Actual source code: fnexp.c

slepc-3.12.2 2020-01-13
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2019, Universitat Politecnica de Valencia, Spain

  6:    This file is part of SLEPc.
  7:    SLEPc is distributed under a 2-clause BSD license (see LICENSE).
  8:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  9: */
 10: /*
 11:    Exponential function  exp(x)
 12: */

 14: #include <slepc/private/fnimpl.h>      /*I "slepcfn.h" I*/
 15: #include <slepcblaslapack.h>

 17: PetscErrorCode FNEvaluateFunction_Exp(FN fn,PetscScalar x,PetscScalar *y)
 18: {
 20:   *y = PetscExpScalar(x);
 21:   return(0);
 22: }

 24: PetscErrorCode FNEvaluateDerivative_Exp(FN fn,PetscScalar x,PetscScalar *y)
 25: {
 27:   *y = PetscExpScalar(x);
 28:   return(0);
 29: }

 31: #define MAX_PADE 6
 32: #define SWAP(a,b,t) {t=a;a=b;b=t;}

 34: PetscErrorCode FNEvaluateFunctionMat_Exp_Pade(FN fn,Mat A,Mat B)
 35: {
 36: #if defined(PETSC_MISSING_LAPACK_GESV) || defined(SLEPC_MISSING_LAPACK_LANGE)
 38:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"GESV/LANGE - Lapack routines are unavailable");
 39: #else
 41:   PetscBLASInt   n,ld,ld2,*ipiv,info,inc=1;
 42:   PetscInt       m,j,k,sexp;
 43:   PetscBool      odd;
 44:   const PetscInt p=MAX_PADE;
 45:   PetscReal      c[MAX_PADE+1],s,*rwork;
 46:   PetscScalar    scale,mone=-1.0,one=1.0,two=2.0,zero=0.0;
 47:   PetscScalar    *Aa,*Ba,*As,*A2,*Q,*P,*W,*aux;

 50:   MatDenseGetArray(A,&Aa);
 51:   MatDenseGetArray(B,&Ba);
 52:   MatGetSize(A,&m,NULL);
 53:   PetscBLASIntCast(m,&n);
 54:   ld  = n;
 55:   ld2 = ld*ld;
 56:   P   = Ba;
 57:   PetscMalloc6(m*m,&Q,m*m,&W,m*m,&As,m*m,&A2,ld,&rwork,ld,&ipiv);
 58:   PetscArraycpy(As,Aa,ld2);

 60:   /* Pade' coefficients */
 61:   c[0] = 1.0;
 62:   for (k=1;k<=p;k++) c[k] = c[k-1]*(p+1-k)/(k*(2*p+1-k));

 64:   /* Scaling */
 65:   s = LAPACKlange_("I",&n,&n,As,&ld,rwork);
 66:   PetscLogFlops(1.0*n*n);
 67:   if (s>0.5) {
 68:     sexp = PetscMax(0,(int)(PetscLogReal(s)/PetscLogReal(2.0))+2);
 69:     scale = PetscPowRealInt(2.0,-sexp);
 70:     PetscStackCallBLAS("BLASscal",BLASscal_(&ld2,&scale,As,&inc));
 71:     PetscLogFlops(1.0*n*n);
 72:   } else sexp = 0;

 74:   /* Horner evaluation */
 75:   PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,As,&ld,As,&ld,&zero,A2,&ld));
 76:   PetscLogFlops(2.0*n*n*n);
 77:   PetscArrayzero(Q,ld2);
 78:   PetscArrayzero(P,ld2);
 79:   for (j=0;j<n;j++) {
 80:     Q[j+j*ld] = c[p];
 81:     P[j+j*ld] = c[p-1];
 82:   }

 84:   odd = PETSC_TRUE;
 85:   for (k=p-1;k>0;k--) {
 86:     if (odd) {
 87:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,Q,&ld,A2,&ld,&zero,W,&ld));
 88:       SWAP(Q,W,aux);
 89:       for (j=0;j<n;j++) Q[j+j*ld] += c[k-1];
 90:       odd = PETSC_FALSE;
 91:     } else {
 92:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,P,&ld,A2,&ld,&zero,W,&ld));
 93:       SWAP(P,W,aux);
 94:       for (j=0;j<n;j++) P[j+j*ld] += c[k-1];
 95:       odd = PETSC_TRUE;
 96:     }
 97:     PetscLogFlops(2.0*n*n*n);
 98:   }
 99:   /*if (odd) {
100:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,Q,&ld,As,&ld,&zero,W,&ld));
101:     SWAP(Q,W,aux);
102:     PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&ld2,&mone,P,&inc,Q,&inc));
103:     PetscStackCallBLAS("LAPACKgesv",LAPACKgesv_(&n,&n,Q,&ld,ipiv,P,&ld,&info));
104:     SlepcCheckLapackInfo("gesv",info);
105:     PetscStackCallBLAS("BLASscal",BLASscal_(&ld2,&two,P,&inc));
106:     for (j=0;j<n;j++) P[j+j*ld] += 1.0;
107:     PetscStackCallBLAS("BLASscal",BLASscal_(&ld2,&mone,P,&inc));
108:   } else {*/
109:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,P,&ld,As,&ld,&zero,W,&ld));
110:     SWAP(P,W,aux);
111:     PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&ld2,&mone,P,&inc,Q,&inc));
112:     PetscStackCallBLAS("LAPACKgesv",LAPACKgesv_(&n,&n,Q,&ld,ipiv,P,&ld,&info));
113:     SlepcCheckLapackInfo("gesv",info);
114:     PetscStackCallBLAS("BLASscal",BLASscal_(&ld2,&two,P,&inc));
115:     for (j=0;j<n;j++) P[j+j*ld] += 1.0;
116:   /*}*/
117:   PetscLogFlops(2.0*n*n*n+2.0*n*n*n/3.0+4.0*n*n);

119:   for (k=1;k<=sexp;k++) {
120:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&one,P,&ld,P,&ld,&zero,W,&ld));
121:     PetscArraycpy(P,W,ld2);
122:   }
123:   if (P!=Ba) { PetscArraycpy(Ba,P,ld2); }
124:   PetscLogFlops(2.0*n*n*n*sexp);

126:   PetscFree6(Q,W,As,A2,rwork,ipiv);
127:   MatDenseRestoreArray(A,&Aa);
128:   MatDenseRestoreArray(B,&Ba);
129:   return(0);
130: #endif
131: }

