Actual source code: fnutil.c
slepc-3.13.0 2020-03-31
1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-2020, 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: Utility subroutines common to several impls
12: */
14: #include <slepc/private/fnimpl.h> /*I "slepcfn.h" I*/
15: #include <slepcblaslapack.h>
17: /*
18: Compute the square root of an upper quasi-triangular matrix T,
19: using Higham's algorithm (LAA 88, 1987). T is overwritten with sqrtm(T).
20: */
21: PetscErrorCode SlepcMatDenseSqrt(PetscBLASInt n,PetscScalar *T,PetscBLASInt ld)
22: {
23: PetscScalar one=1.0,mone=-1.0;
24: PetscReal scal;
25: PetscBLASInt i,j,si,sj,r,ione=1,info;
26: #if !defined(PETSC_USE_COMPLEX)
27: PetscReal alpha,theta,mu,mu2;
28: #endif
31: for (j=0;j<n;j++) {
32: #if defined(PETSC_USE_COMPLEX)
33: sj = 1;
34: T[j+j*ld] = PetscSqrtScalar(T[j+j*ld]);
35: #else
36: sj = (j==n-1 || T[j+1+j*ld] == 0.0)? 1: 2;
37: if (sj==1) {
38: if (T[j+j*ld]<0.0) SETERRQ(PETSC_COMM_SELF,1,"Matrix has a real negative eigenvalue, no real primary square root exists");
39: T[j+j*ld] = PetscSqrtReal(T[j+j*ld]);
40: } else {
41: /* square root of 2x2 block */
42: theta = (T[j+j*ld]+T[j+1+(j+1)*ld])/2.0;
43: mu = (T[j+j*ld]-T[j+1+(j+1)*ld])/2.0;
44: mu2 = -mu*mu-T[j+1+j*ld]*T[j+(j+1)*ld];
45: mu = PetscSqrtReal(mu2);
46: if (theta>0.0) alpha = PetscSqrtReal((theta+PetscSqrtReal(theta*theta+mu2))/2.0);
47: else alpha = mu/PetscSqrtReal(2.0*(-theta+PetscSqrtReal(theta*theta+mu2)));
48: T[j+j*ld] /= 2.0*alpha;
49: T[j+1+(j+1)*ld] /= 2.0*alpha;
50: T[j+(j+1)*ld] /= 2.0*alpha;
51: T[j+1+j*ld] /= 2.0*alpha;
52: T[j+j*ld] += alpha-theta/(2.0*alpha);
53: T[j+1+(j+1)*ld] += alpha-theta/(2.0*alpha);
54: }
55: #endif
56: for (i=j-1;i>=0;i--) {
57: #if defined(PETSC_USE_COMPLEX)
58: si = 1;
59: #else
60: si = (i==0 || T[i+(i-1)*ld] == 0.0)? 1: 2;
61: if (si==2) i--;
62: #endif
63: /* solve Sylvester equation of order si x sj */
64: r = j-i-si;
65: if (r) PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&si,&sj,&r,&mone,T+i+(i+si)*ld,&ld,T+i+si+j*ld,&ld,&one,T+i+j*ld,&ld));
66: PetscStackCallBLAS("LAPACKtrsyl",LAPACKtrsyl_("N","N",&ione,&si,&sj,T+i+i*ld,&ld,T+j+j*ld,&ld,T+i+j*ld,&ld,&scal,&info));
67: SlepcCheckLapackInfo("trsyl",info);
68: if (scal!=1.0) SETERRQ1(PETSC_COMM_SELF,1,"Current implementation cannot handle scale factor %g",scal);
69: }
70: if (sj==2) j++;
71: }
72: return(0);
73: }
75: #define BLOCKSIZE 64
77: /*
78: Schur method for the square root of an upper quasi-triangular matrix T.
79: T is overwritten with sqrtm(T).
80: If firstonly then only the first column of T will contain relevant values.
