Line data Source code
1 : /*
2 : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3 : SLEPc - Scalable Library for Eigenvalue Problem Computations
4 : Copyright (c) 2002-, Universitat Politecnica de Valencia, Spain
5 :
6 : This file is part of SLEPc.
7 : SLEPc is distributed under a 2-clause BSD license (see LICENSE).
8 : - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
9 : */
10 : /*
11 : Exponential function exp(x)
12 : */
13 :
14 : #include <slepc/private/fnimpl.h> /*I "slepcfn.h" I*/
15 : #include <slepcblaslapack.h>
16 :
17 2117 : static PetscErrorCode FNEvaluateFunction_Exp(FN fn,PetscScalar x,PetscScalar *y)
18 : {
19 2117 : PetscFunctionBegin;
20 2117 : *y = PetscExpScalar(x);
21 2117 : PetscFunctionReturn(PETSC_SUCCESS);
22 : }
23 :
24 676 : static PetscErrorCode FNEvaluateDerivative_Exp(FN fn,PetscScalar x,PetscScalar *y)
25 : {
26 676 : PetscFunctionBegin;
27 676 : *y = PetscExpScalar(x);
28 676 : PetscFunctionReturn(PETSC_SUCCESS);
29 : }
30 :
31 : #define MAX_PADE 6
32 : #define SWAP(a,b,t) do {t=a;a=b;b=t;} while (0)
33 :
34 16 : static PetscErrorCode FNEvaluateFunctionMat_Exp_Pade(FN fn,Mat A,Mat B)
35 : {
36 16 : PetscBLASInt n=0,ld,ld2,*ipiv,info,inc=1;
37 16 : PetscInt m,j,k,sexp;
38 16 : PetscBool odd;
39 16 : const PetscInt p=MAX_PADE;
40 16 : PetscReal c[MAX_PADE+1],s,*rwork;
41 16 : PetscScalar scale,mone=-1.0,one=1.0,two=2.0,zero=0.0;
42 16 : PetscScalar *Ba,*As,*A2,*Q,*P,*W,*aux;
43 16 : const PetscScalar *Aa;
44 :
45 16 : PetscFunctionBegin;
46 16 : PetscCall(MatDenseGetArrayRead(A,&Aa));
47 16 : PetscCall(MatDenseGetArray(B,&Ba));
48 16 : PetscCall(MatGetSize(A,&m,NULL));
49 16 : PetscCall(PetscBLASIntCast(m,&n));
50 16 : ld = n;
51 16 : ld2 = ld*ld;
52 16 : P = Ba;
53 16 : PetscCall(PetscMalloc6(m*m,&Q,m*m,&W,m*m,&As,m*m,&A2,ld,&rwork,ld,&ipiv));
54 16 : PetscCall(PetscArraycpy(As,Aa,ld2));
55 :
56 : /* Pade' coefficients */
57 16 : c[0] = 1.0;
58 112 : for (k=1;k<=p;k++) c[k] = c[k-1]*(p+1-k)/(k*(2*p+1-k));
59 :
60 : /* Scaling */
61 16 : s = LAPACKlange_("I",&n,&n,As,&ld,rwork);
62 16 : PetscCall(PetscLogFlops(1.0*n*n));
63 16 : if (s>0.5) {
64 16 : sexp = PetscMax(0,(int)(PetscLogReal(s)/PetscLogReal(2.0))+2);
65 16 : scale = PetscPowRealInt(2.0,-sexp);
66 16 : PetscCallBLAS("BLASscal",BLASscal_(&ld2,&scale,As,&inc));
67 16 : PetscCall(PetscLogFlops(1.0*n*n));
68 : } else sexp = 0;
69 :
70 : /* Horner evaluation */
71 16 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,As,&ld,As,&ld,&zero,A2,&ld));
72 16 : PetscCall(PetscLogFlops(2.0*n*n*n));
73 16 : PetscCall(PetscArrayzero(Q,ld2));
74 16 : PetscCall(PetscArrayzero(P,ld2));
75 616 : for (j=0;j<n;j++) {
76 600 : Q[j+j*ld] = c[p];
77 600 : P[j+j*ld] = c[p-1];
78 : }
79 :
80 : odd = PETSC_TRUE;
81 96 : for (k=p-1;k>0;k--) {
82 80 : if (odd) {
83 48 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,Q,&ld,A2,&ld,&zero,W,&ld));
84 48 : SWAP(Q,W,aux);
85 1848 : for (j=0;j<n;j++) Q[j+j*ld] += c[k-1];
86 : odd = PETSC_FALSE;
87 : } else {
88 32 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,P,&ld,A2,&ld,&zero,W,&ld));
89 32 : SWAP(P,W,aux);
90 1232 : for (j=0;j<n;j++) P[j+j*ld] += c[k-1];
91 : odd = PETSC_TRUE;
92 : }
93 80 : PetscCall(PetscLogFlops(2.0*n*n*n));
94 : }
95 : /*if (odd) {
96 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,Q,&ld,As,&ld,&zero,W,&ld));
97 : SWAP(Q,W,aux);
98 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&ld2,&mone,P,&inc,Q,&inc));
99 : PetscCallBLAS("LAPACKgesv",LAPACKgesv_(&n,&n,Q,&ld,ipiv,P,&ld,&info));
100 : SlepcCheckLapackInfo("gesv",info);
101 : PetscCallBLAS("BLASscal",BLASscal_(&ld2,&two,P,&inc));
102 : for (j=0;j<n;j++) P[j+j*ld] += 1.0;
103 : PetscCallBLAS("BLASscal",BLASscal_(&ld2,&mone,P,&inc));
104 : } else {*/
105 16 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,P,&ld,As,&ld,&zero,W,&ld));
106 16 : SWAP(P,W,aux);
107 16 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&ld2,&mone,P,&inc,Q,&inc));
108 16 : PetscCallBLAS("LAPACKgesv",LAPACKgesv_(&n,&n,Q,&ld,ipiv,P,&ld,&info));
109 16 : SlepcCheckLapackInfo("gesv",info);
110 16 : PetscCallBLAS("BLASscal",BLASscal_(&ld2,&two,P,&inc));
111 616 : for (j=0;j<n;j++) P[j+j*ld] += 1.0;
112 : /*}*/
113 16 : PetscCall(PetscLogFlops(2.0*n*n*n+2.0*n*n*n/3.0+4.0*n*n));
114 :
115 68 : for (k=1;k<=sexp;k++) {
116 52 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,P,&ld,P,&ld,&zero,W,&ld));
117 52 : PetscCall(PetscArraycpy(P,W,ld2));
118 : }
119 16 : if (P!=Ba) PetscCall(PetscArraycpy(Ba,P,ld2));
120 16 : PetscCall(PetscLogFlops(2.0*n*n*n*sexp));
121 :
122 16 : PetscCall(PetscFree6(Q,W,As,A2,rwork,ipiv));
123 16 : PetscCall(MatDenseRestoreArrayRead(A,&Aa));
124 16 : PetscCall(MatDenseRestoreArray(B,&Ba));
125 16 : PetscFunctionReturn(PETSC_SUCCESS);
126 : }
127 :
128 : #if defined(PETSC_HAVE_COMPLEX)
129 : /*
130 : * Set scaling factor (s) and Pade degree (k,m)
131 : */
132 48 : static PetscErrorCode sexpm_params(PetscReal nrm,PetscInt *s,PetscInt *k,PetscInt *m)
133 : {
134 48 : PetscFunctionBegin;
135 48 : if (nrm>1) {
136 32 : if (nrm<200) {*s = 4; *k = 5; *m = *k-1;}
137 4 : else if (nrm<1e4) {*s = 4; *k = 4; *m = *k+1;}
138 0 : else if (nrm<1e6) {*s = 4; *k = 3; *m = *k+1;}
139 0 : else if (nrm<1e9) {*s = 3; *k = 3; *m = *k+1;}
140 0 : else if (nrm<1e11) {*s = 2; *k = 3; *m = *k+1;}
141 0 : else if (nrm<1e12) {*s = 2; *k = 2; *m = *k+1;}
142 0 : else if (nrm<1e14) {*s = 2; *k = 1; *m = *k+1;}
143 0 : else {*s = 1; *k = 1; *m = *k+1;}
144 : } else { /* nrm<1 */
145 16 : if (nrm>0.5) {*s = 4; *k = 4; *m = *k-1;}
146 12 : else if (nrm>0.3) {*s = 3; *k = 4; *m = *k-1;}
147 8 : else if (nrm>0.15) {*s = 2; *k = 4; *m = *k-1;}
148 8 : else if (nrm>0.07) {*s = 1; *k = 4; *m = *k-1;}
149 8 : else if (nrm>0.01) {*s = 0; *k = 4; *m = *k-1;}
150 8 : else if (nrm>3e-4) {*s = 0; *k = 3; *m = *k-1;}
151 8 : else if (nrm>1e-5) {*s = 0; *k = 3; *m = 0;}
152 8 : else if (nrm>1e-8) {*s = 0; *k = 2; *m = 0;}
153 8 : else {*s = 0; *k = 1; *m = 0;}
154 : }
155 48 : PetscFunctionReturn(PETSC_SUCCESS);
156 : }
157 :
158 : /*
159 : * Partial fraction form coefficients.
160 : * If query, the function returns the size necessary to store the coefficients.
161 : */
162 40 : static PetscErrorCode getcoeffs(PetscInt k,PetscInt m,PetscComplex *r,PetscComplex *q,PetscComplex *remain,PetscBool query)
163 : {
164 40 : PetscInt i;
165 40 : const PetscComplex /* m == k+1 */
166 40 : p1r4[5] = {-1.582680186458572e+01 - 2.412564578224361e+01*PETSC_i,
167 40 : -1.582680186458572e+01 + 2.412564578224361e+01*PETSC_i,
168 40 : 1.499984465975511e+02 + 6.804227952202417e+01*PETSC_i,
169 40 : 1.499984465975511e+02 - 6.804227952202417e+01*PETSC_i,
170 : -2.733432894659307e+02 },
171 40 : p1q4[5] = { 3.655694325463550e+00 + 6.543736899360086e+00*PETSC_i,
172 40 : 3.655694325463550e+00 - 6.543736899360086e+00*PETSC_i,
173 40 : 5.700953298671832e+00 + 3.210265600308496e+00*PETSC_i,
174 40 : 5.700953298671832e+00 - 3.210265600308496e+00*PETSC_i,
175 : 6.286704751729261e+00 },
176 40 : p1r3[4] = {-1.130153999597152e+01 + 1.247167585025031e+01*PETSC_i,
177 40 : -1.130153999597152e+01 - 1.247167585025031e+01*PETSC_i,
178 40 : 1.330153999597152e+01 - 6.007173273704750e+01*PETSC_i,
179 40 : 1.330153999597152e+01 + 6.007173273704750e+01*PETSC_i},
180 40 : p1q3[4] = { 3.212806896871536e+00 + 4.773087433276636e+00*PETSC_i,
181 40 : 3.212806896871536e+00 - 4.773087433276636e+00*PETSC_i,
182 40 : 4.787193103128464e+00 + 1.567476416895212e+00*PETSC_i,
183 40 : 4.787193103128464e+00 - 1.567476416895212e+00*PETSC_i},
184 40 : p1r2[3] = { 7.648749087422928e+00 + 4.171640244747463e+00*PETSC_i,
185 40 : 7.648749087422928e+00 - 4.171640244747463e+00*PETSC_i,
186 : -1.829749817484586e+01 },
187 40 : p1q2[3] = { 2.681082873627756e+00 + 3.050430199247411e+00*PETSC_i,
188 40 : 2.681082873627756e+00 - 3.050430199247411e+00*PETSC_i,
189 : 3.637834252744491e+00 },
190 40 : p1r1[2] = { 1.000000000000000e+00 - 3.535533905932738e+00*PETSC_i,
191 40 : 1.000000000000000e+00 + 3.535533905932738e+00*PETSC_i},
192 40 : p1q1[2] = { 2.000000000000000e+00 + 1.414213562373095e+00*PETSC_i,
193 40 : 2.000000000000000e+00 - 1.414213562373095e+00*PETSC_i};
194 40 : const PetscComplex /* m == k-1 */
195 40 : m1r5[4] = {-1.423367961376821e+02 - 1.385465094833037e+01*PETSC_i,
196 40 : -1.423367961376821e+02 + 1.385465094833037e+01*PETSC_i,
197 40 : 2.647367961376822e+02 - 4.814394493714596e+02*PETSC_i,
198 40 : 2.647367961376822e+02 + 4.814394493714596e+02*PETSC_i},
199 40 : m1q5[4] = { 5.203941240131764e+00 + 5.805856841805367e+00*PETSC_i,
200 40 : 5.203941240131764e+00 - 5.805856841805367e+00*PETSC_i,
201 40 : 6.796058759868242e+00 + 1.886649260140217e+00*PETSC_i,
202 40 : 6.796058759868242e+00 - 1.886649260140217e+00*PETSC_i},
203 40 : m1r4[3] = { 2.484269593165883e+01 + 7.460342395992306e+01*PETSC_i,
204 40 : 2.484269593165883e+01 - 7.460342395992306e+01*PETSC_i,
205 : -1.734353918633177e+02 },
206 40 : m1q4[3] = { 4.675757014491557e+00 + 3.913489560603711e+00*PETSC_i,
207 40 : 4.675757014491557e+00 - 3.913489560603711e+00*PETSC_i,
208 : 5.648485971016893e+00 },
209 40 : m1r3[2] = { 2.533333333333333e+01 - 2.733333333333333e+01*PETSC_i,
210 40 : 2.533333333333333e+01 + 2.733333333333333e+01*PETSC_i},
211 40 : m1q3[2] = { 4.000000000000000e+00 + 2.000000000000000e+00*PETSC_i,
212 40 : 4.000000000000000e+00 - 2.000000000000000e+00*PETSC_i};
213 40 : const PetscScalar /* m == k-1 */
214 40 : m1remain5[2] = { 2.000000000000000e-01, 9.800000000000000e+00},
215 40 : m1remain4[2] = {-2.500000000000000e-01, -7.750000000000000e+00},
216 40 : m1remain3[2] = { 3.333333333333333e-01, 5.666666666666667e+00},
217 40 : m1remain2[2] = {-0.5, -3.5},
218 40 : remain3[4] = {1.0/6.0, 1.0/2.0, 1, 1},
219 40 : remain2[3] = {1.0/2.0, 1, 1};
220 :
221 40 : PetscFunctionBegin;
222 40 : if (query) { /* query about buffer's size */
223 20 : if (m==k+1) {
224 2 : *remain = 0;
225 2 : *r = *q = k+1;
226 2 : PetscFunctionReturn(PETSC_SUCCESS); /* quick return */
227 : }
228 18 : if (m==k-1) {
229 18 : *remain = 2;
230 18 : if (k==5) *r = *q = 4;
231 4 : else if (k==4) *r = *q = 3;
232 0 : else if (k==3) *r = *q = 2;
233 0 : else if (k==2) *r = *q = 1;
234 : }
235 18 : if (m==0) {
236 0 : *r = *q = 0;
237 0 : *remain = k+1;
238 : }
239 : } else {
240 20 : if (m==k+1) {
241 2 : if (k==4) {
242 12 : for (i=0;i<5;i++) { r[i] = p1r4[i]; q[i] = p1q4[i]; }
243 0 : } else if (k==3) {
244 0 : for (i=0;i<4;i++) { r[i] = p1r3[i]; q[i] = p1q3[i]; }
245 0 : } else if (k==2) {
246 0 : for (i=0;i<3;i++) { r[i] = p1r2[i]; q[i] = p1q2[i]; }
247 0 : } else if (k==1) {
248 0 : for (i=0;i<2;i++) { r[i] = p1r1[i]; q[i] = p1q1[i]; }
249 : }
250 2 : PetscFunctionReturn(PETSC_SUCCESS); /* quick return */
251 : }
252 18 : if (m==k-1) {
253 18 : if (k==5) {
254 70 : for (i=0;i<4;i++) { r[i] = m1r5[i]; q[i] = m1q5[i]; }
255 42 : for (i=0;i<2;i++) remain[i] = m1remain5[i];
256 4 : } else if (k==4) {
257 16 : for (i=0;i<3;i++) { r[i] = m1r4[i]; q[i] = m1q4[i]; }
258 12 : for (i=0;i<2;i++) remain[i] = m1remain4[i];
259 0 : } else if (k==3) {
260 0 : for (i=0;i<2;i++) { r[i] = m1r3[i]; q[i] = m1q3[i]; remain[i] = m1remain3[i]; }
261 0 : } else if (k==2) {
262 0 : r[0] = -13.5; q[0] = 3;
263 0 : for (i=0;i<2;i++) remain[i] = m1remain2[i];
264 : }
265 : }
266 18 : if (m==0) {
267 0 : r = q = NULL;
268 0 : if (k==3) {
269 0 : for (i=0;i<4;i++) remain[i] = remain3[i];
270 0 : } else if (k==2) {
271 0 : for (i=0;i<3;i++) remain[i] = remain2[i];
272 : }
273 : }
274 : }
275 36 : PetscFunctionReturn(PETSC_SUCCESS);
276 : }
277 :
278 : /*
279 : * Product form coefficients.