133: /*
134:  * Set scaling factor (s) and Pade degree (k,m)
135:  */
136: static PetscErrorCode sexpm_params(PetscReal nrm,PetscInt *s,PetscInt *k,PetscInt *m)
137: {
139:   if (nrm>1) {
140:     if      (nrm<200)  {*s = 4; *k = 5; *m = *k-1;}
141:     else if (nrm<1e4)  {*s = 4; *k = 4; *m = *k+1;}
142:     else if (nrm<1e6)  {*s = 4; *k = 3; *m = *k+1;}
143:     else if (nrm<1e9)  {*s = 3; *k = 3; *m = *k+1;}
144:     else if (nrm<1e11) {*s = 2; *k = 3; *m = *k+1;}
145:     else if (nrm<1e12) {*s = 2; *k = 2; *m = *k+1;}
146:     else if (nrm<1e14) {*s = 2; *k = 1; *m = *k+1;}
147:     else               {*s = 1; *k = 1; *m = *k+1;}
148:   } else { /* nrm<1 */
149:     if       (nrm>0.5)  {*s = 4; *k = 4; *m = *k-1;}
150:     else  if (nrm>0.3)  {*s = 3; *k = 4; *m = *k-1;}
151:     else  if (nrm>0.15) {*s = 2; *k = 4; *m = *k-1;}
152:     else  if (nrm>0.07) {*s = 1; *k = 4; *m = *k-1;}
153:     else  if (nrm>0.01) {*s = 0; *k = 4; *m = *k-1;}
154:     else  if (nrm>3e-4) {*s = 0; *k = 3; *m = *k-1;}
155:     else  if (nrm>1e-5) {*s = 0; *k = 3; *m = 0;}
156:     else  if (nrm>1e-8) {*s = 0; *k = 2; *m = 0;}
157:     else                {*s = 0; *k = 1; *m = 0;}
158:   }
159:   return(0);
160: }

162: #if defined(PETSC_HAVE_COMPLEX)
163: /*
164:  * Partial fraction form coefficients.
165:  * If query, the function returns the size necessary to store the coefficients.
166:  */
167: static PetscErrorCode getcoeffs(PetscInt k,PetscInt m,PetscComplex *r,PetscComplex *q,PetscComplex *remain,PetscBool query)
168: {
169:   PetscInt i;
170:   const PetscComplex /* m == k+1 */
171:     p1r4[5] = {-1.582680186458572e+01 - 2.412564578224361e+01*PETSC_i,
172:                -1.582680186458572e+01 + 2.412564578224361e+01*PETSC_i,
173:                 1.499984465975511e+02 + 6.804227952202417e+01*PETSC_i,
174:                 1.499984465975511e+02 - 6.804227952202417e+01*PETSC_i,
175:                -2.733432894659307e+02                                },
176:     p1q4[5] = { 3.655694325463550e+00 + 6.543736899360086e+00*PETSC_i,
177:                 3.655694325463550e+00 - 6.543736899360086e+00*PETSC_i,
178:                 5.700953298671832e+00 + 3.210265600308496e+00*PETSC_i,
179:                 5.700953298671832e+00 - 3.210265600308496e+00*PETSC_i,
180:                 6.286704751729261e+00                               },
181:     p1r3[4] = {-1.130153999597152e+01 + 1.247167585025031e+01*PETSC_i,
182:                -1.130153999597152e+01 - 1.247167585025031e+01*PETSC_i,
183:                 1.330153999597152e+01 - 6.007173273704750e+01*PETSC_i,
184:                 1.330153999597152e+01 + 6.007173273704750e+01*PETSC_i},
185:     p1q3[4] = { 3.212806896871536e+00 + 4.773087433276636e+00*PETSC_i,
186:                 3.212806896871536e+00 - 4.773087433276636e+00*PETSC_i,
187:                 4.787193103128464e+00 + 1.567476416895212e+00*PETSC_i,
188:                 4.787193103128464e+00 - 1.567476416895212e+00*PETSC_i},
189:     p1r2[3] = { 7.648749087422928e+00 + 4.171640244747463e+00*PETSC_i,
190:                 7.648749087422928e+00 - 4.171640244747463e+00*PETSC_i,
191:                -1.829749817484586e+01                                },
192:     p1q2[3] = { 2.681082873627756e+00 + 3.050430199247411e+00*PETSC_i,
193:                 2.681082873627756e+00 - 3.050430199247411e+00*PETSC_i,
194:                 3.637834252744491e+00                                },
195:     p1r1[2] = { 1.000000000000000e+00 - 3.535533905932738e+00*PETSC_i,
196:                 1.000000000000000e+00 + 3.535533905932738e+00*PETSC_i},
197:     p1q1[2] = { 2.000000000000000e+00 + 1.414213562373095e+00*PETSC_i,
198:                 2.000000000000000e+00 - 1.414213562373095e+00*PETSC_i};
199:   const PetscComplex /* m == k-1 */
200:     m1r5[4] = {-1.423367961376821e+02 - 1.385465094833037e+01*PETSC_i,
201:                -1.423367961376821e+02 + 1.385465094833037e+01*PETSC_i,
202:                 2.647367961376822e+02 - 4.814394493714596e+02*PETSC_i,
203:                 2.647367961376822e+02 + 4.814394493714596e+02*PETSC_i},
204:     m1q5[4] = { 5.203941240131764e+00 + 5.805856841805367e+00*PETSC_i,
205:                 5.203941240131764e+00 - 5.805856841805367e+00*PETSC_i,
206:                 6.796058759868242e+00 + 1.886649260140217e+00*PETSC_i,
207:                 6.796058759868242e+00 - 1.886649260140217e+00*PETSC_i},
208:     m1r4[3] = { 2.484269593165883e+01 + 7.460342395992306e+01*PETSC_i,
209:                 2.484269593165883e+01 - 7.460342395992306e+01*PETSC_i,
210:                -1.734353918633177e+02                                },
211:     m1q4[3] = { 4.675757014491557e+00 + 3.913489560603711e+00*PETSC_i,
212:                 4.675757014491557e+00 - 3.913489560603711e+00*PETSC_i,
213:                 5.648485971016893e+00                                },
214:     m1r3[2] = { 2.533333333333333e+01 - 2.733333333333333e+01*PETSC_i,
215:                 2.533333333333333e+01 + 2.733333333333333e+01*PETSC_i},
216:     m1q3[2] = { 4.000000000000000e+00 + 2.000000000000000e+00*PETSC_i,
217:                 4.000000000000000e+00 - 2.000000000000000e+00*PETSC_i};
218:   const PetscScalar /* m == k-1 */
219:     m1remain5[2] = { 2.000000000000000e-01,  9.800000000000000e+00},
220:     m1remain4[2] = {-2.500000000000000e-01, -7.750000000000000e+00},
221:     m1remain3[2] = { 3.333333333333333e-01,  5.666666666666667e+00},
222:     m1remain2[2] = {-0.5,                   -3.5},
223:     remain3[4] = {1.0/6.0, 1.0/2.0, 1, 1},
224:     remain2[3] = {1.0/2.0, 1, 1};

