Actual source code: ex41.c

slepc-3.20.1 2023-11-27
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  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,(char*)0,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(MatSetUp(A));
 52:   PetscCall(MatMarkovModel(m,A));
 53:   PetscCall(MatCreateVecs(A,NULL,&t));

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

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

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

 66:   /* allow user to change settings at run time */
 67:   PetscCall(EPSSetFromOptions(eps));
 68:   PetscCall(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:   PetscCall(MatCreateVecs(A,&v0,&w0));
 75:   PetscCallMPI(MPI_Comm_rank(PETSC_COMM_WORLD,&rank));
 76:   if (!rank) {
 77:     PetscCall(VecSetValue(v0,0,1.0,INSERT_VALUES));
 78:     PetscCall(VecSetValue(v0,1,1.0,INSERT_VALUES));
 79:     PetscCall(VecSetValue(v0,2,1.0,INSERT_VALUES));
 80:     PetscCall(VecSetValue(w0,0,2.0,INSERT_VALUES));
 81:     PetscCall(VecSetValue(w0,2,0.5,INSERT_VALUES));
 82:   }
 83:   PetscCall(VecAssemblyBegin(v0));
 84:   PetscCall(VecAssemblyBegin(w0));
 85:   PetscCall(VecAssemblyEnd(v0));
 86:   PetscCall(VecAssemblyEnd(w0));
 87:   PetscCall(EPSSetInitialSpace(eps,1,&v0));
 88:   PetscCall(EPSSetLeftInitialSpace(eps,1,&w0));

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

 94:   PetscCall(EPSSolve(eps));

 96:   /*
 97:      Optional: Get some information from the solver and display it
 98:   */
 99:   PetscCall(EPSGetType(eps,&type));
100:   PetscCall(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:   PetscCall(EPSGetConverged(eps,&nconv));
110:   PetscCall(PetscPrintf(PETSC_COMM_WORLD," Number of converged eigenpairs: %" PetscInt_FMT "\n\n",nconv));
111:   PetscCall(PetscMalloc2(nconv,&kr,nconv,&ki));
112:   PetscCall(VecDuplicateVecs(t,nconv,&xr));
113:   PetscCall(VecDuplicateVecs(t,nconv,&xi));
114:   if (twosided) {
115:     PetscCall(VecDuplicateVecs(t,nconv,&yr));
116:     PetscCall(VecDuplicateVecs(t,nconv,&yi));
117:   }

119:   if (nconv>0) {
120:     /*
121:        Display eigenvalues and relative errors
122:     */
123:     PetscCall(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:       PetscCall(EPSGetEigenpair(eps,i,&kr[i],&ki[i],xr[i],xi[i]));
133:       if (twosided) PetscCall(EPSGetLeftEigenvector(eps,i,yr[i],yi[i]));
134:       /*
135:          Compute the residual norms associated to each eigenpair
136:       */
137:       PetscCall(ComputeResidualNorm(A,PETSC_FALSE,kr[i],ki[i],xr[i],xi[i],t,&nrmr));
138:       if (twosided) PetscCall(ComputeResidualNorm(A,PETSC_TRUE,kr[i],ki[i],yr[i],yi[i],t,&nrml));

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

161:   PetscCall(EPSDestroy(&eps));
162:   PetscCall(MatDestroy(&A));
163:   PetscCall(VecDestroy(&v0));
164:   PetscCall(VecDestroy(&w0));
165:   PetscCall(VecDestroy(&t));
166:   PetscCall(PetscFree2(kr,ki));
167:   PetscCall(VecDestroyVecs(nconv,&xr));
168:   PetscCall(VecDestroyVecs(nconv,&xi));
169:   if (twosided) {
170:     PetscCall(VecDestroyVecs(nconv,&yr));
171:     PetscCall(VecDestroyVecs(nconv,&yi));
172:   }
173:   PetscCall(SlepcFinalize());
174:   return 0;
175: }

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

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

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

202:   PetscFunctionBeginUser;
203:   PetscCall(MatGetOwnershipRange(A,&Istart,&Iend));
204:   for (i=1;i<=m;i++) {
205:     jmax = m-i+1;
206:     for (j=1;j<=jmax;j++) {
207:       ix = ix + 1;
208:       if (ix-1<Istart || ix>Iend) continue;  /* compute only owned rows */
209:       if (j!=jmax) {
210:         pd = cst*(PetscReal)(i+j-1);
211:         /* north */
212:         if (i==1) PetscCall(MatSetValue(A,ix-1,ix,2*pd,INSERT_VALUES));
213:         else PetscCall(MatSetValue(A,ix-1,ix,pd,INSERT_VALUES));
214:         /* east */
215:         if (j==1) PetscCall(MatSetValue(A,ix-1,ix+jmax-1,2*pd,INSERT_VALUES));
216:         else PetscCall(MatSetValue(A,ix-1,ix+jmax-1,pd,INSERT_VALUES));
217:       }
218:       /* south */
219:       pu = 0.5 - cst*(PetscReal)(i+j-3);
220:       if (j>1) PetscCall(MatSetValue(A,ix-1,ix-2,pu,INSERT_VALUES));
221:       /* west */
222:       if (i>1) PetscCall(MatSetValue(A,ix-1,ix-jmax-2,pu,INSERT_VALUES));
223:     }
224:   }
225:   PetscCall(MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY));
226:   PetscCall(MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY));
227:   PetscFunctionReturn(PETSC_SUCCESS);
228: }

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

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

247:   PetscFunctionBegin;
248: #if !defined(PETSC_USE_COMPLEX)
249:   if (ki == 0 || PetscAbsScalar(ki) < PetscAbsScalar(kr*PETSC_MACHINE_EPSILON)) {
250: #endif
251:     PetscCall((*matmult)(A,xr,u));
252:     if (PetscAbsScalar(kr) > PETSC_MACHINE_EPSILON) PetscCall(VecAXPY(u,-kr,xr));
253:     PetscCall(VecNorm(u,NORM_2,norm));
254: #if !defined(PETSC_USE_COMPLEX)
255:   } else {
256:     PetscCall((*matmult)(A,xr,u));
257:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
258:       PetscCall(VecAXPY(u,-kr,xr));
259:       PetscCall(VecAXPY(u,ki,xi));
260:     }
261:     PetscCall(VecNorm(u,NORM_2,&nr));
262:     PetscCall((*matmult)(A,xi,u));
263:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
264:       PetscCall(VecAXPY(u,-kr,xi));
265:       PetscCall(VecAXPY(u,-ki,xr));
266:     }
267:     PetscCall(VecNorm(u,NORM_2,&ni));
268:     *norm = SlepcAbsEigenvalue(nr,ni);
269:   }
270: #endif
271:   PetscFunctionReturn(PETSC_SUCCESS);
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//g" | 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 17 -eps_krylovschur_locking 0
287:          requires: !__float128

289: TEST*/