Actual source code: ex12.c
1: /*
2: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
3: SLEPc - Scalable Library for Eigenvalue Problem Computations
4: Copyright (c) 2002-2009, Universidad Politecnica de Valencia, Spain
6: This file is part of SLEPc.
7:
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[] = "Solves the same eigenproblem as in example ex5, but computing also left eigenvectors. "
23: "It is a Markov model of a random walk on a triangular grid. "
24: "A standard nonsymmetric eigenproblem with real eigenvalues. The rightmost eigenvalue is known to be 1.\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: {
40: Vec v0,temp; /* initial vector */
41: Vec *X,*Y; /* right and left eigenvectors */
42: Mat A; /* operator matrix */
43: EPS eps; /* eigenproblem solver context */
44: const EPSType type;
45: PetscReal error1, error2, tol, re, im;
46: PetscScalar kr, ki;
47: PetscInt nev, maxit, i, its, nconv, N, m=15;
49: SlepcInitialize(&argc,&argv,(char*)0,help);
51: PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
52: N = m*(m+1)/2;
53: PetscPrintf(PETSC_COMM_WORLD,"\nMarkov Model, N=%d (m=%d)\n\n",N,m);
55: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
56: Compute the operator matrix that defines the eigensystem, Ax=kx
57: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
59: MatCreate(PETSC_COMM_WORLD,&A);
60: MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);
61: MatSetFromOptions(A);
62: MatMarkovModel( m, A );
64: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
65: Create the eigensolver and set various options
66: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
68: /*
69: Create eigensolver context
70: */
71: EPSCreate(PETSC_COMM_WORLD,&eps);
73: /*
74: Set operators. In this case, it is a standard eigenvalue problem
75: */
76: EPSSetOperators(eps,A,PETSC_NULL);
77: EPSSetProblemType(eps,EPS_NHEP);
79: /*
80: Select a two-sided version of the eigensolver so that left eigenvectors
81: are also computed
82: */
83: EPSSetClass(eps,EPS_TWO_SIDE);
85: /*
86: Set solver parameters at runtime
87: */
88: EPSSetFromOptions(eps);
90: /*
91: Set the initial vector. This is optional, if not done the initial
92: vector is set to random values
93: */
94: MatGetVecs(A,&v0,&temp);
95: VecSet(v0,1.0);
96: MatMult(A,v0,temp);
97: EPSSetInitialVector(eps,v0);
98: EPSSetLeftInitialVector(eps,temp);
100: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
101: Solve the eigensystem
102: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
104: EPSSolve(eps);
105: EPSGetIterationNumber(eps, &its);
106: PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the method: %d\n",its);
108: /*
109: Optional: Get some information from the solver and display it
110: */
111: EPSGetType(eps,&type);
112: PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type);
113: EPSGetDimensions(eps,&nev,PETSC_NULL,PETSC_NULL);
114: PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %d\n",nev);
115: EPSGetTolerances(eps,&tol,&maxit);
116: PetscPrintf(PETSC_COMM_WORLD," Stopping condition: tol=%.4g, maxit=%d\n",tol,maxit);
118: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
119: Display solution and clean up
120: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
122: /*
123: Get number of converged approximate eigenpairs
124: */
125: EPSGetConverged(eps,&nconv);
126: PetscPrintf(PETSC_COMM_WORLD," Number of converged approximate eigenpairs: %d\n\n",nconv);
128: if (nconv>0) {
129: /*
130: Display eigenvalues and relative errors
131: */
132: PetscPrintf(PETSC_COMM_WORLD,
133: " k ||Ax-kx||/||kx|| ||y'A-ky'||/||ky||\n"
134: " ----------------- ------------------ --------------------\n" );
