Date: 2026-05-01T10:02:36+00:00

Actual source code: ex5.c

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

      6:    This file is part of SLEPc.
      7:    SLEPc is distributed under a 2-clause BSD license (see LICENSE).
      8:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      9: */

     11: static char help[] = "Eigenvalue problem associated with a Markov model of a random walk on a triangular grid.\n\n"
     12:   "It is a standard nonsymmetric eigenproblem with real eigenvalues and the rightmost eigenvalue is known to be 1.\n"
     13:   "This example illustrates how the user can set the initial vector.\n\n"
     14:   "The command line options are:\n"
     15:   "  -m <m>, where <m> = number of grid subdivisions in each dimension.\n\n";

     17: #include <slepceps.h>

     19: /*
     20:    User-defined routines
     21: */
     22: PetscErrorCode MatMarkovModel(PetscInt m,Mat A);

     24: int main(int argc,char **argv)
     25: {
     26:   Vec            v0;              /* initial vector */
     27:   Mat            A;               /* operator matrix */
     28:   EPS            eps;             /* eigenproblem solver context */
     29:   EPSType        type;
     30:   EPSStop        stop;
     31:   PetscReal      thres;
     32:   PetscInt       N,m=15,nev;
     33:   PetscMPIInt    rank;
     34:   PetscBool      terse;

     36:   PetscFunctionBeginUser;
     37:   PetscCall(SlepcInitialize(&argc,&argv,NULL,help));

     39:   PetscCall(PetscOptionsGetInt(NULL,NULL,"-m",&m,NULL));
     40:   N = m*(m+1)/2;
     41:   PetscCall(PetscPrintf(PETSC_COMM_WORLD,"\nMarkov Model, N=%" PetscInt_FMT " (m=%" PetscInt_FMT ")\n\n",N,m));

     43:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     44:      Compute the operator matrix that defines the eigensystem, Ax=kx
     45:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     47:   PetscCall(MatCreate(PETSC_COMM_WORLD,&A));
     48:   PetscCall(MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N));
     49:   PetscCall(MatSetFromOptions(A));
     50:   PetscCall(MatMarkovModel(m,A));

     52:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     53:                 Create the eigensolver and set various options
     54:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     56:   /*
     57:      Create eigensolver context
     58:   */
     59:   PetscCall(EPSCreate(PETSC_COMM_WORLD,&eps));

     61:   /*
     62:      Set operators. In this case, it is a standard eigenvalue problem
     63:   */
     64:   PetscCall(EPSSetOperators(eps,A,NULL));
     65:   PetscCall(EPSSetProblemType(eps,EPS_NHEP));

     67:   /*
     68:      Set solver parameters at runtime
     69:   */
     70:   PetscCall(EPSSetFromOptions(eps));

     72:   /*
     73:      Set the initial vector. This is optional, if not done the initial
     74:      vector is set to random values
     75:   */
     76:   PetscCall(MatCreateVecs(A,&v0,NULL));
     77:   PetscCallMPI(MPI_Comm_rank(PETSC_COMM_WORLD,&rank));
     78:   if (!rank) {
     79:     PetscCall(VecSetValue(v0,0,1.0,INSERT_VALUES));
     80:     PetscCall(VecSetValue(v0,1,1.0,INSERT_VALUES));
     81:     PetscCall(VecSetValue(v0,2,1.0,INSERT_VALUES));
     82:   }
     83:   PetscCall(VecAssemblyBegin(v0));
     84:   PetscCall(VecAssemblyEnd(v0));
     85:   PetscCall(EPSSetInitialSpace(eps,1,&v0));

     87:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     88:                       Solve the eigensystem
     89:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     91:   PetscCall(EPSSolve(eps));