227:   if (query) { /* query about buffer's size */
228:     if (m==k+1) {
229:       *remain = 0;
230:       *r = *q = k+1;
231:       return(0); /* quick return */
232:     }
233:     if (m==k-1) {
234:       *remain = 2;
235:       if (k==5) *r = *q = 4;
236:       else if (k==4) *r = *q = 3;
237:       else if (k==3) *r = *q = 2;
238:       else if (k==2) *r = *q = 1;
239:     }
240:     if (m==0) {
241:       *r = *q = 0;
242:       *remain = k+1;
243:     }
244:   } else {
245:     if (m==k+1) {
246:       if (k==4) {
247:         for (i=0;i<5;i++) { r[i] = p1r4[i]; q[i] = p1q4[i]; }
248:       } else if (k==3) {
249:         for (i=0;i<4;i++) { r[i] = p1r3[i]; q[i] = p1q3[i]; }
250:       } else if (k==2) {
251:         for (i=0;i<3;i++) { r[i] = p1r2[i]; q[i] = p1q2[i]; }
252:       } else if (k==1) {
253:         for (i=0;i<2;i++) { r[i] = p1r1[i]; q[i] = p1q1[i]; }
254:       }
255:       return(0); /* quick return */
256:     }
257:     if (m==k-1) {
258:       if (k==5) {
259:         for (i=0;i<4;i++) { r[i] = m1r5[i]; q[i] = m1q5[i]; }
260:         for (i=0;i<2;i++) remain[i] = m1remain5[i];
261:       } else if (k==4) {
262:         for (i=0;i<3;i++) { r[i] = m1r4[i]; q[i] = m1q4[i]; }
263:         for (i=0;i<2;i++) remain[i] = m1remain4[i];
264:       } else if (k==3) {
265:         for (i=0;i<2;i++) { r[i] = m1r3[i]; q[i] = m1q3[i]; remain[i] = m1remain3[i]; }
266:       } else if (k==2) {
267:         r[0] = -13.5; q[0] = 3;
268:         for (i=0;i<2;i++) remain[i] = m1remain2[i];
269:       }
270:     }
271:     if (m==0) {
272:       r = q = 0;
273:       if (k==3) {
274:         for (i=0;i<4;i++) remain[i] = remain3[i];
275:       } else if (k==2) {
276:         for (i=0;i<3;i++) remain[i] = remain2[i];
277:       }
278:     }
279:   }
280:   return(0);
281: }

283: /*
284:  * Product form coefficients.
285:  * If query, the function returns the size necessary to store the coefficients.
286:  */
287: static PetscErrorCode getcoeffsproduct(PetscInt k,PetscInt m,PetscComplex *p,PetscComplex *q,PetscComplex *mult,PetscBool query)
288: {
289:   PetscInt i;
290:   const PetscComplex /* m == k+1 */
291:   p1p4[4] = {-5.203941240131764e+00 + 5.805856841805367e+00*PETSC_i,
292:              -5.203941240131764e+00 - 5.805856841805367e+00*PETSC_i,
293:              -6.796058759868242e+00 + 1.886649260140217e+00*PETSC_i,
294:              -6.796058759868242e+00 - 1.886649260140217e+00*PETSC_i},
295:   p1q4[5] = { 3.655694325463550e+00 + 6.543736899360086e+00*PETSC_i,
296:               3.655694325463550e+00 - 6.543736899360086e+00*PETSC_i,
297:               6.286704751729261e+00                                ,
298:               5.700953298671832e+00 + 3.210265600308496e+00*PETSC_i,
299:               5.700953298671832e+00 - 3.210265600308496e+00*PETSC_i},
300:   p1p3[3] = {-4.675757014491557e+00 + 3.913489560603711e+00*PETSC_i,
301:              -4.675757014491557e+00 - 3.913489560603711e+00*PETSC_i,
302:              -5.648485971016893e+00                                },
303:   p1q3[4] = { 3.212806896871536e+00 + 4.773087433276636e+00*PETSC_i,
304:               3.212806896871536e+00 - 4.773087433276636e+00*PETSC_i,
305:               4.787193103128464e+00 + 1.567476416895212e+00*PETSC_i,
306:               4.787193103128464e+00 - 1.567476416895212e+00*PETSC_i},
307:   p1p2[2] = {-4.00000000000000e+00  + 2.000000000000000e+00*PETSC_i,
308:              -4.00000000000000e+00  - 2.000000000000000e+00*PETSC_i},
309:   p1q2[3] = { 2.681082873627756e+00 + 3.050430199247411e+00*PETSC_i,
310:               2.681082873627756e+00 - 3.050430199247411e+00*PETSC_i,
311:               3.637834252744491e+00                               },
312:   p1q1[2] = { 2.000000000000000e+00 + 1.414213562373095e+00*PETSC_i,
313:               2.000000000000000e+00 - 1.414213562373095e+00*PETSC_i};
314:   const PetscComplex /* m == k-1 */
315:   m1p5[5] = {-3.655694325463550e+00 + 6.543736899360086e+00*PETSC_i,
316:              -3.655694325463550e+00 - 6.543736899360086e+00*PETSC_i,
317:              -6.286704751729261e+00                                ,
318:              -5.700953298671832e+00 + 3.210265600308496e+00*PETSC_i,
319:              -5.700953298671832e+00 - 3.210265600308496e+00*PETSC_i},
320:   m1q5[4] = { 5.203941240131764e+00 + 5.805856841805367e+00*PETSC_i,
321:               5.203941240131764e+00 - 5.805856841805367e+00*PETSC_i,
322:               6.796058759868242e+00 + 1.886649260140217e+00*PETSC_i,
323:               6.796058759868242e+00 - 1.886649260140217e+00*PETSC_i},
324:   m1p4[4] = {-3.212806896871536e+00 + 4.773087433276636e+00*PETSC_i,
325:              -3.212806896871536e+00 - 4.773087433276636e+00*PETSC_i,
326:              -4.787193103128464e+00 + 1.567476416895212e+00*PETSC_i,
327:              -4.787193103128464e+00 - 1.567476416895212e+00*PETSC_i},
328:   m1q4[3] = { 4.675757014491557e+00 + 3.913489560603711e+00*PETSC_i,
329:               4.675757014491557e+00 - 3.913489560603711e+00*PETSC_i,
330:               5.648485971016893e+00                                },
331:   m1p3[3] = {-2.681082873627756e+00 + 3.050430199247411e+00*PETSC_i,
332:              -2.681082873627756e+00 - 3.050430199247411e+00*PETSC_i,
333:              -3.637834252744491e+00                                },
334:   m1q3[2] = { 4.000000000000000e+00 + 2.000000000000000e+00*PETSC_i,
335:               4.000000000000000e+00 - 2.000000000000001e+00*PETSC_i},
336:   m1p2[2] = {-2.000000000000000e+00 + 1.414213562373095e+00*PETSC_i,
337:              -2.000000000000000e+00 - 1.414213562373095e+00*PETSC_i};

