Date: 2021-02-01T14:42:04+00:00

Actual source code: ex41.c
slepc-3.14.2 2021-02-01
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      1: /*
      2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
      3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
      4:    Copyright (c) 2002-2020, 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;

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

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

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

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

     55:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     56:                 Create the eigensolver and set various options
     57:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

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

     90:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
     91:                       Solve the eigensystem
     92:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

     94:   EPSSolve(eps);

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

    102:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    103:                     Display solution and clean up
    104:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    106:   /*
    107:      Get number of converged approximate eigenpairs
    108:   */
    109:   EPSGetConverged(eps,&nconv);
    110:   PetscPrintf(PETSC_COMM_WORLD," Number of converged eigenpairs: %D\n\n",nconv);
    111:   PetscMalloc2(nconv,&kr,nconv,&ki);
    112:   VecDuplicateVecs(t,nconv,&xr);
    113:   VecDuplicateVecs(t,nconv,&xi);
    114:   if (twosided) {
    115:     VecDuplicateVecs(t,nconv,&yr);
    116:     VecDuplicateVecs(t,nconv,&yi);
    117:   }

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

    127:     for (i=0;i<nconv;i++) {
    128:       /*
    129:         Get converged eigenpairs: i-th eigenvalue is stored in kr (real part) and
    130:         ki (imaginary part)
    131:       */
    132:       EPSGetEigenpair(eps,i,&kr[i],&ki[i],xr[i],xi[i]);
    133:       if (twosided) {
    134:         EPSGetLeftEigenvector(eps,i,yr[i],yi[i]);
    135:       }
    136:       /*
    137:          Compute the residual norms associated to each eigenpair
    138:       */
    139:       ComputeResidualNorm(A,PETSC_FALSE,kr[i],ki[i],xr[i],xi[i],t,&nrmr);
    140:       if (twosided) {
    141:         ComputeResidualNorm(A,PETSC_TRUE,kr[i],ki[i],yr[i],yi[i],t,&nrml);
    142:       }

    144: #if defined(PETSC_USE_COMPLEX)
    145:       re = PetscRealPart(kr[i]);
    146:       im = PetscImaginaryPart(kr[i]);
    147: #else
    148:       re = kr[i];
    149:       im = ki[i];
    150: #endif
    151:       if (im!=0.0) {
    152:         PetscPrintf(PETSC_COMM_WORLD," %8f%+8fi %12g %12g\n",(double)re,(double)im,(double)nrmr,(double)nrml);
    153:       } else {
    154:         PetscPrintf(PETSC_COMM_WORLD,"   %12f       %12g       %12g\n",(double)re,(double)nrmr,(double)nrml);
    155:       }
    156:     }
    157:     PetscPrintf(PETSC_COMM_WORLD,"\n");
    158:     /*
    159:        Check bi-orthogonality of eigenvectors
    160:     */
    161:     if (twosided) {
    162:       VecCheckOrthogonality(xr,nconv,yr,nconv,NULL,NULL,&lev);
    163:       if (lev<100*PETSC_MACHINE_EPSILON) {
    164:         PetscPrintf(PETSC_COMM_WORLD,"  Level of bi-orthogonality of eigenvectors < 100*eps\n\n");
    165:       } else {
    166:         PetscPrintf(PETSC_COMM_WORLD,"  Level of bi-orthogonality of eigenvectors: %g\n\n",(double)lev);
    167:       }
    168:     }
    169:   }

    171:   EPSDestroy(&eps);
    172:   MatDestroy(&A);
    173:   VecDestroy(&v0);
    174:   VecDestroy(&w0);
    175:   VecDestroy(&t);
    176:   PetscFree2(kr,ki);
    177:   VecDestroyVecs(nconv,&xr);
    178:   VecDestroyVecs(nconv,&xi);
    179:   if (twosided) {
    180:     VecDestroyVecs(nconv,&yr);
    181:     VecDestroyVecs(nconv,&yi);
    182:   }
    183:   SlepcFinalize();
    184:   return ierr;
    185: }

    187: /*
    188:     Matrix generator for a Markov model of a random walk on a triangular grid.

