Actual source code: fnutil.c
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;
26: #if !PetscDefined(USE_COMPLEX)
27: PetscReal alpha,theta,mu,mu2;
28: #endif
30: PetscFunctionBegin;
31: for (j=0;j<n;j++) {
32: #if PetscDefined(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: PetscCheck(T[j+j*ld]>=0.0,PETSC_COMM_SELF,PETSC_ERR_USER_INPUT,"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 PetscDefined(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) 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));
66: PetscCallLAPACKInfo("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: PetscCheck(scal==1.0,PETSC_COMM_SELF,PETSC_ERR_SUP,"Current implementation cannot handle scale factor %g",(double)scal);
68: }
69: if (sj==2) j++;
70: }
71: PetscFunctionReturn(PETSC_SUCCESS);
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,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 PetscDefined(USE_COMPLEX)
88: PetscReal *rwork;
89: #else
90: PetscReal *wi;
91: #endif
93: PetscFunctionBegin;
94: m = n;
95: nblk = (m+bs-1)/bs;
96: lwork = 5*n;
97: k = firstonly? 1: n;
99: /* compute Schur decomposition A*Q = Q*T */
100: #if !PetscDefined(USE_COMPLEX)
101: PetscCall(PetscMalloc7(m,&wr,m,&wi,m*k,&W,m*m,&Q,lwork,&work,nblk,&s,nblk,&p));
102: PetscCallLAPACKInfo("LAPACKgees",LAPACKgees_("V","N",NULL,&n,T,&ld,&sdim,wr,wi,Q,&ld,work,&lwork,NULL,&info));
103: #else
104: PetscCall(PetscMalloc7(m,&wr,m,&rwork,m*k,&W,m*m,&Q,lwork,&work,nblk,&s,nblk,&p));
105: PetscCallLAPACKInfo("LAPACKgees",LAPACKgees_("V","N",NULL,&n,T,&ld,&sdim,wr,Q,&ld,work,&lwork,rwork,NULL,&info));
106: #endif
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 !PetscDefined(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: PetscCall(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: PetscCallLAPACKInfo("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: PetscCheck(scal==1.0,PETSC_COMM_SELF,PETSC_ERR_SUP,"Current implementation cannot handle scale factor %g",(double)scal);
131: }
132: }
134: /* backtransform B = Q*T*Q' */
135: PetscCallBLAS("BLASgemm",BLASgemm_("N","C",&n,&k,&n,&one,T,&ld,Q,&ld,&zero,W,&ld));
136: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&k,&n,&one,Q,&ld,W,&ld,&zero,T,&ld));
138: /* flop count: Schur decomposition, triangular square root, and backtransform */
139: PetscCall(PetscLogFlops(25.0*n*n*n+n*n*n/3.0+4.0*n*n*k));
141: #if !PetscDefined(USE_COMPLEX)
142: PetscCall(PetscFree7(wr,wi,W,Q,work,s,p));
143: #else
144: PetscCall(PetscFree7(wr,rwork,W,Q,work,s,p));
145: #endif
146: PetscFunctionReturn(PETSC_SUCCESS);
147: }
149: #define DBMAXIT 25
151: /*
152: Computes the principal square root of the matrix T using the product form
153: of the Denman-Beavers iteration.
154: T is overwritten with sqrtm(T) or inv(sqrtm(T)) depending on flag inv.
