Actual source code: fnutil.c
slepc-3.18.2 2023-01-26
1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-, 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>
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: static 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
30: for (j=0;j<n;j++) {
31: #if defined(PETSC_USE_COMPLEX)
32: sj = 1;
33: T[j+j*ld] = PetscSqrtScalar(T[j+j*ld]);
34: #else
35: sj = (j==n-1 || T[j+1+j*ld] == 0.0)? 1: 2;
36: if (sj==1) {
38: T[j+j*ld] = PetscSqrtReal(T[j+j*ld]);
39: } else {
40: /* square root of 2x2 block */
41: theta = (T[j+j*ld]+T[j+1+(j+1)*ld])/2.0;
42: mu = (T[j+j*ld]-T[j+1+(j+1)*ld])/2.0;
43: mu2 = -mu*mu-T[j+1+j*ld]*T[j+(j+1)*ld];
44: mu = PetscSqrtReal(mu2);
45: if (theta>0.0) alpha = PetscSqrtReal((theta+PetscSqrtReal(theta*theta+mu2))/2.0);
46: else alpha = mu/PetscSqrtReal(2.0*(-theta+PetscSqrtReal(theta*theta+mu2)));
47: T[j+j*ld] /= 2.0*alpha;
48: T[j+1+(j+1)*ld] /= 2.0*alpha;
49: T[j+(j+1)*ld] /= 2.0*alpha;
50: T[j+1+j*ld] /= 2.0*alpha;
51: T[j+j*ld] += alpha-theta/(2.0*alpha);
52: T[j+1+(j+1)*ld] += alpha-theta/(2.0*alpha);
53: }
54: #endif
55: for (i=j-1;i>=0;i--) {
56: #if defined(PETSC_USE_COMPLEX)
57: si = 1;
58: #else
59: si = (i==0 || T[i+(i-1)*ld] == 0.0)? 1: 2;
60: if (si==2) i--;
61: #endif
62: /* solve Sylvester equation of order si x sj */
63: r = j-i-si;
64: if (r) PetscCallBLAS("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));
65: PetscCallBLAS("LAPACKtrsyl",LAPACKtrsyl_("N","N",&ione,&si,&sj,T+i+i*ld,&ld,T+j+j*ld,&ld,T+i+j*ld,&ld,&scal,&info));
66: SlepcCheckLapackInfo("trsyl",info);
68: }
69: if (sj==2) j++;
70: }
71: return 0;
72: }
74: #define BLOCKSIZE 64
76: /*
77: Schur method for the square root of an upper quasi-triangular matrix T.
78: T is overwritten with sqrtm(T).
79: If firstonly then only the first column of T will contain relevant values.
80: */
81: PetscErrorCode FNSqrtmSchur(FN fn,PetscBLASInt n,PetscScalar *T,PetscBLASInt ld,PetscBool firstonly)
82: {
83: PetscBLASInt i,j,k,r,ione=1,sdim,lwork,*s,*p,info,bs=BLOCKSIZE;
84: PetscScalar *wr,*W,*Q,*work,one=1.0,zero=0.0,mone=-1.0;
85: PetscInt m,nblk;
86: PetscReal scal;
87: #if defined(PETSC_USE_COMPLEX)
88: PetscReal *rwork;
89: #else
90: PetscReal *wi;
91: #endif
93: m = n;
94: nblk = (m+bs-1)/bs;
95: lwork = 5*n;
96: k = firstonly? 1: n;
98: /* compute Schur decomposition A*Q = Q*T */
99: #if !defined(PETSC_USE_COMPLEX)
100: PetscMalloc7(m,&wr,m,&wi,m*k,&W,m*m,&Q,lwork,&work,nblk,&s,nblk,&p);
101: PetscCallBLAS("LAPACKgees",LAPACKgees_("V","N",NULL,&n,T,&ld,&sdim,wr,wi,Q,&ld,work,&lwork,NULL,&info));
