Actual source code: ex41.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[] = "Illustrates the computation of left eigenvectors.\n\n"
     12:   "The problem is the Markov model as in ex5.c.\n"
     13:   "The command line options are:\n"
     14:   "  -m <m>, where <m> = number of grid subdivisions in each dimension.\n\n";

     16: #include <slepceps.h>

     18: /*
     19:    User-defined routines
     20: */
     21: PetscErrorCode MatMarkovModel(PetscInt,Mat);
     22: PetscErrorCode ComputeResidualNorm(Mat,PetscBool,PetscScalar,PetscScalar,Vec,Vec,Vec,PetscReal*);

     24: int main(int argc,char **argv)
     25: {
     26:   Vec            v0,w0;           /* initial vectors */
     27:   Mat            A;               /* operator matrix */
     28:   EPS            eps;             /* eigenproblem solver context */
     29:   EPSType        type;
     30:   PetscInt       i,N,m=15,nconv;
     31:   PetscBool      twosided;
     32:   PetscReal      nrmr,nrml=0.0,re,im,lev;
     33:   PetscScalar    *kr,*ki;
     34:   Vec            t,*xr,*xi,*yr,*yi;
     35:   PetscMPIInt    rank;

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

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

     44:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     45:      Compute the operator matrix that defines the eigensystem, Ax=kx
     46:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     48:   PetscCall(MatCreate(PETSC_COMM_WORLD,&A));
     49:   PetscCall(MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N));
     50:   PetscCall(MatSetFromOptions(A));
     51:   PetscCall(MatMarkovModel(m,A));
     52:   PetscCall(MatCreateVecs(A,NULL,&t));

     54:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     55:                 Create the eigensolver and set various options
     56:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     58:   PetscCall(EPSCreate(PETSC_COMM_WORLD,&eps));
     59:   PetscCall(EPSSetOperators(eps,A,NULL));
     60:   PetscCall(EPSSetProblemType(eps,EPS_NHEP));

     62:   /* use a two-sided algorithm to compute left eigenvectors as well */
     63:   PetscCall(EPSSetTwoSided(eps,PETSC_TRUE));

     65:   /* allow user to change settings at run time */
     66:   PetscCall(EPSSetFromOptions(eps));
     67:   PetscCall(EPSGetTwoSided(eps,&twosided));

     69:   /*
     70:      Set the initial vectors. This is optional, if not done the initial
     71:      vectors are set to random values
     72:   */
     73:   PetscCall(MatCreateVecs(A,&v0,&w0));
     74:   PetscCallMPI(MPI_Comm_rank(PETSC_COMM_WORLD,&rank));
     75:   if (!rank) {
     76:     PetscCall(VecSetValue(v0,0,1.0,INSERT_VALUES));
     77:     PetscCall(VecSetValue(v0,1,1.0,INSERT_VALUES));
     78:     PetscCall(VecSetValue(v0,2,1.0,INSERT_VALUES));
     79:     PetscCall(VecSetValue(w0,0,2.0,INSERT_VALUES));
     80:     PetscCall(VecSetValue(w0,2,0.5,INSERT_VALUES));
     81:   }
     82:   PetscCall(VecAssemblyBegin(v0));
     83:   PetscCall(VecAssemblyBegin(w0));
     84:   PetscCall(VecAssemblyEnd(v0));
     85:   PetscCall(VecAssemblyEnd(w0));
     86:   PetscCall(EPSSetInitialSpace(eps,1,&v0));
     87:   PetscCall(EPSSetLeftInitialSpace(eps,1,&w0));

     89:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     90:                       Solve the eigensystem
     91:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     93:   PetscCall(EPSSolve(eps));

     95:   /*
     96:      Optional: Get some information from the solver and display it
     97:   */
     98:   PetscCall(EPSGetType(eps,&type));
     99:   PetscCall(PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type));