136: for( i=0; i<nconv; i++ ) {
137: /*
138: Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
139: ki (imaginary part)
140: */
141: EPSGetValue(eps,i,&kr,&ki);
142: /*
143: Compute the relative errors associated to both right and left eigenvectors
144: */
145: EPSComputeRelativeError(eps,i,&error1);
146: EPSComputeRelativeErrorLeft(eps,i,&error2);
148: #ifdef PETSC_USE_COMPLEX
149: re = PetscRealPart(kr);
150: im = PetscImaginaryPart(kr);
151: #else
152: re = kr;
153: im = ki;
154: #endif
155: if (im!=0.0) {
156: PetscPrintf(PETSC_COMM_WORLD," %9f%+9f j %12g%12g\n",re,im,error1,error2);
157: } else {
158: PetscPrintf(PETSC_COMM_WORLD," %12f %12g %12g\n",re,error1,error2);
159: }
160: }
161: PetscPrintf(PETSC_COMM_WORLD,"\n" );
163: VecDuplicateVecs(v0,nconv,&X);
164: VecDuplicateVecs(temp,nconv,&Y);
165: for (i=0;i<nconv;i++) {
166: EPSGetRightVector(eps,i,X[i],PETSC_NULL);
167: EPSGetLeftVector(eps,i,Y[i],PETSC_NULL);
168: }
169: PetscPrintf(PETSC_COMM_WORLD,
170: " Bi-orthogonality <x,y> \n"
171: " ---------------------------------------------------------\n" );
173: SlepcCheckOrthogonality(X,nconv,Y,nconv,PETSC_NULL,PETSC_NULL);
174: PetscPrintf(PETSC_COMM_WORLD,"\n" );
175: VecDestroyVecs(X,nconv);
176: VecDestroyVecs(Y,nconv);
178: }
179:
180: /*
181: Free work space
182: */
183: VecDestroy(v0);
184: VecDestroy(temp);
185: EPSDestroy(eps);
186: MatDestroy(A);
187: SlepcFinalize();
188: return 0;
189: }
193: /*
194: Matrix generator for a Markov model of a random walk on a triangular grid.
196: This subroutine generates a test matrix that models a random walk on a
197: triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a
198: FORTRAN subroutine to calculate the dominant invariant subspaces of a real
199: matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
200: papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
201: (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
202: algorithms. The transpose of the matrix is stochastic and so it is known
203: that one is an exact eigenvalue. One seeks the eigenvector of the transpose
204: associated with the eigenvalue unity. The problem is to calculate the steady
205: state probability distribution of the system, which is the eigevector
206: associated with the eigenvalue one and scaled in such a way that the sum all
207: the components is equal to one.
209: Note: the code will actually compute the transpose of the stochastic matrix
210: that contains the transition probabilities.
211: */
212: PetscErrorCode MatMarkovModel( PetscInt m, Mat A )
213: {
214: const PetscReal cst = 0.5/(PetscReal)(m-1);
215: PetscReal pd, pu;
216: PetscErrorCode ierr;
217: PetscInt i, j, jmax, ix=0, Istart, Iend;
220: MatGetOwnershipRange(A,&Istart,&Iend);
221: for( i=1; i<=m; i++ ) {
222: jmax = m-i+1;
223: for( j=1; j<=jmax; j++ ) {
224: ix = ix + 1;
225: if( ix-1<Istart || ix>Iend ) continue; /* compute only owned rows */
226: if( j!=jmax ) {
227: pd = cst*(PetscReal)(i+j-1);
228: /* north */
229: if( i==1 ) {
230: MatSetValue( A, ix-1, ix, 2*pd, INSERT_VALUES );
231: }
232: else {
233: MatSetValue( A, ix-1, ix, pd, INSERT_VALUES );
234: }
235: /* east */
236: if( j==1 ) {
237: MatSetValue( A, ix-1, ix+jmax-1, 2*pd, INSERT_VALUES );
238: }
239: else {
240: MatSetValue( A, ix-1, ix+jmax-1, pd, INSERT_VALUES );
241: }
242: }
243: /* south */
244: pu = 0.5 - cst*(PetscReal)(i+j-3);
245: if( j>1 ) {
246: MatSetValue( A, ix-1, ix-2, pu, INSERT_VALUES );
247: }
248: /* west */
249: if( i>1 ) {
250: MatSetValue( A, ix-1, ix-jmax-2, pu, INSERT_VALUES );
251: }
252: }
253: }
254: MatAssemblyBegin( A, MAT_FINAL_ASSEMBLY );
255: MatAssemblyEnd( A, MAT_FINAL_ASSEMBLY );
256: return(0);
257: }