102: #else
103: PetscMalloc7(m,&wr,m,&rwork,m*k,&W,m*m,&Q,lwork,&work,nblk,&s,nblk,&p);
104: PetscCallBLAS("LAPACKgees",LAPACKgees_("V","N",NULL,&n,T,&ld,&sdim,wr,Q,&ld,work,&lwork,rwork,NULL,&info));
105: #endif
106: SlepcCheckLapackInfo("gees",info);
108: /* determine block sizes and positions, to avoid cutting 2x2 blocks */
109: j = 0;
110: p[j] = 0;
111: do {
112: s[j] = PetscMin(bs,n-p[j]);
113: #if !defined(PETSC_USE_COMPLEX)
114: if (p[j]+s[j]!=n && T[p[j]+s[j]+(p[j]+s[j]-1)*ld]!=0.0) s[j]++;
115: #endif
116: if (p[j]+s[j]==n) break;
117: j++;
118: p[j] = p[j-1]+s[j-1];
119: } while (1);
120: nblk = j+1;
122: for (j=0;j<nblk;j++) {
123: /* evaluate f(T_jj) */
124: SlepcMatDenseSqrt(s[j],T+p[j]+p[j]*ld,ld);
125: for (i=j-1;i>=0;i--) {
126: /* solve Sylvester equation for block (i,j) */
127: r = p[j]-p[i]-s[i];
128: if (r) PetscCallBLAS("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));
129: PetscCallBLAS("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));
130: SlepcCheckLapackInfo("trsyl",info);
132: }
133: }
135: /* backtransform B = Q*T*Q' */
136: PetscCallBLAS("BLASgemm",BLASgemm_("N","C",&n,&k,&n,&one,T,&ld,Q,&ld,&zero,W,&ld));
137: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&k,&n,&one,Q,&ld,W,&ld,&zero,T,&ld));
139: /* flop count: Schur decomposition, triangular square root, and backtransform */
140: PetscLogFlops(25.0*n*n*n+n*n*n/3.0+4.0*n*n*k);
142: #if !defined(PETSC_USE_COMPLEX)
143: PetscFree7(wr,wi,W,Q,work,s,p);
144: #else
145: PetscFree7(wr,rwork,W,Q,work,s,p);
146: #endif
147: return 0;
148: }
150: #define DBMAXIT 25
152: /*
153: Computes the principal square root of the matrix T using the product form
154: of the Denman-Beavers iteration.
155: T is overwritten with sqrtm(T) or inv(sqrtm(T)) depending on flag inv.
156: */
157: PetscErrorCode FNSqrtmDenmanBeavers(FN fn,PetscBLASInt n,PetscScalar *T,PetscBLASInt ld,PetscBool inv)
158: {
159: PetscScalar *Told,*M=NULL,*invM,*work,work1,prod,alpha;
160: PetscScalar szero=0.0,sone=1.0,smone=-1.0,spfive=0.5,sp25=0.25;
161: PetscReal tol,Mres=0.0,detM,g,reldiff,fnormdiff,fnormT,rwork[1];
162: PetscBLASInt N,i,it,*piv=NULL,info,query=-1,lwork;
163: const PetscBLASInt one=1;
164: PetscBool converged=PETSC_FALSE,scale;
165: unsigned int ftz;
167: N = n*n;
168: tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
169: scale = PetscDefined(USE_REAL_SINGLE)? PETSC_FALSE: PETSC_TRUE;
170: SlepcSetFlushToZero(&ftz);
172: /* query work size */
173: PetscCallBLAS("LAPACKgetri",LAPACKgetri_(&n,M,&ld,piv,&work1,&query,&info));
