Date: 2020-01-13T12:00:51+00:00

Actual source code: ex41.c
slepc-3.12.2 2020-01-13
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2019, 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:   Mat            A;               /* operator matrix */
     27:   EPS            eps;             /* eigenproblem solver context */
     28:   EPSType        type;
     29:   PetscInt       i,N,m=15,nconv;
     30:   PetscBool      twosided;
     31:   PetscReal      nrmr,nrml=0.0,re,im,lev;
     32:   PetscScalar    *kr,*ki;
     33:   Vec            t,*xr,*xi,*yr,*yi;

     36:   SlepcInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;

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

     42:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     43:      Compute the operator matrix that defines the eigensystem, Ax=kx
     44:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

     53:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     54:                 Create the eigensolver and set various options
     55:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

     68:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     69:                       Solve the eigensystem
     70:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     72:   EPSSolve(eps);

     74:   /*
     75:      Optional: Get some information from the solver and display it
     76:   */
     77:   EPSGetType(eps,&type);
     78:   PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type);

     80:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     81:                     Display solution and clean up
     82:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     84:   /*
     85:      Get number of converged approximate eigenpairs
     86:   */
     87:   EPSGetConverged(eps,&nconv);
     88:   PetscPrintf(PETSC_COMM_WORLD," Number of converged eigenpairs: %D\n\n",nconv);
     89:   PetscMalloc2(nconv,&kr,nconv,&ki);
     90:   VecDuplicateVecs(t,nconv,&xr);
     91:   VecDuplicateVecs(t,nconv,&xi);
     92:   if (twosided) {
     93:     VecDuplicateVecs(t,nconv,&yr);
     94:     VecDuplicateVecs(t,nconv,&yi);
     95:   }

     97:   if (nconv>0) {
     98:     /*
     99:        Display eigenvalues and relative errors
    100:     */
    101:     PetscPrintf(PETSC_COMM_WORLD,
    102:          "           k            ||Ax-kx||         ||y'A-y'k||\n"
    103:          "   ---------------- ------------------ ------------------\n");

    105:     for (i=0;i<nconv;i++) {
    106:       /*
    107:         Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
    108:         ki (imaginary part)
    109:       */
    110:       EPSGetEigenpair(eps,i,&kr[i],&ki[i],xr[i],xi[i]);
    111:       if (twosided) {
    112:         EPSGetLeftEigenvector(eps,i,yr[i],yi[i]);
    113:       }
    114:       /*
    115:          Compute the residual norms associated to each eigenpair
    116:       */
    117:       ComputeResidualNorm(A,PETSC_FALSE,kr[i],ki[i],xr[i],xi[i],t,&nrmr);
    118:       if (twosided) {
    119:         ComputeResidualNorm(A,PETSC_TRUE,kr[i],ki[i],yr[i],yi[i],t,&nrml);
    120:       }

    122: #if defined(PETSC_USE_COMPLEX)
    123:       re = PetscRealPart(kr[i]);
    124:       im = PetscImaginaryPart(kr[i]);
    125: #else
    126:       re = kr[i];
    127:       im = ki[i];
    128: #endif
    129:       if (im!=0.0) {
    130:         PetscPrintf(PETSC_COMM_WORLD," %8f%+8fi %12g %12g\n",(double)re,(double)im,(double)nrmr,(double)nrml);
    131:       } else {
    132:         PetscPrintf(PETSC_COMM_WORLD,"   %12f       %12g       %12g\n",(double)re,(double)nrmr,(double)nrml);
    133:       }
    134:     }
    135:     PetscPrintf(PETSC_COMM_WORLD,"\n");
    136:     /*
    137:        Check bi-orthogonality of eigenvectors
    138:     */
    139:     if (twosided) {
    140:       VecCheckOrthogonality(xr,nconv,yr,nconv,NULL,NULL,&lev);
    141:       if (lev<100*PETSC_MACHINE_EPSILON) {
    142:         PetscPrintf(PETSC_COMM_WORLD,"  Level of bi-orthogonality of eigenvectors < 100*eps\n\n");
    143:       } else {
    144:         PetscPrintf(PETSC_COMM_WORLD,"  Level of bi-orthogonality of eigenvectors: %g\n\n",(double)lev);
    145:       }
    146:     }
    147:   }

    149:   EPSDestroy(&eps);
    150:   MatDestroy(&A);
    151:   VecDestroy(&t);
    152:   PetscFree2(kr,ki);
    153:   VecDestroyVecs(nconv,&xr);
    154:   VecDestroyVecs(nconv,&xi);
    155:   if (twosided) {
    156:     VecDestroyVecs(nconv,&yr);
    157:     VecDestroyVecs(nconv,&yi);
    158:   }
    159:   SlepcFinalize();
    160:   return ierr;
    161: }

    163: /*
    164:     Matrix generator for a Markov model of a random walk on a triangular grid.

