Actual source code: ex5.c
2: static char help[] = "Eigenvalue problem associated with a Markov model of a random walk on a triangular grid. "
3: "It is a standard nonsymmetric eigenproblem with real eigenvalues and the rightmost eigenvalue is known to be 1.\n"
4: "This example illustrates how the user can set the initial vector.\n\n"
5: "The command line options are:\n"
6: " -m <m>, where <m> = number of grid subdivisions in each dimension.\n\n";
8: #include slepceps.h
10: /*
11: User-defined routines
12: */
13: PetscErrorCode MatMarkovModel( PetscInt m, Mat A );
17: int main( int argc, char **argv )
18: {
19: Vec v0; /* initial vector */
20: Mat A; /* operator matrix */
21: EPS eps; /* eigenproblem solver context */
22: EPSType type;
23: PetscReal error, tol, re, im;
24: PetscScalar kr, ki;
25: PetscInt N, m=15;
26: int nev, maxit, i, its, nconv;
28:
29: SlepcInitialize(&argc,&argv,(char*)0,help);
31: PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
32: N = m*(m+1)/2;
33: PetscPrintf(PETSC_COMM_WORLD,"\nMarkov Model, N=%d (m=%d)\n\n",N,m);
35: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
36: Compute the operator matrix that defines the eigensystem, Ax=kx
37: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
39: MatCreate(PETSC_COMM_WORLD,&A);
40: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);
41: MatSetFromOptions(A);
42: MatMarkovModel( m, A );
44: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
45: Create the eigensolver and set various options
46: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
48: /*
49: Create eigensolver context
50: */
51: EPSCreate(PETSC_COMM_WORLD,&eps);
53: /*
54: Set operators. In this case, it is a standard eigenvalue problem
55: */
56: EPSSetOperators(eps,A,PETSC_NULL);
57: EPSSetProblemType(eps,EPS_NHEP);
59: /*
60: Set solver parameters at runtime
61: */
62: EPSSetFromOptions(eps);
64: /*
65: Set the initial vector. This is optional, if not done the initial
66: vector is set to random values
67: */
68: MatGetVecs(A,&v0,PETSC_NULL);
69: VecSet(v0,1.0);
70: EPSSetInitialVector(eps,v0);
72: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
73: Solve the eigensystem
74: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
76: EPSSolve(eps);
77: EPSGetIterationNumber(eps, &its);
78: PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the method: %d\n",its);
80: /*
81: Optional: Get some information from the solver and display it
82: */
83: EPSGetType(eps,&type);
84: PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type);
85: EPSGetDimensions(eps,&nev,PETSC_NULL);
86: PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %d\n",nev);
87: EPSGetTolerances(eps,&tol,&maxit);
88: PetscPrintf(PETSC_COMM_WORLD," Stopping condition: tol=%.4g, maxit=%d\n",tol,maxit);
90: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
91: Display solution and clean up
92: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
94: /*
95: Get number of converged approximate eigenpairs
96: */
97: EPSGetConverged(eps,&nconv);
98: PetscPrintf(PETSC_COMM_WORLD," Number of converged approximate eigenpairs: %d\n\n",nconv);
100: if (nconv>0) {
101: /*
102: Display eigenvalues and relative errors
103: */
104: PetscPrintf(PETSC_COMM_WORLD,
105: " k ||Ax-kx||/||kx||\n"
106: " ----------------- ------------------\n" );
108: for( i=0; i<nconv; i++ ) {
109: /*
110: Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
111: ki (imaginary part)
112: */
113: EPSGetEigenpair(eps,i,&kr,&ki,PETSC_NULL,PETSC_NULL);
114: /*
115: Compute the relative error associated to each eigenpair
116: */
117: EPSComputeRelativeError(eps,i,&error);
119: #ifdef PETSC_USE_COMPLEX
120: re = PetscRealPart(kr);
121: im = PetscImaginaryPart(kr);
122: #else
123: re = kr;
124: im = ki;
125: #endif
126: if (im!=0.0) {
127: PetscPrintf(PETSC_COMM_WORLD," %9f%+9f j %12g\n",re,im,error);
128: } else {
129: PetscPrintf(PETSC_COMM_WORLD," %12f %12g\n",re,error);
130: }
131: }
132: PetscPrintf(PETSC_COMM_WORLD,"\n" );
133: }
134:
135: /*
136: Free work space
137: */
138: EPSDestroy(eps);
139: MatDestroy(A);
140: VecDestroy(v0);
141: SlepcFinalize();
142: return 0;
143: }
147: /*
148: Matrix generator for a Markov model of a random walk on a triangular grid.
150: This subroutine generates a test matrix that models a random walk on a
151: triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a
152: FORTRAN subroutine to calculate the dominant invariant subspaces of a real
153: matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
154: papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
155: (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
156: algorithms. The transpose of the matrix is stochastic and so it is known
157: that one is an exact eigenvalue. One seeks the eigenvector of the transpose
158: associated with the eigenvalue unity. The problem is to calculate the steady
159: state probability distribution of the system, which is the eigevector
160: associated with the eigenvalue one and scaled in such a way that the sum all
161: the components is equal to one.
163: Note: the code will actually compute the transpose of the stochastic matrix
164: that contains the transition probabilities.
165: */
166: PetscErrorCode MatMarkovModel( PetscInt m, Mat A )
167: {
168: const PetscReal cst = 0.5/(PetscReal)(m-1);
169: PetscReal pd, pu;
170: PetscErrorCode ierr;
171: PetscInt Istart, Iend, i, j, jmax, ix=0;
174: MatGetOwnershipRange(A,&Istart,&Iend);
175: for( i=1; i<=m; i++ ) {
176: jmax = m-i+1;
177: for( j=1; j<=jmax; j++ ) {
178: ix = ix + 1;
179: if( ix-1<Istart || ix>Iend ) continue; /* compute only owned rows */
180: if( j!=jmax ) {
181: pd = cst*(PetscReal)(i+j-1);
182: /* north */
183: if( i==1 ) {
184: MatSetValue( A, ix-1, ix, 2*pd, INSERT_VALUES );
185: }
186: else {
187: MatSetValue( A, ix-1, ix, pd, INSERT_VALUES );
188: }
189: /* east */
190: if( j==1 ) {
191: MatSetValue( A, ix-1, ix+jmax-1, 2*pd, INSERT_VALUES );
192: }
193: else {
194: MatSetValue( A, ix-1, ix+jmax-1, pd, INSERT_VALUES );
195: }
196: }
197: /* south */
198: pu = 0.5 - cst*(PetscReal)(i+j-3);
199: if( j>1 ) {
200: MatSetValue( A, ix-1, ix-2, pu, INSERT_VALUES );
201: }
202: /* west */
203: if( i>1 ) {
204: MatSetValue( A, ix-1, ix-jmax-2, pu, INSERT_VALUES );
205: }
206: }
207: }
208: MatAssemblyBegin( A, MAT_FINAL_ASSEMBLY );
209: MatAssemblyEnd( A, MAT_FINAL_ASSEMBLY );
210: return(0);
211: }