     93:   /*
     94:      Optional: Get some information from the solver and display it
     95:   */
     96:   PetscCall(EPSGetType(eps,&type));
     97:   PetscCall(PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type));
     98:   PetscCall(EPSGetStoppingTest(eps,&stop));
     99:   if (stop!=EPS_STOP_THRESHOLD) {
    100:     PetscCall(EPSGetDimensions(eps,&nev,NULL,NULL));
    101:     PetscCall(PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %" PetscInt_FMT "\n",nev));
    102:   } else {
    103:     PetscCall(EPSGetThreshold(eps,&thres,NULL));
    104:     PetscCall(PetscPrintf(PETSC_COMM_WORLD," Using threshold: %.4g\n",(double)thres));
    105:   }

    107:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    108:                     Display solution and clean up
    109:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    111:   /* show detailed info unless -terse option is given by user */
    112:   PetscCall(PetscOptionsHasName(NULL,NULL,"-terse",&terse));
    113:   if (terse) PetscCall(EPSErrorView(eps,EPS_ERROR_RELATIVE,NULL));
    114:   else {
    115:     PetscCall(PetscViewerPushFormat(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_ASCII_INFO_DETAIL));
    116:     PetscCall(EPSConvergedReasonView(eps,PETSC_VIEWER_STDOUT_WORLD));
    117:     PetscCall(EPSErrorView(eps,EPS_ERROR_RELATIVE,PETSC_VIEWER_STDOUT_WORLD));
    118:     PetscCall(PetscViewerPopFormat(PETSC_VIEWER_STDOUT_WORLD));
    119:   }
    120:   PetscCall(EPSDestroy(&eps));
    121:   PetscCall(MatDestroy(&A));
    122:   PetscCall(VecDestroy(&v0));
    123:   PetscCall(SlepcFinalize());
    124:   return 0;
    125: }

    127: /*
    128:     Matrix generator for a Markov model of a random walk on a triangular grid.

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

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

    152:   PetscFunctionBeginUser;
    153:   PetscCall(MatGetOwnershipRange(A,&Istart,&Iend));
    154:   for (i=1;i<=m;i++) {
    155:     jmax = m-i+1;
    156:     for (j=1;j<=jmax;j++) {
    157:       ix = ix + 1;
    158:       if (ix-1<Istart || ix>Iend) continue;  /* compute only owned rows */
    159:       if (j!=jmax) {
    160:         pd = cst*(PetscReal)(i+j-1);
    161:         /* north */
    162:         if (i==1) PetscCall(MatSetValue(A,ix-1,ix,2*pd,INSERT_VALUES));
    163:         else PetscCall(MatSetValue(A,ix-1,ix,pd,INSERT_VALUES));
    164:         /* east */
    165:         if (j==1) PetscCall(MatSetValue(A,ix-1,ix+jmax-1,2*pd,INSERT_VALUES));
    166:         else PetscCall(MatSetValue(A,ix-1,ix+jmax-1,pd,INSERT_VALUES));
    167:       }
    168:       /* south */
    169:       pu = 0.5 - cst*(PetscReal)(i+j-3);
    170:       if (j>1) PetscCall(MatSetValue(A,ix-1,ix-2,pu,INSERT_VALUES));
    171:       /* west */
    172:       if (i>1) PetscCall(MatSetValue(A,ix-1,ix-jmax-2,pu,INSERT_VALUES));
    173:     }
    174:   }
    175:   PetscCall(MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY));
    176:   PetscCall(MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY));
    177:   PetscFunctionReturn(PETSC_SUCCESS);
    178: }

    180: /*TEST

    182:    test:
    183:       suffix: 1
    184:       nsize: 2
    185:       args: -eps_largest_real -eps_nev 4 -eps_two_sided {{0 1}} -eps_krylovschur_locking {{0 1}} -ds_parallel synchronized -terse
    186:       filter: sed -e "s/90424/90423/" | sed -e "s/85715/85714/"

    188:    test:
    189:       suffix: 2
    190:       args: -eps_threshold_relative .9 -eps_ncv 10 -terse
    191:       filter: sed -e "s/-0.85714/0.85714/" | sed -e "s/90424/90423/" | sed -e "s/-1.00000, 1.00000/1.00000, -1.00000/" | sed -e "s/-0.97137, 0.97137/0.97137, -0.97137/" | sed -e "s/-0.90423, 0.90423/0.90423, -0.90423/"

    193: TEST*/