Actual source code: hz.c
slepc-3.11.2 2019-07-30
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: */
10: /*
11: HZ iteration for generalized symmetric-indefinite eigenproblem.
12: Based on Matlab code from David Watkins.
14: References:
16: [1] D.S. Watkins, The Matrix Eigenvalue Problem: GR and Krylov Subspace
17: Methods, SIAM, 2007.
19: [2] M.A. Brebner, J. Grad, "Eigenvalues of Ax = lambda Bx for real
20: symmetric matrices A and B computed by reduction to pseudosymmetric
21: form and the HR process", Linear Alg. Appl. 43:99-118, 1982.
22: */
24: #include <slepc/private/dsimpl.h>
25: #include <slepcblaslapack.h>
27: /*
28: Sets up a 2-by-2 matrix to eliminate y in the vector [x y]'.
29: Transformation is rotator if sygn = 1 and hyperbolic if sygn = -1.
30: */
31: static PetscErrorCode UnifiedRotation(PetscReal x,PetscReal y,PetscReal sygn,PetscReal *rot,PetscReal *rcond,PetscBool *swap)
32: {
33: PetscReal nrm,c,s;
36: *swap = PETSC_FALSE;
37: if (y == 0) {
38: rot[0] = 1.0; rot[1] = 0.0; rot[2] = 0.0; rot[3] = 1.0;
39: *rcond = 1.0;
40: } else {
41: nrm = PetscMax(PetscAbs(x),PetscAbs(y));
42: c = x/nrm; s = y/nrm;
43: if (sygn == 1.0) { /* set up a rotator */
44: nrm = PetscSqrtReal(c*c+s*s);
45: c = c/nrm; s = s/nrm;
46: /* rot = [c s; -s c]; */
47: rot[0] = c; rot[1] = -s; rot[2] = s; rot[3] = c;
48: *rcond = 1.0;
49: } else if (sygn == -1) { /* set up a hyperbolic transformation */
50: nrm = c*c-s*s;
51: if (nrm > 0) nrm = PetscSqrtReal(nrm);
52: else if (nrm < 0) {
53: nrm = PetscSqrtReal(-nrm);
54: *swap = PETSC_TRUE;
55: } else SETERRQ(PETSC_COMM_SELF,1,"Breakdown in construction of hyperbolic transformation");
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 = PetscAbs(PetscAbs(s)-PetscAbs(c))/(PetscAbs(s)+PetscAbs(c));
60: } else SETERRQ(PETSC_COMM_SELF,1,"Value of sygn sent to transetup must be 1 or -1");
61: }
62: return(0);
63: }
65: 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)
66: {
68: PetscBLASInt one=1;
69: PetscInt k,jj,ii;
70: PetscBLASInt n_;
71: PetscReal bulge10,bulge20,bulge30,bulge31,bulge41,bulge42;
72: PetscReal sygn,rcond=1.0,worstcond,rot[4],buf[2],t;
73: PetscScalar rtmp;
74: PetscBool swap;
77: *flag = PETSC_FALSE;
78: worstcond = 1.0;
79: PetscBLASIntCast(n,&n_);
81: /* Build initial bulge that sets step in motion */
82: bulge10 = dd[ntop+1]*(aa[ntop]*(aa[ntop] - dd[ntop]*tr) + dt*dd[ntop]*dd[ntop]) + dd[ntop]*bb[ntop]*bb[ntop];
83: bulge20 = bb[ntop]*(dd[ntop+1]*aa[ntop] + dd[ntop]*aa[ntop+1] - dd[ntop]*dd[ntop+1]*tr);
84: bulge30 = bb[ntop]*bb[ntop+1]*dd[ntop];
85: bulge31 = 0.0;
86: bulge41 = 0.0;
87: bulge42 = 0.0;
89: /* Chase the bulge */
90: for (jj=ntop;jj<nn-1;jj++) {
