
117:           if (dd[jj] != dd[jj+1]) {  /* interchange 1st and 2nd */
118:             buf[0] = bulge20; bulge20 = bulge10; bulge10 = buf[0];
119:             buf[0] = aa[jj]; aa[jj] = aa[jj+1]; aa[jj+1] = buf[0];
120:             buf[0] = bb[jj+1]; bb[jj+1] = bulge31; bulge31 = buf[0];
121:             buf[0] = dd[jj]; dd[jj] = dd[jj+1]; dd[jj+1] = buf[0];
122:             for (k=0;k<n;k++) {
123:               rtmp = uu[k+jj*ld]; uu[k+jj*ld] = uu[k+(jj+1)*ld]; uu[k+(jj+1)*ld] = rtmp;
124:             }
125:           } else {  /* interchange 1st and 3rd */
126:             buf[0] = bulge30; bulge30 = bulge10; bulge10 = buf[0];
127:             buf[0] = aa[jj]; aa[jj] = aa[jj+2]; aa[jj+2] = buf[0];
128:             buf[0] = bb[jj]; bb[jj] = bb[jj+1]; bb[jj+1] = buf[0];
129:             buf[0] = dd[jj]; dd[jj] = dd[jj+2]; dd[jj+2] = buf[0];
130:             if (jj + 2 < nn-1) {
131:               bulge41 = bb[jj+2];
132:               bb[jj+2] = 0;
133:             }
134:             for (k=0;k<n;k++) {
135:               rtmp = uu[k+jj*ld]; uu[k+jj*ld] = uu[k+(jj+2)*ld]; uu[k+(jj+2)*ld] = rtmp;
136:             }
137:           }
138:         }

140:         /* Set up transforming matrix rot. */
141:         UnifiedRotation(bulge20,bulge30,1,rot,&rcond,&swap);

143:         /* Apply transforming matrix rot to T. */
144:         bulge20 = rot[0]*bulge20 + rot[2]*bulge30;
145:         buf[0] = rot[0]*bb[jj] + rot[2]*bulge31;
146:         buf[1] = rot[1]*bb[jj] + rot[3]*bulge31;
147:         bb[jj] = buf[0];
148:         bulge31 = buf[1];
149:         buf[0] = rot[0]*rot[0]*aa[jj+1] + 2.0*rot[0]*rot[2]*bb[jj+1] + rot[2]*rot[2]*aa[jj+2];
150:         buf[1] = rot[1]*rot[1]*aa[jj+1] + 2.0*rot[3]*rot[1]*bb[jj+1] + rot[3]*rot[3]*aa[jj+2];
151:         bb[jj+1] = rot[1]*rot[0]*aa[jj+1] + rot[3]*rot[2]*aa[jj+2] + (rot[3]*rot[0] + rot[1]*rot[2])*bb[jj+1];
152:         aa[jj+1] = buf[0];
153:         aa[jj+2] = buf[1];
154:         if (jj + 2 < nn-1) {
155:           bulge42 = bb[jj+2]*rot[2];
156:           bb[jj+2] = bb[jj+2]*rot[3];
157:         }

159:         /* Accumulate transforming matrix */
160:         PetscStackCallBLAS("BLASrot",BLASrot_(&n_,uu+(jj+1)*ld,&one,uu+(jj+2)*ld,&one,&rot[0],&rot[2]));
161:       }

163:       /* Annihilate inner entry bulge20 */
164:       if (bulge20 != 0.0) {

166:         /* Begin by setting up transforming matrix rot */
167:         sygn = dd[jj]*dd[jj+1];
168:         UnifiedRotation(bulge10,bulge20,sygn,rot,&rcond,&swap);
169:         if (rcond<PETSC_MACHINE_EPSILON) {
170:           SETERRQ1(PETSC_COMM_SELF,0,"Transforming matrix is numerically singular rcond=%g",rcond);
171:           *flag = PETSC_TRUE;
172:           return(0);
173:         }
174:         if (rcond < worstcond) worstcond = rcond;

