Actual source code: ex12.c
2: static char help[] = "Solves the same eigenproblem as in example ex5, but computing also left eigenvectors. "
3: "It is a Markov model of a random walk on a triangular grid. "
4: "A standard nonsymmetric eigenproblem with real eigenvalues. The rightmost eigenvalue is known to be 1.\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: {
20: Vec v0,temp; /* initial vector */
21: Vec *X,*Y; /* right and left eigenvectors */
22: Mat A; /* operator matrix */
23: EPS eps; /* eigenproblem solver context */
24: EPSType type;
25: PetscReal error1, error2, tol, re, im;
26: PetscScalar kr, ki;
27: int nev, maxit, i, its, nconv;
28: PetscInt N, m=15;
30: SlepcInitialize(&argc,&argv,(char*)0,help);
32: PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
33: N = m*(m+1)/2;
34: PetscPrintf(PETSC_COMM_WORLD,"\nMarkov Model, N=%d (m=%d)\n\n",N,m);
36: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
37: Compute the operator matrix that defines the eigensystem, Ax=kx
38: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
40: MatCreate(PETSC_COMM_WORLD,&A);
41: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);
42: MatSetFromOptions(A);
43: MatMarkovModel( m, A );
45: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
46: Create the eigensolver and set various options
47: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
49: /*
50: Create eigensolver context
51: */
52: EPSCreate(PETSC_COMM_WORLD,&eps);
54: /*
55: Set operators. In this case, it is a standard eigenvalue problem
56: */
57: EPSSetOperators(eps,A,PETSC_NULL);
58: EPSSetProblemType(eps,EPS_NHEP);
60: /*
61: Select a two-sided version of the eigensolver so that left eigenvectors
62: are also computed
63: */
64: EPSSetClass(eps,EPS_TWO_SIDE);
66: /*
67: Set solver parameters at runtime
68: */
69: EPSSetFromOptions(eps);
71: /*
72: Set the initial vector. This is optional, if not done the initial
73: vector is set to random values
74: */
75: MatGetVecs(A,&v0,&temp);
76: VecSet(v0,1.0);
77: MatMult(A,v0,temp);
78: EPSSetInitialVector(eps,v0);
79: EPSSetLeftInitialVector(eps,temp);
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|| ||y'A-ky'||/||ky||\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: EPSGetValue(eps,i,&kr,&ki);
123: /*
124: Compute the relative errors associated to both right and left eigenvectors
125: */
126: EPSComputeRelativeError(eps,i,&error1);
127: EPSComputeRelativeErrorLeft(eps,i,&error2);
129: #ifdef PETSC_USE_COMPLEX
130: re = PetscRealPart(kr);
131: im = PetscImaginaryPart(kr);
132: #else
133: re = kr;
134: im = ki;
135: #endif
136: if (im!=0.0) {
137: PetscPrintf(PETSC_COMM_WORLD," %9f%+9f j %12g%12g\n",re,im,error1,error2);
138: } else {
139: PetscPrintf(PETSC_COMM_WORLD," %12f %12g %12g\n",re,error1,error2);
140: }
141: }
142: PetscPrintf(PETSC_COMM_WORLD,"\n" );
144: VecDuplicateVecs(v0,nconv,&X);
145: VecDuplicateVecs(temp,nconv,&Y);
146: for (i=0;i<nconv;i++) {
147: EPSGetRightVector(eps,i,X[i],PETSC_NULL);
148: EPSGetLeftVector(eps,i,Y[i],PETSC_NULL);
149: }
150: PetscPrintf(PETSC_COMM_WORLD,
151: " Bi-orthogonality <x,y> \n"
152: " ---------------------------------------------------------\n" );
154: SlepcCheckOrthogonality(X,nconv,Y,nconv,PETSC_NULL,PETSC_NULL);
155: PetscPrintf(PETSC_COMM_WORLD,"\n" );
156: VecDestroyVecs(X,nconv);
157: VecDestroyVecs(Y,nconv);
159: }
160:
161: /*
162: Free work space
163: */
164: VecDestroy(v0);
165: VecDestroy(temp);
166: EPSDestroy(eps);
167: MatDestroy(A);
168: SlepcFinalize();
169: return 0;
170: }
174: /*
175: Matrix generator for a Markov model of a random walk on a triangular grid.
177: This subroutine generates a test matrix that models a random walk on a
178: triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a
179: FORTRAN subroutine to calculate the dominant invariant subspaces of a real
180: matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
181: papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
182: (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
183: algorithms. The transpose of the matrix is stochastic and so it is known
184: that one is an exact eigenvalue. One seeks the eigenvector of the transpose
185: associated with the eigenvalue unity. The problem is to calculate the steady
186: state probability distribution of the system, which is the eigevector
187: associated with the eigenvalue one and scaled in such a way that the sum all
188: the components is equal to one.
190: Note: the code will actually compute the transpose of the stochastic matrix
191: that contains the transition probabilities.
192: */
193: PetscErrorCode MatMarkovModel( PetscInt m, Mat A )
194: {
195: const PetscReal cst = 0.5/(PetscReal)(m-1);
196: PetscReal pd, pu;
197: PetscErrorCode ierr;
198: PetscInt i, j, jmax, ix=0, Istart, Iend;
201: MatGetOwnershipRange(A,&Istart,&Iend);
202: for( i=1; i<=m; i++ ) {
203: jmax = m-i+1;
204: for( j=1; j<=jmax; j++ ) {
205: ix = ix + 1;
206: if( ix-1<Istart || ix>Iend ) continue; /* compute only owned rows */
207: if( j!=jmax ) {
208: pd = cst*(PetscReal)(i+j-1);
209: /* north */
210: if( i==1 ) {
211: MatSetValue( A, ix-1, ix, 2*pd, INSERT_VALUES );
212: }
213: else {
214: MatSetValue( A, ix-1, ix, pd, INSERT_VALUES );
215: }
216: /* east */
217: if( j==1 ) {
218: MatSetValue( A, ix-1, ix+jmax-1, 2*pd, INSERT_VALUES );
219: }
220: else {
221: MatSetValue( A, ix-1, ix+jmax-1, pd, INSERT_VALUES );
222: }
223: }
224: /* south */
225: pu = 0.5 - cst*(PetscReal)(i+j-3);
226: if( j>1 ) {
227: MatSetValue( A, ix-1, ix-2, pu, INSERT_VALUES );
228: }
229: /* west */
230: if( i>1 ) {
231: MatSetValue( A, ix-1, ix-jmax-2, pu, INSERT_VALUES );
232: }
233: }
234: }
235: MatAssemblyBegin( A, MAT_FINAL_ASSEMBLY );
236: MatAssemblyEnd( A, MAT_FINAL_ASSEMBLY );
237: return(0);
238: }