280 : * If query, the function returns the size necessary to store the coefficients.
281 : */
282 40 : static PetscErrorCode getcoeffsproduct(PetscInt k,PetscInt m,PetscComplex *p,PetscComplex *q,PetscComplex *mult,PetscBool query)
283 : {
284 40 : PetscInt i;
285 40 : const PetscComplex /* m == k+1 */
286 40 : p1p4[4] = {-5.203941240131764e+00 + 5.805856841805367e+00*PETSC_i,
287 40 : -5.203941240131764e+00 - 5.805856841805367e+00*PETSC_i,
288 40 : -6.796058759868242e+00 + 1.886649260140217e+00*PETSC_i,
289 40 : -6.796058759868242e+00 - 1.886649260140217e+00*PETSC_i},
290 40 : p1q4[5] = { 3.655694325463550e+00 + 6.543736899360086e+00*PETSC_i,
291 40 : 3.655694325463550e+00 - 6.543736899360086e+00*PETSC_i,
292 : 6.286704751729261e+00 ,
293 40 : 5.700953298671832e+00 + 3.210265600308496e+00*PETSC_i,
294 40 : 5.700953298671832e+00 - 3.210265600308496e+00*PETSC_i},
295 40 : p1p3[3] = {-4.675757014491557e+00 + 3.913489560603711e+00*PETSC_i,
296 40 : -4.675757014491557e+00 - 3.913489560603711e+00*PETSC_i,
297 : -5.648485971016893e+00 },
298 40 : p1q3[4] = { 3.212806896871536e+00 + 4.773087433276636e+00*PETSC_i,
299 40 : 3.212806896871536e+00 - 4.773087433276636e+00*PETSC_i,
300 40 : 4.787193103128464e+00 + 1.567476416895212e+00*PETSC_i,
301 40 : 4.787193103128464e+00 - 1.567476416895212e+00*PETSC_i},
302 40 : p1p2[2] = {-4.00000000000000e+00 + 2.000000000000000e+00*PETSC_i,
303 40 : -4.00000000000000e+00 - 2.000000000000000e+00*PETSC_i},
304 40 : p1q2[3] = { 2.681082873627756e+00 + 3.050430199247411e+00*PETSC_i,
305 40 : 2.681082873627756e+00 - 3.050430199247411e+00*PETSC_i,
306 : 3.637834252744491e+00 },
307 40 : p1q1[2] = { 2.000000000000000e+00 + 1.414213562373095e+00*PETSC_i,
308 40 : 2.000000000000000e+00 - 1.414213562373095e+00*PETSC_i};
309 40 : const PetscComplex /* m == k-1 */
310 40 : m1p5[5] = {-3.655694325463550e+00 + 6.543736899360086e+00*PETSC_i,
311 40 : -3.655694325463550e+00 - 6.543736899360086e+00*PETSC_i,
312 : -6.286704751729261e+00 ,
313 40 : -5.700953298671832e+00 + 3.210265600308496e+00*PETSC_i,
314 40 : -5.700953298671832e+00 - 3.210265600308496e+00*PETSC_i},
315 40 : m1q5[4] = { 5.203941240131764e+00 + 5.805856841805367e+00*PETSC_i,
316 40 : 5.203941240131764e+00 - 5.805856841805367e+00*PETSC_i,
317 40 : 6.796058759868242e+00 + 1.886649260140217e+00*PETSC_i,
318 40 : 6.796058759868242e+00 - 1.886649260140217e+00*PETSC_i},
319 40 : m1p4[4] = {-3.212806896871536e+00 + 4.773087433276636e+00*PETSC_i,
320 40 : -3.212806896871536e+00 - 4.773087433276636e+00*PETSC_i,
321 40 : -4.787193103128464e+00 + 1.567476416895212e+00*PETSC_i,
322 40 : -4.787193103128464e+00 - 1.567476416895212e+00*PETSC_i},
323 40 : m1q4[3] = { 4.675757014491557e+00 + 3.913489560603711e+00*PETSC_i,
324 40 : 4.675757014491557e+00 - 3.913489560603711e+00*PETSC_i,
325 : 5.648485971016893e+00 },
326 40 : m1p3[3] = {-2.681082873627756e+00 + 3.050430199247411e+00*PETSC_i,
327 40 : -2.681082873627756e+00 - 3.050430199247411e+00*PETSC_i,
328 : -3.637834252744491e+00 },
329 40 : m1q3[2] = { 4.000000000000000e+00 + 2.000000000000000e+00*PETSC_i,
330 40 : 4.000000000000000e+00 - 2.000000000000001e+00*PETSC_i},
331 40 : m1p2[2] = {-2.000000000000000e+00 + 1.414213562373095e+00*PETSC_i,
332 40 : -2.000000000000000e+00 - 1.414213562373095e+00*PETSC_i};
333 :
334 40 : PetscFunctionBegin;
335 40 : if (query) {
336 20 : if (m == k+1) {
337 2 : *mult = 1;
338 2 : *p = k;
339 2 : *q = k+1;
340 2 : PetscFunctionReturn(PETSC_SUCCESS);
341 : }
342 18 : if (m==k-1) {
343 18 : *mult = 1;
344 18 : *p = k;
345 18 : *q = k-1;
346 : }
347 : } else {
348 20 : if (m == k+1) {
349 2 : *mult = PetscPowInt(-1,m);
350 2 : *mult *= m;
351 2 : if (k==4) {
352 10 : for (i=0;i<4;i++) { p[i] = p1p4[i]; q[i] = p1q4[i]; }
353 2 : q[4] = p1q4[4];
354 0 : } else if (k==3) {
355 0 : for (i=0;i<3;i++) { p[i] = p1p3[i]; q[i] = p1q3[i]; }
356 0 : q[3] = p1q3[3];
357 0 : } else if (k==2) {
358 0 : for (i=0;i<2;i++) { p[i] = p1p2[i]; q[i] = p1q2[i]; }
359 0 : q[2] = p1q2[2];
360 0 : } else if (k==1) {
361 0 : p[0] = -3;
362 0 : for (i=0;i<2;i++) q[i] = p1q1[i];
363 : }
364 2 : PetscFunctionReturn(PETSC_SUCCESS);
365 : }
366 18 : if (m==k-1) {
367 18 : *mult = PetscPowInt(-1,m);
368 18 : *mult /= k;
369 18 : if (k==5) {
370 70 : for (i=0;i<4;i++) { p[i] = m1p5[i]; q[i] = m1q5[i]; }
371 14 : p[4] = m1p5[4];
372 4 : } else if (k==4) {
373 16 : for (i=0;i<3;i++) { p[i] = m1p4[i]; q[i] = m1q4[i]; }
374 4 : p[3] = m1p4[3];
375 0 : } else if (k==3) {
376 0 : for (i=0;i<2;i++) { p[i] = m1p3[i]; q[i] = m1q3[i]; }
377 0 : p[2] = m1p3[2];
378 0 : } else if (k==2) {
379 0 : for (i=0;i<2;i++) p[i] = m1p2[i];
380 0 : q[0] = 3;
381 : }
382 : }
383 : }
384 36 : PetscFunctionReturn(PETSC_SUCCESS);
385 : }
386 : #endif /* PETSC_HAVE_COMPLEX */
387 :
388 : #if defined(PETSC_USE_COMPLEX)
389 : static PetscErrorCode getisreal(PetscInt n,PetscComplex *a,PetscBool *result)
390 : {
391 : PetscInt i;
392 :
393 : PetscFunctionBegin;
394 : *result=PETSC_TRUE;
395 : for (i=0;i<n&&*result;i++) {
396 : if (PetscImaginaryPartComplex(a[i])) *result=PETSC_FALSE;
397 : }
398 : PetscFunctionReturn(PETSC_SUCCESS);
399 : }
400 : #endif
401 :
402 : /*
403 : * Matrix exponential implementation based on algorithm and matlab code by Stefan Guettel
404 : * and Yuji Nakatsukasa
405 : *
406 : * Stefan Guettel and Yuji Nakatsukasa, "Scaled and Squared Subdiagonal Pade
407 : * Approximation for the Matrix Exponential",
408 : * SIAM J. Matrix Anal. Appl. 37(1):145-170, 2016.