174: PetscBLASIntCast((PetscInt)PetscRealPart(work1),&lwork);
175: PetscMalloc5(lwork,&work,n,&piv,n*n,&Told,n*n,&M,n*n,&invM);
176: PetscArraycpy(M,T,n*n);
178: if (inv) { /* start recurrence with I instead of A */
179: PetscArrayzero(T,n*n);
180: for (i=0;i<n;i++) T[i+i*ld] += 1.0;
181: }
183: for (it=0;it<DBMAXIT && !converged;it++) {
185: if (scale) { /* g = (abs(det(M)))^(-1/(2*n)) */
186: PetscArraycpy(invM,M,n*n);
187: PetscCallBLAS("LAPACKgetrf",LAPACKgetrf_(&n,&n,invM,&ld,piv,&info));
188: SlepcCheckLapackInfo("getrf",info);
189: prod = invM[0];
190: for (i=1;i<n;i++) prod *= invM[i+i*ld];
191: detM = PetscAbsScalar(prod);
192: g = (detM>PETSC_MAX_REAL)? 0.5: PetscPowReal(detM,-1.0/(2.0*n));
193: alpha = g;
194: PetscCallBLAS("BLASscal",BLASscal_(&N,&alpha,T,&one));
195: alpha = g*g;
196: PetscCallBLAS("BLASscal",BLASscal_(&N,&alpha,M,&one));
197: PetscLogFlops(2.0*n*n*n/3.0+2.0*n*n);
198: }
200: PetscArraycpy(Told,T,n*n);
201: PetscArraycpy(invM,M,n*n);
203: PetscCallBLAS("LAPACKgetrf",LAPACKgetrf_(&n,&n,invM,&ld,piv,&info));
204: SlepcCheckLapackInfo("getrf",info);
205: PetscCallBLAS("LAPACKgetri",LAPACKgetri_(&n,invM,&ld,piv,work,&lwork,&info));
206: SlepcCheckLapackInfo("getri",info);
207: PetscLogFlops(2.0*n*n*n/3.0+4.0*n*n*n/3.0);
209: for (i=0;i<n;i++) invM[i+i*ld] += 1.0;
210: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&spfive,Told,&ld,invM,&ld,&szero,T,&ld));
211: for (i=0;i<n;i++) invM[i+i*ld] -= 1.0;
213: PetscCallBLAS("BLASaxpy",BLASaxpy_(&N,&sone,invM,&one,M,&one));
214: PetscCallBLAS("BLASscal",BLASscal_(&N,&sp25,M,&one));
215: for (i=0;i<n;i++) M[i+i*ld] -= 0.5;
216: PetscLogFlops(2.0*n*n*n+2.0*n*n);
218: Mres = LAPACKlange_("F",&n,&n,M,&n,rwork);
219: for (i=0;i<n;i++) M[i+i*ld] += 1.0;
221: if (scale) {
222: /* reldiff = norm(T - Told,'fro')/norm(T,'fro') */
223: PetscCallBLAS("BLASaxpy",BLASaxpy_(&N,&smone,T,&one,Told,&one));
224: fnormdiff = LAPACKlange_("F",&n,&n,Told,&n,rwork);
225: fnormT = LAPACKlange_("F",&n,&n,T,&n,rwork);
226: PetscLogFlops(7.0*n*n);
227: reldiff = fnormdiff/fnormT;
228: PetscInfo(fn,"it: %" PetscBLASInt_FMT " reldiff: %g scale: %g tol*scale: %g\n",it,(double)reldiff,(double)g,(double)(tol*g));
229: if (reldiff<1e-2) scale = PETSC_FALSE; /* Switch off scaling */
230: }
232: if (Mres<=tol) converged = PETSC_TRUE;
233: }
236: PetscFree5(work,piv,Told,M,invM);
237: SlepcResetFlushToZero(&ftz);
238: return 0;
239: }
241: #define NSMAXIT 50
243: /*
244: Computes the principal square root of the matrix A using the Newton-Schulz iteration.
245: T is overwritten with sqrtm(T) or inv(sqrtm(T)) depending on flag inv.
