Date: 2013-09-02T10:05:25+00:00

Actual source code: test9.c

      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2013, Universitat Politecnica de Valencia, Spain

      6:    This file is part of SLEPc.

      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);
     34: PetscErrorCode MyEigenSort(PetscScalar ar,PetscScalar ai,PetscScalar br,PetscScalar bi,PetscInt *r,void *ctx);

     38: int main(int argc,char **argv)
     39: {
     40:   Vec            v0;              /* initial vector */
     41:   Mat            A;               /* operator matrix */
     42:   EPS            eps;             /* eigenproblem solver context */
     43:   EPSType        type;
     44:   PetscReal      tol=1000*PETSC_MACHINE_EPSILON;
     45:   PetscInt       N,m=15,nev;
     46:   PetscScalar    origin=0.0;

     49:   SlepcInitialize(&argc,&argv,(char*)0,help);

     51:   PetscOptionsGetInt(NULL,"-m",&m,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:   MatSetUp(A);
     63:   MatMarkovModel(m,A);

     65:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     66:                 Create the eigensolver and set various options
     67:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     69:   /*
     70:      Create eigensolver context
     71:   */
     72:   EPSCreate(PETSC_COMM_WORLD,&eps);

     74:   /*
     75:      Set operators. In this case, it is a standard eigenvalue problem
     76:   */
     77:   EPSSetOperators(eps,A,NULL);
     78:   EPSSetProblemType(eps,EPS_NHEP);
     79:   EPSSetTolerances(eps,tol,PETSC_DECIDE);

     81:   /*
     82:      Set the custom comparing routine in order to obtain the eigenvalues
     83:      closest to the target on the right only
     84:   */
     85:   EPSSetEigenvalueComparison(eps,MyEigenSort,&origin);

     88:   /*
     89:      Set solver parameters at runtime
     90:   */
     91:   EPSSetFromOptions(eps);

     93:   /*
     94:      Set the initial vector. This is optional, if not done the initial
     95:      vector is set to random values
     96:   */
     97:   MatGetVecs(A,&v0,NULL);
     98:   VecSet(v0,1.0);
     99:   EPSSetInitialSpace(eps,1,&v0);

    101:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    102:                       Solve the eigensystem
    103:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    105:   EPSSolve(eps);

    107:   /*
    108:      Optional: Get some information from the solver and display it
    109:   */
    110:   EPSGetType(eps,&type);
    111:   PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type);
    112:   EPSGetDimensions(eps,&nev,NULL,NULL);
    113:   PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %D\n",nev);

    115:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    116:                     Display solution and clean up
    117:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    119:   EPSPrintSolution(eps,NULL);
    120:   EPSDestroy(&eps);
    121:   MatDestroy(&A);
    122:   VecDestroy(&v0);
    123:   SlepcFinalize();
    124:   return 0;
    125: }

    129: /*
    130:     Matrix generator for a Markov model of a random walk on a triangular grid.

    132:     This subroutine generates a test matrix that models a random walk on a
    133:     triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a
    134:     FORTRAN subroutine to calculate the dominant invariant subspaces of a real
    135:     matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
    136:     papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
    137:     (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
    138:     algorithms. The transpose of the matrix  is stochastic and so it is known
    139:     that one is an exact eigenvalue. One seeks the eigenvector of the transpose
    140:     associated with the eigenvalue unity. The problem is to calculate the steady
    141:     state probability distribution of the system, which is the eigevector
    142:     associated with the eigenvalue one and scaled in such a way that the sum all
    143:     the components is equal to one.

    145:     Note: the code will actually compute the transpose of the stochastic matrix
    146:     that contains the transition probabilities.
    147: */
    148: PetscErrorCode MatMarkovModel(PetscInt m,Mat A)
    149: {
    150:   const PetscReal cst = 0.5/(PetscReal)(m-1);
    151:   PetscReal       pd,pu;
    152:   PetscInt        Istart,Iend,i,j,jmax,ix=0;
    153:   PetscErrorCode  ierr;

    156:   MatGetOwnershipRange(A,&Istart,&Iend);
    157:   for (i=1;i<=m;i++) {
    158:     jmax = m-i+1;
    159:     for (j=1;j<=jmax;j++) {
    160:       ix = ix + 1;
    161:       if (ix-1<Istart || ix>Iend) continue;  /* compute only owned rows */
    162:       if (j!=jmax) {
    163:         pd = cst*(PetscReal)(i+j-1);
    164:         /* north */
    165:         if (i==1) {
    166:           MatSetValue(A,ix-1,ix,2*pd,INSERT_VALUES);
    167:         } else {
    168:           MatSetValue(A,ix-1,ix,pd,INSERT_VALUES);
    169:         }
    170:         /* east */
    171:         if (j==1) {
    172:           MatSetValue(A,ix-1,ix+jmax-1,2*pd,INSERT_VALUES);
    173:         } else {
    174:           MatSetValue(A,ix-1,ix+jmax-1,pd,INSERT_VALUES);
    175:         }
    176:       }
    177:       /* south */
    178:       pu = 0.5 - cst*(PetscReal)(i+j-3);
    179:       if (j>1) {
    180:         MatSetValue(A,ix-1,ix-2,pu,INSERT_VALUES);
    181:       }
    182:       /* west */
    183:       if (i>1) {
    184:         MatSetValue(A,ix-1,ix-jmax-2,pu,INSERT_VALUES);
    185:       }
    186:     }
    187:   }
    188:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    189:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    190:   return(0);
    191: }

    195: /*
    196:     Function for user-defined eigenvalue ordering criterion.

    198:     Given two eigenvalues ar+i*ai and br+i*bi, the subroutine must choose
    199:     one of them as the preferred one according to the criterion.
    200:     In this example, the preferred value is the one furthest to the origin.
    201: */
    202: PetscErrorCode MyEigenSort(PetscScalar ar,PetscScalar ai,PetscScalar br,PetscScalar bi,PetscInt *r,void *ctx)
    203: {
    204:   PetscScalar origin = *(PetscScalar*)ctx;
    205:   PetscReal   d;

    208:   d = (SlepcAbsEigenvalue(br-origin,bi) - SlepcAbsEigenvalue(ar-origin,ai))/PetscMax(SlepcAbsEigenvalue(ar-origin,ai),SlepcAbsEigenvalue(br-origin,bi));
    209:   *r = d > PETSC_SQRT_MACHINE_EPSILON ? 1 : (d < -PETSC_SQRT_MACHINE_EPSILON ? -1 : PetscSign(PetscRealPart(br)));
    210:   return(0);
    211: }