155: */
156: PetscErrorCode FNSqrtmDenmanBeavers(FN fn,PetscBLASInt n,PetscScalar *T,PetscBLASInt ld,PetscBool inv)
157: {
158: PetscScalar *Told,*M=NULL,*invM,*work,work1,prod,alpha;
159: PetscScalar szero=0.0,sone=1.0,smone=-1.0,spfive=0.5,sp25=0.25;
160: PetscReal tol,Mres=0.0,detM,g,reldiff,fnormdiff,fnormT,rwork[1];
161: PetscBLASInt N,i,it,*piv=NULL,query=-1,lwork;
162: const PetscBLASInt one=1;
163: PetscBool converged=PETSC_FALSE,scale;
164: unsigned int ftz;
166: PetscFunctionBegin;
167: N = n*n;
168: tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
169: scale = PetscDefined(USE_REAL_SINGLE)? PETSC_FALSE: PETSC_TRUE;
170: PetscCall(SlepcSetFlushToZero(&ftz));
172: /* query work size */
173: PetscCallLAPACKInfo("LAPACKgetri",LAPACKgetri_(&n,M,&ld,piv,&work1,&query,&info));
174: PetscCall(PetscBLASIntCast((PetscInt)PetscRealPart(work1),&lwork));
175: PetscCall(PetscMalloc5(lwork,&work,n,&piv,n*n,&Told,n*n,&M,n*n,&invM));
176: PetscCall(PetscArraycpy(M,T,n*n));
178: if (inv) { /* start recurrence with I instead of A */
179: PetscCall(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: PetscCall(PetscArraycpy(invM,M,n*n));
187: PetscCallLAPACKInfo("LAPACKgetrf",LAPACKgetrf_(&n,&n,invM,&ld,piv,&info));
188: prod = invM[0];
189: for (i=1;i<n;i++) prod *= invM[i+i*ld];
190: detM = PetscAbsScalar(prod);
191: g = (detM>PETSC_MAX_REAL)? 0.5: PetscPowReal(detM,-1.0/(2.0*n));
192: alpha = g;
193: PetscCallBLAS("BLASscal",BLASscal_(&N,&alpha,T,&one));
194: alpha = g*g;
195: PetscCallBLAS("BLASscal",BLASscal_(&N,&alpha,M,&one));
196: PetscCall(PetscLogFlops(2.0*n*n*n/3.0+2.0*n*n));
197: }
199: PetscCall(PetscArraycpy(Told,T,n*n));
200: PetscCall(PetscArraycpy(invM,M,n*n));
202: PetscCallLAPACKInfo("LAPACKgetrf",LAPACKgetrf_(&n,&n,invM,&ld,piv,&info));
203: PetscCallLAPACKInfo("LAPACKgetri",LAPACKgetri_(&n,invM,&ld,piv,work,&lwork,&info));
204: PetscCall(PetscLogFlops(2.0*n*n*n/3.0+4.0*n*n*n/3.0));
206: for (i=0;i<n;i++) invM[i+i*ld] += 1.0;
207: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&spfive,Told,&ld,invM,&ld,&szero,T,&ld));
208: for (i=0;i<n;i++) invM[i+i*ld] -= 1.0;
210: PetscCallBLAS("BLASaxpy",BLASaxpy_(&N,&sone,invM,&one,M,&one));
211: PetscCallBLAS("BLASscal",BLASscal_(&N,&sp25,M,&one));
212: for (i=0;i<n;i++) M[i+i*ld] -= 0.5;
213: PetscCall(PetscLogFlops(2.0*n*n*n+2.0*n*n));
215: Mres = LAPACKlange_("F",&n,&n,M,&n,rwork);
216: for (i=0;i<n;i++) M[i+i*ld] += 1.0;
218: if (scale) {
219: /* reldiff = norm(T - Told,'fro')/norm(T,'fro') */
220: PetscCallBLAS("BLASaxpy",BLASaxpy_(&N,&smone,T,&one,Told,&one));
221: fnormdiff = LAPACKlange_("F",&n,&n,Told,&n,rwork);
222: fnormT = LAPACKlange_("F",&n,&n,T,&n,rwork);
223: PetscCall(PetscLogFlops(7.0*n*n));
224: reldiff = fnormdiff/fnormT;
225: PetscCall(PetscInfo(fn,"it: %" PetscBLASInt_FMT " reldiff: %g scale: %g tol*scale: %g\n",it,(double)reldiff,(double)g,(double)(tol*g)));
226: if (reldiff<1e-2) scale = PETSC_FALSE; /* Switch off scaling */
227: }
229: if (Mres<=tol) converged = PETSC_TRUE;
230: }
232: PetscCheck(Mres<=tol,PETSC_COMM_SELF,PETSC_ERR_LIB,"SQRTM not converged after %d iterations",DBMAXIT);
233: PetscCall(PetscFree5(work,piv,Told,M,invM));
234: PetscCall(SlepcResetFlushToZero(&ftz));
235: PetscFunctionReturn(PETSC_SUCCESS);
236: }
238: #define NSMAXIT 50
240: /*
241: Computes the principal square root of the matrix A using the Newton-Schulz iteration.