246: */
247: PetscErrorCode FNSqrtmNewtonSchulz(FN fn,PetscBLASInt n,PetscScalar *A,PetscBLASInt ld,PetscBool inv)
248: {
249: PetscScalar *Y=A,*Yold,*Z,*Zold,*M;
250: PetscScalar szero=0.0,sone=1.0,smone=-1.0,spfive=0.5,sthree=3.0;
251: PetscReal sqrtnrm,tol,Yres=0.0,nrm,rwork[1],done=1.0;
252: PetscBLASInt info,i,it,N,one=1,zero=0;
253: PetscBool converged=PETSC_FALSE;
254: unsigned int ftz;
256: N = n*n;
257: tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
258: SlepcSetFlushToZero(&ftz);
260: PetscMalloc4(N,&Yold,N,&Z,N,&Zold,N,&M);
262: /* scale */
263: PetscArraycpy(Z,A,N);
264: for (i=0;i<n;i++) Z[i+i*ld] -= 1.0;
265: nrm = LAPACKlange_("fro",&n,&n,Z,&n,rwork);
266: sqrtnrm = PetscSqrtReal(nrm);
267: PetscCallBLAS("LAPACKlascl",LAPACKlascl_("G",&zero,&zero,&nrm,&done,&N,&one,A,&N,&info));
268: SlepcCheckLapackInfo("lascl",info);
269: tol *= nrm;
270: PetscInfo(fn,"||I-A||_F = %g, new tol: %g\n",(double)nrm,(double)tol);
271: PetscLogFlops(2.0*n*n);
273: /* Z = I */
274: PetscArrayzero(Z,N);
275: for (i=0;i<n;i++) Z[i+i*ld] = 1.0;
277: for (it=0;it<NSMAXIT && !converged;it++) {
278: /* Yold = Y, Zold = Z */
279: PetscArraycpy(Yold,Y,N);
280: PetscArraycpy(Zold,Z,N);
282: /* M = (3*I-Zold*Yold) */
283: PetscArrayzero(M,N);
284: for (i=0;i<n;i++) M[i+i*ld] = sthree;
285: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&smone,Zold,&ld,Yold,&ld,&sone,M,&ld));
287: /* Y = (1/2)*Yold*M, Z = (1/2)*M*Zold */
288: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&spfive,Yold,&ld,M,&ld,&szero,Y,&ld));
289: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&spfive,M,&ld,Zold,&ld,&szero,Z,&ld));
291: /* reldiff = norm(Y-Yold,'fro')/norm(Y,'fro') */
292: PetscCallBLAS("BLASaxpy",BLASaxpy_(&N,&smone,Y,&one,Yold,&one));
293: Yres = LAPACKlange_("fro",&n,&n,Yold,&n,rwork);
295: if (Yres<=tol) converged = PETSC_TRUE;
296: PetscInfo(fn,"it: %" PetscBLASInt_FMT " res: %g\n",it,(double)Yres);
298: PetscLogFlops(6.0*n*n*n+2.0*n*n);
299: }
303: /* undo scaling */
304: if (inv) {
305: PetscArraycpy(A,Z,N);
306: PetscCallBLAS("LAPACKlascl",LAPACKlascl_("G",&zero,&zero,&sqrtnrm,&done,&N,&one,A,&N,&info));
307: } else PetscCallBLAS("LAPACKlascl",LAPACKlascl_("G",&zero,&zero,&done,&sqrtnrm,&N,&one,A,&N,&info));
308: SlepcCheckLapackInfo("lascl",info);
310: PetscFree4(Yold,Z,Zold,M);
311: SlepcResetFlushToZero(&ftz);
312: return 0;
313: }
315: #if defined(PETSC_HAVE_CUDA)
316: #include "../src/sys/classes/fn/impls/cuda/fnutilcuda.h"
317: #include <slepccublas.h>
319: /*
320: * Matrix square root by Newton-Schulz iteration. CUDA version.
321: * Computes the principal square root of the matrix A using the
322: * Newton-Schulz iteration. A is overwritten with sqrtm(A).