92: /* Check for trivial bulge */
93: if (jj>ntop && PetscMax(PetscMax(PetscAbs(bulge10),PetscAbs(bulge20)),PetscAbs(bulge30))<PETSC_MACHINE_EPSILON*(PetscAbs(aa[jj]) + PetscAbs(aa[jj+1]))) {
94: bb[jj-1] = 0.0; /* deflate and move on */
96: } else { /* carry out the step */
98: /* Annihilate tip entry bulge30 */
99: if (bulge30 != 0.0) {
101: /* Make an interchange if necessary to ensure that the
102: first transformation is othogonal, not hyperbolic. */
103: if (dd[jj+1] != dd[jj+2]) { /* make an interchange */
104: if (dd[jj] != dd[jj+1]) { /* interchange 1st and 2nd */
105: buf[0] = bulge20; bulge20 = bulge10; bulge10 = buf[0];
106: buf[0] = aa[jj]; aa[jj] = aa[jj+1]; aa[jj+1] = buf[0];
107: buf[0] = bb[jj+1]; bb[jj+1] = bulge31; bulge31 = buf[0];
108: buf[0] = dd[jj]; dd[jj] = dd[jj+1]; dd[jj+1] = buf[0];
109: for (k=0;k<n;k++) {
110: rtmp = uu[k+jj*ld]; uu[k+jj*ld] = uu[k+(jj+1)*ld]; uu[k+(jj+1)*ld] = rtmp;
111: }
112: } else { /* interchange 1st and 3rd */
113: buf[0] = bulge30; bulge30 = bulge10; bulge10 = buf[0];
114: buf[0] = aa[jj]; aa[jj] = aa[jj+2]; aa[jj+2] = buf[0];
115: buf[0] = bb[jj]; bb[jj] = bb[jj+1]; bb[jj+1] = buf[0];
116: buf[0] = dd[jj]; dd[jj] = dd[jj+2]; dd[jj+2] = buf[0];
117: if (jj + 2 < nn-1) {
118: bulge41 = bb[jj+2];
119: bb[jj+2] = 0;
120: }
121: for (k=0;k<n;k++) {
122: rtmp = uu[k+jj*ld]; uu[k+jj*ld] = uu[k+(jj+2)*ld]; uu[k+(jj+2)*ld] = rtmp;
123: }
124: }
125: }
127: /* Set up transforming matrix rot. */
128: UnifiedRotation(bulge20,bulge30,1,rot,&rcond,&swap);
130: /* Apply transforming matrix rot to T. */
131: bulge20 = rot[0]*bulge20 + rot[2]*bulge30;
132: buf[0] = rot[0]*bb[jj] + rot[2]*bulge31;
133: buf[1] = rot[1]*bb[jj] + rot[3]*bulge31;
134: bb[jj] = buf[0];
135: bulge31 = buf[1];
136: 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];
137: 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];
138: 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];
139: aa[jj+1] = buf[0];
140: aa[jj+2] = buf[1];
141: if (jj + 2 < nn-1) {
142: bulge42 = bb[jj+2]*rot[2];
143: bb[jj+2] = bb[jj+2]*rot[3];
144: }
146: /* Accumulate transforming matrix */
147: PetscStackCallBLAS("BLASrot",BLASMIXEDrot_(&n_,uu+(jj+1)*ld,&one,uu+(jj+2)*ld,&one,&rot[0],&rot[2]));
148: }
150: /* Annihilate inner entry bulge20 */
151: if (bulge20 != 0.0) {
153: /* Begin by setting up transforming matrix rot */
154: sygn = dd[jj]*dd[jj+1];
155: UnifiedRotation(bulge10,bulge20,sygn,rot,&rcond,&swap);
156: if (rcond<PETSC_MACHINE_EPSILON) {
157: *flag = PETSC_TRUE;
158: return(0);
159: }
160: if (rcond < worstcond) worstcond = rcond;
162: /* Apply transforming matrix rot to T */
163: if (jj > ntop) bb[jj-1] = rot[0]*bulge10 + rot[2]*bulge20;