242: T is overwritten with sqrtm(T) or inv(sqrtm(T)) depending on flag inv.
243: */
244: PetscErrorCode FNSqrtmNewtonSchulz(FN fn,PetscBLASInt n,PetscScalar *A,PetscBLASInt ld,PetscBool inv)
245: {
246: PetscScalar *Y=A,*Yold,*Z,*Zold,*M;
247: PetscScalar szero=0.0,sone=1.0,smone=-1.0,spfive=0.5,sthree=3.0;
248: PetscReal sqrtnrm,tol,Yres=0.0,nrm,rwork[1],done=1.0;
249: PetscBLASInt i,it,N,one=1,zero=0;
250: PetscBool converged=PETSC_FALSE;
251: unsigned int ftz;
253: PetscFunctionBegin;
254: N = n*n;
255: tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
256: PetscCall(SlepcSetFlushToZero(&ftz));
258: PetscCall(PetscMalloc4(N,&Yold,N,&Z,N,&Zold,N,&M));
260: /* scale */
261: PetscCall(PetscArraycpy(Z,A,N));
262: for (i=0;i<n;i++) Z[i+i*ld] -= 1.0;
263: nrm = LAPACKlange_("fro",&n,&n,Z,&n,rwork);
264: sqrtnrm = PetscSqrtReal(nrm);
265: PetscCallLAPACKInfo("LAPACKlascl",LAPACKlascl_("G",&zero,&zero,&nrm,&done,&N,&one,A,&N,&info));
266: tol *= nrm;
267: PetscCall(PetscInfo(fn,"||I-A||_F = %g, new tol: %g\n",(double)nrm,(double)tol));
268: PetscCall(PetscLogFlops(2.0*n*n));
270: /* Z = I */
271: PetscCall(PetscArrayzero(Z,N));
272: for (i=0;i<n;i++) Z[i+i*ld] = 1.0;
274: for (it=0;it<NSMAXIT && !converged;it++) {
275: /* Yold = Y, Zold = Z */
276: PetscCall(PetscArraycpy(Yold,Y,N));
277: PetscCall(PetscArraycpy(Zold,Z,N));
279: /* M = (3*I-Zold*Yold) */
280: PetscCall(PetscArrayzero(M,N));
281: for (i=0;i<n;i++) M[i+i*ld] = sthree;
282: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&smone,Zold,&ld,Yold,&ld,&sone,M,&ld));
284: /* Y = (1/2)*Yold*M, Z = (1/2)*M*Zold */
285: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&spfive,Yold,&ld,M,&ld,&szero,Y,&ld));
286: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&spfive,M,&ld,Zold,&ld,&szero,Z,&ld));
288: /* reldiff = norm(Y-Yold,'fro')/norm(Y,'fro') */
289: PetscCallBLAS("BLASaxpy",BLASaxpy_(&N,&smone,Y,&one,Yold,&one));
290: Yres = LAPACKlange_("fro",&n,&n,Yold,&n,rwork);
291: PetscCheck(!PetscIsNanReal(Yres),PETSC_COMM_SELF,PETSC_ERR_FP,"The computed norm is not-a-number");
292: if (Yres<=tol) converged = PETSC_TRUE;
293: PetscCall(PetscInfo(fn,"it: %" PetscBLASInt_FMT " res: %g\n",it,(double)Yres));
295: PetscCall(PetscLogFlops(6.0*n*n*n+2.0*n*n));
296: }
298: PetscCheck(Yres<=tol,PETSC_COMM_SELF,PETSC_ERR_LIB,"SQRTM not converged after %d iterations",NSMAXIT);
300: /* undo scaling */
301: if (inv) {
302: PetscCall(PetscArraycpy(A,Z,N));
303: PetscCallLAPACKInfo("LAPACKlascl",LAPACKlascl_("G",&zero,&zero,&sqrtnrm,&done,&N,&one,A,&N,&info));
304: } else PetscCallLAPACKInfo("LAPACKlascl",LAPACKlascl_("G",&zero,&zero,&done,&sqrtnrm,&N,&one,A,&N,&info));
306: PetscCall(PetscFree4(Yold,Z,Zold,M));
307: PetscCall(SlepcResetFlushToZero(&ftz));
308: PetscFunctionReturn(PETSC_SUCCESS);
309: }
311: #if PetscDefined(HAVE_CUDA)
312: #include "../src/sys/classes/fn/impls/cuda/fnutilcuda.h"
313: #include <slepccupmblas.h>
315: /*
316: * Matrix square root by Newton-Schulz iteration. CUDA version.