323: */
324: PetscErrorCode FNSqrtmNewtonSchulz_CUDA(FN fn,PetscBLASInt n,PetscScalar *d_A,PetscBLASInt ld,PetscBool inv)
325: {
326: PetscScalar *d_Yold,*d_Z,*d_Zold,*d_M,alpha;
327: PetscReal nrm,sqrtnrm,tol,Yres=0.0;
328: const PetscScalar szero=0.0,sone=1.0,smone=-1.0,spfive=0.5,sthree=3.0;
329: PetscInt it;
330: PetscBLASInt N;
331: const PetscBLASInt one=1;
332: PetscBool converged=PETSC_FALSE;
333: cublasHandle_t cublasv2handle;
335: PetscDeviceInitialize(PETSC_DEVICE_CUDA); /* For CUDA event timers */
336: PetscCUBLASGetHandle(&cublasv2handle);
337: N = n*n;
338: tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
340: cudaMalloc((void **)&d_Yold,sizeof(PetscScalar)*N);
341: cudaMalloc((void **)&d_Z,sizeof(PetscScalar)*N);
342: cudaMalloc((void **)&d_Zold,sizeof(PetscScalar)*N);
343: cudaMalloc((void **)&d_M,sizeof(PetscScalar)*N);
345: PetscLogGpuTimeBegin();
347: /* Z = I; */
348: cudaMemset(d_Z,0,sizeof(PetscScalar)*N);
349: set_diagonal(n,d_Z,ld,sone);
351: /* scale */
352: cublasXaxpy(cublasv2handle,N,&smone,d_A,one,d_Z,one);
353: cublasXnrm2(cublasv2handle,N,d_Z,one,&nrm);
354: sqrtnrm = PetscSqrtReal(nrm);
355: alpha = 1.0/nrm;
356: cublasXscal(cublasv2handle,N,&alpha,d_A,one);
357: tol *= nrm;
358: PetscInfo(fn,"||I-A||_F = %g, new tol: %g\n",(double)nrm,(double)tol);
359: PetscLogGpuFlops(2.0*n*n);
361: /* Z = I; */
362: cudaMemset(d_Z,0,sizeof(PetscScalar)*N);
363: set_diagonal(n,d_Z,ld,sone);
365: for (it=0;it<NSMAXIT && !converged;it++) {
366: /* Yold = Y, Zold = Z */
367: cudaMemcpy(d_Yold,d_A,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice);
368: cudaMemcpy(d_Zold,d_Z,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice);
370: /* M = (3*I - Zold*Yold) */
371: cudaMemset(d_M,0,sizeof(PetscScalar)*N);
372: set_diagonal(n,d_M,ld,sthree);
373: cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&smone,d_Zold,ld,d_Yold,ld,&sone,d_M,ld);
375: /* Y = (1/2) * Yold * M, Z = (1/2) * M * Zold */
376: cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&spfive,d_Yold,ld,d_M,ld,&szero,d_A,ld);
377: cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&spfive,d_M,ld,d_Zold,ld,&szero,d_Z,ld);
379: /* reldiff = norm(Y-Yold,'fro')/norm(Y,'fro') */
380: cublasXaxpy(cublasv2handle,N,&smone,d_A,one,d_Yold,one);
381: cublasXnrm2(cublasv2handle,N,d_Yold,one,&Yres);
383: if (Yres<=tol) converged = PETSC_TRUE;
384: PetscInfo(fn,"it: %" PetscInt_FMT " res: %g\n",it,(double)Yres);
386: PetscLogGpuFlops(6.0*n*n*n+2.0*n*n);
387: }
391: /* undo scaling */
392: if (inv) {
393: alpha = 1.0/sqrtnrm;
394: cublasXscal(cublasv2handle,N,&alpha,d_Z,one);
395: cudaMemcpy(d_A,d_Z,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice);
396: } else {
397: alpha = sqrtnrm;
398: cublasXscal(cublasv2handle,N,&alpha,d_A,one);
399: }
401: PetscLogGpuTimeEnd();
402: cudaFree(d_Yold);
403: cudaFree(d_Z);
404: cudaFree(d_Zold);
405: cudaFree(d_M);
406: return 0;
407: }
409: #if defined(PETSC_HAVE_MAGMA)
410: #include <slepcmagma.h>
412: /*
413: * Matrix square root by product form of Denman-Beavers iteration. CUDA version.