176:         /* Apply transforming matrix rot to T */
177:         if (jj > ntop) bb[jj-1] = rot[0]*bulge10 + rot[2]*bulge20;
178:         buf[0] = rot[0]*rot[0]*aa[jj] + 2*rot[0]*rot[2]*bb[jj] + rot[2]*rot[2]*aa[jj+1];
179:         buf[1] = rot[1]*rot[1]*aa[jj] + 2*rot[3]*rot[1]*bb[jj] + rot[3]*rot[3]*aa[jj+1];
180:         bb[jj] = rot[1]*rot[0]*aa[jj] + rot[3]*rot[2]*aa[jj+1] + (rot[3]*rot[0] + rot[1]*rot[2])*bb[jj];
181:         aa[jj] = buf[0];
182:         aa[jj+1] = buf[1];
183:         if (jj + 1 < nn-1) {
184:           /* buf = [ bulge31 bb(jj+1) ] * rot' */
185:           buf[0] = rot[0]*bulge31 + rot[2]*bb[jj+1];
186:           buf[1] = rot[1]*bulge31 + rot[3]*bb[jj+1];
187:           bulge31 = buf[0];
188:           bb[jj+1] = buf[1];
189:         }
190:         if (jj + 2 < nn-1) {
191:           /* buf = [bulge41 bulge42] * rot' */
192:           buf[0] = rot[0]*bulge41 + rot[2]*bulge42;
193:           buf[1] = rot[1]*bulge41 + rot[3]*bulge42;
194:           bulge41 = buf[0];
195:           bulge42 = buf[1];
196:         }

198:         /* Apply transforming matrix rot to D */
199:         if (swap == 1) {
200:           buf[0] = dd[jj]; dd[jj] = dd[jj+1]; dd[jj+1] = buf[0];
201:         }

203:         /* Accumulate transforming matrix, uu(jj:jj+1,:) = rot*uu(jj:jj+1,:) */
204:         if (sygn==1) {
205:           PetscStackCallBLAS("BLASrot",BLASrot_(&n_,uu+jj*ld,&one,uu+(jj+1)*ld,&one,&rot[0],&rot[2]));
206:         } else {
207:           if (PetscAbsReal(rot[0])>PetscAbsReal(rot[1])) { /* Type I */
208:             t = rot[1]/rot[0];
209:             for (ii=0;ii<n;ii++) {
210:               uu[jj*ld+ii] = rot[0]*uu[jj*ld+ii] + rot[1]*uu[(jj+1)*ld+ii];
211:               uu[(jj+1)*ld+ii] = t*uu[jj*ld+ii] + uu[(jj+1)*ld+ii]/rot[0];
212:             }
213:           } else { /* Type II */
214:             t = rot[0]/rot[1];
215:             for (ii=0;ii<n;ii++) {
216:               rtmp = uu[jj*ld+ii];
217:               uu[jj*ld+ii] = rot[0]*uu[jj*ld+ii] + rot[1]*uu[(jj+1)*ld+ii];
218:               uu[(jj+1)*ld+ii] = t*uu[jj*ld+ii] + rtmp/rot[1];
219:             }
220:           }
221:         }
222:       }
223:     }

225:     /* Adjust bulge for next step */
226:     bulge10 = bb[jj];
227:     bulge20 = bulge31;
228:     bulge30 = bulge41;
229:     bulge31 = bulge42;
230:     bulge41 = 0.0;
231:     bulge42 = 0.0;
232:   }
233:   return(0);
234: }

238: static PetscErrorCode HZIteration(PetscBLASInt nn,PetscBLASInt cgd,PetscReal *aa,PetscReal *bb,PetscReal *dd,PetscScalar *uu,PetscBLASInt ld)
239: {
241:   PetscBLASInt   j2,one=1;
242:   PetscInt       its,nits,nstop,jj,ntop,nbot,ntry;
243:   PetscReal      htr,det,dis,dif,tn,kt,c,s,tr,dt;
244:   PetscBool      flag=PETSC_FALSE;

247:   its = 0;
248:   nbot = nn-1;
249:   nits = 0;
250:   nstop = 40*(nn - cgd);