81: */
82: PetscErrorCode SlepcSqrtmSchur(PetscBLASInt n,PetscScalar *T,PetscBLASInt ld,PetscBool firstonly)
83: {
85: PetscBLASInt i,j,k,r,ione=1,sdim,lwork,*s,*p,info,bs=BLOCKSIZE;
86: PetscScalar *wr,*W,*Q,*work,one=1.0,zero=0.0,mone=-1.0;
87: PetscInt m,nblk;
88: PetscReal scal;
89: #if defined(PETSC_USE_COMPLEX)
90: PetscReal *rwork;
91: #else
92: PetscReal *wi;
93: #endif
96: m = n;
97: nblk = (m+bs-1)/bs;
98: lwork = 5*n;
99: k = firstonly? 1: n;
101: /* compute Schur decomposition A*Q = Q*T */
102: #if !defined(PETSC_USE_COMPLEX)
103: PetscMalloc7(m,&wr,m,&wi,m*k,&W,m*m,&Q,lwork,&work,nblk,&s,nblk,&p);
104: PetscStackCallBLAS("LAPACKgees",LAPACKgees_("V","N",NULL,&n,T,&ld,&sdim,wr,wi,Q,&ld,work,&lwork,NULL,&info));
105: #else
106: PetscMalloc7(m,&wr,m,&rwork,m*k,&W,m*m,&Q,lwork,&work,nblk,&s,nblk,&p);
107: PetscStackCallBLAS("LAPACKgees",LAPACKgees_("V","N",NULL,&n,T,&ld,&sdim,wr,Q,&ld,work,&lwork,rwork,NULL,&info));
108: #endif
109: SlepcCheckLapackInfo("gees",info);
111: /* determine block sizes and positions, to avoid cutting 2x2 blocks */
112: j = 0;
113: p[j] = 0;
114: do {
115: s[j] = PetscMin(bs,n-p[j]);
116: #if !defined(PETSC_USE_COMPLEX)
117: if (p[j]+s[j]!=n && T[p[j]+s[j]+(p[j]+s[j]-1)*ld]!=0.0) s[j]++;
118: #endif
119: if (p[j]+s[j]==n) break;
120: j++;
121: p[j] = p[j-1]+s[j-1];
122: } while (1);
123: nblk = j+1;
125: for (j=0;j<nblk;j++) {
126: /* evaluate f(T_jj) */
127: SlepcMatDenseSqrt(s[j],T+p[j]+p[j]*ld,ld);
128: for (i=j-1;i>=0;i--) {
129: /* solve Sylvester equation for block (i,j) */
130: r = p[j]-p[i]-s[i];
131: if (r) PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",s+i,s+j,&r,&mone,T+p[i]+(p[i]+s[i])*ld,&ld,T+p[i]+s[i]+p[j]*ld,&ld,&one,T+p[i]+p[j]*ld,&ld));
132: PetscStackCallBLAS("LAPACKtrsyl",LAPACKtrsyl_("N","N",&ione,s+i,s+j,T+p[i]+p[i]*ld,&ld,T+p[j]+p[j]*ld,&ld,T+p[i]+p[j]*ld,&ld,&scal,&info));
133: SlepcCheckLapackInfo("trsyl",info);
134: if (scal!=1.0) SETERRQ1(PETSC_COMM_SELF,1,"Current implementation cannot handle scale factor %g",scal);
135: }
136: }
138: /* backtransform B = Q*T*Q' */
139: PetscStackCallBLAS("BLASgemm",BLASgemm_("N","C",&n,&k,&n,&one,T,&ld,Q,&ld,&zero,W,&ld));
140: PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&k,&n,&one,Q,&ld,W,&ld,&zero,T,&ld));
142: /* flop count: Schur decomposition, triangular square root, and backtransform */
143: PetscLogFlops(25.0*n*n*n+n*n*n/3.0+4.0*n*n*k);
145: #if !defined(PETSC_USE_COMPLEX)
146: PetscFree7(wr,wi,W,Q,work,s,p);
147: #else
148: PetscFree7(wr,rwork,W,Q,work,s,p);
149: #endif
150: return(0);
151: }
153: #define DBMAXIT 25
155: /*
156: Computes the principal square root of the matrix T using the product form
157: of the Denman-Beavers iteration.