414: * Computes the principal square root of the matrix T using the product form
415: * of the Denman-Beavers iteration. T is overwritten with sqrtm(T).
416: */
417: PetscErrorCode FNSqrtmDenmanBeavers_CUDAm(FN fn,PetscBLASInt n,PetscScalar *d_T,PetscBLASInt ld,PetscBool inv)
418: {
419: PetscScalar *d_Told,*d_M,*d_invM,*d_work,prod,szero=0.0,sone=1.0,smone=-1.0,spfive=0.5,sneg_pfive=-0.5,sp25=0.25,alpha;
420: PetscReal tol,Mres=0.0,detM,g,reldiff,fnormdiff,fnormT;
421: PetscInt it,lwork,nb;
422: PetscBLASInt N,one=1,*piv=NULL;
423: PetscBool converged=PETSC_FALSE,scale;
424: cublasHandle_t cublasv2handle;
426: PetscDeviceInitialize(PETSC_DEVICE_CUDA); /* For CUDA event timers */
427: PetscCUBLASGetHandle(&cublasv2handle);
428: SlepcMagmaInit();
429: N = n*n;
430: scale = PetscDefined(USE_REAL_SINGLE)? PETSC_FALSE: PETSC_TRUE;
431: tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
433: /* query work size */
434: nb = magma_get_xgetri_nb(n);
435: lwork = nb*n;
436: PetscMalloc1(n,&piv);
437: cudaMalloc((void **)&d_work,sizeof(PetscScalar)*lwork);
438: cudaMalloc((void **)&d_Told,sizeof(PetscScalar)*N);
439: cudaMalloc((void **)&d_M,sizeof(PetscScalar)*N);
440: cudaMalloc((void **)&d_invM,sizeof(PetscScalar)*N);
442: PetscLogGpuTimeBegin();
443: cudaMemcpy(d_M,d_T,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice);
444: if (inv) { /* start recurrence with I instead of A */
445: cudaMemset(d_T,0,sizeof(PetscScalar)*N);
446: set_diagonal(n,d_T,ld,1.0);
447: }
449: for (it=0;it<DBMAXIT && !converged;it++) {
451: if (scale) { /* g = (abs(det(M)))^(-1/(2*n)); */
452: cudaMemcpy(d_invM,d_M,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice);
453: PetscCallMAGMA(magma_xgetrf_gpu,n,n,d_invM,ld,piv);
454: mult_diagonal(n,d_invM,ld,&prod);
455: detM = PetscAbsScalar(prod);
456: g = (detM>PETSC_MAX_REAL)? 0.5: PetscPowReal(detM,-1.0/(2.0*n));
457: alpha = g;
458: cublasXscal(cublasv2handle,N,&alpha,d_T,one);
459: alpha = g*g;
460: cublasXscal(cublasv2handle,N,&alpha,d_M,one);
461: PetscLogGpuFlops(2.0*n*n*n/3.0+2.0*n*n);
462: }
464: cudaMemcpy(d_Told,d_T,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice);
465: cudaMemcpy(d_invM,d_M,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice);
467: PetscCallMAGMA(magma_xgetrf_gpu,n,n,d_invM,ld,piv);
468: PetscCallMAGMA(magma_xgetri_gpu,n,d_invM,ld,piv,d_work,lwork);
469: PetscLogGpuFlops(2.0*n*n*n/3.0+4.0*n*n*n/3.0);
471: shift_diagonal(n,d_invM,ld,sone);
472: cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&spfive,d_Told,ld,d_invM,ld,&szero,d_T,ld);
473: shift_diagonal(n,d_invM,ld,smone);
