Actual source code: ex5.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[] = "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:   const EPSType  type;
 43:   PetscReal            error, tol, re, im;
 44:   PetscScalar          kr, ki;
 45:   PetscInt             N, m=15, nev, maxit, i, its, nconv;
 47: 
 48:   SlepcInitialize(&argc,&argv,(char*)0,help);

 50:   PetscOptionsGetInt(PETSC_NULL,"-m",&m,PETSC_NULL);
 51:   N = m*(m+1)/2;
 52:   PetscPrintf(PETSC_COMM_WORLD,"\nMarkov Model, N=%d (m=%d)\n\n",N,m);

 54:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
 55:      Compute the operator matrix that defines the eigensystem, Ax=kx
 56:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

 58:   MatCreate(PETSC_COMM_WORLD,&A);
 59:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);
 60:   MatSetFromOptions(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,PETSC_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:   MatGetVecs(A,&v0,PETSC_NULL);
 88:   VecSet(v0,1.0);
 89:   EPSSetInitialVector(eps,v0);

 91:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
 92:                       Solve the eigensystem
 93:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

 95:   EPSSolve(eps);
 96:   EPSGetIterationNumber(eps, &its);
 97:   PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the method: %d\n",its);

 99:   /*
100:      Optional: Get some information from the solver and display it
101:   */
102:   EPSGetType(eps,&type);
103:   PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type);
104:   EPSGetDimensions(eps,&nev,PETSC_NULL,PETSC_NULL);
105:   PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %d\n",nev);
106:   EPSGetTolerances(eps,&tol,&maxit);
107:   PetscPrintf(PETSC_COMM_WORLD," Stopping condition: tol=%.4g, maxit=%d\n",tol,maxit);

109:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
110:                     Display solution and clean up
111:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

113:   /* 
114:      Get number of converged approximate eigenpairs
115:   */
116:   EPSGetConverged(eps,&nconv);
117:   PetscPrintf(PETSC_COMM_WORLD," Number of converged approximate eigenpairs: %d\n\n",nconv);

119:   if (nconv>0) {
120:     /*
121:        Display eigenvalues and relative errors
122:     */
123:     PetscPrintf(PETSC_COMM_WORLD,
124:          "           k          ||Ax-kx||/||kx||\n"
125:          "   ----------------- ------------------\n" );

127:     for( i=0; i<nconv; i++ ) {
128:       /* 
129:         Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
130:         ki (imaginary part)
131:       */
132:       EPSGetEigenpair(eps,i,&kr,&ki,PETSC_NULL,PETSC_NULL);
133:       /*
134:          Compute the relative error associated to each eigenpair
135:       */
136:       EPSComputeRelativeError(eps,i,&error);

138: #ifdef PETSC_USE_COMPLEX
139:       re = PetscRealPart(kr);
140:       im = PetscImaginaryPart(kr);
141: #else
142:       re = kr;
143:       im = ki;
144: #endif 
145:       if (im!=0.0) {
146:         PetscPrintf(PETSC_COMM_WORLD," %9f%+9f j %12g\n",re,im,error);
147:       } else {
148:         PetscPrintf(PETSC_COMM_WORLD,"   %12f       %12g\n",re,error);
149:       }
150:     }
151:     PetscPrintf(PETSC_COMM_WORLD,"\n" );
152:   }
153: 
154:   /* 
155:      Free work space
156:   */
157:   EPSDestroy(eps);
158:   MatDestroy(A);
159:   VecDestroy(v0);
160:   SlepcFinalize();
161:   return 0;
162: }

166: /*
167:     Matrix generator for a Markov model of a random walk on a triangular grid.

169:     This subroutine generates a test matrix that models a random walk on a 
170:     triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a 
171:     FORTRAN subroutine to calculate the dominant invariant subspaces of a real
172:     matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
173:     papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
174:     (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
175:     algorithms. The transpose of the matrix  is stochastic and so it is known 
176:     that one is an exact eigenvalue. One seeks the eigenvector of the transpose 
177:     associated with the eigenvalue unity. The problem is to calculate the steady
178:     state probability distribution of the system, which is the eigevector 
179:     associated with the eigenvalue one and scaled in such a way that the sum all
180:     the components is equal to one.

182:     Note: the code will actually compute the transpose of the stochastic matrix
183:     that contains the transition probabilities.
184: */
185: PetscErrorCode MatMarkovModel( PetscInt m, Mat A )
186: {
187:   const PetscReal cst = 0.5/(PetscReal)(m-1);
188:   PetscReal       pd, pu;
189:   PetscErrorCode  ierr;
190:   PetscInt        Istart, Iend, i, j, jmax, ix=0;

193:   MatGetOwnershipRange(A,&Istart,&Iend);
194:   for( i=1; i<=m; i++ ) {
195:     jmax = m-i+1;
196:     for( j=1; j<=jmax; j++ ) {
197:       ix = ix + 1;
198:       if( ix-1<Istart || ix>Iend ) continue;  /* compute only owned rows */
199:       if( j!=jmax ) {
200:         pd = cst*(PetscReal)(i+j-1);
201:         /* north */
202:         if( i==1 ) {
203:           MatSetValue( A, ix-1, ix, 2*pd, INSERT_VALUES );
204:         }
205:         else {
206:           MatSetValue( A, ix-1, ix, pd, INSERT_VALUES );
207:         }
208:         /* east */
209:         if( j==1 ) {
210:           MatSetValue( A, ix-1, ix+jmax-1, 2*pd, INSERT_VALUES );
211:         }
212:         else {
213:           MatSetValue( A, ix-1, ix+jmax-1, pd, INSERT_VALUES );
214:         }
215:       }
216:       /* south */
217:       pu = 0.5 - cst*(PetscReal)(i+j-3);
218:       if( j>1 ) {
219:         MatSetValue( A, ix-1, ix-2, pu, INSERT_VALUES );
220:       }
221:       /* west */
222:       if( i>1 ) {
223:         MatSetValue( A, ix-1, ix-jmax-2, pu, INSERT_VALUES );
224:       }
225:     }
226:   }
227:   MatAssemblyBegin( A, MAT_FINAL_ASSEMBLY );
228:   MatAssemblyEnd( A, MAT_FINAL_ASSEMBLY );
229:   return(0);
230: }