164: buf[0] = rot[0]*rot[0]*aa[jj] + 2*rot[0]*rot[2]*bb[jj] + rot[2]*rot[2]*aa[jj+1];
165: buf[1] = rot[1]*rot[1]*aa[jj] + 2*rot[3]*rot[1]*bb[jj] + rot[3]*rot[3]*aa[jj+1];
166: 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];
167: aa[jj] = buf[0];
168: aa[jj+1] = buf[1];
169: if (jj + 1 < nn-1) {
170: /* buf = [ bulge31 bb(jj+1) ] * rot' */
171: buf[0] = rot[0]*bulge31 + rot[2]*bb[jj+1];
172: buf[1] = rot[1]*bulge31 + rot[3]*bb[jj+1];
173: bulge31 = buf[0];
174: bb[jj+1] = buf[1];
175: }
176: if (jj + 2 < nn-1) {
177: /* buf = [bulge41 bulge42] * rot' */
178: buf[0] = rot[0]*bulge41 + rot[2]*bulge42;
179: buf[1] = rot[1]*bulge41 + rot[3]*bulge42;
180: bulge41 = buf[0];
181: bulge42 = buf[1];
182: }
184: /* Apply transforming matrix rot to D */
185: if (swap == 1) {
186: buf[0] = dd[jj]; dd[jj] = dd[jj+1]; dd[jj+1] = buf[0];
187: }
189: /* Accumulate transforming matrix, uu(jj:jj+1,:) = rot*uu(jj:jj+1,:) */
190: if (sygn==1) {
191: PetscStackCallBLAS("BLASrot",BLASMIXEDrot_(&n_,uu+jj*ld,&one,uu+(jj+1)*ld,&one,&rot[0],&rot[2]));
192: } else {
193: if (PetscAbsReal(rot[0])>PetscAbsReal(rot[1])) { /* Type I */
194: t = rot[1]/rot[0];
195: for (ii=0;ii<n;ii++) {
196: uu[jj*ld+ii] = rot[0]*uu[jj*ld+ii] + rot[1]*uu[(jj+1)*ld+ii];
197: uu[(jj+1)*ld+ii] = t*uu[jj*ld+ii] + uu[(jj+1)*ld+ii]/rot[0];
198: }
199: } else { /* Type II */
200: t = rot[0]/rot[1];
201: for (ii=0;ii<n;ii++) {
202: rtmp = uu[jj*ld+ii];
203: uu[jj*ld+ii] = rot[0]*uu[jj*ld+ii] + rot[1]*uu[(jj+1)*ld+ii];
204: uu[(jj+1)*ld+ii] = t*uu[jj*ld+ii] + rtmp/rot[1];
205: }
206: }
207: }
208: }
209: }
211: /* Adjust bulge for next step */
212: bulge10 = bb[jj];
213: bulge20 = bulge31;
214: bulge30 = bulge41;
215: bulge31 = bulge42;
216: bulge41 = 0.0;
217: bulge42 = 0.0;
218: }
219: return(0);
220: }
222: static PetscErrorCode HZIteration(PetscBLASInt nn,PetscBLASInt cgd,PetscReal *aa,PetscReal *bb,PetscReal *dd,PetscScalar *uu,PetscBLASInt ld)
223: {
225: PetscBLASInt j2,one=1;
226: PetscInt its,nits,nstop,jj,ntop,nbot,ntry;
227: PetscReal htr,det,dis,dif,tn,kt,c,s,tr,dt;
228: PetscBool flag=PETSC_FALSE;
231: its = 0;
232: nbot = nn-1;
233: nits = 0;
234: nstop = 40*(nn - cgd);
236: while (nbot >= cgd && nits < nstop) {
238: /* Check for zeros on the subdiagonal */
239: jj = nbot - 1;
240: while (jj>=cgd && PetscAbs(bb[jj])>PETSC_MACHINE_EPSILON*(PetscAbs(aa[jj])+PetscAbs(aa[jj+1]))) jj = jj-1;
241: if (jj>=cgd) bb[jj]=0;
242: ntop = jj + 1; /* starting point for step */
243: if (ntop == nbot) { /* isolate single eigenvalue */
244: nbot = ntop - 1;
245: its = 0;
246: } else if (ntop+1 == nbot) { /* isolate pair of eigenvalues */
247: htr = 0.5*(aa[ntop]*dd[ntop] + aa[nbot]*dd[nbot]);