340:   if (query) {
341:     if (m == k+1) {
342:       *mult = 1;
343:       *p = k;
344:       *q = k+1;
345:       return(0);
346:     }
347:     if (m==k-1) {
348:       *mult = 1;
349:       *p = k;
350:       *q = k-1;
351:     }
352:   } else {
353:     if (m == k+1) {
354:       *mult = PetscPowInt(-1,m);
355:       *mult *= m;
356:       if (k==4) {
357:         for (i=0;i<4;i++) { p[i] = p1p4[i]; q[i] = p1q4[i]; }
358:         q[4] = p1q4[4];
359:       } else if (k==3) {
360:         for (i=0;i<3;i++) { p[i] = p1p3[i]; q[i] = p1q3[i]; }
361:         q[3] = p1q3[3];
362:       } else if (k==2) {
363:         for (i=0;i<2;i++) { p[i] = p1p2[i]; q[i] = p1q2[i]; }
364:         q[2] = p1q2[2];
365:       } else if (k==1) {
366:         p[0] = -3;
367:         for (i=0;i<2;i++) q[i] = p1q1[i];
368:       }
369:       return(0);
370:     }
371:     if (m==k-1) {
372:       *mult = PetscPowInt(-1,m);
373:       *mult /= k;
374:       if (k==5) {
375:         for (i=0;i<4;i++) { p[i] = m1p5[i]; q[i] = m1q5[i]; }
376:         p[4] = m1p5[4];
377:       } else if (k==4) {
378:         for (i=0;i<3;i++) { p[i] = m1p4[i]; q[i] = m1q4[i]; }
379:         p[3] = m1p4[3];
380:       } else if (k==3) {
381:         for (i=0;i<2;i++) { p[i] = m1p3[i]; q[i] = m1q3[i]; }
382:         p[2] = m1p3[2];
383:       } else if (k==2) {
384:         for (i=0;i<2;i++) p[i] = m1p2[i];
385:         q[0] = 3;
386:       }
387:     }
388:   }
389:   return(0);
390: }
391: #endif /* PETSC_HAVE_COMPLEX */

393: #if defined(PETSC_USE_COMPLEX)
394: static PetscErrorCode getisreal(PetscInt n,PetscComplex *a,PetscBool *result)
395: {
396:   PetscInt i;

399:   *result=PETSC_TRUE;
400:   for (i=0;i<n&&*result;i++) {
401:     if (PetscImaginaryPartComplex(a[i])) *result=PETSC_FALSE;
402:   }
403:   return(0);
404: }
405: #endif

407: /*
408:  * Matrix exponential implementation based on algorithm and matlab code by Stefan Guettel
409:  * and Yuji Nakatsukasa
410:  *
411:  *     Stefan Guettel and Yuji Nakatsukasa, "Scaled and Squared Subdiagonal Pade
412:  *     Approximation for the Matrix Exponential",
413:  *     SIAM J. Matrix Anal. Appl. 37(1):145-170, 2016.
414:  *     https://doi.org/10.1137/15M1027553
415:  */
416: PetscErrorCode FNEvaluateFunctionMat_Exp_GuettelNakatsukasa(FN fn,Mat A,Mat B)
417: {
418: #if !defined(PETSC_HAVE_COMPLEX)
420:   SETERRQ(PETSC_COMM_SELF,1,"This function requires C99 or C++ complex support");
421: #elif defined(PETSC_MISSING_LAPACK_GEEV) || defined(PETSC_MISSING_LAPACK_GESV) || defined(SLEPC_MISSING_LAPACK_LANGE)
423:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"GEEV/GESV/LANGE - Lapack routines are unavailable");
424: #else
425:   PetscInt       i,j,n_,s,k,m,mod;
426:   PetscBLASInt   n,n2,irsize,rsizediv2,ipsize,iremainsize,query=-1,info,*piv,minlen,lwork,one=1;
427:   PetscReal      nrm,shift;
428: #if defined(PETSC_USE_COMPLEX)
429:   PetscReal      *rwork=NULL;
430: #endif
431:   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;
432:   PetscScalar    *Aa,*Ba,*Ba2,*sMaux,*wr,*wi,expshift,sone=1.0,szero=0.0,*work,work1,*saux;
434:   PetscBool      isreal;
435: #if defined(PETSC_HAVE_ESSL)
436:   PetscScalar    sdummy;
437:   PetscBLASInt   idummy,io=0;
438:   PetscScalar    *wri;
439: #endif

442:   MatGetSize(A,&n_,NULL);
443:   PetscBLASIntCast(n_,&n);
444:   MatDenseGetArray(A,&Aa);
445:   MatDenseGetArray(B,&Ba);
446:   Ba2 = Ba;
447:   PetscBLASIntCast(n*n,&n2);

449:   PetscMalloc2(n2,&sMaux,n2,&Maux);
450:   Maux2 = Maux;
451:   PetscMalloc2(n,&wr,n,&wi);
452:   PetscArraycpy(sMaux,Aa,n2);
453:   /* estimate rightmost eigenvalue and shift A with it */
454: #if !defined(PETSC_HAVE_ESSL)
455: #if !defined(PETSC_USE_COMPLEX)
456:   PetscStackCallBLAS("LAPACKgeev",LAPACKgeev_("N","N",&n,sMaux,&n,wr,wi,NULL,&n,NULL,&n,&work1,&query,&info));
457:   SlepcCheckLapackInfo("geev",info);
458:   PetscBLASIntCast((PetscInt)PetscRealPart(work1),&lwork);
459:   PetscMalloc1(lwork,&work);
460:   PetscStackCallBLAS("LAPACKgeev",LAPACKgeev_("N","N",&n,sMaux,&n,wr,wi,NULL,&n,NULL,&n,work,&lwork,&info));
461:   PetscFree(work);
462: #else
463:   PetscArraycpy(Maux,Aa,n2);
464:   PetscStackCallBLAS("LAPACKgeev",LAPACKgeev_("N","N",&n,Maux,&n,wr,NULL,&n,NULL,&n,&work1,&query,rwork,&info));
465:   SlepcCheckLapackInfo("geev",info);
466:   PetscBLASIntCast((PetscInt)PetscRealPart(work1),&lwork);
467:   PetscMalloc2(2*n,&rwork,lwork,&work);
468:   PetscStackCallBLAS("LAPACKgeev",LAPACKgeev_("N","N",&n,Maux,&n,wr,NULL,&n,NULL,&n,work,&lwork,rwork,&info));
469:   PetscFree2(rwork,work);
470: #endif
471:   SlepcCheckLapackInfo("geev",info);
472: #else /* defined(PETSC_HAVE_ESSL) */
473:   PetscBLASIntCast(3*n,&lwork);
474:   PetscMalloc2(lwork,&work,2*n,&wri);
475:   PetscStackCallBLAS("LAPACKgeev",LAPACKgeev_(&io,Maux,&n,wri,&sdummy,&idummy,&idummy,&n,work,&lwork));
476: #if !defined(PETSC_USE_COMPLEX)
477:   for (i=0;i<n;i++) {
478:     wr[i] = wri[2*i];
479:     wi[i] = wri[2*i+1];
480:   }
481: #else
482:   for (i=0;i<n;i++) wr[i] = wri[i];
483: #endif
484:   PetscFree2(work,wri);
485: #endif
486:   PetscLogFlops(25.0*n*n*n+(n*n*n)/3.0+1.0*n*n*n);

488:   shift = PetscRealPart(wr[0]);
489:   for (i=1;i<n;i++) {
490:     if (PetscRealPart(wr[i]) > shift) shift = PetscRealPart(wr[i]);
491:   }
492:   PetscFree2(wr,wi);
493:   /* shift so that largest real part is (about) 0 */
494:   PetscArraycpy(sMaux,Aa,n2);
495:   for (i=0;i<n;i++) {
496:     sMaux[i+i*n] -= shift;
497:   }
498:   PetscLogFlops(1.0*n);
499: #if defined(PETSC_USE_COMPLEX)
500:   PetscArraycpy(Maux,Aa,n2);
501:   for (i=0;i<n;i++) {
502:     Maux[i+i*n] -= shift;
503:   }
504:   PetscLogFlops(1.0*n);
505: #endif