    166:     This subroutine generates a test matrix that models a random walk on a
    167:     triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a
    168:     FORTRAN subroutine to calculate the dominant invariant subspaces of a real
    169:     matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
    170:     papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
    171:     (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
    172:     algorithms. The transpose of the matrix  is stochastic and so it is known
    173:     that one is an exact eigenvalue. One seeks the eigenvector of the transpose
    174:     associated with the eigenvalue unity. The problem is to calculate the steady
    175:     state probability distribution of the system, which is the eigevector
    176:     associated with the eigenvalue one and scaled in such a way that the sum all
    177:     the components is equal to one.

    179:     Note: the code will actually compute the transpose of the stochastic matrix
    180:     that contains the transition probabilities.
    181: */
    182: PetscErrorCode MatMarkovModel(PetscInt m,Mat A)
    183: {
    184:   const PetscReal cst = 0.5/(PetscReal)(m-1);
    185:   PetscReal       pd,pu;
    186:   PetscInt        Istart,Iend,i,j,jmax,ix=0;
    187:   PetscErrorCode  ierr;

    190:   MatGetOwnershipRange(A,&Istart,&Iend);
    191:   for (i=1;i<=m;i++) {
    192:     jmax = m-i+1;
    193:     for (j=1;j<=jmax;j++) {
    194:       ix = ix + 1;
    195:       if (ix-1<Istart || ix>Iend) continue;  /* compute only owned rows */
    196:       if (j!=jmax) {
    197:         pd = cst*(PetscReal)(i+j-1);
    198:         /* north */
    199:         if (i==1) {
    200:           MatSetValue(A,ix-1,ix,2*pd,INSERT_VALUES);
    201:         } else {
    202:           MatSetValue(A,ix-1,ix,pd,INSERT_VALUES);
    203:         }
    204:         /* east */
    205:         if (j==1) {
    206:           MatSetValue(A,ix-1,ix+jmax-1,2*pd,INSERT_VALUES);
    207:         } else {
    208:           MatSetValue(A,ix-1,ix+jmax-1,pd,INSERT_VALUES);
    209:         }
    210:       }
    211:       /* south */
    212:       pu = 0.5 - cst*(PetscReal)(i+j-3);
    213:       if (j>1) {
    214:         MatSetValue(A,ix-1,ix-2,pu,INSERT_VALUES);
    215:       }
    216:       /* west */
    217:       if (i>1) {
    218:         MatSetValue(A,ix-1,ix-jmax-2,pu,INSERT_VALUES);
    219:       }
    220:     }
    221:   }
    222:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    223:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    224:   return(0);
    225: }

    227: /*
    228:    ComputeResidualNorm - Computes the norm of the residual vector
    229:    associated with an eigenpair.

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

    246: #if !defined(PETSC_USE_COMPLEX)
    247:   if (ki == 0 || PetscAbsScalar(ki) < PetscAbsScalar(kr*PETSC_MACHINE_EPSILON)) {
    248: #endif
    249:     (*matmult)(A,xr,u);
    250:     if (PetscAbsScalar(kr) > PETSC_MACHINE_EPSILON) {
    251:       VecAXPY(u,-kr,xr);
    252:     }
    253:     VecNorm(u,NORM_2,norm);
    254: #if !defined(PETSC_USE_COMPLEX)
    255:   } else {
    256:     (*matmult)(A,xr,u);
    257:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
    258:       VecAXPY(u,-kr,xr);
    259:       VecAXPY(u,ki,xi);
    260:     }
    261:     VecNorm(u,NORM_2,&nr);
    262:     (*matmult)(A,xi,u);
    263:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
    264:       VecAXPY(u,-kr,xi);
    265:       VecAXPY(u,-ki,xr);
    266:     }
    267:     VecNorm(u,NORM_2,&ni);
    268:     *norm = SlepcAbsEigenvalue(nr,ni);
    269:   }
    270: #endif
    271:   return(0);
    272: }

    274: /*TEST

    276:    testset:
    277:       args: -st_type sinvert -eps_target 1.1 -eps_nev 4
    278:       filter: grep -v method | sed -e "s/[+-]0.0*i//" | sed -e "s/[0-9]\.[0-9]*e[+-]\([0-9]*\)/removed/g"
    279:       requires: !single
    280:       output_file: output/ex41_1.out
    281:       test:
    282:          suffix: 1
    283:          args: -eps_type {{power krylovschur}}
    284:       test:
    285:          suffix: 1_balance
    286:          args: -eps_balance {{oneside twoside}} -eps_ncv 16

    288: TEST*/