158: T is overwritten with sqrtm(T) or inv(sqrtm(T)) depending on flag inv.
159: */
160: PetscErrorCode SlepcSqrtmDenmanBeavers(PetscBLASInt n,PetscScalar *T,PetscBLASInt ld,PetscBool inv)
161: {
162: PetscScalar *Told,*M=NULL,*invM,*work,work1,prod,alpha;
163: PetscScalar szero=0.0,sone=1.0,smone=-1.0,spfive=0.5,sp25=0.25;
164: PetscReal tol,Mres=0.0,detM,g,reldiff,fnormdiff,fnormT,rwork[1];
165: PetscBLASInt N,i,it,*piv=NULL,info,query=-1,lwork;
166: const PetscBLASInt one=1;
167: PetscBool converged=PETSC_FALSE,scale=PETSC_FALSE;
168: PetscErrorCode ierr;
169: unsigned int ftz;
172: N = n*n;
173: tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
174: SlepcSetFlushToZero(&ftz);
176: /* query work size */
177: PetscStackCallBLAS("LAPACKgetri",LAPACKgetri_(&n,M,&ld,piv,&work1,&query,&info));
178: PetscBLASIntCast((PetscInt)PetscRealPart(work1),&lwork);
179: PetscMalloc5(lwork,&work,n,&piv,n*n,&Told,n*n,&M,n*n,&invM);
180: PetscArraycpy(M,T,n*n);
182: if (inv) { /* start recurrence with I instead of A */
183: PetscArrayzero(T,n*n);
184: for (i=0;i<n;i++) T[i+i*ld] += 1.0;
185: }
187: for (it=0;it<DBMAXIT && !converged;it++) {
189: if (scale) { /* g = (abs(det(M)))^(-1/(2*n)) */
190: PetscArraycpy(invM,M,n*n);
191: PetscStackCallBLAS("LAPACKgetrf",LAPACKgetrf_(&n,&n,invM,&ld,piv,&info));
192: SlepcCheckLapackInfo("getrf",info);
193: prod = invM[0];
194: for (i=1;i<n;i++) prod *= invM[i+i*ld];
195: detM = PetscAbsScalar(prod);
196: g = PetscPowReal(detM,-1.0/(2.0*n));
197: alpha = g;
198: PetscStackCallBLAS("BLASscal",BLASscal_(&N,&alpha,T,&one));
199: alpha = g*g;
200: PetscStackCallBLAS("BLASscal",BLASscal_(&N,&alpha,M,&one));
201: PetscLogFlops(2.0*n*n*n/3.0+2.0*n*n);
202: }
204: PetscArraycpy(Told,T,n*n);
205: PetscArraycpy(invM,M,n*n);
207: PetscStackCallBLAS("LAPACKgetrf",LAPACKgetrf_(&n,&n,invM,&ld,piv,&info));
208: SlepcCheckLapackInfo("getrf",info);
209: PetscStackCallBLAS("LAPACKgetri",LAPACKgetri_(&n,invM,&ld,piv,work,&lwork,&info));
210: SlepcCheckLapackInfo("getri",info);
211: PetscLogFlops(2.0*n*n*n/3.0+4.0*n*n*n/3.0);
213: for (i=0;i<n;i++) invM[i+i*ld] += 1.0;
214: PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&spfive,Told,&ld,invM,&ld,&szero,T,&ld));
215: for (i=0;i<n;i++) invM[i+i*ld] -= 1.0;
217: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&N,&sone,invM,&one,M,&one));
218: PetscStackCallBLAS("BLASscal",BLASscal_(&N,&sp25,M,&one));
219: for (i=0;i<n;i++) M[i+i*ld] -= 0.5;
220: PetscLogFlops(2.0*n*n*n+2.0*n*n);
222: Mres = LAPACKlange_("F",&n,&n,M,&n,rwork);
