1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-2016, Universitat Politecnica de Valencia, Spain
6: This file is part of SLEPc.
8: SLEPc is free software: you can redistribute it and/or modify it under the
9: terms of version 3 of the GNU Lesser General Public License as published by
10: the Free Software Foundation.
12: SLEPc is distributed in the hope that it will be useful, but WITHOUT ANY
13: WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
14: FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for
15: more details.
17: You should have received a copy of the GNU Lesser General Public License
18: along with SLEPc. If not, see <http://www.gnu.org/licenses/>.
19: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
20: */
22: static char help[] = "Eigenvalue problem associated with a Markov model of a random walk on a triangular grid. "
23: "It is a standard nonsymmetric eigenproblem with real eigenvalues and the rightmost eigenvalue is known to be 1.\n"
24: "This example illustrates how the user can set the initial vector.\n\n"
25: "The command line options are:\n"
26: " -m <m>, where <m> = number of grid subdivisions in each dimension.\n\n";
28: #include <slepceps.h>
30: /*
31: User-defined routines
32: */
33: PetscErrorCode MatMarkovModel(PetscInt m,Mat A);
37: int main(int argc,char **argv) 38: {
39: Vec v0; /* initial vector */
40: Mat A; /* operator matrix */
41: EPS eps; /* eigenproblem solver context */
42: EPSType type;
43: PetscInt N,m=15,nev;
44: PetscBool terse;
47: SlepcInitialize(&argc,&argv,(char*)0,help);
49: PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL);
50: N = m*(m+1)/2;
51: PetscPrintf(PETSC_COMM_WORLD,"\nMarkov Model, N=%D (m=%D)\n\n",N,m);
53: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
54: Compute the operator matrix that defines the eigensystem, Ax=kx
55: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
57: MatCreate(PETSC_COMM_WORLD,&A);
58: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);
59: MatSetFromOptions(A);
60: MatSetUp(A);
61: MatMarkovModel(m,A);
63: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
64: Create the eigensolver and set various options
65: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
67: /*
68: Create eigensolver context
69: */
70: EPSCreate(PETSC_COMM_WORLD,&eps);
72: /*
73: Set operators. In this case, it is a standard eigenvalue problem
74: */
75: EPSSetOperators(eps,A,NULL);
76: EPSSetProblemType(eps,EPS_NHEP);
78: /*
79: Set solver parameters at runtime
80: */
81: EPSSetFromOptions(eps);
83: /*
84: Set the initial vector. This is optional, if not done the initial
85: vector is set to random values
86: */
87: MatCreateVecs(A,&v0,NULL);
88: VecSet(v0,1.0);
89: EPSSetInitialSpace(eps,1,&v0);
91: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
92: Solve the eigensystem
93: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
95: EPSSolve(eps);
97: /*
98: Optional: Get some information from the solver and display it
99: */
100: EPSGetType(eps,&type);
101: PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type);
102: EPSGetDimensions(eps,&nev,NULL,NULL);
103: PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %D\n",nev);
105: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
106: Display solution and clean up
107: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
109: /* show detailed info unless -terse option is given by user */
110: PetscOptionsHasName(NULL,NULL,"-terse",&terse);
111: if (terse) {
112: EPSErrorView(eps,EPS_ERROR_RELATIVE,NULL);
113: } else {
114: PetscViewerPushFormat(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_ASCII_INFO_DETAIL);
115: EPSReasonView(eps,PETSC_VIEWER_STDOUT_WORLD);
116: EPSErrorView(eps,EPS_ERROR_RELATIVE,PETSC_VIEWER_STDOUT_WORLD);
117: PetscViewerPopFormat(PETSC_VIEWER_STDOUT_WORLD);
118: }
119: EPSDestroy(&eps);
120: MatDestroy(&A);
121: VecDestroy(&v0);
122: SlepcFinalize();
123: return ierr;
124: }
128: /*
129: Matrix generator for a Markov model of a random walk on a triangular grid.
131: This subroutine generates a test matrix that models a random walk on a
132: triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a
133: FORTRAN subroutine to calculate the dominant invariant subspaces of a real
134: matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
135: papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
136: (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
137: algorithms. The transpose of the matrix is stochastic and so it is known
138: that one is an exact eigenvalue. One seeks the eigenvector of the transpose
139: associated with the eigenvalue unity. The problem is to calculate the steady
140: state probability distribution of the system, which is the eigevector
141: associated with the eigenvalue one and scaled in such a way that the sum all
142: the components is equal to one.
144: Note: the code will actually compute the transpose of the stochastic matrix
145: that contains the transition probabilities.
146: */
147: PetscErrorCode MatMarkovModel(PetscInt m,Mat A)148: {
149: const PetscReal cst = 0.5/(PetscReal)(m-1);
150: PetscReal pd,pu;
151: PetscInt Istart,Iend,i,j,jmax,ix=0;
152: PetscErrorCode ierr;
155: MatGetOwnershipRange(A,&Istart,&Iend);
156: for (i=1;i<=m;i++) {
157: jmax = m-i+1;
158: for (j=1;j<=jmax;j++) {
159: ix = ix + 1;
160: if (ix-1<Istart || ix>Iend) continue; /* compute only owned rows */
161: if (j!=jmax) {
162: pd = cst*(PetscReal)(i+j-1);
163: /* north */
164: if (i==1) {
165: MatSetValue(A,ix-1,ix,2*pd,INSERT_VALUES);
166: } else {
167: MatSetValue(A,ix-1,ix,pd,INSERT_VALUES);
168: }
169: /* east */
170: if (j==1) {
171: MatSetValue(A,ix-1,ix+jmax-1,2*pd,INSERT_VALUES);
172: } else {
173: MatSetValue(A,ix-1,ix+jmax-1,pd,INSERT_VALUES);
174: }
175: }
176: /* south */
177: pu = 0.5 - cst*(PetscReal)(i+j-3);
178: if (j>1) {
179: MatSetValue(A,ix-1,ix-2,pu,INSERT_VALUES);
180: }
181: /* west */
182: if (i>1) {
183: MatSetValue(A,ix-1,ix-jmax-2,pu,INSERT_VALUES);
184: }
185: }
186: }
187: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
188: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
189: return(0);
190: }