507:   /* estimate norm(A) and select the scaling factor */
508:   nrm = LAPACKlange_("O",&n,&n,sMaux,&n,NULL);
509:   PetscLogFlops(1.0*n*n);
510:   sexpm_params(nrm,&s,&k,&m);
511:   if (s==0 && k==1 && m==0) { /* exp(A) = I+A to eps! */
512:     expshift = PetscExpReal(shift);
513:     for (i=0;i<n;i++) sMaux[i+i*n] += 1.0;
514:     PetscStackCallBLAS("BLASscal",BLASscal_(&n2,&expshift,sMaux,&one));
515:     PetscLogFlops(1.0*(n+n2));
516:     PetscArraycpy(Ba,sMaux,n2);
517:     PetscFree2(sMaux,Maux);
518:     MatDenseRestoreArray(A,&Aa);
519:     MatDenseRestoreArray(B,&Ba);
520:     return(0); /* quick return */
521:   }

523:   PetscMalloc4(n2,&expmA,n2,&As,n2,&RR,n,&piv);
524:   expmA2 = expmA; RR2 = RR;
525:   /* scale matrix */
526: #if !defined(PETSC_USE_COMPLEX)
527:   for (i=0;i<n2;i++) {
528:     As[i] = sMaux[i];
529:   }
530: #else
531:   PetscArraycpy(As,sMaux,n2);
532: #endif
533:   scale = 1.0/PetscPowRealInt(2.0,s);
534:   PetscStackCallBLAS("BLASCOMPLEXscal",BLASCOMPLEXscal_(&n2,&scale,As,&one));
535:   SlepcLogFlopsComplex(1.0*n2);

537:   /* evaluate Pade approximant (partial fraction or product form) */
538:   if (fn->method==3 || !m) { /* partial fraction */
539:     getcoeffs(k,m,&rsize,&psize,&remainsize,PETSC_TRUE);
540:     PetscBLASIntCast((PetscInt)PetscRealPartComplex(rsize),&irsize);
541:     PetscBLASIntCast((PetscInt)PetscRealPartComplex(psize),&ipsize);
542:     PetscBLASIntCast((PetscInt)PetscRealPartComplex(remainsize),&iremainsize);
543:     PetscMalloc3(irsize,&r,ipsize,&p,iremainsize,&remainterm);
544:     getcoeffs(k,m,r,p,remainterm,PETSC_FALSE);

546:     PetscArrayzero(expmA,n2);
547: #if !defined(PETSC_USE_COMPLEX)
548:     isreal = PETSC_TRUE;
549: #else
550:     getisreal(n2,Maux,&isreal);
551: #endif
552:     if (isreal) {
553:       rsizediv2 = irsize/2;
554:       for (i=0;i<rsizediv2;i++) { /* use partial fraction to get R(As) */
555:         PetscArraycpy(Maux,As,n2);
556:         PetscArrayzero(RR,n2);
557:         for (j=0;j<n;j++) {
558:           Maux[j+j*n] -= p[2*i];
559:           RR[j+j*n] = r[2*i];
560:         }
561:         PetscStackCallBLAS("LAPACKCOMPLEXgesv",LAPACKCOMPLEXgesv_(&n,&n,Maux,&n,piv,RR,&n,&info));
562:         SlepcCheckLapackInfo("gesv",info);
563:         for (j=0;j<n2;j++) {
564:           expmA[j] += RR[j] + PetscConj(RR[j]);
565:         }
566:         /* loop(n) + gesv + loop(n2) */
567:         SlepcLogFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)+2.0*n2);
568:       }

570:       mod = ipsize % 2;
571:       if (mod) {
572:         PetscArraycpy(Maux,As,n2);
573:         PetscArrayzero(RR,n2);
574:         for (j=0;j<n;j++) {
575:           Maux[j+j*n] -= p[ipsize-1];
576:           RR[j+j*n] = r[irsize-1];
577:         }
578:         PetscStackCallBLAS("LAPACKCOMPLEXgesv",LAPACKCOMPLEXgesv_(&n,&n,Maux,&n,piv,RR,&n,&info));
579:         SlepcCheckLapackInfo("gesv",info);
580:         for (j=0;j<n2;j++) {
581:           expmA[j] += RR[j];
582:         }
583:         SlepcLogFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)+1.0*n2);
584:       }
585:     } else { /* complex */
586:       for (i=0;i<irsize;i++) { /* use partial fraction to get R(As) */
587:         PetscArraycpy(Maux,As,n2);
588:         PetscArrayzero(RR,n2);
589:         for (j=0;j<n;j++) {
590:           Maux[j+j*n] -= p[i];
591:           RR[j+j*n] = r[i];
592:         }
593:         PetscStackCallBLAS("LAPACKCOMPLEXgesv",LAPACKCOMPLEXgesv_(&n,&n,Maux,&n,piv,RR,&n,&info));
594:         SlepcCheckLapackInfo("gesv",info);
595:         for (j=0;j<n2;j++) {
596:           expmA[j] += RR[j];
597:         }
598:         SlepcLogFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n)+1.0*n2);
599:       }
600:     }
601:     for (i=0;i<iremainsize;i++) {
602:       if (!i) {
603:         PetscArrayzero(RR,n2);
604:         for (j=0;j<n;j++) {
605:           RR[j+j*n] = remainterm[iremainsize-1];
606:         }
607:       } else {
608:         PetscArraycpy(RR,As,n2);
609:         for (j=1;j<i;j++) {
610:           PetscStackCallBLAS("BLASCOMPLEXgemm",BLASCOMPLEXgemm_("N","N",&n,&n,&n,&cone,RR,&n,RR,&n,&czero,Maux,&n));
611:           SWAP(RR,Maux,aux);
612:           SlepcLogFlopsComplex(2.0*n*n*n);
613:         }
614:         PetscStackCallBLAS("BLASCOMPLEXscal",BLASCOMPLEXscal_(&n2,&remainterm[iremainsize-1-i],RR,&one));
615:         SlepcLogFlopsComplex(1.0*n2);
616:       }
617:       for (j=0;j<n2;j++) {
618:         expmA[j] += RR[j];
619:       }
620:       SlepcLogFlopsComplex(1.0*n2);
621:     }
622:     PetscFree3(r,p,remainterm);
623:   } else { /* product form, default */
624:     getcoeffsproduct(k,m,&rsize,&psize,&mult,PETSC_TRUE);
625:     PetscBLASIntCast((PetscInt)PetscRealPartComplex(rsize),&irsize);
626:     PetscBLASIntCast((PetscInt)PetscRealPartComplex(psize),&ipsize);
627:     PetscMalloc2(irsize,&rootp,ipsize,&rootq);
628:     getcoeffsproduct(k,m,rootp,rootq,&mult,PETSC_FALSE);