317: * Computes the principal square root of the matrix A using the
318: * Newton-Schulz iteration. A is overwritten with sqrtm(A).
319: */
320: PetscErrorCode FNSqrtmNewtonSchulz_CUDA(FN fn,PetscBLASInt n,PetscScalar *d_A,PetscBLASInt ld,PetscBool inv)
321: {
322: PetscScalar *d_Yold,*d_Z,*d_Zold,*d_M,alpha;
323: PetscReal nrm,sqrtnrm,tol,Yres=0.0;
324: const PetscScalar szero=0.0,sone=1.0,smone=-1.0,spfive=0.5,sthree=3.0;
325: PetscInt it;
326: PetscBLASInt N;
327: const PetscBLASInt one=1;
328: PetscBool converged=PETSC_FALSE;
329: cublasHandle_t cublasv2handle;
331: PetscFunctionBegin;
332: PetscCall(PetscDeviceInitialize(PETSC_DEVICE_CUDA)); /* For CUDA event timers */
333: PetscCall(PetscCUBLASGetHandle(&cublasv2handle));
334: N = n*n;
335: tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
337: PetscCallCUDA(cudaMalloc((void **)&d_Yold,sizeof(PetscScalar)*N));
338: PetscCallCUDA(cudaMalloc((void **)&d_Z,sizeof(PetscScalar)*N));
339: PetscCallCUDA(cudaMalloc((void **)&d_Zold,sizeof(PetscScalar)*N));
340: PetscCallCUDA(cudaMalloc((void **)&d_M,sizeof(PetscScalar)*N));
342: PetscCall(PetscLogGpuTimeBegin());
344: /* Z = I; */
345: PetscCallCUDA(cudaMemset(d_Z,0,sizeof(PetscScalar)*N));
346: PetscCall(set_diagonal(n,d_Z,ld,sone));
348: /* scale */
349: PetscCallCUBLAS(cublasXaxpy(cublasv2handle,N,&smone,d_A,one,d_Z,one));
350: PetscCallCUBLAS(cublasXnrm2(cublasv2handle,N,d_Z,one,&nrm));
351: sqrtnrm = PetscSqrtReal(nrm);
352: alpha = 1.0/nrm;
353: PetscCallCUBLAS(cublasXscal(cublasv2handle,N,&alpha,d_A,one));
354: tol *= nrm;
355: PetscCall(PetscInfo(fn,"||I-A||_F = %g, new tol: %g\n",(double)nrm,(double)tol));
356: PetscCall(PetscLogGpuFlops(2.0*n*n));
358: /* Z = I; */
359: PetscCallCUDA(cudaMemset(d_Z,0,sizeof(PetscScalar)*N));
360: PetscCall(set_diagonal(n,d_Z,ld,sone));
362: for (it=0;it<NSMAXIT && !converged;it++) {
363: /* Yold = Y, Zold = Z */
364: PetscCallCUDA(cudaMemcpy(d_Yold,d_A,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice));