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

    203:     Note: the code will actually compute the transpose of the stochastic matrix
    204:     that contains the transition probabilities.
    205: */
    206: PetscErrorCode MatMarkovModel(PetscInt m,Mat A)
    207: {
    208:   const PetscReal cst = 0.5/(PetscReal)(m-1);
    209:   PetscReal       pd,pu;
    210:   PetscInt        Istart,Iend,i,j,jmax,ix=0;
    211:   PetscErrorCode  ierr;

    214:   MatGetOwnershipRange(A,&Istart,&Iend);
    215:   for (i=1;i<=m;i++) {
    216:     jmax = m-i+1;
    217:     for (j=1;j<=jmax;j++) {
    218:       ix = ix + 1;
    219:       if (ix-1<Istart || ix>Iend) continue;  /* compute only owned rows */
    220:       if (j!=jmax) {
    221:         pd = cst*(PetscReal)(i+j-1);
    222:         /* north */
    223:         if (i==1) {
    224:           MatSetValue(A,ix-1,ix,2*pd,INSERT_VALUES);
    225:         } else {
    226:           MatSetValue(A,ix-1,ix,pd,INSERT_VALUES);
    227:         }
    228:         /* east */
    229:         if (j==1) {
    230:           MatSetValue(A,ix-1,ix+jmax-1,2*pd,INSERT_VALUES);
    231:         } else {
    232:           MatSetValue(A,ix-1,ix+jmax-1,pd,INSERT_VALUES);
    233:         }
    234:       }
    235:       /* south */
    236:       pu = 0.5 - cst*(PetscReal)(i+j-3);
    237:       if (j>1) {
    238:         MatSetValue(A,ix-1,ix-2,pu,INSERT_VALUES);
    239:       }
    240:       /* west */
    241:       if (i>1) {
    242:         MatSetValue(A,ix-1,ix-jmax-2,pu,INSERT_VALUES);
    243:       }
    244:     }
    245:   }
    246:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
    247:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
    248:   return(0);
    249: }

    251: /*
    252:    ComputeResidualNorm - Computes the norm of the residual vector
    253:    associated with an eigenpair.

    255:    Input Parameters:
    256:      trans - whether A' must be used instead of A
    257:      kr,ki - eigenvalue
    258:      xr,xi - eigenvector
    259:      u     - work vector
    260: */
    261: PetscErrorCode ComputeResidualNorm(Mat A,PetscBool trans,PetscScalar kr,PetscScalar ki,Vec xr,Vec xi,Vec u,PetscReal *norm)
    262: {
    264: #if !defined(PETSC_USE_COMPLEX)
    265:   PetscReal      ni,nr;
    266: #endif
    267:   PetscErrorCode (*matmult)(Mat,Vec,Vec) = trans? MatMultTranspose: MatMult;

    270: #if !defined(PETSC_USE_COMPLEX)
    271:   if (ki == 0 || PetscAbsScalar(ki) < PetscAbsScalar(kr*PETSC_MACHINE_EPSILON)) {
    272: #endif
    273:     (*matmult)(A,xr,u);
    274:     if (PetscAbsScalar(kr) > PETSC_MACHINE_EPSILON) {
    275:       VecAXPY(u,-kr,xr);
    276:     }
    277:     VecNorm(u,NORM_2,norm);
    278: #if !defined(PETSC_USE_COMPLEX)
    279:   } else {
    280:     (*matmult)(A,xr,u);
    281:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
    282:       VecAXPY(u,-kr,xr);
    283:       VecAXPY(u,ki,xi);
    284:     }
    285:     VecNorm(u,NORM_2,&nr);
    286:     (*matmult)(A,xi,u);
    287:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
    288:       VecAXPY(u,-kr,xi);
    289:       VecAXPY(u,-ki,xr);
    290:     }
    291:     VecNorm(u,NORM_2,&ni);
    292:     *norm = SlepcAbsEigenvalue(nr,ni);
    293:   }
    294: #endif
    295:   return(0);
    296: }

    298: /*TEST

    300:    testset:
    301:       args: -st_type sinvert -eps_target 1.1 -eps_nev 4
    302:       filter: grep -v method | sed -e "s/[+-]0\.0*i//g" | sed -e "s/[0-9]\.[0-9]*e[+-]\([0-9]*\)/removed/g"
    303:       requires: !single
    304:       output_file: output/ex41_1.out
    305:       test:
    306:          suffix: 1
    307:          args: -eps_type {{power krylovschur}}
    308:       test:
    309:          suffix: 1_balance
    310:          args: -eps_balance {{oneside twoside}} -eps_ncv 18 -eps_krylovschur_locking 0

    312: TEST*/