248: det = dd[ntop]*dd[nbot]*(aa[ntop]*aa[nbot]-bb[ntop]*bb[ntop]);
249: dis = htr*htr - det;
250: if (dis > 0) { /* distinct real eigenvalues */
251: if (dd[ntop] == dd[nbot]) { /* separate the eigenvalues by a Jacobi rotator */
252: dif = aa[ntop]-aa[nbot];
253: if (2.0*PetscAbs(bb[ntop])<=dif) {
254: tn = 2*bb[ntop]/dif;
255: tn = tn/(1.0 + PetscSqrtScalar(1.0+tn*tn));
256: } else {
257: kt = dif/(2.0*bb[ntop]);
258: tn = PetscSign(kt)/(PetscAbs(kt)+PetscSqrtScalar(1.0+kt*kt));
259: }
260: c = 1.0/PetscSqrtScalar(1.0 + tn*tn);
261: s = c*tn;
262: aa[ntop] = aa[ntop] + tn*bb[ntop];
263: aa[nbot] = aa[nbot] - tn*bb[ntop];
264: bb[ntop] = 0;
265: j2 = nn-cgd;
266: PetscStackCallBLAS("BLASrot",BLASMIXEDrot_(&j2,uu+ntop*ld+cgd,&one,uu+nbot*ld+cgd,&one,&c,&s));
267: }
268: }
269: nbot = ntop - 1;
270: } else { /* Do an HZ iteration */
271: its = its + 1;
272: nits = nits + 1;
273: tr = aa[nbot-1]*dd[nbot-1] + aa[nbot]*dd[nbot];
274: dt = dd[nbot-1]*dd[nbot]*(aa[nbot-1]*aa[nbot]-bb[nbot-1]*bb[nbot-1]);
275: for (ntry=1;ntry<=6;ntry++) {
276: HZStep(ntop,nbot+1,tr,dt,aa,bb,dd,uu,nn,ld,&flag);
277: if (!flag) break;
278: else if (ntry == 6) SETERRQ(PETSC_COMM_SELF,1,"Unable to complete hz step on six tries");
279: else {
280: tr = 0.9*tr; dt = 0.81*dt;
281: }
282: }
283: }
284: }
285: return(0);
286: }
288: PetscErrorCode DSSolve_GHIEP_HZ(DS ds,PetscScalar *wr,PetscScalar *wi)
289: {
291: PetscInt off;
292: PetscBLASInt n1,ld;
293: PetscScalar *A,*B,*Q;
294: PetscReal *d,*e,*s;
295: #if defined(PETSC_USE_COMPLEX)
296: PetscInt i;
297: #endif
300: #if !defined(PETSC_USE_COMPLEX)
302: #endif
303: PetscBLASIntCast(ds->ld,&ld);
304: n1 = ds->n - ds->l;
305: off = ds->l + ds->l*ld;
306: A = ds->mat[DS_MAT_A];
307: B = ds->mat[DS_MAT_B];
308: Q = ds->mat[DS_MAT_Q];
309: d = ds->rmat[DS_MAT_T];
310: e = ds->rmat[DS_MAT_T] + ld;
311: s = ds->rmat[DS_MAT_D];
312: /* Quick return */
313: if (n1 == 1) {
314: *(Q+off) = 1;
315: if (ds->compact) {
316: wr[ds->l] = d[ds->l]/s[ds->l]; wi[ds->l] = 0.0;
317: } else {
318: d[ds->l] = PetscRealPart(A[off]); s[ds->l] = PetscRealPart(B[off]);
319: wr[ds->l] = d[ds->l]/s[ds->l]; wi[ds->l] = 0.0;
320: }
321: return(0);
322: }
323: /* Reduce to pseudotriadiagonal form */
324: DSIntermediate_GHIEP(ds);
325: HZIteration(ds->n,ds->l,d,e,s,Q,ld);
326: if (!ds->compact) {
327: DSSwitchFormat_GHIEP(ds,PETSC_FALSE);
328: }
329: /* Undo from diagonal the blocks whith real eigenvalues*/
330: DSGHIEPRealBlocks(ds);
332: /* Recover eigenvalues from diagonal */
333: DSGHIEPComplexEigs(ds,0,ds->n,wr,wi);
334: #if defined(PETSC_USE_COMPLEX)
335: if (wi) {
336: for (i=ds->l;i<ds->n;i++) wi[i] = 0.0;
337: }
338: #endif
339: ds->t = ds->n;
340: return(0);
341: }