409 : * https://doi.org/10.1137/15M1027553
410 : */
411 48 : static PetscErrorCode FNEvaluateFunctionMat_Exp_GuettelNakatsukasa(FN fn,Mat A,Mat B)
412 : {
413 : #if !defined(PETSC_HAVE_COMPLEX)
414 : PetscFunctionBegin;
415 : SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"This function requires C99 or C++ complex support");
416 : #else
417 48 : PetscInt i,j,n_,s,k,m,mod;
418 48 : PetscBLASInt n=0,n2=0,irsize=0,rsizediv2,ipsize=0,iremainsize=0,info,*piv,minlen,lwork=0,one=1;
419 48 : PetscReal nrm,shift=0.0;
420 : #if defined(PETSC_USE_COMPLEX)
421 : PetscReal *rwork=NULL;
422 : #endif
423 48 : PetscComplex *As,*RR,*RR2,*expmA,*expmA2,*Maux,*Maux2,rsize,*r,psize,*p,remainsize,*remainterm,*rootp,*rootq,mult=0.0,scale,cone=1.0,czero=0.0,*aux;
424 48 : PetscScalar *Ba,*Ba2,*sMaux,*wr,*wi,expshift,sone=1.0,szero=0.0,*saux;
425 48 : const PetscScalar *Aa;
426 48 : PetscBool isreal,flg;
427 48 : PetscBLASInt query=-1;
428 48 : PetscScalar work1,*work;
429 :
430 48 : PetscFunctionBegin;
431 48 : PetscCall(MatGetSize(A,&n_,NULL));
432 48 : PetscCall(PetscBLASIntCast(n_,&n));
433 48 : PetscCall(MatDenseGetArrayRead(A,&Aa));
434 48 : PetscCall(MatDenseGetArray(B,&Ba));
435 48 : Ba2 = Ba;
436 48 : PetscCall(PetscBLASIntCast(n*n,&n2));
437 :
438 48 : PetscCall(PetscMalloc2(n2,&sMaux,n2,&Maux));
439 48 : Maux2 = Maux;
440 48 : PetscCall(PetscOptionsGetReal(NULL,NULL,"-fn_expm_estimated_eig",&shift,&flg));
441 48 : if (!flg) {
442 48 : PetscCall(PetscMalloc2(n,&wr,n,&wi));
443 48 : PetscCall(PetscArraycpy(sMaux,Aa,n2));
444 : /* estimate rightmost eigenvalue and shift A with it */
445 : #if !defined(PETSC_USE_COMPLEX)
446 48 : PetscCallBLAS("LAPACKgeev",LAPACKgeev_("N","N",&n,sMaux,&n,wr,wi,NULL,&n,NULL,&n,&work1,&query,&info));
447 48 : SlepcCheckLapackInfo("geev",info);
448 48 : PetscCall(PetscBLASIntCast((PetscInt)work1,&lwork));
449 48 : PetscCall(PetscMalloc1(lwork,&work));
450 48 : PetscCallBLAS("LAPACKgeev",LAPACKgeev_("N","N",&n,sMaux,&n,wr,wi,NULL,&n,NULL,&n,work,&lwork,&info));
451 48 : PetscCall(PetscFree(work));
452 : #else
453 : PetscCall(PetscArraycpy(Maux,Aa,n2));
454 : PetscCallBLAS("LAPACKgeev",LAPACKgeev_("N","N",&n,Maux,&n,wr,NULL,&n,NULL,&n,&work1,&query,rwork,&info));
455 : SlepcCheckLapackInfo("geev",info);
456 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPart(work1),&lwork));
457 : PetscCall(PetscMalloc2(2*n,&rwork,lwork,&work));
458 : PetscCallBLAS("LAPACKgeev",LAPACKgeev_("N","N",&n,Maux,&n,wr,NULL,&n,NULL,&n,work,&lwork,rwork,&info));
459 : PetscCall(PetscFree2(rwork,work));
460 : #endif
461 48 : SlepcCheckLapackInfo("geev",info);
462 48 : PetscCall(PetscLogFlops(25.0*n*n*n+(n*n*n)/3.0+1.0*n*n*n));
463 :
464 48 : shift = PetscRealPart(wr[0]);
465 1360 : for (i=1;i<n;i++) {
466 1312 : if (PetscRealPart(wr[i]) > shift) shift = PetscRealPart(wr[i]);
467 : }
468 48 : PetscCall(PetscFree2(wr,wi));
469 : }
470 : /* shift so that largest real part is (about) 0 */
471 48 : PetscCall(PetscArraycpy(sMaux,Aa,n2));
472 48 : if (shift) {
473 1408 : for (i=0;i<n;i++) sMaux[i+i*n] -= shift;
474 48 : PetscCall(PetscLogFlops(1.0*n));
475 : }
476 : #if defined(PETSC_USE_COMPLEX)
477 : PetscCall(PetscArraycpy(Maux,Aa,n2));
478 : if (shift) {
479 : for (i=0;i<n;i++) Maux[i+i*n] -= shift;
480 : PetscCall(PetscLogFlops(1.0*n));
481 : }
482 : #endif
483 :
484 : /* estimate norm(A) and select the scaling factor */
485 48 : nrm = LAPACKlange_("O",&n,&n,sMaux,&n,NULL);
486 48 : PetscCall(PetscLogFlops(1.0*n*n));
487 48 : PetscCall(sexpm_params(nrm,&s,&k,&m));
488 48 : if (s==0 && k==1 && m==0) { /* exp(A) = I+A to eps! */
489 8 : if (shift) expshift = PetscExpReal(shift);
490 88 : for (i=0;i<n;i++) sMaux[i+i*n] += 1.0;
491 8 : if (shift) {
492 8 : PetscCallBLAS("BLASscal",BLASscal_(&n2,&expshift,sMaux,&one));
493 8 : PetscCall(PetscLogFlops(1.0*(n+n2)));
494 0 : } else PetscCall(PetscLogFlops(1.0*n));
495 8 : PetscCall(PetscArraycpy(Ba,sMaux,n2));
496 8 : PetscCall(PetscFree2(sMaux,Maux));
497 8 : PetscCall(MatDenseRestoreArrayRead(A,&Aa));
498 8 : PetscCall(MatDenseRestoreArray(B,&Ba));
499 8 : PetscFunctionReturn(PETSC_SUCCESS); /* quick return */
500 : }
501 :
502 40 : PetscCall(PetscMalloc4(n2,&expmA,n2,&As,n2,&RR,n,&piv));
503 40 : expmA2 = expmA; RR2 = RR;
504 : /* scale matrix */
505 : #if !defined(PETSC_USE_COMPLEX)
506 118440 : for (i=0;i<n2;i++) {
507 118400 : As[i] = sMaux[i];
508 : }
509 : #else
510 : PetscCall(PetscArraycpy(As,sMaux,n2));
511 : #endif
512 40 : scale = 1.0/PetscPowRealInt(2.0,s);
513 40 : PetscCallBLAS("BLASCOMPLEXscal",BLASCOMPLEXscal_(&n2,&scale,As,&one));
514 40 : PetscCall(SlepcLogFlopsComplex(1.0*n2));
515 :
516 : /* evaluate Pade approximant (partial fraction or product form) */
517 40 : if (fn->method==3 || !m) { /* partial fraction */
518 20 : PetscCall(getcoeffs(k,m,&rsize,&psize,&remainsize,PETSC_TRUE));
519 20 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPartComplex(rsize),&irsize));
520 20 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPartComplex(psize),&ipsize));
521 20 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPartComplex(remainsize),&iremainsize));
522 20 : PetscCall(PetscMalloc3(irsize,&r,ipsize,&p,iremainsize,&remainterm));
523 20 : PetscCall(getcoeffs(k,m,r,p,remainterm,PETSC_FALSE));
524 :
525 20 : PetscCall(PetscArrayzero(expmA,n2));
526 : #if !defined(PETSC_USE_COMPLEX)
527 20 : isreal = PETSC_TRUE;
528 : #else
529 : PetscCall(getisreal(n2,Maux,&isreal));
530 : #endif
531 20 : if (isreal) {
532 20 : rsizediv2 = irsize/2;
533 56 : for (i=0;i<rsizediv2;i++) { /* use partial fraction to get R(As) */
534 36 : PetscCall(PetscArraycpy(Maux,As,n2));
535 36 : PetscCall(PetscArrayzero(RR,n2));
536 1276 : for (j=0;j<n;j++) {
537 1240 : Maux[j+j*n] -= p[2*i];
538 1240 : RR[j+j*n] = r[2*i];
539 : }
540 36 : PetscCallBLAS("LAPACKCOMPLEXgesv",LAPACKCOMPLEXgesv_(&n,&n,Maux,&n,piv,RR,&n,&info));
541 36 : SlepcCheckLapackInfo("gesv",info);
542 118036 : for (j=0;j<n2;j++) {
543 118000 : expmA[j] += RR[j] + PetscConj(RR[j]);
544 : }
545 : /* loop(n) + gesv + loop(n2) */
546 36 : PetscCall(SlepcLogFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)+2.0*n2));
547 : }
548 :
549 20 : mod = ipsize % 2;
550 20 : if (mod) {
551 6 : PetscCall(PetscArraycpy(Maux,As,n2));
552 6 : PetscCall(PetscArrayzero(RR,n2));
553 66 : for (j=0;j<n;j++) {
554 60 : Maux[j+j*n] -= p[ipsize-1];
555 60 : RR[j+j*n] = r[irsize-1];
556 : }
557 6 : PetscCallBLAS("LAPACKCOMPLEXgesv",LAPACKCOMPLEXgesv_(&n,&n,Maux,&n,piv,RR,&n,&info));
558 6 : SlepcCheckLapackInfo("gesv",info);
559 606 : for (j=0;j<n2;j++) {
560 600 : expmA[j] += RR[j];
561 : }
562 6 : PetscCall(SlepcLogFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)+1.0*n2));
563 : }
564 : } else { /* complex */
565 : for (i=0;i<irsize;i++) { /* use partial fraction to get R(As) */
566 : PetscCall(PetscArraycpy(Maux,As,n2));
567 : PetscCall(PetscArrayzero(RR,n2));
568 : for (j=0;j<n;j++) {
569 : Maux[j+j*n] -= p[i];
570 : RR[j+j*n] = r[i];
571 : }
572 : PetscCallBLAS("LAPACKCOMPLEXgesv",LAPACKCOMPLEXgesv_(&n,&n,Maux,&n,piv,RR,&n,&info));
573 : SlepcCheckLapackInfo("gesv",info);
574 : for (j=0;j<n2;j++) {
575 : expmA[j] += RR[j];
576 : }
577 : PetscCall(SlepcLogFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)+1.0*n2));
578 : }
579 : }
580 56 : for (i=0;i<iremainsize;i++) {
581 36 : if (!i) {
582 18 : PetscCall(PetscArrayzero(RR,n2));
583 638 : for (j=0;j<n;j++) {
584 620 : RR[j+j*n] = remainterm[iremainsize-1];
585 : }
586 : } else {
587 18 : PetscCall(PetscArraycpy(RR,As,n2));
588 18 : for (j=1;j<i;j++) {
589 0 : PetscCallBLAS("BLASCOMPLEXgemm",BLASCOMPLEXgemm_("N","N",&n,&n,&n,&cone,RR,&n,RR,&n,&czero,Maux,&n));
590 0 : SWAP(RR,Maux,aux);
591 0 : PetscCall(SlepcLogFlopsComplex(2.0*n*n*n));
592 : }
593 18 : PetscCallBLAS("BLASCOMPLEXscal",BLASCOMPLEXscal_(&n2,&remainterm[iremainsize-1-i],RR,&one));
594 18 : PetscCall(SlepcLogFlopsComplex(1.0*n2));
595 : }
596 118036 : for (j=0;j<n2;j++) {
597 118000 : expmA[j] += RR[j];
598 : }
599 36 : PetscCall(SlepcLogFlopsComplex(1.0*n2));
600 : }
601 20 : PetscCall(PetscFree3(r,p,remainterm));
602 : } else { /* product form, default */
603 20 : PetscCall(getcoeffsproduct(k,m,&rsize,&psize,&mult,PETSC_TRUE));
604 20 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPartComplex(rsize),&irsize));
605 20 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPartComplex(psize),&ipsize));
606 20 : PetscCall(PetscMalloc2(irsize,&rootp,ipsize,&rootq));
607 20 : PetscCall(getcoeffsproduct(k,m,rootp,rootq,&mult,PETSC_FALSE));
608 :
609 20 : PetscCall(PetscArrayzero(expmA,n2));
610 660 : for (i=0;i<n;i++) { /* initialize */
611 640 : expmA[i+i*n] = 1.0;
612 : }
613 20 : minlen = PetscMin(irsize,ipsize);
614 96 : for (i=0;i<minlen;i++) {
615 76 : PetscCall(PetscArraycpy(RR,As,n2));
616 2596 : for (j=0;j<n;j++) {
617 2520 : RR[j+j*n] -= rootp[i];
618 : }
619 76 : PetscCallBLAS("BLASCOMPLEXgemm",BLASCOMPLEXgemm_("N","N",&n,&n,&n,&cone,RR,&n,expmA,&n,&czero,Maux,&n));
620 76 : SWAP(expmA,Maux,aux);
621 76 : PetscCall(PetscArraycpy(RR,As,n2));
622 2596 : for (j=0;j<n;j++) {
623 2520 : RR[j+j*n] -= rootq[i];
624 : }
625 76 : PetscCallBLAS("LAPACKCOMPLEXgesv",LAPACKCOMPLEXgesv_(&n,&n,RR,&n,piv,expmA,&n,&info));
626 76 : SlepcCheckLapackInfo("gesv",info);
627 : /* loop(n) + gemm + loop(n) + gesv */
628 76 : PetscCall(SlepcLogFlopsComplex(1.0*n+(2.0*n*n*n)+1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)));
629 : }
630 : /* extra numerator */
631 38 : for (i=minlen;i<irsize;i++) {
632 18 : PetscCall(PetscArraycpy(RR,As,n2));
633 638 : for (j=0;j<n;j++) {
634 620 : RR[j+j*n] -= rootp[i];
635 : }
636 18 : PetscCallBLAS("BLASCOMPLEXgemm",BLASCOMPLEXgemm_("N","N",&n,&n,&n,&cone,RR,&n,expmA,&n,&czero,Maux,&n));
637 18 : SWAP(expmA,Maux,aux);
638 18 : PetscCall(SlepcLogFlopsComplex(1.0*n+2.0*n*n*n));
639 : }
640 : /* extra denominator */
641 22 : for (i=minlen;i<ipsize;i++) {
642 2 : PetscCall(PetscArraycpy(RR,As,n2));
643 22 : for (j=0;j<n;j++) RR[j+j*n] -= rootq[i];
644 2 : PetscCallBLAS("LAPACKCOMPLEXgesv",LAPACKCOMPLEXgesv_(&n,&n,RR,&n,piv,expmA,&n,&info));
645 2 : SlepcCheckLapackInfo("gesv",info);
646 2 : PetscCall(SlepcLogFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)));
647 : }
648 20 : PetscCallBLAS("BLASCOMPLEXscal",BLASCOMPLEXscal_(&n2,&mult,expmA,&one));
649 20 : PetscCall(SlepcLogFlopsComplex(1.0*n2));
650 20 : PetscCall(PetscFree2(rootp,rootq));
651 : }
652 :
653 : #if !defined(PETSC_USE_COMPLEX)
654 118440 : for (i=0;i<n2;i++) {
655 118400 : Ba2[i] = PetscRealPartComplex(expmA[i]);
656 : }
657 : #else
658 : PetscCall(PetscArraycpy(Ba2,expmA,n2));
659 : #endif
660 :
661 : /* perform repeated squaring */
662 196 : for (i=0;i<s;i++) { /* final squaring */
663 156 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,Ba2,&n,Ba2,&n,&szero,sMaux,&n));
664 156 : SWAP(Ba2,sMaux,saux);
665 156 : PetscCall(PetscLogFlops(2.0*n*n*n));
666 : }
667 40 : if (Ba2!=Ba) {
668 4 : PetscCall(PetscArraycpy(Ba,Ba2,n2));
669 4 : sMaux = Ba2;
670 : }
671 40 : if (shift) {
672 40 : expshift = PetscExpReal(shift);
673 40 : PetscCallBLAS("BLASscal",BLASscal_(&n2,&expshift,Ba,&one));
674 40 : PetscCall(PetscLogFlops(1.0*n2));
675 : }
676 :
677 : /* restore pointers */
678 40 : Maux = Maux2; expmA = expmA2; RR = RR2;
679 40 : PetscCall(PetscFree2(sMaux,Maux));
680 40 : PetscCall(PetscFree4(expmA,As,RR,piv));
681 40 : PetscCall(MatDenseRestoreArrayRead(A,&Aa));