223: for (i=0;i<n;i++) M[i+i*ld] += 1.0;
225: if (scale) {
226: /* reldiff = norm(T - Told,'fro')/norm(T,'fro') */
227: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&N,&smone,T,&one,Told,&one));
228: fnormdiff = LAPACKlange_("F",&n,&n,Told,&n,rwork);
229: fnormT = LAPACKlange_("F",&n,&n,T,&n,rwork);
230: PetscLogFlops(7.0*n*n);
231: reldiff = fnormdiff/fnormT;
232: PetscInfo4(NULL,"it: %D reldiff: %g scale: %g tol*scale: %g\n",it,(double)reldiff,(double)g,(double)tol*g);
233: if (reldiff<1e-2) scale = PETSC_FALSE; /* Switch off scaling */
234: }
236: if (Mres<=tol) converged = PETSC_TRUE;
237: }
239: if (Mres>tol) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"SQRTM not converged after %d iterations",DBMAXIT);
240: PetscFree5(work,piv,Told,M,invM);
241: SlepcResetFlushToZero(&ftz);
242: return(0);
243: }
245: #define NSMAXIT 50
247: /*
248: Computes the principal square root of the matrix A using the Newton-Schulz iteration.
249: T is overwritten with sqrtm(T) or inv(sqrtm(T)) depending on flag inv.
250: */
251: PetscErrorCode SlepcSqrtmNewtonSchulz(PetscBLASInt n,PetscScalar *A,PetscBLASInt ld,PetscBool inv)
252: {
253: PetscScalar *Y=A,*Yold,*Z,*Zold,*M,alpha,sqrtnrm;
254: PetscScalar szero=0.0,sone=1.0,smone=-1.0,spfive=0.5,sthree=3.0;
255: PetscReal tol,Yres=0.0,nrm,rwork[1];
256: PetscBLASInt i,it,N;
257: const PetscBLASInt one=1;
258: PetscBool converged=PETSC_FALSE;
259: PetscErrorCode ierr;
260: unsigned int ftz;
263: N = n*n;
264: tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
265: SlepcSetFlushToZero(&ftz);
267: PetscMalloc4(N,&Yold,N,&Z,N,&Zold,N,&M);
269: /* scale A so that ||I-A|| < 1 */
270: PetscArraycpy(Z,A,N);
271: for (i=0;i<n;i++) Z[i+i*ld] -= 1.0;
272: nrm = LAPACKlange_("fro",&n,&n,Z,&n,rwork);
273: sqrtnrm = PetscSqrtReal(nrm);
274: alpha = 1.0/nrm;
275: PetscStackCallBLAS("BLASscal",BLASscal_(&N,&alpha,A,&one));
276: tol *= nrm;
277: PetscInfo2(NULL,"||I-A||_F = %g, new tol: %g\n",(double)nrm,(double)tol);
278: PetscLogFlops(2.0*n*n);
280: /* Z = I */
281: PetscArrayzero(Z,N);
282: for (i=0;i<n;i++) Z[i+i*ld] = 1.0;
284: for (it=0;it<NSMAXIT && !converged;it++) {
285: /* Yold = Y, Zold = Z */
286: PetscArraycpy(Yold,Y,N);
287: PetscArraycpy(Zold,Z,N);
289: /* M = (3*I-Zold*Yold) */
290: PetscArrayzero(M,N);
291: for (i=0;i<n;i++) M[i+i*ld] = sthree;
292: PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&smone,Zold,&ld,Yold,&ld,&sone,M,&ld));
294: /* Y = (1/2)*Yold*M, Z = (1/2)*M*Zold */
295: PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&spfive,Yold,&ld,M,&ld,&szero,Y,&ld));