475: cublasXaxpy(cublasv2handle,N,&sone,d_invM,one,d_M,one);
476: cublasXscal(cublasv2handle,N,&sp25,d_M,one);
477: shift_diagonal(n,d_M,ld,sneg_pfive);
478: PetscLogGpuFlops(2.0*n*n*n+2.0*n*n);
480: cublasXnrm2(cublasv2handle,N,d_M,one,&Mres);
481: shift_diagonal(n,d_M,ld,sone);
483: if (scale) {
484: /* reldiff = norm(T - Told,'fro')/norm(T,'fro'); */
485: cublasXaxpy(cublasv2handle,N,&smone,d_T,one,d_Told,one);
486: cublasXnrm2(cublasv2handle,N,d_Told,one,&fnormdiff);
487: cublasXnrm2(cublasv2handle,N,d_T,one,&fnormT);
488: PetscLogGpuFlops(7.0*n*n);
489: reldiff = fnormdiff/fnormT;
490: PetscInfo(fn,"it: %" PetscInt_FMT " reldiff: %g scale: %g tol*scale: %g\n",it,(double)reldiff,(double)g,(double)tol*g);
491: if (reldiff<1e-2) scale = PETSC_FALSE; /* Switch to no scaling. */
492: }
494: PetscInfo(fn,"it: %" PetscInt_FMT " Mres: %g\n",it,(double)Mres);
495: if (Mres<=tol) converged = PETSC_TRUE;
496: }
499: PetscLogGpuTimeEnd();
500: PetscFree(piv);
501: cudaFree(d_work);
502: cudaFree(d_Told);
503: cudaFree(d_M);
504: cudaFree(d_invM);
505: return 0;
506: }
507: #endif /* PETSC_HAVE_MAGMA */
509: #endif /* PETSC_HAVE_CUDA */
511: #define ITMAX 5
512: #define SWAP(a,b,t) {t=a;a=b;b=t;}
514: /*
515: Estimate norm(A^m,1) by block 1-norm power method (required workspace is 11*n)
516: */
517: static PetscErrorCode SlepcNormEst1(PetscBLASInt n,PetscScalar *A,PetscInt m,PetscScalar *work,PetscRandom rand,PetscReal *nrm)
518: {
519: PetscScalar *X,*Y,*Z,*S,*S_old,*aux,val,sone=1.0,szero=0.0;
520: PetscReal est=0.0,est_old,vals[2]={0.0,0.0},*zvals,maxzval[2],raux;
521: PetscBLASInt i,j,t=2,it=0,ind[2],est_j=0,m1;
523: X = work;
524: Y = work + 2*n;
525: Z = work + 4*n;
526: S = work + 6*n;
527: S_old = work + 8*n;
528: zvals = (PetscReal*)(work + 10*n);
530: for (i=0;i<n;i++) { /* X has columns of unit 1-norm */
531: X[i] = 1.0/n;
532: PetscRandomGetValue(rand,&val);
533: if (PetscRealPart(val) < 0.5) X[i+n] = -1.0/n;
534: else X[i+n] = 1.0/n;
535: }
536: for (i=0;i<t*n;i++) S[i] = 0.0;
537: ind[0] = 0; ind[1] = 0;
538: est_old = 0;
539: while (1) {
540: it++;
541: for (j=0;j<m;j++) { /* Y = A^m*X */
542: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&t,&n,&sone,A,&n,X,&n,&szero,Y,&n));
543: if (j<m-1) SWAP(X,Y,aux);
544: }
545: for (j=0;j<t;j++) { /* vals[j] = norm(Y(:,j),1) */
546: vals[j] = 0.0;
547: for (i=0;i<n;i++) vals[j] += PetscAbsScalar(Y[i+j*n]);
548: }
549: if (vals[0]<vals[1]) {
550: SWAP(vals[0],vals[1],raux);
551: m1 = 1;
552: } else m1 = 0;
553: est = vals[0];
554: if (est>est_old || it==2) est_j = ind[m1];
555: if (it>=2 && est<=est_old) {
556: est = est_old;
557: break;
558: }
559: est_old = est;