    101:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    102:                     Display solution and clean up
    103:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    105:   /*
    106:      Get number of converged approximate eigenpairs
    107:   */
    108:   PetscCall(EPSGetConverged(eps,&nconv));
    109:   PetscCall(PetscPrintf(PETSC_COMM_WORLD," Number of converged eigenpairs: %" PetscInt_FMT "\n\n",nconv));
    110:   PetscCall(PetscMalloc2(nconv,&kr,nconv,&ki));
    111:   PetscCall(VecDuplicateVecs(t,nconv,&xr));
    112:   PetscCall(VecDuplicateVecs(t,nconv,&xi));
    113:   if (twosided) {
    114:     PetscCall(VecDuplicateVecs(t,nconv,&yr));
    115:     PetscCall(VecDuplicateVecs(t,nconv,&yi));
    116:   }

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

    126:     for (i=0;i<nconv;i++) {
    127:       /*
    128:         Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
    129:         ki (imaginary part)
    130:       */
    131:       PetscCall(EPSGetEigenpair(eps,i,&kr[i],&ki[i],xr[i],xi[i]));
    132:       if (twosided) PetscCall(EPSGetLeftEigenvector(eps,i,yr[i],yi[i]));
    133:       /*
    134:          Compute the residual norms associated to each eigenpair
    135:       */
    136:       PetscCall(ComputeResidualNorm(A,PETSC_FALSE,kr[i],ki[i],xr[i],xi[i],t,&nrmr));
    137:       if (twosided) PetscCall(ComputeResidualNorm(A,PETSC_TRUE,kr[i],ki[i],yr[i],yi[i],t,&nrml));

    139: #if defined(PETSC_USE_COMPLEX)
    140:       re = PetscRealPart(kr[i]);
    141:       im = PetscImaginaryPart(kr[i]);
    142: #else
    143:       re = kr[i];
    144:       im = ki[i];
    145: #endif
    146:       if (im!=0.0) PetscCall(PetscPrintf(PETSC_COMM_WORLD," %8f%+8fi %12g %12g\n",(double)re,(double)im,(double)nrmr,(double)nrml));
    147:       else PetscCall(PetscPrintf(PETSC_COMM_WORLD,"   %12f       %12g       %12g\n",(double)re,(double)nrmr,(double)nrml));
    148:     }
    149:     PetscCall(PetscPrintf(PETSC_COMM_WORLD,"\n"));
    150:     /*
    151:        Check bi-orthogonality of eigenvectors
    152:     */
    153:     if (twosided) {
    154:       PetscCall(VecCheckOrthogonality(xr,nconv,yr,nconv,NULL,NULL,&lev));
    155:       if (lev<100*PETSC_MACHINE_EPSILON) PetscCall(PetscPrintf(PETSC_COMM_WORLD,"  Level of bi-orthogonality of eigenvectors < 100*eps\n\n"));
    156:       else PetscCall(PetscPrintf(PETSC_COMM_WORLD,"  Level of bi-orthogonality of eigenvectors: %g\n\n",(double)lev));
    157:     }
    158:   }

    160:   PetscCall(EPSDestroy(&eps));
    161:   PetscCall(MatDestroy(&A));
    162:   PetscCall(VecDestroy(&v0));
    163:   PetscCall(VecDestroy(&w0));
    164:   PetscCall(VecDestroy(&t));
    165:   PetscCall(PetscFree2(kr,ki));
    166:   PetscCall(VecDestroyVecs(nconv,&xr));
    167:   PetscCall(VecDestroyVecs(nconv,&xi));
    168:   if (twosided) {
    169:     PetscCall(VecDestroyVecs(nconv,&yr));
    170:     PetscCall(VecDestroyVecs(nconv,&yi));
    171:   }
    172:   PetscCall(SlepcFinalize());
    173:   return 0;
    174: }

    176: /*
    177:     Matrix generator for a Markov model of a random walk on a triangular grid.

