Actual source code: ex12.c

  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2010, 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,w0;           /* 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:      Request also left eigenvectors 
 76:   */
 77:   EPSSetOperators(eps,A,PETSC_NULL);
 78:   EPSSetProblemType(eps,EPS_NHEP);
 79:   EPSSetLeftVectorsWanted(eps,PETSC_TRUE);

 81:   /*
 82:      Set solver parameters at runtime
 83:   */
 84:   EPSSetFromOptions(eps);

 86:   /*
 87:      Set the initial vector. This is optional, if not done the initial
 88:      vector is set to random values
 89:   */
 90:   MatGetVecs(A,&v0,&w0);
 91:   VecSet(v0,1.0);
 92:   MatMult(A,v0,w0);
 93:   EPSSetInitialSpace(eps,1,&v0);
 94:   EPSSetInitialSpaceLeft(eps,1,&w0);

 96:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
 97:                       Solve the eigensystem
 98:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

100:   EPSSolve(eps);
101:   EPSGetIterationNumber(eps, &its);
102:   PetscPrintf(PETSC_COMM_WORLD," Number of iterations of the method: %d\n",its);

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

114:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 
115:                     Display solution and clean up
116:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

118:   /* 
119:      Get number of converged approximate eigenpairs
120:   */
121:   EPSGetConverged(eps,&nconv);
122:   PetscPrintf(PETSC_COMM_WORLD," Number of converged approximate eigenpairs: %d\n\n",nconv);

124:   if (nconv>0) {
125:     /*
126:        Display eigenvalues and relative errors
127:     */
128:     PetscPrintf(PETSC_COMM_WORLD,
129:          "           k          ||Ax-kx||/||kx||   ||y'A-ky'||/||ky||\n"
130:          "   ----------------- ------------------ --------------------\n" );

132:     for( i=0; i<nconv; i++ ) {
133:       /* 
134:         Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
135:         ki (imaginary part)
136:       */
137:       EPSGetEigenvalue(eps,i,&kr,&ki);
138:       /*
139:          Compute the relative errors associated to both right and left eigenvectors
140:       */
141:       EPSComputeRelativeError(eps,i,&error1);
142:       EPSComputeRelativeErrorLeft(eps,i,&error2);

144: #ifdef PETSC_USE_COMPLEX
145:       re = PetscRealPart(kr);
146:       im = PetscImaginaryPart(kr);
147: #else
148:       re = kr;
149:       im = ki;
150: #endif 
151:       if (im!=0.0) {
152:         PetscPrintf(PETSC_COMM_WORLD," %9f%+9f j %12g%12g\n",re,im,error1,error2);
153:       } else {
154:         PetscPrintf(PETSC_COMM_WORLD,"   %12f       %12g       %12g\n",re,error1,error2);
155:       }
156:     }
157:     PetscPrintf(PETSC_COMM_WORLD,"\n" );

159:     VecDuplicateVecs(v0,nconv,&X);
160:     VecDuplicateVecs(w0,nconv,&Y);
161:     for (i=0;i<nconv;i++) {
162:       EPSGetEigenvector(eps,i,X[i],PETSC_NULL);
163:       EPSGetEigenvectorLeft(eps,i,Y[i],PETSC_NULL);
164:     }
165:     PetscPrintf(PETSC_COMM_WORLD,
166:          "                   Bi-orthogonality <x,y>                   \n"
167:          "   ---------------------------------------------------------\n" );

169:     SlepcCheckOrthogonality(X,nconv,Y,nconv,PETSC_NULL,PETSC_NULL);
170:     PetscPrintf(PETSC_COMM_WORLD,"\n" );
171:     VecDestroyVecs(X,nconv);
172:     VecDestroyVecs(Y,nconv);

174:   }
175: 
176:   /* 
177:      Free work space
178:   */
179:   VecDestroy(v0);
180:   VecDestroy(w0);
181:   EPSDestroy(eps);
182:   MatDestroy(A);
183:   SlepcFinalize();
184:   return 0;
185: }

189: /*
190:     Matrix generator for a Markov model of a random walk on a triangular grid.

192:     This subroutine generates a test matrix that models a random walk on a 
193:     triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a 
194:     FORTRAN subroutine to calculate the dominant invariant subspaces of a real
195:     matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
196:     papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
197:     (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
198:     algorithms. The transpose of the matrix  is stochastic and so it is known 
199:     that one is an exact eigenvalue. One seeks the eigenvector of the transpose 
200:     associated with the eigenvalue unity. The problem is to calculate the steady
201:     state probability distribution of the system, which is the eigevector 
202:     associated with the eigenvalue one and scaled in such a way that the sum all
203:     the components is equal to one.

205:     Note: the code will actually compute the transpose of the stochastic matrix
206:     that contains the transition probabilities.
207: */
208: PetscErrorCode MatMarkovModel( PetscInt m, Mat A )
209: {
210:   const PetscReal cst = 0.5/(PetscReal)(m-1);
211:   PetscReal       pd, pu;
212:   PetscErrorCode  ierr;
213:   PetscInt        i, j, jmax, ix=0, Istart, Iend;

216:   MatGetOwnershipRange(A,&Istart,&Iend);
217:   for( i=1; i<=m; i++ ) {
218:     jmax = m-i+1;
219:     for( j=1; j<=jmax; j++ ) {
220:       ix = ix + 1;
221:       if( ix-1<Istart || ix>Iend ) continue;  /* compute only owned rows */
222:       if( j!=jmax ) {
223:         pd = cst*(PetscReal)(i+j-1);
224:         /* north */
225:         if( i==1 ) {
226:           MatSetValue( A, ix-1, ix, 2*pd, INSERT_VALUES );
227:         }
228:         else {
229:           MatSetValue( A, ix-1, ix, pd, INSERT_VALUES );
230:         }
231:         /* east */
232:         if( j==1 ) {
233:           MatSetValue( A, ix-1, ix+jmax-1, 2*pd, INSERT_VALUES );
234:         }
235:         else {
236:           MatSetValue( A, ix-1, ix+jmax-1, pd, INSERT_VALUES );
237:         }
238:       }
239:       /* south */
240:       pu = 0.5 - cst*(PetscReal)(i+j-3);
241:       if( j>1 ) {
242:         MatSetValue( A, ix-1, ix-2, pu, INSERT_VALUES );
243:       }
244:       /* west */
245:       if( i>1 ) {
246:         MatSetValue( A, ix-1, ix-jmax-2, pu, INSERT_VALUES );
247:       }
248:     }
249:   }
250:   MatAssemblyBegin( A, MAT_FINAL_ASSEMBLY );
251:   MatAssemblyEnd( A, MAT_FINAL_ASSEMBLY );
252:   return(0);
253: }