682 40 : PetscCall(MatDenseRestoreArray(B,&Ba));
683 40 : PetscFunctionReturn(PETSC_SUCCESS);
684 : #endif
685 : }
686 :
687 : #define SMALLN 100
688 :
689 : /*
690 : * Function needed to compute optimal parameters (required workspace is 3*n*n)
691 : */
692 432 : static PetscInt ell(PetscBLASInt n,PetscScalar *A,PetscReal coeff,PetscInt m,PetscScalar *work,PetscRandom rand)
693 : {
694 432 : PetscScalar *Ascaled=work;
695 432 : PetscReal nrm,alpha,beta,rwork[1];
696 432 : PetscInt t;
697 432 : PetscBLASInt i,j;
698 :
699 432 : PetscFunctionBegin;
700 432 : beta = PetscPowReal(coeff,1.0/(2*m+1));
701 11282 : for (i=0;i<n;i++)
702 654502 : for (j=0;j<n;j++)
703 643652 : Ascaled[i+j*n] = beta*PetscAbsScalar(A[i+j*n]);
704 432 : nrm = LAPACKlange_("O",&n,&n,A,&n,rwork);
705 432 : PetscCall(PetscLogFlops(2.0*n*n));
706 432 : PetscCall(SlepcNormAm(n,Ascaled,2*m+1,work+n*n,rand,&alpha));
707 432 : alpha /= nrm;
708 432 : t = PetscMax((PetscInt)PetscCeilReal(PetscLogReal(2.0*alpha/PETSC_MACHINE_EPSILON)/PetscLogReal(2.0)/(2*m)),0);
709 432 : PetscFunctionReturn(t);
710 : }
711 :
712 : /*
713 : * Compute scaling parameter (s) and order of Pade approximant (m) (required workspace is 4*n*n)
714 : */
715 432 : static PetscErrorCode expm_params(PetscInt n,PetscScalar **Apowers,PetscInt *s,PetscInt *m,PetscScalar *work)
716 : {
717 432 : PetscScalar sfactor,sone=1.0,szero=0.0,*A=Apowers[0],*Ascaled;
718 432 : PetscReal d4,d6,d8,d10,eta1,eta3,eta4,eta5,rwork[1];
719 432 : PetscBLASInt n_=0,n2,one=1;
720 432 : PetscRandom rand;
721 432 : const PetscReal coeff[5] = { 9.92063492063492e-06, 9.94131285136576e-11, /* backward error function */
722 : 2.22819456055356e-16, 1.69079293431187e-22, 8.82996160201868e-36 };
723 432 : const PetscReal theta[5] = { 1.495585217958292e-002, /* m = 3 */
724 : 2.539398330063230e-001, /* m = 5 */
725 : 9.504178996162932e-001, /* m = 7 */
726 : 2.097847961257068e+000, /* m = 9 */
727 : 5.371920351148152e+000 }; /* m = 13 */
728 :
729 432 : PetscFunctionBegin;
730 432 : *s = 0;
731 432 : *m = 13;
732 432 : PetscCall(PetscBLASIntCast(n,&n_));
733 432 : PetscCall(PetscRandomCreate(PETSC_COMM_SELF,&rand));
734 432 : d4 = PetscPowReal(LAPACKlange_("O",&n_,&n_,Apowers[2],&n_,rwork),1.0/4.0);
735 432 : if (d4==0.0) { /* safeguard for the case A = 0 */
736 0 : *m = 3;
737 0 : goto done;
738 : }
739 432 : d6 = PetscPowReal(LAPACKlange_("O",&n_,&n_,Apowers[3],&n_,rwork),1.0/6.0);
740 432 : PetscCall(PetscLogFlops(2.0*n*n));
741 432 : eta1 = PetscMax(d4,d6);
742 432 : if (eta1<=theta[0] && !ell(n_,A,coeff[0],3,work,rand)) {
743 82 : *m = 3;
744 82 : goto done;
745 : }
746 350 : if (eta1<=theta[1] && !ell(n_,A,coeff[1],5,work,rand)) {
747 111 : *m = 5;
748 111 : goto done;
749 : }
750 239 : if (n<SMALLN) {
751 225 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[2],&n_,Apowers[2],&n_,&szero,work,&n_));
752 225 : d8 = PetscPowReal(LAPACKlange_("O",&n_,&n_,work,&n_,rwork),1.0/8.0);
753 225 : PetscCall(PetscLogFlops(2.0*n*n*n+1.0*n*n));
754 : } else {
755 14 : PetscCall(SlepcNormAm(n_,Apowers[2],2,work,rand,&d8));
756 14 : d8 = PetscPowReal(d8,1.0/8.0);
757 : }
758 239 : eta3 = PetscMax(d6,d8);
759 239 : if (eta3<=theta[2] && !ell(n_,A,coeff[2],7,work,rand)) {
760 13 : *m = 7;
761 13 : goto done;
762 : }
763 226 : if (eta3<=theta[3] && !ell(n_,A,coeff[3],9,work,rand)) {
764 18 : *m = 9;
765 18 : goto done;
766 : }
767 208 : if (n<SMALLN) {
768 194 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[2],&n_,Apowers[3],&n_,&szero,work,&n_));
769 194 : d10 = PetscPowReal(LAPACKlange_("O",&n_,&n_,work,&n_,rwork),1.0/10.0);
770 194 : PetscCall(PetscLogFlops(2.0*n*n*n+1.0*n*n));
771 : } else {
772 14 : PetscCall(SlepcNormAm(n_,Apowers[1],5,work,rand,&d10));
773 14 : d10 = PetscPowReal(d10,1.0/10.0);
774 : }
775 208 : eta4 = PetscMax(d8,d10);
776 208 : eta5 = PetscMin(eta3,eta4);
777 208 : *s = PetscMax((PetscInt)PetscCeilReal(PetscLogReal(eta5/theta[4])/PetscLogReal(2.0)),0);
778 208 : if (*s) {
779 74 : Ascaled = work+3*n*n;
780 74 : n2 = n_*n_;
781 74 : PetscCallBLAS("BLAScopy",BLAScopy_(&n2,A,&one,Ascaled,&one));
782 74 : sfactor = PetscPowRealInt(2.0,-(*s));
783 74 : PetscCallBLAS("BLASscal",BLASscal_(&n2,&sfactor,Ascaled,&one));
784 74 : PetscCall(PetscLogFlops(1.0*n*n));
785 : } else Ascaled = A;
786 208 : *s += ell(n_,Ascaled,coeff[4],13,work,rand);
787 432 : done:
788 432 : PetscCall(PetscRandomDestroy(&rand));
789 432 : PetscFunctionReturn(PETSC_SUCCESS);
790 : }
791 :
792 : /*
793 : * Matrix exponential implementation based on algorithm and matlab code by N. Higham and co-authors
794 : *
795 : * N. J. Higham, "The scaling and squaring method for the matrix exponential
796 : * revisited", SIAM J. Matrix Anal. Appl. 26(4):1179-1193, 2005.
797 : */
798 432 : PetscErrorCode FNEvaluateFunctionMat_Exp_Higham(FN fn,Mat A,Mat B)
799 : {
800 432 : PetscBLASInt n_=0,n2,*ipiv,info,one=1;
801 432 : PetscInt n,m,j,s;
802 432 : PetscScalar scale,smone=-1.0,sone=1.0,stwo=2.0,szero=0.0;
803 432 : PetscScalar *Ba,*Apowers[5],*Q,*P,*W,*work,*aux;
804 432 : const PetscScalar *Aa,*c;
805 432 : const PetscScalar c3[4] = { 120, 60, 12, 1 };
806 432 : const PetscScalar c5[6] = { 30240, 15120, 3360, 420, 30, 1 };
807 432 : const PetscScalar c7[8] = { 17297280, 8648640, 1995840, 277200, 25200, 1512, 56, 1 };
808 432 : const PetscScalar c9[10] = { 17643225600.0, 8821612800.0, 2075673600, 302702400, 30270240,
809 : 2162160, 110880, 3960, 90, 1 };
810 432 : const PetscScalar c13[14] = { 64764752532480000.0, 32382376266240000.0, 7771770303897600.0,
811 : 1187353796428800.0, 129060195264000.0, 10559470521600.0,
812 : 670442572800.0, 33522128640.0, 1323241920.0,
813 : 40840800, 960960, 16380, 182, 1 };
814 :
815 432 : PetscFunctionBegin;
816 432 : PetscCall(MatDenseGetArrayRead(A,&Aa));
817 432 : PetscCall(MatDenseGetArray(B,&Ba));
818 432 : PetscCall(MatGetSize(A,&n,NULL));
819 432 : PetscCall(PetscBLASIntCast(n,&n_));
820 432 : n2 = n_*n_;
821 432 : PetscCall(PetscMalloc2(8*n*n,&work,n,&ipiv));
822 :
823 : /* Matrix powers */
824 432 : Apowers[0] = work; /* Apowers[0] = A */
825 432 : Apowers[1] = Apowers[0] + n*n; /* Apowers[1] = A^2 */
826 432 : Apowers[2] = Apowers[1] + n*n; /* Apowers[2] = A^4 */
827 432 : Apowers[3] = Apowers[2] + n*n; /* Apowers[3] = A^6 */
828 432 : Apowers[4] = Apowers[3] + n*n; /* Apowers[4] = A^8 */
829 :
830 432 : PetscCall(PetscArraycpy(Apowers[0],Aa,n2));
831 432 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[0],&n_,Apowers[0],&n_,&szero,Apowers[1],&n_));
832 432 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[1],&n_,Apowers[1],&n_,&szero,Apowers[2],&n_));
833 432 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[1],&n_,Apowers[2],&n_,&szero,Apowers[3],&n_));
834 432 : PetscCall(PetscLogFlops(6.0*n*n*n));
835 :
836 : /* Compute scaling parameter and order of Pade approximant */
837 432 : PetscCall(expm_params(n,Apowers,&s,&m,Apowers[4]));
838 :
839 432 : if (s) { /* rescale */
840 390 : for (j=0;j<4;j++) {
841 312 : scale = PetscPowRealInt(2.0,-PetscMax(2*j,1)*s);
842 312 : PetscCallBLAS("BLASscal",BLASscal_(&n2,&scale,Apowers[j],&one));
843 : }
844 78 : PetscCall(PetscLogFlops(4.0*n*n));
845 : }
846 :
847 : /* Evaluate the Pade approximant */
848 432 : switch (m) {
849 : case 3: c = c3; break;
850 111 : case 5: c = c5; break;
851 13 : case 7: c = c7; break;
852 18 : case 9: c = c9; break;
853 208 : case 13: c = c13; break;
854 0 : default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong value of m %" PetscInt_FMT,m);
855 : }
856 432 : P = Ba;
857 432 : Q = Apowers[4] + n*n;
858 432 : W = Q + n*n;
859 432 : switch (m) {
860 224 : case 3:
861 : case 5:
862 : case 7:
863 : case 9:
864 224 : if (m==9) PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[1],&n_,Apowers[3],&n_,&szero,Apowers[4],&n_));
865 224 : PetscCall(PetscArrayzero(P,n2));
866 224 : PetscCall(PetscArrayzero(Q,n2));
867 2526 : for (j=0;j<n;j++) {
868 2302 : P[j+j*n] = c[1];
869 2302 : Q[j+j*n] = c[0];
870 : }
871 639 : for (j=m;j>=3;j-=2) {
872 415 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[j],Apowers[(j+1)/2-1],&one,P,&one));
873 415 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[j-1],Apowers[(j+1)/2-1],&one,Q,&one));
874 415 : PetscCall(PetscLogFlops(4.0*n*n));
875 : }
876 224 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[0],&n_,P,&n_,&szero,W,&n_));
877 224 : PetscCall(PetscLogFlops(2.0*n*n*n));
878 432 : SWAP(P,W,aux);
879 : break;
880 208 : case 13:
881 : /* P = A*(Apowers[3]*(c[13]*Apowers[3] + c[11]*Apowers[2] + c[9]*Apowers[1])
882 : + c[7]*Apowers[3] + c[5]*Apowers[2] + c[3]*Apowers[1] + c[1]*I) */
883 208 : PetscCallBLAS("BLAScopy",BLAScopy_(&n2,Apowers[3],&one,P,&one));
884 208 : PetscCallBLAS("BLASscal",BLASscal_(&n2,&c[13],P,&one));
885 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[11],Apowers[2],&one,P,&one));
886 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[9],Apowers[1],&one,P,&one));
887 208 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[3],&n_,P,&n_,&szero,W,&n_));
888 208 : PetscCall(PetscLogFlops(5.0*n*n+2.0*n*n*n));
889 208 : PetscCall(PetscArrayzero(P,n2));
890 8756 : for (j=0;j<n;j++) P[j+j*n] = c[1];
891 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[7],Apowers[3],&one,P,&one));
892 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[5],Apowers[2],&one,P,&one));
893 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[3],Apowers[1],&one,P,&one));
894 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&sone,P,&one,W,&one));
895 208 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[0],&n_,W,&n_,&szero,P,&n_));
896 208 : PetscCall(PetscLogFlops(7.0*n*n+2.0*n*n*n));
897 : /* Q = Apowers[3]*(c[12]*Apowers[3] + c[10]*Apowers[2] + c[8]*Apowers[1])
898 : + c[6]*Apowers[3] + c[4]*Apowers[2] + c[2]*Apowers[1] + c[0]*I */
899 208 : PetscCallBLAS("BLAScopy",BLAScopy_(&n2,Apowers[3],&one,Q,&one));
900 208 : PetscCallBLAS("BLASscal",BLASscal_(&n2,&c[12],Q,&one));
901 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[10],Apowers[2],&one,Q,&one));
902 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[8],Apowers[1],&one,Q,&one));
903 208 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[3],&n_,Q,&n_,&szero,W,&n_));
904 208 : PetscCall(PetscLogFlops(5.0*n*n+2.0*n*n*n));
905 208 : PetscCall(PetscArrayzero(Q,n2));
906 8756 : for (j=0;j<n;j++) Q[j+j*n] = c[0];
907 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[6],Apowers[3],&one,Q,&one));
908 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[4],Apowers[2],&one,Q,&one));
909 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[2],Apowers[1],&one,Q,&one));
910 208 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&sone,W,&one,Q,&one));
911 208 : PetscCall(PetscLogFlops(7.0*n*n));
912 : break;
913 0 : default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong value of m %" PetscInt_FMT,m);
914 : }
915 432 : PetscCallBLAS("BLASaxpy",BLASaxpy_(&n2,&smone,P,&one,Q,&one));
916 432 : PetscCallBLAS("LAPACKgesv",LAPACKgesv_(&n_,&n_,Q,&n_,ipiv,P,&n_,&info));
917 432 : SlepcCheckLapackInfo("gesv",info);
918 432 : PetscCallBLAS("BLASscal",BLASscal_(&n2,&stwo,P,&one));
919 11282 : for (j=0;j<n;j++) P[j+j*n] += 1.0;
920 432 : PetscCall(PetscLogFlops(2.0*n*n*n/3.0+4.0*n*n));
921 :
922 : /* Squaring */
923 1092 : for (j=1;j<=s;j++) {
924 660 : PetscCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,P,&n_,P,&n_,&szero,W,&n_));