296: PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&spfive,M,&ld,Zold,&ld,&szero,Z,&ld));
298: /* reldiff = norm(Y-Yold,'fro')/norm(Y,'fro') */
299: PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&N,&smone,Y,&one,Yold,&one));
300: Yres = LAPACKlange_("fro",&n,&n,Yold,&n,rwork);
301: PetscIsNanReal(Yres);
302: if (Yres<=tol) converged = PETSC_TRUE;
303: PetscInfo2(NULL,"it: %D res: %g\n",it,(double)Yres);
305: PetscLogFlops(6.0*n*n*n+2.0*n*n);
306: }
308: if (Yres>tol) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_LIB,"SQRTM not converged after %d iterations",NSMAXIT);
310: /* undo scaling */
311: if (inv) {
312: PetscArraycpy(A,Z,N);
313: sqrtnrm = 1.0/sqrtnrm;
314: PetscStackCallBLAS("BLASscal",BLASscal_(&N,&sqrtnrm,A,&one));
315: } else PetscStackCallBLAS("BLASscal",BLASscal_(&N,&sqrtnrm,A,&one));
317: PetscFree4(Yold,Z,Zold,M);
318: SlepcResetFlushToZero(&ftz);
319: return(0);
320: }
322: #define ITMAX 5
323: #define SWAP(a,b,t) {t=a;a=b;b=t;}
325: /*
326: Estimate norm(A^m,1) by block 1-norm power method (required workspace is 11*n)
327: */
328: static PetscErrorCode SlepcNormEst1(PetscBLASInt n,PetscScalar *A,PetscInt m,PetscScalar *work,PetscRandom rand,PetscReal *nrm)
329: {
330: PetscScalar *X,*Y,*Z,*S,*S_old,*aux,val,sone=1.0,szero=0.0;
331: PetscReal est=0.0,est_old,vals[2]={0.0,0.0},*zvals,maxzval[2],raux;
332: PetscBLASInt i,j,t=2,it=0,ind[2],est_j=0,m1;
336: X = work;
337: Y = work + 2*n;
338: Z = work + 4*n;
339: S = work + 6*n;
340: S_old = work + 8*n;
341: zvals = (PetscReal*)(work + 10*n);
343: for (i=0;i<n;i++) { /* X has columns of unit 1-norm */
344: X[i] = 1.0/n;
345: PetscRandomGetValue(rand,&val);
346: if (PetscRealPart(val) < 0.5) X[i+n] = -1.0/n;
347: else X[i+n] = 1.0/n;
348: }
349: for (i=0;i<t*n;i++) S[i] = 0.0;
350: ind[0] = 0; ind[1] = 0;
351: est_old = 0;
352: while (1) {
353: it++;
354: for (j=0;j<m;j++) { /* Y = A^m*X */
355: PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&t,&n,&sone,A,&n,X,&n,&szero,Y,&n));
356: if (j<m-1) SWAP(X,Y,aux);
357: }
358: for (j=0;j<t;j++) { /* vals[j] = norm(Y(:,j),1) */
359: vals[j] = 0.0;
360: for (i=0;i<n;i++) vals[j] += PetscAbsScalar(Y[i+j*n]);
361: }
362: if (vals[0]<vals[1]) {
363: SWAP(vals[0],vals[1],raux);
364: m1 = 1;
365: } else m1 = 0;
366: est = vals[0];
367: if (est>est_old || it==2) est_j = ind[m1];
368: if (it>=2 && est<=est_old) {
369: est = est_old;
370: break;
371: }
372: est_old = est;
373: if (it>ITMAX) break;
374: SWAP(S,S_old,aux);
375: for (i=0;i<t*n;i++) { /* S = sign(Y) */
