Actual source code: ex5.c
1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-2007, Universidad Politecnica de Valencia, Spain
6: This file is part of SLEPc. See the README file for conditions of use
7: and additional information.
8: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
9: */
11: static char help[] = "Eigenvalue problem associated with a Markov model of a random walk on a triangular grid. "
12: "It is a standard nonsymmetric eigenproblem with real eigenvalues and the rightmost eigenvalue is known to be 1.\n"
13: "This example illustrates how the user can set the initial vector.\n\n"
14: "The command line options are:\n"
15: " -m <m>, where <m> = number of grid subdivisions in each dimension.\n\n";
17: #include slepceps.h
19: /*
20: User-defined routines
21: */
22: PetscErrorCode MatMarkovModel( PetscInt m, Mat A );
26: int main( int argc, char **argv )
27: {
28: Vec v0; /* initial vector */
29: Mat A; /* operator matrix */
30: EPS eps; /* eigenproblem solver context */
31: EPSType type;
32: PetscReal error, tol, re, im;
33: PetscScalar kr, ki;
34: PetscInt N, m=15;
35: int nev, maxit, i, its, nconv;
37:
38: SlepcInitialize(&argc,&argv,(char*)0,help);
40: PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
41: N = m*(m+1)/2;
42: PetscPrintf(PETSC_COMM_WORLD,"\nMarkov Model, N=%d (m=%d)\n\n",N,m);
44: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
45: Compute the operator matrix that defines the eigensystem, Ax=kx
46: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
48: MatCreate(PETSC_COMM_WORLD,&A);
49: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);
50: MatSetFromOptions(A);
51: MatMarkovModel( m, A );
53: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
54: Create the eigensolver and set various options
55: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
57: /*
58: Create eigensolver context
59: */
60: EPSCreate(PETSC_COMM_WORLD,&eps);
62: /*
63: Set operators. In this case, it is a standard eigenvalue problem
64: */
65: EPSSetOperators(eps,A,PETSC_NULL);
66: EPSSetProblemType(eps,EPS_NHEP);
68: /*
69: Set solver parameters at runtime
70: */
71: EPSSetFromOptions(eps);
73: /*
74: Set the initial vector. This is optional, if not done the initial
75: vector is set to random values
76: */
77: MatGetVecs(A,&v0,PETSC_NULL);
78: VecSet(v0,1.0);
79: EPSSetInitialVector(eps,v0);
81: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
82: Solve the eigensystem
83: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
85: EPSSolve(eps);
86: EPSGetIterationNumber(eps, &its);
87: PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the method: %d\n",its);
89: /*
90: Optional: Get some information from the solver and display it
91: */
92: EPSGetType(eps,&type);
93: PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type);
94: EPSGetDimensions(eps,&nev,PETSC_NULL);
95: PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %d\n",nev);
96: EPSGetTolerances(eps,&tol,&maxit);
97: PetscPrintf(PETSC_COMM_WORLD," Stopping condition: tol=%.4g, maxit=%d\n",tol,maxit);
99: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
100: Display solution and clean up
101: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
103: /*
104: Get number of converged approximate eigenpairs
105: */
106: EPSGetConverged(eps,&nconv);
107: PetscPrintf(PETSC_COMM_WORLD," Number of converged approximate eigenpairs: %d\n\n",nconv);
109: if (nconv>0) {
110: /*
111: Display eigenvalues and relative errors
112: */
113: PetscPrintf(PETSC_COMM_WORLD,
114: " k ||Ax-kx||/||kx||\n"
115: " ----------------- ------------------\n" );
117: for( i=0; i<nconv; i++ ) {
118: /*
119: Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
120: ki (imaginary part)
121: */
122: EPSGetEigenpair(eps,i,&kr,&ki,PETSC_NULL,PETSC_NULL);
123: /*
124: Compute the relative error associated to each eigenpair
125: */
126: EPSComputeRelativeError(eps,i,&error);
128: #ifdef PETSC_USE_COMPLEX
129: re = PetscRealPart(kr);
130: im = PetscImaginaryPart(kr);
131: #else
132: re = kr;
133: im = ki;
134: #endif
135: if (im!=0.0) {
136: PetscPrintf(PETSC_COMM_WORLD," %9f%+9f j %12g\n",re,im,error);
137: } else {
138: PetscPrintf(PETSC_COMM_WORLD," %12f %12g\n",re,error);
139: }
140: }
141: PetscPrintf(PETSC_COMM_WORLD,"\n" );
142: }
143:
144: /*
145: Free work space
146: */
147: EPSDestroy(eps);
148: MatDestroy(A);
149: VecDestroy(v0);
150: SlepcFinalize();
151: return 0;
152: }
156: /*
157: Matrix generator for a Markov model of a random walk on a triangular grid.
159: This subroutine generates a test matrix that models a random walk on a
160: triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a
161: FORTRAN subroutine to calculate the dominant invariant subspaces of a real
162: matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
163: papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
164: (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
165: algorithms. The transpose of the matrix is stochastic and so it is known
166: that one is an exact eigenvalue. One seeks the eigenvector of the transpose
167: associated with the eigenvalue unity. The problem is to calculate the steady
168: state probability distribution of the system, which is the eigevector
169: associated with the eigenvalue one and scaled in such a way that the sum all
170: the components is equal to one.
172: Note: the code will actually compute the transpose of the stochastic matrix
173: that contains the transition probabilities.
174: */
175: PetscErrorCode MatMarkovModel( PetscInt m, Mat A )
176: {
177: const PetscReal cst = 0.5/(PetscReal)(m-1);
178: PetscReal pd, pu;
179: PetscErrorCode ierr;
180: PetscInt Istart, Iend, i, j, jmax, ix=0;
183: MatGetOwnershipRange(A,&Istart,&Iend);
184: for( i=1; i<=m; i++ ) {
185: jmax = m-i+1;
186: for( j=1; j<=jmax; j++ ) {
187: ix = ix + 1;
188: if( ix-1<Istart || ix>Iend ) continue; /* compute only owned rows */
189: if( j!=jmax ) {
190: pd = cst*(PetscReal)(i+j-1);
191: /* north */
192: if( i==1 ) {
193: MatSetValue( A, ix-1, ix, 2*pd, INSERT_VALUES );
194: }
195: else {
196: MatSetValue( A, ix-1, ix, pd, INSERT_VALUES );
197: }
198: /* east */
199: if( j==1 ) {
200: MatSetValue( A, ix-1, ix+jmax-1, 2*pd, INSERT_VALUES );
201: }
202: else {
203: MatSetValue( A, ix-1, ix+jmax-1, pd, INSERT_VALUES );
204: }
205: }
206: /* south */
207: pu = 0.5 - cst*(PetscReal)(i+j-3);
208: if( j>1 ) {
209: MatSetValue( A, ix-1, ix-2, pu, INSERT_VALUES );
210: }
211: /* west */
212: if( i>1 ) {
213: MatSetValue( A, ix-1, ix-jmax-2, pu, INSERT_VALUES );
214: }
215: }
216: }
217: MatAssemblyBegin( A, MAT_FINAL_ASSEMBLY );
218: MatAssemblyEnd( A, MAT_FINAL_ASSEMBLY );
219: return(0);
220: }