630:     PetscArrayzero(expmA,n2);
631:     for (i=0;i<n;i++) { /* initialize */
632:       expmA[i+i*n] = 1.0;
633:     }
634:     minlen = PetscMin(irsize,ipsize);
635:     for (i=0;i<minlen;i++) {
636:       PetscArraycpy(RR,As,n2);
637:       for (j=0;j<n;j++) {
638:         RR[j+j*n] -= rootp[i];
639:       }
640:       PetscStackCallBLAS("BLASCOMPLEXgemm",BLASCOMPLEXgemm_("N","N",&n,&n,&n,&cone,RR,&n,expmA,&n,&czero,Maux,&n));
641:       SWAP(expmA,Maux,aux);
642:       PetscArraycpy(RR,As,n2);
643:       for (j=0;j<n;j++) {
644:         RR[j+j*n] -= rootq[i];
645:       }
646:       PetscStackCallBLAS("LAPACKCOMPLEXgesv",LAPACKCOMPLEXgesv_(&n,&n,RR,&n,piv,expmA,&n,&info));
647:       SlepcCheckLapackInfo("gesv",info);
648:       /* loop(n) + gemm + loop(n) + gesv */
649:       SlepcLogFlopsComplex(1.0*n+(2.0*n*n*n)+1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n));
650:     }
651:     /* extra numerator */
652:     for (i=minlen;i<irsize;i++) {
653:       PetscArraycpy(RR,As,n2);
654:       for (j=0;j<n;j++) {
655:         RR[j+j*n] -= rootp[i];
656:       }
657:       PetscStackCallBLAS("BLASCOMPLEXgemm",BLASCOMPLEXgemm_("N","N",&n,&n,&n,&cone,RR,&n,expmA,&n,&czero,Maux,&n));
658:       SWAP(expmA,Maux,aux);
659:       SlepcLogFlopsComplex(1.0*n+2.0*n*n*n);
660:     }
661:     /* extra denominator */
662:     for (i=minlen;i<ipsize;i++) {
663:       PetscArraycpy(RR,As,n2);
664:       for (j=0;j<n;j++) RR[j+j*n] -= rootq[i];
665:       PetscStackCallBLAS("LAPACKCOMPLEXgesv",LAPACKCOMPLEXgesv_(&n,&n,RR,&n,piv,expmA,&n,&info));
666:       SlepcCheckLapackInfo("gesv",info);
667:       SlepcLogFlopsComplex(1.0*n+(2.0*n*n*n/3.0+2.0*n*n*n));
668:     }
669:     PetscStackCallBLAS("BLASCOMPLEXscal",BLASCOMPLEXscal_(&n2,&mult,expmA,&one));
670:     SlepcLogFlopsComplex(1.0*n2);
671:     PetscFree2(rootp,rootq);
672:   }

674: #if !defined(PETSC_USE_COMPLEX)
675:   for (i=0;i<n2;i++) {
676:     Ba2[i] = PetscRealPartComplex(expmA[i]);
677:   }
678: #else
679:   PetscArraycpy(Ba2,expmA,n2);
680: #endif

682:   /* perform repeated squaring */
683:   for (i=0;i<s;i++) { /* final squaring */
684:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n,&n,&n,&sone,Ba2,&n,Ba2,&n,&szero,sMaux,&n));
685:     SWAP(Ba2,sMaux,saux);
686:     PetscLogFlops(2.0*n*n*n);
687:   }
688:   if (Ba2!=Ba) {
689:     PetscArraycpy(Ba,Ba2,n2);
690:     sMaux = Ba2;
691:   }
692:   expshift = PetscExpReal(shift);
693:   PetscStackCallBLAS("BLASscal",BLASscal_(&n2,&expshift,Ba,&one));
694:   PetscLogFlops(1.0*n2);

696:   /* restore pointers */
697:   Maux = Maux2; expmA = expmA2; RR = RR2;
698:   PetscFree2(sMaux,Maux);
699:   PetscFree4(expmA,As,RR,piv);
700:   MatDenseRestoreArray(A,&Aa);
701:   MatDenseRestoreArray(B,&Ba);
702:   return(0);
703: #endif
704: }

706: #define SMALLN 100

708: /*
709:  * Function needed to compute optimal parameters (required workspace is 3*n*n)
710:  */
711: static PetscInt ell(PetscBLASInt n,PetscScalar *A,PetscReal coeff,PetscInt m,PetscScalar *work,PetscRandom rand)
712: {
713:   PetscScalar    *Ascaled=work;
714:   PetscReal      nrm,alpha,beta,rwork[1];
715:   PetscInt       t;
716:   PetscBLASInt   i,j;

720:   beta = PetscPowReal(coeff,1.0/(2*m+1));
721:   for (i=0;i<n;i++)
722:     for (j=0;j<n;j++)
723:       Ascaled[i+j*n] = beta*PetscAbsScalar(A[i+j*n]);
724:   nrm = LAPACKlange_("O",&n,&n,A,&n,rwork);
725:   PetscLogFlops(2.0*n*n);
726:   SlepcNormAm(n,Ascaled,2*m+1,work+n*n,rand,&alpha);
727:   alpha /= nrm;
728:   t = PetscMax((PetscInt)PetscCeilReal(PetscLogReal(2.0*alpha/PETSC_MACHINE_EPSILON)/PetscLogReal(2.0)/(2*m)),0);
729:   PetscFunctionReturn(t);
730: }

732: /*
733:  * Compute scaling parameter (s) and order of Pade approximant (m)  (required workspace is 4*n*n)
734:  */
735: static PetscErrorCode expm_params(PetscInt n,PetscScalar **Apowers,PetscInt *s,PetscInt *m,PetscScalar *work)
736: {
737:   PetscErrorCode  ierr;
738:   PetscScalar     sfactor,sone=1.0,szero=0.0,*A=Apowers[0],*Ascaled;
739:   PetscReal       d4,d6,d8,d10,eta1,eta3,eta4,eta5,rwork[1];
740:   PetscBLASInt    n_,n2,one=1;
741:   PetscRandom     rand;
742:   const PetscReal coeff[5] = { 9.92063492063492e-06, 9.94131285136576e-11,  /* backward error function */
743:                                2.22819456055356e-16, 1.69079293431187e-22, 8.82996160201868e-36 };
744:   const PetscReal theta[5] = { 1.495585217958292e-002,    /* m = 3  */
745:                                2.539398330063230e-001,    /* m = 5  */
746:                                9.504178996162932e-001,    /* m = 7  */
747:                                2.097847961257068e+000,    /* m = 9  */
748:                                5.371920351148152e+000 };  /* m = 13 */