925 660 : SWAP(P,W,aux);
926 : }
927 432 : if (P!=Ba) PetscCall(PetscArraycpy(Ba,P,n2));
928 432 : PetscCall(PetscLogFlops(2.0*n*n*n*s));
929 :
930 432 : PetscCall(PetscFree2(work,ipiv));
931 432 : PetscCall(MatDenseRestoreArrayRead(A,&Aa));
932 432 : PetscCall(MatDenseRestoreArray(B,&Ba));
933 432 : PetscFunctionReturn(PETSC_SUCCESS);
934 : }
935 :
936 : #if defined(PETSC_HAVE_CUDA)
937 : #include "../src/sys/classes/fn/impls/cuda/fnutilcuda.h"
938 : #include <slepccublas.h>
939 :
940 : PetscErrorCode FNEvaluateFunctionMat_Exp_Pade_CUDA(FN fn,Mat A,Mat B)
941 : {
942 : PetscBLASInt n=0,ld,ld2,*d_ipiv,*d_info,info,one=1;
943 : PetscInt m,k,sexp;
944 : PetscBool odd;
945 : const PetscInt p=MAX_PADE;
946 : PetscReal c[MAX_PADE+1],s;
947 : PetscScalar scale,smone=-1.0,sone=1.0,stwo=2.0,szero=0.0;
948 : const PetscScalar *Aa;
949 : PetscScalar *d_Ba,*d_As,*d_A2,*d_Q,*d_P,*d_W,*aux,**ppP,**d_ppP,**ppQ,**d_ppQ;
950 : cublasHandle_t cublasv2handle;
951 :
952 : PetscFunctionBegin;
953 : PetscCall(PetscDeviceInitialize(PETSC_DEVICE_CUDA)); /* For CUDA event timers */
954 : PetscCall(PetscCUBLASGetHandle(&cublasv2handle));
955 : PetscCall(MatGetSize(A,&m,NULL));
956 : PetscCall(PetscBLASIntCast(m,&n));
957 : ld = n;
958 : ld2 = ld*ld;
959 : if (A==B) {
960 : PetscCallCUDA(cudaMalloc((void **)&d_As,sizeof(PetscScalar)*m*m));
961 : PetscCall(MatDenseCUDAGetArrayRead(A,&Aa));
962 : PetscCallCUDA(cudaMemcpy(d_As,Aa,sizeof(PetscScalar)*ld2,cudaMemcpyDeviceToDevice));
963 : PetscCall(MatDenseCUDARestoreArrayRead(A,&Aa));
964 : } else PetscCall(MatDenseCUDAGetArrayRead(A,(const PetscScalar**)&d_As));
965 : PetscCall(MatDenseCUDAGetArrayWrite(B,&d_Ba));
966 :
967 : PetscCallCUDA(cudaMalloc((void **)&d_Q,sizeof(PetscScalar)*m*m));
968 : PetscCallCUDA(cudaMalloc((void **)&d_W,sizeof(PetscScalar)*m*m));
969 : PetscCallCUDA(cudaMalloc((void **)&d_A2,sizeof(PetscScalar)*m*m));
970 : PetscCallCUDA(cudaMalloc((void **)&d_ipiv,sizeof(PetscBLASInt)*ld));
971 : PetscCallCUDA(cudaMalloc((void **)&d_info,sizeof(PetscBLASInt)));
972 : PetscCallCUDA(cudaMalloc((void **)&d_ppP,sizeof(PetscScalar*)));
973 : PetscCallCUDA(cudaMalloc((void **)&d_ppQ,sizeof(PetscScalar*)));
974 :
975 : PetscCall(PetscMalloc1(1,&ppP));
976 : PetscCall(PetscMalloc1(1,&ppQ));
977 :
978 : d_P = d_Ba;
979 : PetscCall(PetscLogGpuTimeBegin());
980 :
981 : /* Pade' coefficients */
982 : c[0] = 1.0;
983 : for (k=1;k<=p;k++) c[k] = c[k-1]*(p+1-k)/(k*(2*p+1-k));
984 :
985 : /* Scaling */
986 : PetscCallCUBLAS(cublasXnrm2(cublasv2handle,ld2,d_As,one,&s));
987 : if (s>0.5) {
988 : sexp = PetscMax(0,(int)(PetscLogReal(s)/PetscLogReal(2.0))+2);
989 : scale = PetscPowRealInt(2.0,-sexp);
990 : PetscCallCUBLAS(cublasXscal(cublasv2handle,ld2,&scale,d_As,one));
991 : PetscCall(PetscLogGpuFlops(1.0*n*n));
992 : } else sexp = 0;
993 :
994 : /* Horner evaluation */
995 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_As,ld,d_As,ld,&szero,d_A2,ld));
996 : PetscCall(PetscLogGpuFlops(2.0*n*n*n));
997 : PetscCallCUDA(cudaMemset(d_Q,0,sizeof(PetscScalar)*ld2));
998 : PetscCallCUDA(cudaMemset(d_P,0,sizeof(PetscScalar)*ld2));
999 : PetscCall(set_diagonal(n,d_Q,ld,c[p]));
1000 : PetscCall(set_diagonal(n,d_P,ld,c[p-1]));
1001 :
1002 : odd = PETSC_TRUE;
1003 : for (k=p-1;k>0;k--) {
1004 : if (odd) {
1005 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_Q,ld,d_A2,ld,&szero,d_W,ld));
1006 : SWAP(d_Q,d_W,aux);
1007 : PetscCall(shift_diagonal(n,d_Q,ld,c[k-1]));
1008 : odd = PETSC_FALSE;
1009 : } else {
1010 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_P,ld,d_A2,ld,&szero,d_W,ld));
1011 : SWAP(d_P,d_W,aux);
1012 : PetscCall(shift_diagonal(n,d_P,ld,c[k-1]));
1013 : odd = PETSC_TRUE;
1014 : }
1015 : PetscCall(PetscLogGpuFlops(2.0*n*n*n));
1016 : }
1017 : if (odd) {
1018 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_Q,ld,d_As,ld,&szero,d_W,ld));
1019 : SWAP(d_Q,d_W,aux);
1020 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,ld2,&smone,d_P,one,d_Q,one));
1021 :
1022 : ppQ[0] = d_Q;
1023 : ppP[0] = d_P;
1024 : PetscCallCUDA(cudaMemcpy(d_ppQ,ppQ,sizeof(PetscScalar*),cudaMemcpyHostToDevice));
1025 : PetscCallCUDA(cudaMemcpy(d_ppP,ppP,sizeof(PetscScalar*),cudaMemcpyHostToDevice));
1026 :
1027 : PetscCallCUBLAS(cublasXgetrfBatched(cublasv2handle,n,d_ppQ,ld,d_ipiv,d_info,one));
1028 : PetscCallCUDA(cudaMemcpy(&info,d_info,sizeof(PetscBLASInt),cudaMemcpyDeviceToHost));
1029 : PetscCheck(info>=0,PETSC_COMM_SELF,PETSC_ERR_LIB,"LAPACKgetrf: Illegal value on argument %" PetscBLASInt_FMT,PetscAbsInt(info));
1030 : PetscCheck(info<=0,PETSC_COMM_SELF,PETSC_ERR_MAT_LU_ZRPVT,"LAPACKgetrf: Matrix is singular. U(%" PetscBLASInt_FMT ",%" PetscBLASInt_FMT ") is zero",info,info);
1031 : PetscCallCUBLAS(cublasXgetrsBatched(cublasv2handle,CUBLAS_OP_N,n,n,(const PetscScalar **)d_ppQ,ld,d_ipiv,d_ppP,ld,&info,one));
1032 : PetscCheck(info>=0,PETSC_COMM_SELF,PETSC_ERR_LIB,"LAPACKgetri: Illegal value on argument %" PetscBLASInt_FMT,PetscAbsInt(info));
1033 : PetscCheck(info<=0,PETSC_COMM_SELF,PETSC_ERR_MAT_LU_ZRPVT,"LAPACKgetri: Matrix is singular. U(%" PetscBLASInt_FMT ",%" PetscBLASInt_FMT ") is zero",info,info);
1034 : PetscCallCUBLAS(cublasXscal(cublasv2handle,ld2,&stwo,d_P,one));
1035 : PetscCall(shift_diagonal(n,d_P,ld,sone));
1036 : PetscCallCUBLAS(cublasXscal(cublasv2handle,ld2,&smone,d_P,one));
1037 : } else {
1038 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_P,ld,d_As,ld,&szero,d_W,ld));
1039 : SWAP(d_P,d_W,aux);
1040 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,ld2,&smone,d_P,one,d_Q,one));
1041 :
1042 : ppQ[0] = d_Q;
1043 : ppP[0] = d_P;
1044 : PetscCallCUDA(cudaMemcpy(d_ppQ,ppQ,sizeof(PetscScalar*),cudaMemcpyHostToDevice));
1045 : PetscCallCUDA(cudaMemcpy(d_ppP,ppP,sizeof(PetscScalar*),cudaMemcpyHostToDevice));
1046 :
1047 : PetscCallCUBLAS(cublasXgetrfBatched(cublasv2handle,n,d_ppQ,ld,d_ipiv,d_info,one));
1048 : PetscCallCUDA(cudaMemcpy(&info,d_info,sizeof(PetscBLASInt),cudaMemcpyDeviceToHost));
1049 : PetscCheck(info>=0,PETSC_COMM_SELF,PETSC_ERR_LIB,"LAPACKgetrf: Illegal value on argument %" PetscBLASInt_FMT,PetscAbsInt(info));
1050 : PetscCheck(info<=0,PETSC_COMM_SELF,PETSC_ERR_MAT_LU_ZRPVT,"LAPACKgetrf: Matrix is singular. U(%" PetscBLASInt_FMT ",%" PetscBLASInt_FMT ") is zero",info,info);
1051 : PetscCallCUBLAS(cublasXgetrsBatched(cublasv2handle,CUBLAS_OP_N,n,n,(const PetscScalar **)d_ppQ,ld,d_ipiv,d_ppP,ld,&info,one));
1052 : PetscCheck(info>=0,PETSC_COMM_SELF,PETSC_ERR_LIB,"LAPACKgetri: Illegal value on argument %" PetscBLASInt_FMT,PetscAbsInt(info));
1053 : PetscCheck(info<=0,PETSC_COMM_SELF,PETSC_ERR_MAT_LU_ZRPVT,"LAPACKgetri: Matrix is singular. U(%" PetscBLASInt_FMT ",%" PetscBLASInt_FMT ") is zero",info,info);
1054 : PetscCallCUBLAS(cublasXscal(cublasv2handle,ld2,&stwo,d_P,one));
1055 : PetscCall(shift_diagonal(n,d_P,ld,sone));
1056 : }
1057 : PetscCall(PetscLogGpuFlops(2.0*n*n*n+2.0*n*n*n/3.0+4.0*n*n));
1058 :
1059 : for (k=1;k<=sexp;k++) {
1060 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_P,ld,d_P,ld,&szero,d_W,ld));
1061 : PetscCallCUDA(cudaMemcpy(d_P,d_W,sizeof(PetscScalar)*ld2,cudaMemcpyDeviceToDevice));
1062 : }
1063 : PetscCall(PetscLogGpuFlops(2.0*n*n*n*sexp));
1064 :
1065 : PetscCall(PetscLogGpuTimeEnd());
1066 : PetscCallCUDA(cudaFree(d_Q));
1067 : PetscCallCUDA(cudaFree(d_W));
1068 : PetscCallCUDA(cudaFree(d_A2));
1069 : PetscCallCUDA(cudaFree(d_ipiv));
1070 : PetscCallCUDA(cudaFree(d_info));
1071 : PetscCallCUDA(cudaFree(d_ppP));
1072 : PetscCallCUDA(cudaFree(d_ppQ));
1073 :
1074 : PetscCall(PetscFree(ppP));
1075 : PetscCall(PetscFree(ppQ));
1076 :
1077 : if (d_P!=d_Ba) PetscCallCUDA(cudaMemcpy(d_Ba,d_P,sizeof(PetscScalar)*ld2,cudaMemcpyDeviceToDevice));
1078 : if (A!=B) {
1079 : if (s>0.5) { /* undo scaling */
1080 : scale = 1.0/scale;
1081 : PetscCallCUBLAS(cublasXscal(cublasv2handle,ld2,&scale,d_As,one));
1082 : }
1083 : PetscCall(MatDenseCUDARestoreArrayRead(A,(const PetscScalar**)&d_As));
1084 : } else PetscCallCUDA(cudaFree(d_As));
1085 : PetscCall(MatDenseCUDARestoreArrayWrite(B,&d_Ba));
1086 : PetscFunctionReturn(PETSC_SUCCESS);
1087 : }
1088 :
1089 : #if defined(PETSC_HAVE_MAGMA)
1090 : #include <slepcmagma.h>
1091 :
1092 : PetscErrorCode FNEvaluateFunctionMat_Exp_Pade_CUDAm(FN fn,Mat A,Mat B)
1093 : {
1094 : PetscBLASInt n=0,ld,ld2,*piv,one=1;
1095 : PetscInt m,k,sexp;
1096 : PetscBool odd;
1097 : const PetscInt p=MAX_PADE;
1098 : PetscReal c[MAX_PADE+1],s;
1099 : PetscScalar scale,smone=-1.0,sone=1.0,stwo=2.0,szero=0.0;
1100 : const PetscScalar *Aa;
1101 : PetscScalar *d_Ba,*d_As,*d_A2,*d_Q,*d_P,*d_W,*aux;
1102 : cublasHandle_t cublasv2handle;
1103 :
1104 : PetscFunctionBegin;
1105 : PetscCall(PetscDeviceInitialize(PETSC_DEVICE_CUDA)); /* For CUDA event timers */
1106 : PetscCall(PetscCUBLASGetHandle(&cublasv2handle));
1107 : PetscCall(SlepcMagmaInit());
1108 : PetscCall(MatGetSize(A,&m,NULL));
1109 : PetscCall(PetscBLASIntCast(m,&n));
1110 : ld = n;
1111 : ld2 = ld*ld;
1112 : if (A==B) {
1113 : PetscCallCUDA(cudaMalloc((void **)&d_As,sizeof(PetscScalar)*m*m));
1114 : PetscCall(MatDenseCUDAGetArrayRead(A,&Aa));
1115 : PetscCallCUDA(cudaMemcpy(d_As,Aa,sizeof(PetscScalar)*ld2,cudaMemcpyDeviceToDevice));
1116 : PetscCall(MatDenseCUDARestoreArrayRead(A,&Aa));
1117 : } else PetscCall(MatDenseCUDAGetArrayRead(A,(const PetscScalar**)&d_As));
1118 : PetscCall(MatDenseCUDAGetArrayWrite(B,&d_Ba));
1119 :
1120 : PetscCallCUDA(cudaMalloc((void **)&d_Q,sizeof(PetscScalar)*m*m));
1121 : PetscCallCUDA(cudaMalloc((void **)&d_W,sizeof(PetscScalar)*m*m));
1122 : PetscCallCUDA(cudaMalloc((void **)&d_A2,sizeof(PetscScalar)*m*m));
1123 :
1124 : PetscCall(PetscMalloc1(n,&piv));
1125 :
1126 : d_P = d_Ba;
1127 : PetscCall(PetscLogGpuTimeBegin());
1128 :
1129 : /* Pade' coefficients */
1130 : c[0] = 1.0;
1131 : for (k=1;k<=p;k++) c[k] = c[k-1]*(p+1-k)/(k*(2*p+1-k));
1132 :
1133 : /* Scaling */
1134 : PetscCallCUBLAS(cublasXnrm2(cublasv2handle,ld2,d_As,one,&s));
1135 : PetscCall(PetscLogGpuFlops(1.0*n*n));
1136 :
1137 : if (s>0.5) {
1138 : sexp = PetscMax(0,(int)(PetscLogReal(s)/PetscLogReal(2.0))+2);
1139 : scale = PetscPowRealInt(2.0,-sexp);
1140 : PetscCallCUBLAS(cublasXscal(cublasv2handle,ld2,&scale,d_As,one));
1141 : PetscCall(PetscLogGpuFlops(1.0*n*n));
1142 : } else sexp = 0;
1143 :
1144 : /* Horner evaluation */
1145 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_As,ld,d_As,ld,&szero,d_A2,ld));
1146 : PetscCall(PetscLogGpuFlops(2.0*n*n*n));
1147 : PetscCallCUDA(cudaMemset(d_Q,0,sizeof(PetscScalar)*ld2));
1148 : PetscCallCUDA(cudaMemset(d_P,0,sizeof(PetscScalar)*ld2));
1149 : PetscCall(set_diagonal(n,d_Q,ld,c[p]));
1150 : PetscCall(set_diagonal(n,d_P,ld,c[p-1]));
1151 :
1152 : odd = PETSC_TRUE;
1153 : for (k=p-1;k>0;k--) {
1154 : if (odd) {
1155 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_Q,ld,d_A2,ld,&szero,d_W,ld));
1156 : SWAP(d_Q,d_W,aux);
1157 : PetscCall(shift_diagonal(n,d_Q,ld,c[k-1]));
1158 : odd = PETSC_FALSE;
1159 : } else {
1160 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_P,ld,d_A2,ld,&szero,d_W,ld));
1161 : SWAP(d_P,d_W,aux);
1162 : PetscCall(shift_diagonal(n,d_P,ld,c[k-1]));
1163 : odd = PETSC_TRUE;
1164 : }
1165 : PetscCall(PetscLogGpuFlops(2.0*n*n*n));
1166 : }
1167 : if (odd) {
1168 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_Q,ld,d_As,ld,&szero,d_W,ld));
1169 : SWAP(d_Q,d_W,aux);