252:   while (nbot >= cgd && nits < nstop) {

254:     /* Check for zeros on the subdiagonal */
255:     jj = nbot - 1;
256:     while (jj>=cgd && PetscAbs(bb[jj])>PETSC_MACHINE_EPSILON*(PetscAbs(aa[jj])+PetscAbs(aa[jj+1]))) jj = jj-1;
257:     if (jj>=cgd) bb[jj]=0;
258:     ntop = jj + 1;  /* starting point for step */
259:     if (ntop == nbot) {  /* isolate single eigenvalue */
260:       nbot = ntop - 1;
261:       its = 0;
262:     } else if (ntop+1 == nbot) {  /* isolate pair of eigenvalues */
263:       htr = 0.5*(aa[ntop]*dd[ntop] + aa[nbot]*dd[nbot]);
264:       det = dd[ntop]*dd[nbot]*(aa[ntop]*aa[nbot]-bb[ntop]*bb[ntop]);
265:       dis = htr*htr - det;
266:       if (dis > 0) {  /* distinct real eigenvalues */
267:         if (dd[ntop] == dd[nbot]) {  /* separate the eigenvalues by a Jacobi rotator */
268:           dif = aa[ntop]-aa[nbot];
269:           if (2.0*PetscAbs(bb[ntop])<=dif) {
270:             tn = 2*bb[ntop]/dif;
271:             tn = tn/(1.0 + PetscSqrtScalar(1.0+tn*tn));
272:           } else {
273:             kt = dif/(2.0*bb[ntop]);
274:             tn = PetscSign(kt)/(PetscAbs(kt)+PetscSqrtScalar(1.0+kt*kt));
275:           }
276:           c = 1.0/PetscSqrtScalar(1.0 + tn*tn);
277:           s = c*tn;
278:           aa[ntop] = aa[ntop] + tn*bb[ntop];
279:           aa[nbot] = aa[nbot] - tn*bb[ntop];
280:           bb[ntop] = 0;
281:           j2 = nn-cgd;
282:           PetscStackCallBLAS("BLASrot",BLASrot_(&j2,uu+ntop*ld+cgd,&one,uu+nbot*ld+cgd,&one,&c,&s));
283:         }
284:       }
285:       nbot = ntop - 1;
286:     } else {  /* Do an HZ iteration */
287:       its = its + 1;
288:       nits = nits + 1;
289:       tr = aa[nbot-1]*dd[nbot-1] + aa[nbot]*dd[nbot];
290:       dt = dd[nbot-1]*dd[nbot]*(aa[nbot-1]*aa[nbot]-bb[nbot-1]*bb[nbot-1]);
291:       for (ntry=1;ntry<=6;ntry++) {
292:         HZStep(ntop,nbot+1,tr,dt,aa,bb,dd,uu,nn,ld,&flag);
293:         if (!flag) break;
294:         else if (ntry == 6) SETERRQ(PETSC_COMM_SELF,1,"Unable to complete hz step on six tries");
295:         else {
296:           tr = 0.9*tr; dt = 0.81*dt;
297:         }
298:       }
299:     }
300:   }
301:   return(0);
302: }

306: PetscErrorCode DSSolve_GHIEP_HZ(DS ds,PetscScalar *wr,PetscScalar *wi)
307: {
309:   PetscInt       off;
310:   PetscBLASInt   n1,ld;
311:   PetscScalar    *A,*B,*Q;
312:   PetscReal      *d,*e,*s;
313: #if defined(PETSC_USE_COMPLEX)
314:   PetscInt       i;
315: #endif

318: #if !defined(PETSC_USE_COMPLEX)
320: #endif
321:   PetscBLASIntCast(ds->ld,&ld);
322:   n1  = ds->n - ds->l;
323:   off = ds->l + ds->l*ld;
324:   A   = ds->mat[DS_MAT_A];
325:   B   = ds->mat[DS_MAT_B];
326:   Q   = ds->mat[DS_MAT_Q];
327:   d   = ds->rmat[DS_MAT_T];
328:   e   = ds->rmat[DS_MAT_T] + ld;
329:   s   = ds->rmat[DS_MAT_D];
330:   /* Quick return */
331:   if (n1 == 1) {
332:     *(Q+off) = 1;
333:     if (ds->compact) {
334:       wr[ds->l] = d[ds->l]/s[ds->l]; wi[ds->l] = 0.0;
335:     } else {
336:       d[ds->l] = PetscRealPart(A[off]); s[ds->l] = PetscRealPart(B[off]);
337:       wr[ds->l] = d[ds->l]/s[ds->l]; wi[ds->l] = 0.0;
338:     }
339:     return(0);
340:   }
341:   /* Reduce to pseudotriadiagonal form */
342:   DSIntermediate_GHIEP(ds);
343:   HZIteration(ds->n,ds->l,d,e,s,Q,ld);
344:   if (!ds->compact) {
345:     DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
346:   }
347:   /* Undo from diagonal the blocks whith real eigenvalues*/
348:   DSGHIEPRealBlocks(ds);

350:   /* Recover eigenvalues from diagonal */
351:   DSGHIEPComplexEigs(ds,0,ds->n,wr,wi);
352: #if defined(PETSC_USE_COMPLEX)
353:   if (wi) {
354:     for (i=ds->l;i<ds->n;i++) wi[i] = 0.0;
355:   }
356: #endif
357:   ds->t = ds->n;
358:   return(0);
359: }