376: S[i] = (PetscRealPart(Y[i]) < 0.0)? -1.0: 1.0;
377: }
378: for (j=0;j<m;j++) { /* Z = (A^T)^m*S */
379: PetscStackCallBLAS("BLASgemm",BLASgemm_("C","N",&n,&t,&n,&sone,A,&n,S,&n,&szero,Z,&n));
380: if (j<m-1) SWAP(S,Z,aux);
381: }
382: maxzval[0] = -1; maxzval[1] = -1;
383: ind[0] = 0; ind[1] = 0;
384: for (i=0;i<n;i++) { /* zvals[i] = norm(Z(i,:),inf) */
385: zvals[i] = PetscMax(PetscAbsScalar(Z[i+0*n]),PetscAbsScalar(Z[i+1*n]));
386: if (zvals[i]>maxzval[0]) {
387: maxzval[0] = zvals[i];
388: ind[0] = i;
389: } else if (zvals[i]>maxzval[1]) {
390: maxzval[1] = zvals[i];
391: ind[1] = i;
392: }
393: }
394: if (it>=2 && maxzval[0]==zvals[est_j]) break;
395: for (i=0;i<t*n;i++) X[i] = 0.0;
396: for (j=0;j<t;j++) X[ind[j]+j*n] = 1.0;
397: }
398: *nrm = est;
399: /* Flop count is roughly (it * 2*m * t*gemv) = 4*its*m*t*n*n */
400: PetscLogFlops(4.0*it*m*t*n*n);
401: return(0);
402: }
404: #define SMALLN 100
406: /*
407: Estimate norm(A^m,1) (required workspace is 2*n*n)
408: */
409: PetscErrorCode SlepcNormAm(PetscBLASInt n,PetscScalar *A,PetscInt m,PetscScalar *work,PetscRandom rand,PetscReal *nrm)
410: {
411: PetscScalar *v=work,*w=work+n*n,*aux,sone=1.0,szero=0.0;
412: PetscReal rwork[1],tmp;
413: PetscBLASInt i,j,one=1;
414: PetscBool isrealpos=PETSC_TRUE;
418: if (n<SMALLN) { /* compute matrix power explicitly */
419: if (m==1) {
420: *nrm = LAPACKlange_("O",&n,&n,A,&n,rwork);
421: PetscLogFlops(1.0*n*n);
422: } else { /* m>=2 */
423: PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,A,&n,A,&n,&szero,v,&n));
424: for (j=0;j<m-2;j++) {
425: PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,A,&n,v,&n,&szero,w,&n));
426: SWAP(v,w,aux);
427: }
428: *nrm = LAPACKlange_("O",&n,&n,v,&n,rwork);
429: PetscLogFlops(2.0*n*n*n*(m-1)+1.0*n*n);
430: }
431: } else {
432: for (i=0;i<n;i++)
433: for (j=0;j<n;j++)
434: #if defined(PETSC_USE_COMPLEX)
435: if (PetscRealPart(A[i+j*n])<0.0 || PetscImaginaryPart(A[i+j*n])!=0.0) { isrealpos = PETSC_FALSE; break; }
436: #else
437: if (A[i+j*n]<0.0) { isrealpos = PETSC_FALSE; break; }
438: #endif
439: if (isrealpos) { /* for positive matrices only */
440: for (i=0;i<n;i++) v[i] = 1.0;
441: for (j=0;j<m;j++) { /* w = A'*v */
442: PetscStackCallBLAS("BLASgemv",BLASgemv_("C",&n,&n,&sone,A,&n,v,&one,&szero,w,&one));
443: SWAP(v,w,aux);
444: }
445: PetscLogFlops(2.0*n*n*m);
446: *nrm = 0.0;
447: for (i=0;i<n;i++) if ((tmp = PetscAbsScalar(v[i])) > *nrm) *nrm = tmp; /* norm(v,inf) */
448: } else {
449: SlepcNormEst1(n,A,m,work,rand,nrm);
450: }
451: }
452: return(0);
453: }