1170 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,ld2,&smone,d_P,one,d_Q,one));
1171 : PetscCallMAGMA(magma_xgesv_gpu,n,n,d_Q,ld,piv,d_P,ld);
1172 : PetscCallCUBLAS(cublasXscal(cublasv2handle,ld2,&stwo,d_P,one));
1173 : PetscCall(shift_diagonal(n,d_P,ld,sone));
1174 : PetscCallCUBLAS(cublasXscal(cublasv2handle,ld2,&smone,d_P,one));
1175 : } else {
1176 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_P,ld,d_As,ld,&szero,d_W,ld));
1177 : SWAP(d_P,d_W,aux);
1178 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,ld2,&smone,d_P,one,d_Q,one));
1179 : PetscCallMAGMA(magma_xgesv_gpu,n,n,d_Q,ld,piv,d_P,ld);
1180 : PetscCallCUBLAS(cublasXscal(cublasv2handle,ld2,&stwo,d_P,one));
1181 : PetscCall(shift_diagonal(n,d_P,ld,sone));
1182 : }
1183 : PetscCall(PetscLogGpuFlops(2.0*n*n*n+2.0*n*n*n/3.0+4.0*n*n));
1184 :
1185 : for (k=1;k<=sexp;k++) {
1186 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_P,ld,d_P,ld,&szero,d_W,ld));
1187 : PetscCallCUDA(cudaMemcpy(d_P,d_W,sizeof(PetscScalar)*ld2,cudaMemcpyDeviceToDevice));
1188 : }
1189 : PetscCall(PetscLogGpuFlops(2.0*n*n*n*sexp));
1190 :
1191 : PetscCall(PetscLogGpuTimeEnd());
1192 : PetscCallCUDA(cudaFree(d_Q));
1193 : PetscCallCUDA(cudaFree(d_W));
1194 : PetscCallCUDA(cudaFree(d_A2));
1195 : PetscCall(PetscFree(piv));
1196 :
1197 : if (d_P!=d_Ba) PetscCallCUDA(cudaMemcpy(d_Ba,d_P,sizeof(PetscScalar)*ld2,cudaMemcpyDeviceToDevice));
1198 : if (A!=B) {
1199 : if (s>0.5) { /* undo scaling */
1200 : scale = 1.0/scale;
1201 : PetscCallCUBLAS(cublasXscal(cublasv2handle,ld2,&scale,d_As,one));
1202 : }
1203 : PetscCall(MatDenseCUDARestoreArrayRead(A,(const PetscScalar**)&d_As));
1204 : } else PetscCallCUDA(cudaFree(d_As));
1205 : PetscCall(MatDenseCUDARestoreArrayWrite(B,&d_Ba));
1206 : PetscFunctionReturn(PETSC_SUCCESS);
1207 : }
1208 :
1209 : /*
1210 : * Matrix exponential implementation based on algorithm and matlab code by N. Higham and co-authors
1211 : *
1212 : * N. J. Higham, "The scaling and squaring method for the matrix exponential
1213 : * revisited", SIAM J. Matrix Anal. Appl. 26(4):1179-1193, 2005.
1214 : */
1215 : PetscErrorCode FNEvaluateFunctionMat_Exp_Higham_CUDAm(FN fn,Mat A,Mat B)
1216 : {
1217 : PetscBLASInt n_=0,n2,*ipiv,one=1;
1218 : PetscInt n,m,j,s;
1219 : PetscScalar scale,smone=-1.0,sone=1.0,stwo=2.0,szero=0.0;
1220 : PetscScalar *d_Ba,*Apowers[5],*d_Apowers[5],*d_Q,*d_P,*d_W,*work,*d_work,*aux;
1221 : const PetscScalar *Aa,*c;
1222 : const PetscScalar c3[4] = { 120, 60, 12, 1 };
1223 : const PetscScalar c5[6] = { 30240, 15120, 3360, 420, 30, 1 };
1224 : const PetscScalar c7[8] = { 17297280, 8648640, 1995840, 277200, 25200, 1512, 56, 1 };
1225 : const PetscScalar c9[10] = { 17643225600, 8821612800, 2075673600, 302702400, 30270240,
1226 : 2162160, 110880, 3960, 90, 1 };
1227 : const PetscScalar c13[14] = { 64764752532480000, 32382376266240000, 7771770303897600,
1228 : 1187353796428800, 129060195264000, 10559470521600,
1229 : 670442572800, 33522128640, 1323241920,
1230 : 40840800, 960960, 16380, 182, 1 };
1231 : cublasHandle_t cublasv2handle;
1232 :
1233 : PetscFunctionBegin;
1234 : PetscCall(PetscDeviceInitialize(PETSC_DEVICE_CUDA)); /* For CUDA event timers */
1235 : PetscCall(PetscCUBLASGetHandle(&cublasv2handle));
1236 : PetscCall(SlepcMagmaInit());
1237 : PetscCall(MatGetSize(A,&n,NULL));
1238 : PetscCall(PetscBLASIntCast(n,&n_));
1239 : n2 = n_*n_;
1240 : PetscCall(PetscMalloc2(8*n*n,&work,n,&ipiv));
1241 : /* Matrix powers */
1242 : Apowers[0] = work; /* Apowers[0] = A */
1243 : Apowers[1] = Apowers[0] + n*n; /* Apowers[1] = A^2 */
1244 : Apowers[2] = Apowers[1] + n*n; /* Apowers[2] = A^4 */
1245 : Apowers[3] = Apowers[2] + n*n; /* Apowers[3] = A^6 */
1246 : Apowers[4] = Apowers[3] + n*n; /* Apowers[4] = A^8 */
1247 : if (A==B) {
1248 : PetscCallCUDA(cudaMalloc((void**)&d_work,7*n*n*sizeof(PetscScalar)));
1249 : d_Apowers[0] = d_work; /* d_Apowers[0] = A */
1250 : d_Apowers[1] = d_Apowers[0] + n*n; /* d_Apowers[1] = A^2 */
1251 : PetscCall(MatDenseCUDAGetArrayRead(A,&Aa));
1252 : PetscCallCUDA(cudaMemcpy(d_Apowers[0],Aa,n2*sizeof(PetscScalar),cudaMemcpyDeviceToDevice));
1253 : PetscCall(MatDenseCUDARestoreArrayRead(A,&Aa));
1254 : } else {
1255 : PetscCallCUDA(cudaMalloc((void**)&d_work,6*n*n*sizeof(PetscScalar)));
1256 : PetscCall(MatDenseCUDAGetArrayRead(A,(const PetscScalar**)&d_Apowers[0]));
1257 : d_Apowers[1] = d_work; /* d_Apowers[1] = A^2 */
1258 : }
1259 : PetscCall(MatDenseCUDAGetArrayWrite(B,&d_Ba));
1260 : d_Apowers[2] = d_Apowers[1] + n*n; /* d_Apowers[2] = A^4 */
1261 : d_Apowers[3] = d_Apowers[2] + n*n; /* d_Apowers[3] = A^6 */
1262 : d_Apowers[4] = d_Apowers[3] + n*n; /* d_Apowers[4] = A^8 */
1263 : d_Q = d_Apowers[4] + n*n;
1264 : d_W = d_Q + n*n;
1265 :
1266 : PetscCall(PetscLogGpuTimeBegin());
1267 :
1268 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n_,n_,n_,&sone,d_Apowers[0],n_,d_Apowers[0],n_,&szero,d_Apowers[1],n_));
1269 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n_,n_,n_,&sone,d_Apowers[1],n_,d_Apowers[1],n_,&szero,d_Apowers[2],n_));
1270 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n_,n_,n_,&sone,d_Apowers[1],n_,d_Apowers[2],n_,&szero,d_Apowers[3],n_));
1271 : PetscCall(PetscLogGpuFlops(6.0*n*n*n));
1272 :
1273 : PetscCallCUDA(cudaMemcpy(Apowers[0],d_Apowers[0],n2*sizeof(PetscScalar),cudaMemcpyDeviceToHost));
1274 : PetscCallCUDA(cudaMemcpy(Apowers[1],d_Apowers[1],3*n2*sizeof(PetscScalar),cudaMemcpyDeviceToHost));
1275 : PetscCall(PetscLogGpuToCpu(4*n2*sizeof(PetscScalar)));
1276 : /* Compute scaling parameter and order of Pade approximant */
1277 : PetscCall(expm_params(n,Apowers,&s,&m,Apowers[4]));
1278 :
1279 : if (s) { /* rescale */
1280 : for (j=0;j<4;j++) {
1281 : scale = PetscPowRealInt(2.0,-PetscMax(2*j,1)*s);
1282 : PetscCallCUBLAS(cublasXscal(cublasv2handle,n2,&scale,d_Apowers[j],one));
1283 : }
1284 : PetscCall(PetscLogGpuFlops(4.0*n*n));
1285 : }
1286 :
1287 : /* Evaluate the Pade approximant */
1288 : switch (m) {
1289 : case 3: c = c3; break;
1290 : case 5: c = c5; break;
1291 : case 7: c = c7; break;
1292 : case 9: c = c9; break;
1293 : case 13: c = c13; break;
1294 : default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong value of m %" PetscInt_FMT,m);
1295 : }
1296 : d_P = d_Ba;
1297 : switch (m) {
1298 : case 3:
1299 : case 5:
1300 : case 7:
1301 : case 9:
1302 : if (m==9) PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n_,n_,n_,&sone,d_Apowers[1],n_,d_Apowers[3],n_,&szero,d_Apowers[4],n_));
1303 : PetscCallCUDA(cudaMemset(d_P,0,sizeof(PetscScalar)*n2));
1304 : PetscCallCUDA(cudaMemset(d_Q,0,sizeof(PetscScalar)*n2));
1305 : PetscCall(set_diagonal(n,d_P,n,c[1]));
1306 : PetscCall(set_diagonal(n,d_Q,n,c[0]));
1307 : for (j=m;j>=3;j-=2) {
1308 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[j],d_Apowers[(j+1)/2-1],one,d_P,one));
1309 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[j-1],d_Apowers[(j+1)/2-1],one,d_Q,one));
1310 : PetscCall(PetscLogGpuFlops(4.0*n*n));
1311 : }
1312 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n_,n_,n_,&sone,d_Apowers[0],n_,d_P,n_,&szero,d_W,n_));
1313 : PetscCall(PetscLogGpuFlops(2.0*n*n*n));
1314 : SWAP(d_P,d_W,aux);
1315 : break;
1316 : case 13:
1317 : /* P = A*(Apowers[3]*(c[13]*Apowers[3] + c[11]*Apowers[2] + c[9]*Apowers[1])
1318 : + c[7]*Apowers[3] + c[5]*Apowers[2] + c[3]*Apowers[1] + c[1]*I) */
1319 : PetscCallCUDA(cudaMemcpy(d_P,d_Apowers[3],n2*sizeof(PetscScalar),cudaMemcpyDeviceToDevice));
1320 : PetscCallCUBLAS(cublasXscal(cublasv2handle,n2,&c[13],d_P,one));
1321 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[11],d_Apowers[2],one,d_P,one));
1322 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[9],d_Apowers[1],one,d_P,one));
1323 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n_,n_,n_,&sone,d_Apowers[3],n_,d_P,n_,&szero,d_W,n_));
1324 : PetscCall(PetscLogGpuFlops(5.0*n*n+2.0*n*n*n));
1325 :
1326 : PetscCallCUDA(cudaMemset(d_P,0,sizeof(PetscScalar)*n2));
1327 : PetscCall(set_diagonal(n,d_P,n,c[1]));
1328 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[7],d_Apowers[3],one,d_P,one));
1329 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[5],d_Apowers[2],one,d_P,one));
1330 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[3],d_Apowers[1],one,d_P,one));
1331 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&sone,d_P,one,d_W,one));
1332 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n_,n_,n_,&sone,d_Apowers[0],n_,d_W,n_,&szero,d_P,n_));
1333 : PetscCall(PetscLogGpuFlops(7.0*n*n+2.0*n*n*n));
1334 : /* Q = Apowers[3]*(c[12]*Apowers[3] + c[10]*Apowers[2] + c[8]*Apowers[1])
1335 : + c[6]*Apowers[3] + c[4]*Apowers[2] + c[2]*Apowers[1] + c[0]*I */
1336 : PetscCallCUDA(cudaMemcpy(d_Q,d_Apowers[3],n2*sizeof(PetscScalar),cudaMemcpyDeviceToDevice));
1337 : PetscCallCUBLAS(cublasXscal(cublasv2handle,n2,&c[12],d_Q,one));
1338 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[10],d_Apowers[2],one,d_Q,one));
1339 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[8],d_Apowers[1],one,d_Q,one));
1340 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n_,n_,n_,&sone,d_Apowers[3],n_,d_Q,n_,&szero,d_W,n_));
1341 : PetscCall(PetscLogGpuFlops(5.0*n*n+2.0*n*n*n));
1342 : PetscCallCUDA(cudaMemset(d_Q,0,sizeof(PetscScalar)*n2));
1343 : PetscCall(set_diagonal(n,d_Q,n,c[0]));
1344 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[6],d_Apowers[3],one,d_Q,one));
1345 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[4],d_Apowers[2],one,d_Q,one));
1346 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&c[2],d_Apowers[1],one,d_Q,one));
1347 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&sone,d_W,one,d_Q,one));
1348 : PetscCall(PetscLogGpuFlops(7.0*n*n));
1349 : break;
1350 : default: SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong value of m %" PetscInt_FMT,m);
1351 : }
1352 : PetscCallCUBLAS(cublasXaxpy(cublasv2handle,n2,&smone,d_P,one,d_Q,one));
1353 :
1354 : PetscCallMAGMA(magma_xgesv_gpu,n_,n_,d_Q,n_,ipiv,d_P,n_);
1355 :
1356 : PetscCallCUBLAS(cublasXscal(cublasv2handle,n2,&stwo,d_P,one));
1357 : PetscCall(shift_diagonal(n,d_P,n,sone));
1358 : PetscCall(PetscLogGpuFlops(2.0*n*n*n/3.0+4.0*n*n));
1359 :
1360 : /* Squaring */
1361 : for (j=1;j<=s;j++) {
1362 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n_,n_,n_,&sone,d_P,n_,d_P,n_,&szero,d_W,n_));
1363 : SWAP(d_P,d_W,aux);
1364 : }
1365 : PetscCall(PetscLogGpuFlops(2.0*n*n*n*s));
1366 : PetscCall(PetscLogGpuTimeEnd());
1367 :
1368 : PetscCall(PetscFree2(work,ipiv));
1369 : if (d_P!=d_Ba) PetscCallCUDA(cudaMemcpy(d_Ba,d_P,n2*sizeof(PetscScalar),cudaMemcpyDeviceToDevice));
1370 : if (A!=B) {
1371 : if (s>0.5) { /* undo scaling */
1372 : scale = 1.0/PetscPowRealInt(2.0,-s);
1373 : PetscCallCUBLAS(cublasXscal(cublasv2handle,n2,&scale,d_Apowers[0],one));
1374 : }
1375 : PetscCall(MatDenseCUDARestoreArrayRead(A,(const PetscScalar**)&d_Apowers[0]));
1376 : }
1377 : PetscCall(MatDenseCUDARestoreArrayWrite(B,&d_Ba));
1378 : PetscCallCUDA(cudaFree(d_work));
1379 : PetscFunctionReturn(PETSC_SUCCESS);
1380 : }
1381 :
1382 : /*
1383 : * Matrix exponential implementation based on algorithm and matlab code by Stefan Guettel
1384 : * and Yuji Nakatsukasa
1385 : *
1386 : * Stefan Guettel and Yuji Nakatsukasa, "Scaled and Squared Subdiagonal Pade'
1387 : * Approximation for the Matrix Exponential",
1388 : * SIAM J. Matrix Anal. Appl. 37(1):145-170, 2016.