365: PetscCallCUDA(cudaMemcpy(d_Zold,d_Z,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice));
367: /* M = (3*I - Zold*Yold) */
368: PetscCallCUDA(cudaMemset(d_M,0,sizeof(PetscScalar)*N));
369: PetscCall(set_diagonal(n,d_M,ld,sthree));
370: PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&smone,d_Zold,ld,d_Yold,ld,&sone,d_M,ld));
372: /* Y = (1/2) * Yold * M, Z = (1/2) * M * Zold */
373: PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&spfive,d_Yold,ld,d_M,ld,&szero,d_A,ld));
374: PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&spfive,d_M,ld,d_Zold,ld,&szero,d_Z,ld));
376: /* reldiff = norm(Y-Yold,'fro')/norm(Y,'fro') */
377: PetscCallCUBLAS(cublasXaxpy(cublasv2handle,N,&smone,d_A,one,d_Yold,one));
378: PetscCallCUBLAS(cublasXnrm2(cublasv2handle,N,d_Yold,one,&Yres));
379: PetscCheck(!PetscIsNanReal(Yres),PETSC_COMM_SELF,PETSC_ERR_FP,"The computed norm is not-a-number");
380: if (Yres<=tol) converged = PETSC_TRUE;
381: PetscCall(PetscInfo(fn,"it: %" PetscInt_FMT " res: %g\n",it,(double)Yres));
383: PetscCall(PetscLogGpuFlops(6.0*n*n*n+2.0*n*n));
384: }
386: PetscCheck(Yres<=tol,PETSC_COMM_SELF,PETSC_ERR_LIB,"SQRTM not converged after %d iterations", NSMAXIT);
388: /* undo scaling */
389: if (inv) {
390: alpha = 1.0/sqrtnrm;
391: PetscCallCUBLAS(cublasXscal(cublasv2handle,N,&alpha,d_Z,one));
392: PetscCallCUDA(cudaMemcpy(d_A,d_Z,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice));
393: } else {
394: alpha = sqrtnrm;
395: PetscCallCUBLAS(cublasXscal(cublasv2handle,N,&alpha,d_A,one));
396: }
398: PetscCall(PetscLogGpuTimeEnd());
399: PetscCallCUDA(cudaFree(d_Yold));
400: PetscCallCUDA(cudaFree(d_Z));
401: PetscCallCUDA(cudaFree(d_Zold));
402: PetscCallCUDA(cudaFree(d_M));
403: PetscFunctionReturn(PETSC_SUCCESS);
404: }
406: #if PetscDefined(HAVE_MAGMA)
407: #include <slepcmagma.h>
409: /*
410: * Matrix square root by product form of Denman-Beavers iteration. CUDA version.
411: * Computes the principal square root of the matrix T using the product form
412: * of the Denman-Beavers iteration. T is overwritten with sqrtm(T).