751:   *s = 0;
752:   *m = 13;
753:   PetscBLASIntCast(n,&n_);
754:   PetscRandomCreate(PETSC_COMM_SELF,&rand);
755:   d4 = PetscPowReal(LAPACKlange_("O",&n_,&n_,Apowers[2],&n_,rwork),1.0/4.0);
756:   if (d4==0.0) { /* safeguard for the case A = 0 */
757:     *m = 3;
758:     goto done;
759:   }
760:   d6 = PetscPowReal(LAPACKlange_("O",&n_,&n_,Apowers[3],&n_,rwork),1.0/6.0);
761:   PetscLogFlops(2.0*n*n);
762:   eta1 = PetscMax(d4,d6);
763:   if (eta1<=theta[0] && !ell(n_,A,coeff[0],3,work,rand)) {
764:     *m = 3;
765:     goto done;
766:   }
767:   if (eta1<=theta[1] && !ell(n_,A,coeff[1],5,work,rand)) {
768:     *m = 5;
769:     goto done;
770:   }
771:   if (n<SMALLN) {
772:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[2],&n_,Apowers[2],&n_,&szero,work,&n_));
773:     d8 = PetscPowReal(LAPACKlange_("O",&n_,&n_,work,&n_,rwork),1.0/8.0);
774:     PetscLogFlops(2.0*n*n*n+1.0*n*n);
775:   } else {
776:     SlepcNormAm(n_,Apowers[2],2,work,rand,&d8);
777:     d8 = PetscPowReal(d8,1.0/8.0);
778:   }
779:   eta3 = PetscMax(d6,d8);
780:   if (eta3<=theta[2] && !ell(n_,A,coeff[2],7,work,rand)) {
781:     *m = 7;
782:     goto done;
783:   }
784:   if (eta3<=theta[3] && !ell(n_,A,coeff[3],9,work,rand)) {
785:     *m = 9;
786:     goto done;
787:   }
788:   if (n<SMALLN) {
789:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[2],&n_,Apowers[3],&n_,&szero,work,&n_));
790:     d10 = PetscPowReal(LAPACKlange_("O",&n_,&n_,work,&n_,rwork),1.0/10.0);
791:     PetscLogFlops(2.0*n*n*n+1.0*n*n);
792:   } else {
793:     SlepcNormAm(n_,Apowers[1],5,work,rand,&d10);
794:     d10 = PetscPowReal(d10,1.0/10.0);
795:   }
796:   eta4 = PetscMax(d8,d10);
797:   eta5 = PetscMin(eta3,eta4);
798:   *s = PetscMax((PetscInt)PetscCeilReal(PetscLogReal(eta5/theta[4])/PetscLogReal(2.0)),0);
799:   if (*s) {
800:     Ascaled = work+3*n*n;
801:     n2 = n_*n_;
802:     PetscStackCallBLAS("BLAScopy",BLAScopy_(&n2,A,&one,Ascaled,&one));
803:     sfactor = PetscPowRealInt(2.0,-(*s));
804:     PetscStackCallBLAS("BLASscal",BLASscal_(&n2,&sfactor,Ascaled,&one));
805:     PetscLogFlops(1.0*n*n);
806:   } else Ascaled = A;
807:   *s += ell(n_,Ascaled,coeff[4],13,work,rand);
808: done:
809:   PetscRandomDestroy(&rand);
810:   return(0);
811: }

813: /*
814:  * Matrix exponential implementation based on algorithm and matlab code by N. Higham and co-authors
815:  *
816:  *     N. J. Higham, "The scaling and squaring method for the matrix exponential
817:  *     revisited", SIAM J. Matrix Anal. Appl. 26(4):1179-1193, 2005.
818:  */
819: PetscErrorCode FNEvaluateFunctionMat_Exp_Higham(FN fn,Mat A,Mat B)
820: {
821: #if defined(PETSC_MISSING_LAPACK_GESV) || defined(SLEPC_MISSING_LAPACK_LANGE)
823:   SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"GESV/LANGE - Lapack routines are unavailable");
824: #else
825:   PetscErrorCode    ierr;
826:   PetscBLASInt      n_,n2,*ipiv,info,one=1;
827:   PetscInt          n,m,j,s;
828:   PetscScalar       scale,smone=-1.0,sone=1.0,stwo=2.0,szero=0.0;
829:   PetscScalar       *Aa,*Ba,*Apowers[5],*Q,*P,*W,*work,*aux;
830:   const PetscScalar *c;
831:   const PetscScalar c3[4]   = { 120, 60, 12, 1 };
832:   const PetscScalar c5[6]   = { 30240, 15120, 3360, 420, 30, 1 };
833:   const PetscScalar c7[8]   = { 17297280, 8648640, 1995840, 277200, 25200, 1512, 56, 1 };
834:   const PetscScalar c9[10]  = { 17643225600, 8821612800, 2075673600, 302702400, 30270240,
835:                                 2162160, 110880, 3960, 90, 1 };
836:   const PetscScalar c13[14] = { 64764752532480000, 32382376266240000, 7771770303897600,
837:                                 1187353796428800,  129060195264000,   10559470521600,
838:                                 670442572800,      33522128640,       1323241920,
839:                                 40840800,          960960,            16380,  182,  1 };

842:   MatDenseGetArray(A,&Aa);
843:   MatDenseGetArray(B,&Ba);
844:   MatGetSize(A,&n,NULL);
845:   PetscBLASIntCast(n,&n_);
846:   n2 = n_*n_;
847:   PetscMalloc2(8*n*n,&work,n,&ipiv);

849:   /* Matrix powers */
850:   Apowers[0] = work;                  /* Apowers[0] = A   */
851:   Apowers[1] = Apowers[0] + n*n;      /* Apowers[1] = A^2 */
852:   Apowers[2] = Apowers[1] + n*n;      /* Apowers[2] = A^4 */
853:   Apowers[3] = Apowers[2] + n*n;      /* Apowers[3] = A^6 */
854:   Apowers[4] = Apowers[3] + n*n;      /* Apowers[4] = A^8 */

856:   PetscArraycpy(Apowers[0],Aa,n2);
857:   PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[0],&n_,Apowers[0],&n_,&szero,Apowers[1],&n_));
858:   PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[1],&n_,Apowers[1],&n_,&szero,Apowers[2],&n_));
859:   PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[1],&n_,Apowers[2],&n_,&szero,Apowers[3],&n_));
860:   PetscLogFlops(6.0*n*n*n);

862:   /* Compute scaling parameter and order of Pade approximant */
863:   expm_params(n,Apowers,&s,&m,Apowers[4]);

865:   if (s) { /* rescale */
866:     for (j=0;j<4;j++) {
867:       scale = PetscPowRealInt(2.0,-PetscMax(2*j,1)*s);
868:       PetscStackCallBLAS("BLASscal",BLASscal_(&n2,&scale,Apowers[j],&one));
869:     }
870:     PetscLogFlops(4.0*n*n);
871:   }