1389 : * https://doi.org/10.1137/15M1027553
1390 : */
1391 : PetscErrorCode FNEvaluateFunctionMat_Exp_GuettelNakatsukasa_CUDAm(FN fn,Mat A,Mat B)
1392 : {
1393 : PetscInt i,j,n_,s,k,m,mod;
1394 : PetscBLASInt n=0,n2=0,irsize=0,rsizediv2,ipsize=0,iremainsize=0,query=-1,*piv,minlen,lwork=0,one=1;
1395 : PetscReal nrm,shift=0.0,rone=1.0,rzero=0.0;
1396 : #if defined(PETSC_USE_COMPLEX)
1397 : PetscReal *rwork=NULL;
1398 : #endif
1399 : PetscComplex *d_As,*d_RR,*d_RR2,*d_expmA,*d_expmA2,*d_Maux,*d_Maux2,rsize,*r,psize,*p,remainsize,*remainterm,*rootp,*rootq,mult=0.0,scale,cone=1.0,czero=0.0,*aux;
1400 : PetscScalar *d_Aa,*d_Ba,*d_Ba2,*Maux,*d_sMaux,*wr,*wi,expshift,sone=1.0,szero=0.0,*work,work1,*saux;
1401 : const PetscScalar *Aa;
1402 : PetscBool isreal,*d_isreal,flg;
1403 : cublasHandle_t cublasv2handle;
1404 :
1405 : PetscFunctionBegin;
1406 : PetscCall(PetscDeviceInitialize(PETSC_DEVICE_CUDA)); /* For CUDA event timers */
1407 : PetscCall(PetscCUBLASGetHandle(&cublasv2handle));
1408 : PetscCall(SlepcMagmaInit());
1409 : PetscCall(MatGetSize(A,&n_,NULL));
1410 : PetscCall(PetscBLASIntCast(n_,&n));
1411 : PetscCall(PetscBLASIntCast(n*n,&n2));
1412 :
1413 : if (A==B) {
1414 : PetscCallCUDA(cudaMalloc((void **)&d_Aa,sizeof(PetscScalar)*n2));
1415 : PetscCall(MatDenseCUDAGetArrayRead(A,&Aa));
1416 : PetscCallCUDA(cudaMemcpy(d_Aa,Aa,sizeof(PetscScalar)*n2,cudaMemcpyDeviceToDevice));
1417 : PetscCall(MatDenseCUDARestoreArrayRead(A,&Aa));
1418 : } else PetscCall(MatDenseCUDAGetArrayRead(A,(const PetscScalar**)&d_Aa));
1419 : PetscCall(MatDenseCUDAGetArrayWrite(B,&d_Ba));
1420 : d_Ba2 = d_Ba;
1421 :
1422 : PetscCallCUDA(cudaMalloc((void **)&d_isreal,sizeof(PetscBool)));
1423 : PetscCallCUDA(cudaMalloc((void **)&d_sMaux,sizeof(PetscScalar)*n2));
1424 : PetscCallCUDA(cudaMalloc((void **)&d_Maux,sizeof(PetscComplex)*n2));
1425 :
1426 : PetscCall(PetscLogGpuTimeBegin());
1427 : d_Maux2 = d_Maux;
1428 : PetscCall(PetscOptionsGetReal(NULL,NULL,"-fn_expm_estimated_eig",&shift,&flg));
1429 : if (!flg) {
1430 : PetscCall(PetscMalloc2(n,&wr,n,&wi));
1431 : /* estimate rightmost eigenvalue and shift A with it */
1432 : PetscCall(PetscMalloc1(n2,&Maux));
1433 : PetscCall(MatDenseGetArrayRead(A,&Aa));
1434 : PetscCall(PetscArraycpy(Maux,Aa,n2));
1435 : PetscCall(MatDenseRestoreArrayRead(A,&Aa));
1436 : #if !defined(PETSC_USE_COMPLEX)
1437 : PetscCallMAGMA(magma_xgeev,MagmaNoVec,MagmaNoVec,n,Maux,n,wr,wi,NULL,n,NULL,n,&work1,query);
1438 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPart(work1),&lwork));
1439 : PetscCall(PetscMalloc1(lwork,&work));
1440 : PetscCallMAGMA(magma_xgeev,MagmaNoVec,MagmaNoVec,n,Maux,n,wr,wi,NULL,n,NULL,n,work,lwork);
1441 : PetscCall(PetscFree(work));
1442 : #else
1443 : PetscCallMAGMA(magma_xgeev,MagmaNoVec,MagmaNoVec,n,Maux,n,wr,NULL,n,NULL,n,&work1,query,rwork);
1444 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPart(work1),&lwork));
1445 : PetscCall(PetscMalloc2(2*n,&rwork,lwork,&work));
1446 : PetscCallMAGMA(magma_xgeev,MagmaNoVec,MagmaNoVec,n,Maux,n,wr,NULL,n,NULL,n,work,lwork,rwork);
1447 : PetscCall(PetscFree2(rwork,work));
1448 : #endif
1449 : PetscCall(PetscFree(Maux));
1450 : PetscCall(PetscLogGpuFlops(25.0*n*n*n+(n*n*n)/3.0+1.0*n*n*n));
1451 :
1452 : shift = PetscRealPart(wr[0]);
1453 : for (i=1;i<n;i++) {
1454 : if (PetscRealPart(wr[i]) > shift) shift = PetscRealPart(wr[i]);
1455 : }
1456 : PetscCall(PetscFree2(wr,wi));
1457 : }
1458 : /* shift so that largest real part is (about) 0 */
1459 : PetscCallCUDA(cudaMemcpy(d_sMaux,d_Aa,sizeof(PetscScalar)*n2,cudaMemcpyDeviceToDevice));
1460 : if (shift) {
1461 : PetscCall(shift_diagonal(n,d_sMaux,n,-shift));
1462 : PetscCall(PetscLogGpuFlops(1.0*n));
1463 : }
1464 : #if defined(PETSC_USE_COMPLEX)
1465 : PetscCallCUDA(cudaMemcpy(d_Maux,d_Aa,sizeof(PetscScalar)*n2,cudaMemcpyDeviceToDevice));
1466 : if (shift) {
1467 : PetscCall(shift_diagonal(n,d_Maux,n,-shift));
1468 : PetscCall(PetscLogGpuFlops(1.0*n));
1469 : }
1470 : #endif
1471 : if (A!=B) PetscCall(MatDenseCUDARestoreArrayRead(A,(const PetscScalar**)&d_Aa));
1472 : else PetscCallCUDA(cudaFree(d_Aa));
1473 :
1474 : /* estimate norm(A) and select the scaling factor */
1475 : PetscCallCUBLAS(cublasXnrm2(cublasv2handle,n2,d_sMaux,one,&nrm));
1476 : PetscCall(PetscLogGpuFlops(2.0*n*n));
1477 : PetscCall(sexpm_params(nrm,&s,&k,&m));
1478 : if (s==0 && k==1 && m==0) { /* exp(A) = I+A to eps! */
1479 : if (shift) expshift = PetscExpReal(shift);
1480 : PetscCall(shift_Cdiagonal(n,d_Maux,n,rone,rzero));
1481 : if (shift) {
1482 : PetscCallCUBLAS(cublasXscal(cublasv2handle,n2,&expshift,d_sMaux,one));
1483 : PetscCall(PetscLogGpuFlops(1.0*(n+n2)));
1484 : } else PetscCall(PetscLogGpuFlops(1.0*n));
1485 : PetscCallCUDA(cudaMemcpy(d_Ba,d_sMaux,sizeof(PetscScalar)*n2,cudaMemcpyDeviceToDevice));
1486 : PetscCallCUDA(cudaFree(d_isreal));
1487 : PetscCallCUDA(cudaFree(d_sMaux));
1488 : PetscCallCUDA(cudaFree(d_Maux));
1489 : PetscCall(MatDenseCUDARestoreArrayWrite(B,&d_Ba));
1490 : PetscFunctionReturn(PETSC_SUCCESS); /* quick return */
1491 : }
1492 :
1493 : PetscCallCUDA(cudaMalloc((void **)&d_expmA,sizeof(PetscComplex)*n2));
1494 : PetscCallCUDA(cudaMalloc((void **)&d_As,sizeof(PetscComplex)*n2));
1495 : PetscCallCUDA(cudaMalloc((void **)&d_RR,sizeof(PetscComplex)*n2));
1496 : d_expmA2 = d_expmA; d_RR2 = d_RR;
1497 : PetscCall(PetscMalloc1(n,&piv));
1498 : /* scale matrix */
1499 : #if !defined(PETSC_USE_COMPLEX)
1500 : PetscCall(copy_array2D_S2C(n,n,d_As,n,d_sMaux,n));
1501 : #else
1502 : PetscCallCUDA(cudaMemcpy(d_As,d_sMaux,sizeof(PetscScalar)*n2,cudaMemcpyDeviceToDevice));
1503 : #endif
1504 : scale = 1.0/PetscPowRealInt(2.0,s);
1505 : PetscCallCUBLAS(cublasXCscal(cublasv2handle,n2,(const cuComplex *)&scale,(cuComplex *)d_As,one));
1506 : PetscCall(SlepcLogGpuFlopsComplex(1.0*n2));
1507 :
1508 : /* evaluate Pade approximant (partial fraction or product form) */
1509 : if (fn->method==3 || !m) { /* partial fraction */
1510 : PetscCall(getcoeffs(k,m,&rsize,&psize,&remainsize,PETSC_TRUE));
1511 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPartComplex(rsize),&irsize));
1512 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPartComplex(psize),&ipsize));
1513 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPartComplex(remainsize),&iremainsize));
1514 : PetscCall(PetscMalloc3(irsize,&r,ipsize,&p,iremainsize,&remainterm));
1515 : PetscCall(getcoeffs(k,m,r,p,remainterm,PETSC_FALSE));
1516 :
1517 : PetscCallCUDA(cudaMemset(d_expmA,0,sizeof(PetscComplex)*n2));
1518 : #if !defined(PETSC_USE_COMPLEX)
1519 : isreal = PETSC_TRUE;
1520 : #else
1521 : PetscCall(getisreal_array2D(n,n,d_Maux,n,d_isreal));
1522 : PetscCallCUDA(cudaMemcpy(&isreal,d_isreal,sizeof(PetscBool),cudaMemcpyDeviceToHost));
1523 : #endif
1524 : if (isreal) {
1525 : rsizediv2 = irsize/2;
1526 : for (i=0;i<rsizediv2;i++) { /* use partial fraction to get R(As) */
1527 : PetscCallCUDA(cudaMemcpy(d_Maux,d_As,sizeof(PetscComplex)*n2,cudaMemcpyDeviceToDevice));
1528 : PetscCallCUDA(cudaMemset(d_RR,0,sizeof(PetscComplex)*n2));
1529 : PetscCall(shift_Cdiagonal(n,d_Maux,n,-PetscRealPartComplex(p[2*i]),-PetscImaginaryPartComplex(p[2*i])));
1530 : PetscCall(set_Cdiagonal(n,d_RR,n,PetscRealPartComplex(r[2*i]),PetscImaginaryPartComplex(r[2*i])));
1531 : PetscCallMAGMA(magma_Cgesv_gpu,n,n,d_Maux,n,piv,d_RR,n);
1532 : PetscCall(add_array2D_Conj(n,n,d_RR,n));
1533 : PetscCallCUBLAS(cublasXCaxpy(cublasv2handle,n2,&cone,d_RR,one,d_expmA,one));
1534 : /* shift(n) + gesv + axpy(n2) */
1535 : PetscCall(SlepcLogGpuFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)+2.0*n2));
1536 : }
1537 :
1538 : mod = ipsize % 2;
1539 : if (mod) {
1540 : PetscCallCUDA(cudaMemcpy(d_Maux,d_As,sizeof(PetscComplex)*n2,cudaMemcpyDeviceToDevice));
1541 : PetscCallCUDA(cudaMemset(d_RR,0,sizeof(PetscComplex)*n2));
1542 : PetscCall(shift_Cdiagonal(n,d_Maux,n,-PetscRealPartComplex(p[ipsize-1]),-PetscImaginaryPartComplex(p[ipsize-1])));
1543 : PetscCall(set_Cdiagonal(n,d_RR,n,PetscRealPartComplex(r[irsize-1]),PetscImaginaryPartComplex(r[irsize-1])));
1544 : PetscCallMAGMA(magma_Cgesv_gpu,n,n,d_Maux,n,piv,d_RR,n);
1545 : PetscCallCUBLAS(cublasXCaxpy(cublasv2handle,n2,&cone,d_RR,one,d_expmA,one));
1546 : PetscCall(SlepcLogGpuFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)+1.0*n2));
1547 : }
1548 : } else { /* complex */
1549 : for (i=0;i<irsize;i++) { /* use partial fraction to get R(As) */
1550 : PetscCallCUDA(cudaMemcpy(d_Maux,d_As,sizeof(PetscComplex)*n2,cudaMemcpyDeviceToDevice));
1551 : PetscCallCUDA(cudaMemset(d_RR,0,sizeof(PetscComplex)*n2));