413: */
414: PetscErrorCode FNSqrtmDenmanBeavers_CUDAm(FN fn,PetscBLASInt n,PetscScalar *d_T,PetscBLASInt ld,PetscBool inv)
415: {
416: 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;
417: PetscReal tol,Mres=0.0,detM,g,reldiff,fnormdiff,fnormT;
418: PetscInt it,lwork,nb;
419: PetscBLASInt N,one=1,*piv=NULL;
420: PetscBool converged=PETSC_FALSE,scale;
421: cublasHandle_t cublasv2handle;
423: PetscFunctionBegin;
424: PetscCall(PetscDeviceInitialize(PETSC_DEVICE_CUDA)); /* For CUDA event timers */
425: PetscCall(PetscCUBLASGetHandle(&cublasv2handle));
426: PetscCall(SlepcMagmaInit());
427: N = n*n;
428: scale = PetscDefined(USE_REAL_SINGLE)? PETSC_FALSE: PETSC_TRUE;
429: tol = PetscSqrtReal((PetscReal)n)*PETSC_MACHINE_EPSILON/2;
431: /* query work size */
432: nb = magma_get_xgetri_nb(n);
433: lwork = nb*n;
434: PetscCall(PetscMalloc1(n,&piv));
435: PetscCallCUDA(cudaMalloc((void **)&d_work,sizeof(PetscScalar)*lwork));
436: PetscCallCUDA(cudaMalloc((void **)&d_Told,sizeof(PetscScalar)*N));
437: PetscCallCUDA(cudaMalloc((void **)&d_M,sizeof(PetscScalar)*N));
438: PetscCallCUDA(cudaMalloc((void **)&d_invM,sizeof(PetscScalar)*N));
440: PetscCall(PetscLogGpuTimeBegin());
441: PetscCallCUDA(cudaMemcpy(d_M,d_T,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice));
442: if (inv) { /* start recurrence with I instead of A */
443: PetscCallCUDA(cudaMemset(d_T,0,sizeof(PetscScalar)*N));
444: PetscCall(set_diagonal(n,d_T,ld,1.0));
445: }
447: for (it=0;it<DBMAXIT && !converged;it++) {
449: if (scale) { /* g = (abs(det(M)))^(-1/(2*n)); */
450: PetscCallCUDA(cudaMemcpy(d_invM,d_M,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice));
451: PetscCallMAGMA(magma_xgetrf_gpu,n,n,d_invM,ld,piv);
452: PetscCall(mult_diagonal(n,d_invM,ld,&prod));
453: detM = PetscAbsScalar(prod);
454: g = (detM>PETSC_MAX_REAL)? 0.5: PetscPowReal(detM,-1.0/(2.0*n));
455: alpha = g;
456: PetscCallCUBLAS(cublasXscal(cublasv2handle,N,&alpha,d_T,one));
457: alpha = g*g;
458: PetscCallCUBLAS(cublasXscal(cublasv2handle,N,&alpha,d_M,one));
459: PetscCall(PetscLogGpuFlops(2.0*n*n*n/3.0+2.0*n*n));
460: }
462: PetscCallCUDA(cudaMemcpy(d_Told,d_T,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice));
463: PetscCallCUDA(cudaMemcpy(d_invM,d_M,sizeof(PetscScalar)*N,cudaMemcpyDeviceToDevice));
465: PetscCallMAGMA(magma_xgetrf_gpu,n,n,d_invM,ld,piv);
466: PetscCallMAGMA(magma_xgetri_gpu,n,d_invM,ld,piv,d_work,lwork);
467: PetscCall(PetscLogGpuFlops(2.0*n*n*n/3.0+4.0*n*n*n/3.0));
469: PetscCall(shift_diagonal(n,d_invM,ld,sone));
470: PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&spfive,d_Told,ld,d_invM,ld,&szero,d_T,ld));
471: PetscCall(shift_diagonal(n,d_invM,ld,smone));
473: PetscCallCUBLAS(cublasXaxpy(cublasv2handle,N,&sone,d_invM,one,d_M,one));
474: PetscCallCUBLAS(cublasXscal(cublasv2handle,N,&sp25,d_M,one));
