Actual source code: mfnexpokit.c
slepc-3.15.1 2021-05-28
1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-2021, 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: SLEPc matrix function solver: "expokit"
13: Method: Arnoldi method tailored for the matrix exponential
15: Algorithm:
17: Uses Arnoldi relations to compute exp(t_step*A)*v_last for
18: several time steps.
20: References:
22: [1] R. Sidje, "Expokit: a software package for computing matrix
23: exponentials", ACM Trans. Math. Softw. 24(1):130-156, 1998.
24: */
26: #include <slepc/private/mfnimpl.h>
28: PetscErrorCode MFNSetUp_Expokit(MFN mfn)
29: {
31: PetscInt N;
32: PetscBool isexp;
35: MatGetSize(mfn->A,&N,NULL);
36: if (mfn->ncv==PETSC_DEFAULT) mfn->ncv = PetscMin(30,N);
37: if (mfn->max_it==PETSC_DEFAULT) mfn->max_it = 100;
38: MFNAllocateSolution(mfn,2);
40: PetscObjectTypeCompare((PetscObject)mfn->fn,FNEXP,&isexp);
41: if (!isexp) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"This solver only supports the exponential function");
42: return(0);
43: }
45: PetscErrorCode MFNSolve_Expokit(MFN mfn,Vec b,Vec x)
46: {
47: PetscErrorCode ierr;
48: PetscInt mxstep,mxrej,m,mb,ld,i,j,ireject,mx,k1;
49: Vec v,r;
50: Mat M=NULL,K=NULL;
51: FN fn;
52: PetscScalar *H,*B,*F,*betaF,t,sgn,sfactor;
53: const PetscScalar *pK;
54: PetscReal anorm,avnorm,tol,err_loc,rndoff,t_out,t_new,t_now,t_step;
55: PetscReal xm,fact,s,p1,p2,beta,beta2,gamma,delta;
56: PetscBool breakdown;
59: m = mfn->ncv;
60: tol = mfn->tol;
61: mxstep = mfn->max_it;
62: mxrej = 10;
63: gamma = 0.9;
64: delta = 1.2;
65: mb = m;
66: FNGetScale(mfn->fn,&t,&sfactor);
67: FNDuplicate(mfn->fn,PetscObjectComm((PetscObject)mfn->fn),&fn);
68: FNSetScale(fn,1.0,1.0);
69: t_out = PetscAbsScalar(t);
70: t_now = 0.0;
71: MatNorm(mfn->A,NORM_INFINITY,&anorm);
72: rndoff = anorm*PETSC_MACHINE_EPSILON;
74: k1 = 2;
75: xm = 1.0/(PetscReal)m;
76: beta = mfn->bnorm;
77: fact = PetscPowRealInt((m+1)/2.72,m+1)*PetscSqrtReal(2.0*PETSC_PI*(m+1));
78: t_new = (1.0/anorm)*PetscPowReal((fact*tol)/(4.0*beta*anorm),xm);
79: s = PetscPowReal(10.0,PetscFloorReal(PetscLog10Real(t_new))-1);
80: t_new = PetscCeilReal(t_new/s)*s;
81: sgn = t/PetscAbsScalar(t);
83: VecCopy(b,x);
84: ld = m+2;
85: PetscCalloc3(m+1,&betaF,ld*ld,&H,ld*ld,&B);
87: while (mfn->reason == MFN_CONVERGED_ITERATING) {
88: mfn->its++;
89: if (PetscIsInfOrNanReal(t_new)) t_new = PETSC_MAX_REAL;
90: t_step = PetscMin(t_out-t_now,t_new);
91: BVInsertVec(mfn->V,0,x);
92: BVScaleColumn(mfn->V,0,1.0/beta);
