Date: 2013-09-02T10:05:43+00:00

Actual source code: ex12.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[] = "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: {
     39:   Vec            v0,w0;           /* initial vector */
     40:   Vec            *X,*Y;           /* right and left eigenvectors */
     41:   Mat            A;               /* operator matrix */
     42:   EPS            eps;             /* eigenproblem solver context */
     43:   EPSType        type;
     44:   PetscReal      error1,error2,tol,re,im;
     45:   PetscScalar    kr,ki;
     46:   PetscInt       nev,maxit,i,its,nconv,N,m=15;

     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:      Request also left eigenvectors
     77:   */
     78:   EPSSetOperators(eps,A,NULL);
     79:   EPSSetProblemType(eps,EPS_NHEP);
     80:   EPSSetLeftVectorsWanted(eps,PETSC_TRUE);

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

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

     97:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     98:                       Solve the eigensystem
     99:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

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

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

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

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

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

    170:     SlepcCheckOrthogonality(X,nconv,Y,nconv,NULL,NULL,NULL);
    171:     PetscPrintf(PETSC_COMM_WORLD,"\n");
    172:     VecDestroyVecs(nconv,&X);
    173:     VecDestroyVecs(nconv,&Y);

    175:   }

    177:   /*
    178:      Free work space
    179:   */
    180:   VecDestroy(&v0);
    181:   VecDestroy(&w0);
    182:   EPSDestroy(&eps);
    183:   MatDestroy(&A);
    184:   SlepcFinalize();
    185:   return 0;
    186: }

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

    193:     This subroutine generates a test matrix that models a random walk on a
    194:     triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a
    195:     FORTRAN subroutine to calculate the dominant invariant subspaces of a real
    196:     matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
    197:     papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
    198:     (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
    199:     algorithms. The transpose of the matrix  is stochastic and so it is known
    200:     that one is an exact eigenvalue. One seeks the eigenvector of the transpose
    201:     associated with the eigenvalue unity. The problem is to calculate the steady
    202:     state probability distribution of the system, which is the eigevector
    203:     associated with the eigenvalue one and scaled in such a way that the sum all
    204:     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:   PetscInt        i,j,jmax,ix=0,Istart,Iend;
    213:   PetscErrorCode  ierr;

    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:         } else {
    228:           MatSetValue(A,ix-1,ix,pd,INSERT_VALUES);
    229:         }
    230:         /* east */
    231:         if (j==1) {
    232:           MatSetValue(A,ix-1,ix+jmax-1,2*pd,INSERT_VALUES);
    233:         } else {
    234:           MatSetValue(A,ix-1,ix+jmax-1,pd,INSERT_VALUES);
    235:         }
    236:       }
    237:       /* south */
    238:       pu = 0.5 - cst*(PetscReal)(i+j-3);
    239:       if (j>1) {
    240:         MatSetValue(A,ix-1,ix-2,pu,INSERT_VALUES);
    241:       }
    242:       /* west */
    243:       if (i>1) {
    244:         MatSetValue(A,ix-1,ix-jmax-2,pu,INSERT_VALUES);
    245:       }
    246:     }
    247:   }
    248:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    249:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    250:   return(0);
    251: }