    179:     This subroutine generates a test matrix that models a random walk on a
    180:     triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a
    181:     FORTRAN subroutine to calculate the dominant invariant subspaces of a real
    182:     matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
    183:     papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
    184:     (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
    185:     algorithms. The transpose of the matrix  is stochastic and so it is known
    186:     that one is an exact eigenvalue. One seeks the eigenvector of the transpose
    187:     associated with the eigenvalue unity. The problem is to calculate the steady
    188:     state probability distribution of the system, which is the eigevector
    189:     associated with the eigenvalue one and scaled in such a way that the sum all
    190:     the components is equal to one.

    192:     Note: the code will actually compute the transpose of the stochastic matrix
    193:     that contains the transition probabilities.
    194: */
    195: PetscErrorCode MatMarkovModel(PetscInt m,Mat A)
    196: {
    197:   const PetscReal cst = 0.5/(PetscReal)(m-1);
    198:   PetscReal       pd,pu;
    199:   PetscInt        Istart,Iend,i,j,jmax,ix=0;

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

    229: /*
    230:    ComputeResidualNorm - Computes the norm of the residual vector
    231:    associated with an eigenpair.

    233:    Input Parameters:
    234:      trans - whether A' must be used instead of A
    235:      kr,ki - eigenvalue
    236:      xr,xi - eigenvector
    237:      u     - work vector
    238: */
    239: PetscErrorCode ComputeResidualNorm(Mat A,PetscBool trans,PetscScalar kr,PetscScalar ki,Vec xr,Vec xi,Vec u,PetscReal *norm)
    240: {
    241: #if !defined(PETSC_USE_COMPLEX)
    242:   PetscReal      ni,nr;
    243: #endif
    244:   PetscErrorCode (*matmult)(Mat,Vec,Vec) = trans? MatMultTranspose: MatMult;

    246:   PetscFunctionBegin;
    247: #if !defined(PETSC_USE_COMPLEX)
    248:   if (ki == 0 || PetscAbsScalar(ki) < PetscAbsScalar(kr*PETSC_MACHINE_EPSILON)) {
    249: #endif
    250:     PetscCall((*matmult)(A,xr,u));
    251:     if (PetscAbsScalar(kr) > PETSC_MACHINE_EPSILON) PetscCall(VecAXPY(u,-kr,xr));
    252:     PetscCall(VecNorm(u,NORM_2,norm));
    253: #if !defined(PETSC_USE_COMPLEX)
    254:   } else {
    255:     PetscCall((*matmult)(A,xr,u));
    256:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
    257:       PetscCall(VecAXPY(u,-kr,xr));
    258:       PetscCall(VecAXPY(u,ki,xi));
    259:     }
    260:     PetscCall(VecNorm(u,NORM_2,&nr));
    261:     PetscCall((*matmult)(A,xi,u));
    262:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
    263:       PetscCall(VecAXPY(u,-kr,xi));
    264:       PetscCall(VecAXPY(u,-ki,xr));
    265:     }
    266:     PetscCall(VecNorm(u,NORM_2,&ni));
    267:     *norm = SlepcAbsEigenvalue(nr,ni);
    268:   }
    269: #endif
    270:   PetscFunctionReturn(PETSC_SUCCESS);
    271: }

    273: /*TEST

    275:    testset:
    276:       args: -st_type sinvert -eps_target 1.1 -eps_nev 4
    277:       filter: grep -v method | sed -e "s/[+-]0\.0*i//g" | sed -e "s/[0-9]\.[0-9]*e[+-]\([0-9]*\)/removed/g"
    278:       requires: !single
    279:       output_file: output/ex41_1.out
    280:       test:
    281:          suffix: 1
    282:          args: -eps_type {{power krylovschur}}
    283:       test:
    284:          suffix: 1_balance
    285:          args: -eps_balance {{oneside twoside}} -eps_ncv 17 -eps_krylovschur_locking 0
    286:          requires: !__float128

    288: TEST*/