93: BVMatArnoldi(mfn->V,mfn->transpose_solve?mfn->AT:mfn->A,H,ld,0,&mb,&beta2,&breakdown);
94: if (breakdown) {
95: k1 = 0;
96: t_step = t_out-t_now;
97: }
98: if (k1!=0) {
99: H[m+1+ld*m] = 1.0;
100: BVGetColumn(mfn->V,m,&v);
101: BVGetColumn(mfn->V,m+1,&r);
102: MatMult(mfn->transpose_solve?mfn->AT:mfn->A,v,r);
103: BVRestoreColumn(mfn->V,m,&v);
104: BVRestoreColumn(mfn->V,m+1,&r);
105: BVNormColumn(mfn->V,m+1,NORM_2,&avnorm);
106: }
107: PetscArraycpy(B,H,ld*ld);
109: ireject = 0;
110: while (ireject <= mxrej) {
111: mx = mb + k1;
112: for (i=0;i<mx;i++) {
113: for (j=0;j<mx;j++) {
114: H[i+j*ld] = sgn*B[i+j*ld]*t_step;
115: }
116: }
117: MFN_CreateDenseMat(mx,&M);
118: MFN_CreateDenseMat(mx,&K);
119: MatDenseGetArray(M,&F);
120: for (j=0;j<mx;j++) {
121: PetscArraycpy(F+j*mx,H+j*ld,mx);
122: }
123: MatDenseRestoreArray(M,&F);
124: FNEvaluateFunctionMat(fn,M,K);
126: if (k1==0) {
127: err_loc = tol;
128: break;
129: } else {
130: MatDenseGetArrayRead(K,&pK);
131: p1 = PetscAbsScalar(beta*pK[m]);
132: p2 = PetscAbsScalar(beta*pK[m+1]*avnorm);
133: MatDenseRestoreArrayRead(K,&pK);
134: if (p1 > 10*p2) {
135: err_loc = p2;
136: xm = 1.0/(PetscReal)m;
137: } else if (p1 > p2) {
138: err_loc = (p1*p2)/(p1-p2);
139: xm = 1.0/(PetscReal)m;
140: } else {
141: err_loc = p1;
142: xm = 1.0/(PetscReal)(m-1);
143: }
144: }
145: if (err_loc <= delta*t_step*tol) break;
146: else {
147: t_step = gamma*t_step*PetscPowReal(t_step*tol/err_loc,xm);
148: s = PetscPowReal(10.0,PetscFloorReal(PetscLog10Real(t_step))-1);
149: t_step = PetscCeilReal(t_step/s)*s;
150: ireject = ireject+1;
151: }
152: }
154: mx = mb + PetscMax(0,k1-1);
155: MatDenseGetArrayRead(K,&pK);
156: for (j=0;j<mx;j++) betaF[j] = beta*pK[j];
157: MatDenseRestoreArrayRead(K,&pK);
158: BVSetActiveColumns(mfn->V,0,mx);
159: BVMultVec(mfn->V,1.0,0.0,x,betaF);
160: VecNorm(x,NORM_2,&beta);
162: t_now = t_now+t_step;
163: if (t_now>=t_out) mfn->reason = MFN_CONVERGED_TOL;
164: else {
165: t_new = gamma*t_step*PetscPowReal((t_step*tol)/err_loc,xm);
166: s = PetscPowReal(10.0,PetscFloorReal(PetscLog10Real(t_new))-1);
167: t_new = PetscCeilReal(t_new/s)*s;
168: }
169: err_loc = PetscMax(err_loc,rndoff);
170: if (mfn->its==mxstep) mfn->reason = MFN_DIVERGED_ITS;
171: MFNMonitor(mfn,mfn->its,err_loc);
172: }
173: VecScale(x,sfactor);
175: MatDestroy(&M);
176: MatDestroy(&K);
177: FNDestroy(&fn);
178: PetscFree3(betaF,H,B);
179: return(0);
180: }
182: SLEPC_EXTERN PetscErrorCode MFNCreate_Expokit(MFN mfn)
183: {
185: mfn->ops->solve = MFNSolve_Expokit;
186: mfn->ops->setup = MFNSetUp_Expokit;
187: return(0);
188: }