475: PetscCall(shift_diagonal(n,d_M,ld,sneg_pfive));
476: PetscCall(PetscLogGpuFlops(2.0*n*n*n+2.0*n*n));
478: PetscCallCUBLAS(cublasXnrm2(cublasv2handle,N,d_M,one,&Mres));
479: PetscCall(shift_diagonal(n,d_M,ld,sone));
481: if (scale) {
482: /* reldiff = norm(T - Told,'fro')/norm(T,'fro'); */
483: PetscCallCUBLAS(cublasXaxpy(cublasv2handle,N,&smone,d_T,one,d_Told,one));
484: PetscCallCUBLAS(cublasXnrm2(cublasv2handle,N,d_Told,one,&fnormdiff));
485: PetscCallCUBLAS(cublasXnrm2(cublasv2handle,N,d_T,one,&fnormT));
486: PetscCall(PetscLogGpuFlops(7.0*n*n));
487: reldiff = fnormdiff/fnormT;
488: PetscCall(PetscInfo(fn,"it: %" PetscInt_FMT " reldiff: %g scale: %g tol*scale: %g\n",it,(double)reldiff,(double)g,(double)tol*g));
489: if (reldiff<1e-2) scale = PETSC_FALSE; /* Switch to no scaling. */
490: }
492: PetscCall(PetscInfo(fn,"it: %" PetscInt_FMT " Mres: %g\n",it,(double)Mres));
493: if (Mres<=tol) converged = PETSC_TRUE;
494: }
496: PetscCheck(Mres<=tol,PETSC_COMM_SELF,PETSC_ERR_LIB,"SQRTM not converged after %d iterations", DBMAXIT);
497: PetscCall(PetscLogGpuTimeEnd());
498: PetscCall(PetscFree(piv));
499: PetscCallCUDA(cudaFree(d_work));
500: PetscCallCUDA(cudaFree(d_Told));
501: PetscCallCUDA(cudaFree(d_M));
502: PetscCallCUDA(cudaFree(d_invM));
503: PetscFunctionReturn(PETSC_SUCCESS);
504: }
505: #endif /* PETSC_HAVE_MAGMA */
507: #endif /* PETSC_HAVE_CUDA */
509: #define ITMAX 5
511: /*
512: Estimate norm(A^m,1) by block 1-norm power method (required workspace is 11*n)
513: */
514: static PetscErrorCode SlepcNormEst1(PetscBLASInt n,PetscScalar *A,PetscInt m,PetscScalar *work,PetscRandom rand,PetscReal *nrm)
515: {
516: PetscScalar *X,*Y,*Z,*S,*S_old,*aux,val,sone=1.0,szero=0.0;
517: PetscReal est=0.0,est_old,vals[2]={0.0,0.0},*zvals,maxzval[2],raux;
518: PetscBLASInt i,j,t=2,it=0,ind[2],est_j=0,m1;
520: PetscFunctionBegin;
521: X = work;
522: Y = work + 2*n;
523: Z = work + 4*n;
524: S = work + 6*n;
525: S_old = work + 8*n;
526: zvals = (PetscReal*)(work + 10*n);
528: for (i=0;i<n;i++) { /* X has columns of unit 1-norm */
529: X[i] = 1.0/n;
530: PetscCall(PetscRandomGetValue(rand,&val));
531: if (PetscRealPart(val) < 0.5) X[i+n] = -1.0/n;
532: else X[i+n] = 1.0/n;
533: }
534: for (i=0;i<t*n;i++) S[i] = 0.0;
535: ind[0] = 0; ind[1] = 0;
536: est_old = 0;
537: while (1) {
538: it++;
539: for (j=0;j<m;j++) { /* Y = A^m*X */
540: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&t,&n,&sone,A,&n,X,&n,&szero,Y,&n));
541: if (j<m-1) SlepcSwap(X,Y,aux);
542: }
543: for (j=0;j<t;j++) { /* vals[j] = norm(Y(:,j),1) */
544: vals[j] = 0.0;
545: for (i=0;i<n;i++) vals[j] += PetscAbsScalar(Y[i+j*n]);
546: }
547: if (vals[0]<vals[1]) {
548: SlepcSwap(vals[0],vals[1],raux);
549: m1 = 1;
550: } else m1 = 0;
551: est = vals[0];
552: if (est>est_old || it==2) est_j = ind[m1];
553: if (it>=2 && est<=est_old) {