560: if (it>ITMAX) break;
561: SWAP(S,S_old,aux);
562: for (i=0;i<t*n;i++) { /* S = sign(Y) */
563: S[i] = (PetscRealPart(Y[i]) < 0.0)? -1.0: 1.0;
564: }
565: for (j=0;j<m;j++) { /* Z = (A^T)^m*S */
566: PetscCallBLAS("BLASgemm",BLASgemm_("C","N",&n,&t,&n,&sone,A,&n,S,&n,&szero,Z,&n));
567: if (j<m-1) SWAP(S,Z,aux);
568: }
569: maxzval[0] = -1; maxzval[1] = -1;
570: ind[0] = 0; ind[1] = 0;
571: for (i=0;i<n;i++) { /* zvals[i] = norm(Z(i,:),inf) */
572: zvals[i] = PetscMax(PetscAbsScalar(Z[i+0*n]),PetscAbsScalar(Z[i+1*n]));
573: if (zvals[i]>maxzval[0]) {
574: maxzval[0] = zvals[i];
575: ind[0] = i;
576: } else if (zvals[i]>maxzval[1]) {
577: maxzval[1] = zvals[i];
578: ind[1] = i;
579: }
580: }
581: if (it>=2 && maxzval[0]==zvals[est_j]) break;
582: for (i=0;i<t*n;i++) X[i] = 0.0;
583: for (j=0;j<t;j++) X[ind[j]+j*n] = 1.0;
584: }
585: *nrm = est;
586: /* Flop count is roughly (it * 2*m * t*gemv) = 4*its*m*t*n*n */
587: PetscLogFlops(4.0*it*m*t*n*n);
588: return 0;
589: }
591: #define SMALLN 100
593: /*
594: Estimate norm(A^m,1) (required workspace is 2*n*n)
595: */
596: PetscErrorCode SlepcNormAm(PetscBLASInt n,PetscScalar *A,PetscInt m,PetscScalar *work,PetscRandom rand,PetscReal *nrm)
597: {
598: PetscScalar *v=work,*w=work+n*n,*aux,sone=1.0,szero=0.0;
599: PetscReal rwork[1],tmp;
600: PetscBLASInt i,j,one=1;
601: PetscBool isrealpos=PETSC_TRUE;
603: if (n<SMALLN) { /* compute matrix power explicitly */
604: if (m==1) {
605: *nrm = LAPACKlange_("O",&n,&n,A,&n,rwork);
606: PetscLogFlops(1.0*n*n);
607: } else { /* m>=2 */
608: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,A,&n,A,&n,&szero,v,&n));
609: for (j=0;j<m-2;j++) {
610: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,A,&n,v,&n,&szero,w,&n));
611: SWAP(v,w,aux);
612: }
613: *nrm = LAPACKlange_("O",&n,&n,v,&n,rwork);
614: PetscLogFlops(2.0*n*n*n*(m-1)+1.0*n*n);
615: }
616: } else {
617: for (i=0;i<n;i++)
618: for (j=0;j<n;j++)
619: #if defined(PETSC_USE_COMPLEX)
620: if (PetscRealPart(A[i+j*n])<0.0 || PetscImaginaryPart(A[i+j*n])!=0.0) { isrealpos = PETSC_FALSE; break; }
621: #else
622: if (A[i+j*n]<0.0) { isrealpos = PETSC_FALSE; break; }
623: #endif
624: if (isrealpos) { /* for positive matrices only */
625: for (i=0;i<n;i++) v[i] = 1.0;
626: for (j=0;j<m;j++) { /* w = A'*v */
627: PetscCallBLAS("BLASgemv",BLASgemv_("C",&n,&n,&sone,A,&n,v,&one,&szero,w,&one));
628: SWAP(v,w,aux);
629: }
630: PetscLogFlops(2.0*n*n*m);
631: *nrm = 0.0;
632: for (i=0;i<n;i++) if ((tmp = PetscAbsScalar(v[i])) > *nrm) *nrm = tmp; /* norm(v,inf) */
633: } else SlepcNormEst1(n,A,m,work,rand,nrm);
634: }
635: return 0;
636: }