1552 : PetscCall(shift_Cdiagonal(n,d_Maux,n,-PetscRealPartComplex(p[i]),-PetscImaginaryPartComplex(p[i])));
1553 : PetscCall(set_Cdiagonal(n,d_RR,n,PetscRealPartComplex(r[i]),PetscImaginaryPartComplex(r[i])));
1554 : PetscCallMAGMA(magma_Cgesv_gpu,n,n,d_Maux,n,piv,d_RR,n);
1555 : PetscCallCUBLAS(cublasXCaxpy(cublasv2handle,n2,&cone,d_RR,one,d_expmA,one));
1556 : PetscCall(SlepcLogGpuFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)+1.0*n2));
1557 : }
1558 : }
1559 : for (i=0;i<iremainsize;i++) {
1560 : if (!i) {
1561 : PetscCallCUDA(cudaMemset(d_RR,0,sizeof(PetscComplex)*n2));
1562 : PetscCall(set_Cdiagonal(n,d_RR,n,PetscRealPartComplex(remainterm[iremainsize-1]),PetscImaginaryPartComplex(remainterm[iremainsize-1])));
1563 : } else {
1564 : PetscCallCUDA(cudaMemcpy(d_RR,d_As,sizeof(PetscComplex)*n2,cudaMemcpyDeviceToDevice));
1565 : for (j=1;j<i;j++) {
1566 : PetscCallCUBLAS(cublasXCgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&cone,d_RR,n,d_RR,n,&czero,d_Maux,n));
1567 : SWAP(d_RR,d_Maux,aux);
1568 : PetscCall(SlepcLogGpuFlopsComplex(2.0*n*n*n));
1569 : }
1570 : PetscCallCUBLAS(cublasXCscal(cublasv2handle,n2,&remainterm[iremainsize-1-i],d_RR,one));
1571 : PetscCall(SlepcLogGpuFlopsComplex(1.0*n2));
1572 : }
1573 : PetscCallCUBLAS(cublasXCaxpy(cublasv2handle,n2,&cone,d_RR,one,d_expmA,one));
1574 : PetscCall(SlepcLogGpuFlopsComplex(1.0*n2));
1575 : }
1576 : PetscCall(PetscFree3(r,p,remainterm));
1577 : } else { /* product form, default */
1578 : PetscCall(getcoeffsproduct(k,m,&rsize,&psize,&mult,PETSC_TRUE));
1579 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPartComplex(rsize),&irsize));
1580 : PetscCall(PetscBLASIntCast((PetscInt)PetscRealPartComplex(psize),&ipsize));
1581 : PetscCall(PetscMalloc2(irsize,&rootp,ipsize,&rootq));
1582 : PetscCall(getcoeffsproduct(k,m,rootp,rootq,&mult,PETSC_FALSE));
1583 :
1584 : PetscCallCUDA(cudaMemset(d_expmA,0,sizeof(PetscComplex)*n2));
1585 : PetscCall(set_Cdiagonal(n,d_expmA,n,rone,rzero)); /* initialize */
1586 : minlen = PetscMin(irsize,ipsize);
1587 : for (i=0;i<minlen;i++) {
1588 : PetscCallCUDA(cudaMemcpy(d_RR,d_As,sizeof(PetscComplex)*n2,cudaMemcpyDeviceToDevice));
1589 : PetscCall(shift_Cdiagonal(n,d_RR,n,-PetscRealPartComplex(rootp[i]),-PetscImaginaryPartComplex(rootp[i])));
1590 : PetscCallCUBLAS(cublasXCgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&cone,d_RR,n,d_expmA,n,&czero,d_Maux,n));
1591 : SWAP(d_expmA,d_Maux,aux);
1592 : PetscCallCUDA(cudaMemcpy(d_RR,d_As,sizeof(PetscComplex)*n2,cudaMemcpyDeviceToDevice));
1593 : PetscCall(shift_Cdiagonal(n,d_RR,n,-PetscRealPartComplex(rootq[i]),-PetscImaginaryPartComplex(rootq[i])));
1594 : PetscCallMAGMA(magma_Cgesv_gpu,n,n,d_RR,n,piv,d_expmA,n);
1595 : /* shift(n) + gemm + shift(n) + gesv */
1596 : PetscCall(SlepcLogGpuFlopsComplex(1.0*n+(2.0*n*n*n)+1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)));
1597 : }
1598 : /* extra enumerator */
1599 : for (i=minlen;i<irsize;i++) {
1600 : PetscCallCUDA(cudaMemcpy(d_RR,d_As,sizeof(PetscComplex)*n2,cudaMemcpyDeviceToDevice));
1601 : PetscCall(shift_Cdiagonal(n,d_RR,n,-PetscRealPartComplex(rootp[i]),-PetscImaginaryPartComplex(rootp[i])));
1602 : PetscCallCUBLAS(cublasXCgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&cone,d_RR,n,d_expmA,n,&czero,d_Maux,n));
1603 : SWAP(d_expmA,d_Maux,aux);
1604 : PetscCall(SlepcLogGpuFlopsComplex(1.0*n+2.0*n*n*n));
1605 : }
1606 : /* extra denominator */
1607 : for (i=minlen;i<ipsize;i++) {
1608 : PetscCallCUDA(cudaMemcpy(d_RR,d_As,sizeof(PetscComplex)*n2,cudaMemcpyDeviceToDevice));
1609 : PetscCall(shift_Cdiagonal(n,d_RR,n,-PetscRealPartComplex(rootq[i]),-PetscImaginaryPartComplex(rootq[i])));
1610 : PetscCallMAGMA(magma_Cgesv_gpu,n,n,d_RR,n,piv,d_expmA,n);
1611 : PetscCall(SlepcLogGpuFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)));
1612 : }
1613 : PetscCallCUBLAS(cublasXCscal(cublasv2handle,n2,&mult,d_expmA,one));
1614 : PetscCall(SlepcLogGpuFlopsComplex(1.0*n2));
1615 : PetscCall(PetscFree2(rootp,rootq));
1616 : }
1617 :
1618 : #if !defined(PETSC_USE_COMPLEX)
1619 : PetscCall(copy_array2D_C2S(n,n,d_Ba2,n,d_expmA,n));
1620 : #else
1621 : PetscCallCUDA(cudaMemcpy(d_Ba2,d_expmA,sizeof(PetscScalar)*n2,cudaMemcpyDeviceToDevice));
1622 : #endif
1623 :
1624 : /* perform repeated squaring */
1625 : for (i=0;i<s;i++) { /* final squaring */
1626 : PetscCallCUBLAS(cublasXgemm(cublasv2handle,CUBLAS_OP_N,CUBLAS_OP_N,n,n,n,&sone,d_Ba2,n,d_Ba2,n,&szero,d_sMaux,n));
1627 : SWAP(d_Ba2,d_sMaux,saux);
1628 : PetscCall(PetscLogGpuFlops(2.0*n*n*n));
1629 : }
1630 : if (d_Ba2!=d_Ba) {
1631 : PetscCallCUDA(cudaMemcpy(d_Ba,d_Ba2,sizeof(PetscScalar)*n2,cudaMemcpyDeviceToDevice));
1632 : d_sMaux = d_Ba2;
1633 : }
1634 : if (shift) {
1635 : expshift = PetscExpReal(shift);
1636 : PetscCallCUBLAS(cublasXscal(cublasv2handle,n2,&expshift,d_Ba,one));
1637 : PetscCall(PetscLogGpuFlops(1.0*n2));
1638 : }
1639 :
1640 : PetscCall(PetscLogGpuTimeEnd());
1641 :
1642 : /* restore pointers */
1643 : d_Maux = d_Maux2; d_expmA = d_expmA2; d_RR = d_RR2;
1644 : PetscCall(MatDenseCUDARestoreArrayWrite(B,&d_Ba));
1645 : PetscCallCUDA(cudaFree(d_isreal));
1646 : PetscCallCUDA(cudaFree(d_sMaux));
1647 : PetscCallCUDA(cudaFree(d_Maux));
1648 : PetscCallCUDA(cudaFree(d_expmA));
1649 : PetscCallCUDA(cudaFree(d_As));
1650 : PetscCallCUDA(cudaFree(d_RR));
1651 : PetscCall(PetscFree(piv));
1652 : PetscFunctionReturn(PETSC_SUCCESS);
1653 : }
1654 : #endif /* PETSC_HAVE_MAGMA */
1655 : #endif /* PETSC_HAVE_CUDA */
1656 :
1657 29 : static PetscErrorCode FNView_Exp(FN fn,PetscViewer viewer)
1658 : {
1659 29 : PetscBool isascii;
1660 29 : char str[50];
1661 29 : const char *methodname[] = {
1662 : "scaling & squaring, [m/m] Pade approximant (Higham)",
1663 : "scaling & squaring, [6/6] Pade approximant",
1664 : "scaling & squaring, subdiagonal Pade approximant (product form)",
1665 : "scaling & squaring, subdiagonal Pade approximant (partial fraction)"
1666 : };
1667 29 : const int nmeth=PETSC_STATIC_ARRAY_LENGTH(methodname);
1668 :
1669 29 : PetscFunctionBegin;
1670 29 : PetscCall(PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii));
1671 29 : if (isascii) {
1672 29 : if (fn->beta==(PetscScalar)1.0) {
1673 24 : if (fn->alpha==(PetscScalar)1.0) PetscCall(PetscViewerASCIIPrintf(viewer," exponential: exp(x)\n"));
1674 : else {
1675 10 : PetscCall(SlepcSNPrintfScalar(str,sizeof(str),fn->alpha,PETSC_TRUE));
1676 10 : PetscCall(PetscViewerASCIIPrintf(viewer," exponential: exp(%s*x)\n",str));
1677 : }
1678 : } else {
1679 5 : PetscCall(SlepcSNPrintfScalar(str,sizeof(str),fn->beta,PETSC_TRUE));
1680 5 : if (fn->alpha==(PetscScalar)1.0) PetscCall(PetscViewerASCIIPrintf(viewer," exponential: %s*exp(x)\n",str));
1681 : else {
1682 5 : PetscCall(PetscViewerASCIIPrintf(viewer," exponential: %s",str));
1683 5 : PetscCall(PetscViewerASCIIUseTabs(viewer,PETSC_FALSE));
1684 5 : PetscCall(SlepcSNPrintfScalar(str,sizeof(str),fn->alpha,PETSC_TRUE));
1685 5 : PetscCall(PetscViewerASCIIPrintf(viewer,"*exp(%s*x)\n",str));
1686 5 : PetscCall(PetscViewerASCIIUseTabs(viewer,PETSC_TRUE));
1687 : }
1688 : }
1689 29 : if (fn->method<nmeth) PetscCall(PetscViewerASCIIPrintf(viewer," computing matrix functions with: %s\n",methodname[fn->method]));
1690 : }
1691 29 : PetscFunctionReturn(PETSC_SUCCESS);
1692 : }
1693 :
1694 93 : SLEPC_EXTERN PetscErrorCode FNCreate_Exp(FN fn)
1695 : {
1696 93 : PetscFunctionBegin;
1697 93 : fn->ops->evaluatefunction = FNEvaluateFunction_Exp;
1698 93 : fn->ops->evaluatederivative = FNEvaluateDerivative_Exp;
1699 93 : fn->ops->evaluatefunctionmat[0] = FNEvaluateFunctionMat_Exp_Higham;
1700 93 : fn->ops->evaluatefunctionmat[1] = FNEvaluateFunctionMat_Exp_Pade;
1701 93 : fn->ops->evaluatefunctionmat[2] = FNEvaluateFunctionMat_Exp_GuettelNakatsukasa; /* product form */
1702 93 : fn->ops->evaluatefunctionmat[3] = FNEvaluateFunctionMat_Exp_GuettelNakatsukasa; /* partial fraction */
1703 : #if defined(PETSC_HAVE_CUDA)
1704 : fn->ops->evaluatefunctionmatcuda[1] = FNEvaluateFunctionMat_Exp_Pade_CUDA;
1705 : #if defined(PETSC_HAVE_MAGMA)
1706 : fn->ops->evaluatefunctionmatcuda[0] = FNEvaluateFunctionMat_Exp_Higham_CUDAm;
1707 : fn->ops->evaluatefunctionmatcuda[1] = FNEvaluateFunctionMat_Exp_Pade_CUDAm;
1708 : fn->ops->evaluatefunctionmatcuda[2] = FNEvaluateFunctionMat_Exp_GuettelNakatsukasa_CUDAm; /* product form */
1709 : fn->ops->evaluatefunctionmatcuda[3] = FNEvaluateFunctionMat_Exp_GuettelNakatsukasa_CUDAm; /* partial fraction */
1710 : #endif
1711 : #endif
1712 93 : fn->ops->view = FNView_Exp;
1713 93 : PetscFunctionReturn(PETSC_SUCCESS);
1714 : }
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