554: est = est_old;
555: break;
556: }
557: est_old = est;
558: if (it>ITMAX) break;
559: SlepcSwap(S,S_old,aux);
560: for (i=0;i<t*n;i++) { /* S = sign(Y) */
561: S[i] = (PetscRealPart(Y[i]) < 0.0)? -1.0: 1.0;
562: }
563: for (j=0;j<m;j++) { /* Z = (A^T)^m*S */
564: PetscCallBLAS("BLASgemm",BLASgemm_("C","N",&n,&t,&n,&sone,A,&n,S,&n,&szero,Z,&n));
565: if (j<m-1) SlepcSwap(S,Z,aux);
566: }
567: maxzval[0] = -1; maxzval[1] = -1;
568: ind[0] = 0; ind[1] = 0;
569: for (i=0;i<n;i++) { /* zvals[i] = norm(Z(i,:),inf) */
570: zvals[i] = PetscMax(PetscAbsScalar(Z[i+0*n]),PetscAbsScalar(Z[i+1*n]));
571: if (zvals[i]>maxzval[0]) {
572: maxzval[0] = zvals[i];
573: ind[0] = i;
574: } else if (zvals[i]>maxzval[1]) {
575: maxzval[1] = zvals[i];
576: ind[1] = i;
577: }
578: }
579: if (it>=2 && maxzval[0]==zvals[est_j]) break;
580: for (i=0;i<t*n;i++) X[i] = 0.0;
581: for (j=0;j<t;j++) X[ind[j]+j*n] = 1.0;
582: }
583: *nrm = est;
584: /* Flop count is roughly (it * 2*m * t*gemv) = 4*its*m*t*n*n */
585: PetscCall(PetscLogFlops(4.0*it*m*t*n*n));
586: PetscFunctionReturn(PETSC_SUCCESS);
587: }
589: #define SMALLN 100
591: /*
592: Estimate norm(A^m,1) (required workspace is 2*n*n)
593: */
594: PetscErrorCode SlepcNormAm(PetscBLASInt n,PetscScalar *A,PetscInt m,PetscScalar *work,PetscRandom rand,PetscReal *nrm)
595: {
596: PetscScalar *v=work,*w=work+n*n,*aux,sone=1.0,szero=0.0;
597: PetscReal rwork[1],tmp;
598: PetscBLASInt i,j,one=1;
599: PetscBool isrealpos=PETSC_TRUE;
601: PetscFunctionBegin;
602: if (n<SMALLN) { /* compute matrix power explicitly */
603: if (m==1) {
604: *nrm = LAPACKlange_("O",&n,&n,A,&n,rwork);
605: PetscCall(PetscLogFlops(1.0*n*n));
606: } else { /* m>=2 */
607: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,A,&n,A,&n,&szero,v,&n));
608: for (j=0;j<m-2;j++) {
609: PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,A,&n,v,&n,&szero,w,&n));
610: SlepcSwap(v,w,aux);
611: }
612: *nrm = LAPACKlange_("O",&n,&n,v,&n,rwork);
613: PetscCall(PetscLogFlops(2.0*n*n*n*(m-1)+1.0*n*n));
614: }
615: } else {
616: for (i=0;i<n;i++)
617: for (j=0;j<n;j++)
618: #if PetscDefined(USE_COMPLEX)
619: if (PetscRealPart(A[i+j*n])<0.0 || PetscImaginaryPart(A[i+j*n])!=0.0) { isrealpos = PETSC_FALSE; break; }
620: #else
621: if (A[i+j*n]<0.0) { isrealpos = PETSC_FALSE; break; }
622: #endif
623: if (isrealpos) { /* for positive matrices only */
624: for (i=0;i<n;i++) v[i] = 1.0;
625: for (j=0;j<m;j++) { /* w = A'*v */
626: PetscCallBLAS("BLASgemv",BLASgemv_("C",&n,&n,&sone,A,&n,v,&one,&szero,w,&one));
627: SlepcSwap(v,w,aux);
628: }
629: PetscCall(PetscLogFlops(2.0*n*n*m));
630: *nrm = 0.0;
631: for (i=0;i<n;i++) if ((tmp = PetscAbsScalar(v[i])) > *nrm) *nrm = tmp; /* norm(v,inf) */
632: } else PetscCall(SlepcNormEst1(n,A,m,work,rand,nrm));
633: }
634: PetscFunctionReturn(PETSC_SUCCESS);
635: }