873:   /* Evaluate the Pade approximant */
874:   switch (m) {
875:     case 3:  c = c3;  break;
876:     case 5:  c = c5;  break;
877:     case 7:  c = c7;  break;
878:     case 9:  c = c9;  break;
879:     case 13: c = c13; break;
880:     default: SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong value of m %d",m);
881:   }
882:   P = Ba;
883:   Q = Apowers[4] + n*n;
884:   W = Q + n*n;
885:   switch (m) {
886:     case 3:
887:     case 5:
888:     case 7:
889:     case 9:
890:       if (m==9) PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[1],&n_,Apowers[3],&n_,&szero,Apowers[4],&n_));
891:       PetscArrayzero(P,n2);
892:       PetscArrayzero(Q,n2);
893:       for (j=0;j<n;j++) {
894:         P[j+j*n] = c[1];
895:         Q[j+j*n] = c[0];
896:       }
897:       for (j=m;j>=3;j-=2) {
898:         PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[j],Apowers[(j+1)/2-1],&one,P,&one));
899:         PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[j-1],Apowers[(j+1)/2-1],&one,Q,&one));
900:         PetscLogFlops(4.0*n*n);
901:       }
902:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[0],&n_,P,&n_,&szero,W,&n_));
903:       PetscLogFlops(2.0*n*n*n);
904:       SWAP(P,W,aux);
905:       break;
906:     case 13:
907:       /*  P = A*(Apowers[3]*(c[13]*Apowers[3] + c[11]*Apowers[2] + c[9]*Apowers[1])
908:               + c[7]*Apowers[3] + c[5]*Apowers[2] + c[3]*Apowers[1] + c[1]*I)       */
909:       PetscStackCallBLAS("BLAScopy",BLAScopy_(&n2,Apowers[3],&one,P,&one));
910:       PetscStackCallBLAS("BLASscal",BLASscal_(&n2,&c[13],P,&one));
911:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[11],Apowers[2],&one,P,&one));
912:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[9],Apowers[1],&one,P,&one));
913:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[3],&n_,P,&n_,&szero,W,&n_));
914:       PetscLogFlops(5.0*n*n+2.0*n*n*n);
915:       PetscArrayzero(P,n2);
916:       for (j=0;j<n;j++) P[j+j*n] = c[1];
917:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[7],Apowers[3],&one,P,&one));
918:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[5],Apowers[2],&one,P,&one));
919:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[3],Apowers[1],&one,P,&one));
920:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&sone,P,&one,W,&one));
921:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[0],&n_,W,&n_,&szero,P,&n_));
922:       PetscLogFlops(7.0*n*n+2.0*n*n*n);
923:       /*  Q = Apowers[3]*(c[12]*Apowers[3] + c[10]*Apowers[2] + c[8]*Apowers[1])
924:               + c[6]*Apowers[3] + c[4]*Apowers[2] + c[2]*Apowers[1] + c[0]*I        */
925:       PetscStackCallBLAS("BLAScopy",BLAScopy_(&n2,Apowers[3],&one,Q,&one));
926:       PetscStackCallBLAS("BLASscal",BLASscal_(&n2,&c[12],Q,&one));
927:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[10],Apowers[2],&one,Q,&one));
928:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[8],Apowers[1],&one,Q,&one));
929:       PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,Apowers[3],&n_,Q,&n_,&szero,W,&n_));
930:       PetscLogFlops(5.0*n*n+2.0*n*n*n);
931:       PetscArrayzero(Q,n2);
932:       for (j=0;j<n;j++) Q[j+j*n] = c[0];
933:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[6],Apowers[3],&one,Q,&one));
934:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[4],Apowers[2],&one,Q,&one));
935:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&c[2],Apowers[1],&one,Q,&one));
936:       PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&sone,W,&one,Q,&one));
937:       PetscLogFlops(7.0*n*n);
938:       break;
939:     default: SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Wrong value of m %d",m);
940:   }
941:   PetscStackCallBLAS("BLASaxpy",BLASaxpy_(&n2,&smone,P,&one,Q,&one));
942:   PetscStackCallBLAS("LAPACKgesv",LAPACKgesv_(&n_,&n_,Q,&n_,ipiv,P,&n_,&info));
943:   SlepcCheckLapackInfo("gesv",info);
944:   PetscStackCallBLAS("BLASscal",BLASscal_(&n2,&stwo,P,&one));
945:   for (j=0;j<n;j++) P[j+j*n] += 1.0;
946:   PetscLogFlops(2.0*n*n*n/3.0+4.0*n*n);

948:   /* Squaring */
949:   for (j=1;j<=s;j++) {
950:     PetscStackCallBLAS("BLASgemm",BLASgemm_("N","N",&n_,&n_,&n_,&sone,P,&n_,P,&n_,&szero,W,&n_));
951:     SWAP(P,W,aux);
952:   }
953:   if (P!=Ba) { PetscArraycpy(Ba,P,n2); }
954:   PetscLogFlops(2.0*n*n*n*s);

956:   PetscFree2(work,ipiv);
957:   MatDenseRestoreArray(A,&Aa);
958:   MatDenseRestoreArray(B,&Ba);
959:   return(0);
960: #endif
961: }

963: PetscErrorCode FNView_Exp(FN fn,PetscViewer viewer)
964: {
966:   PetscBool      isascii;
967:   char           str[50];
968:   const char     *methodname[] = {
969:                   "scaling & squaring, [m/m] Pade approximant (Higham)",
970:                   "scaling & squaring, [6/6] Pade approximant",
971:                   "scaling & squaring, subdiagonal Pade approximant (product form)",
972:                   "scaling & squaring, subdiagonal Pade approximant (partial fraction)"
973:   };
974:   const int      nmeth=sizeof(methodname)/sizeof(methodname[0]);

977:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);
978:   if (isascii) {
979:     if (fn->beta==(PetscScalar)1.0) {
980:       if (fn->alpha==(PetscScalar)1.0) {
981:         PetscViewerASCIIPrintf(viewer,"  Exponential: exp(x)\n");
982:       } else {
983:         SlepcSNPrintfScalar(str,50,fn->alpha,PETSC_TRUE);
984:         PetscViewerASCIIPrintf(viewer,"  Exponential: exp(%s*x)\n",str);
985:       }
986:     } else {
987:       SlepcSNPrintfScalar(str,50,fn->beta,PETSC_TRUE);
988:       if (fn->alpha==(PetscScalar)1.0) {
989:         PetscViewerASCIIPrintf(viewer,"  Exponential: %s*exp(x)\n",str);
990:       } else {
991:         PetscViewerASCIIPrintf(viewer,"  Exponential: %s",str);
992:         PetscViewerASCIIUseTabs(viewer,PETSC_FALSE);
993:         SlepcSNPrintfScalar(str,50,fn->alpha,PETSC_TRUE);
994:         PetscViewerASCIIPrintf(viewer,"*exp(%s*x)\n",str);
995:         PetscViewerASCIIUseTabs(viewer,PETSC_TRUE);
996:       }
997:     }
998:     if (fn->method<nmeth) {
999:       PetscViewerASCIIPrintf(viewer,"  computing matrix functions with: %s\n",methodname[fn->method]);
1000:     }
1001:   }
1002:   return(0);
1003: }

1005: SLEPC_EXTERN PetscErrorCode FNCreate_Exp(FN fn)
1006: {
1008:   fn->ops->evaluatefunction       = FNEvaluateFunction_Exp;
1009:   fn->ops->evaluatederivative     = FNEvaluateDerivative_Exp;
1010:   fn->ops->evaluatefunctionmat[0] = FNEvaluateFunctionMat_Exp_Higham;
1011:   fn->ops->evaluatefunctionmat[1] = FNEvaluateFunctionMat_Exp_Pade;
1012:   fn->ops->evaluatefunctionmat[2] = FNEvaluateFunctionMat_Exp_GuettelNakatsukasa; /* product form */
1013:   fn->ops->evaluatefunctionmat[3] = FNEvaluateFunctionMat_Exp_GuettelNakatsukasa; /* partial fraction */
1014:   fn->ops->view                   = FNView_Exp;
1015:   return(0);
1016: }