Actual source code: ex41.c

slepc-3.17.1 2022-04-11
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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:   SlepcInitialize(&argc,&argv,(char*)0,help);

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

 43:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 44:      Compute the operator matrix that defines the eigensystem, Ax=kx
 45:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

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

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

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

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

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

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

 93:   EPSSolve(eps);

 95:   /*
 96:      Optional: Get some information from the solver and display it
 97:   */
 98:   EPSGetType(eps,&type);
 99:   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:   EPSGetConverged(eps,&nconv);
109:   PetscPrintf(PETSC_COMM_WORLD," Number of converged eigenpairs: %" PetscInt_FMT "\n\n",nconv);
110:   PetscMalloc2(nconv,&kr,nconv,&ki);
111:   VecDuplicateVecs(t,nconv,&xr);
112:   VecDuplicateVecs(t,nconv,&xi);
113:   if (twosided) {
114:     VecDuplicateVecs(t,nconv,&yr);
115:     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:       EPSGetEigenpair(eps,i,&kr[i],&ki[i],xr[i],xi[i]);
132:       if (twosided) EPSGetLeftEigenvector(eps,i,yr[i],yi[i]);
133:       /*
134:          Compute the residual norms associated to each eigenpair
135:       */
136:       ComputeResidualNorm(A,PETSC_FALSE,kr[i],ki[i],xr[i],xi[i],t,&nrmr);
137:       if (twosided) 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) PetscPrintf(PETSC_COMM_WORLD," %8f%+8fi %12g %12g\n",(double)re,(double)im,(double)nrmr,(double)nrml);
147:       else PetscPrintf(PETSC_COMM_WORLD,"   %12f       %12g       %12g\n",(double)re,(double)nrmr,(double)nrml);
148:     }
149:     PetscPrintf(PETSC_COMM_WORLD,"\n");
150:     /*
151:        Check bi-orthogonality of eigenvectors
152:     */
153:     if (twosided) {
154:       VecCheckOrthogonality(xr,nconv,yr,nconv,NULL,NULL,&lev);
155:       if (lev<100*PETSC_MACHINE_EPSILON) PetscPrintf(PETSC_COMM_WORLD,"  Level of bi-orthogonality of eigenvectors < 100*eps\n\n");
156:       else PetscPrintf(PETSC_COMM_WORLD,"  Level of bi-orthogonality of eigenvectors: %g\n\n",(double)lev);
157:     }
158:   }

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

202:   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) MatSetValue(A,ix-1,ix,2*pd,INSERT_VALUES);
212:         else MatSetValue(A,ix-1,ix,pd,INSERT_VALUES);
213:         /* east */
214:         if (j==1) MatSetValue(A,ix-1,ix+jmax-1,2*pd,INSERT_VALUES);
215:         else 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) MatSetValue(A,ix-1,ix-2,pu,INSERT_VALUES);
220:       /* west */
221:       if (i>1) MatSetValue(A,ix-1,ix-jmax-2,pu,INSERT_VALUES);
222:     }
223:   }
224:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
225:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
226:   PetscFunctionReturn(0);
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: #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) VecAXPY(u,-kr,xr);
251:     VecNorm(u,NORM_2,norm);
252: #if !defined(PETSC_USE_COMPLEX)
253:   } else {
254:     (*matmult)(A,xr,u);
255:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
256:       VecAXPY(u,-kr,xr);
257:       VecAXPY(u,ki,xi);
258:     }
259:     VecNorm(u,NORM_2,&nr);
260:     (*matmult)(A,xi,u);
261:     if (SlepcAbsEigenvalue(kr,ki) > PETSC_MACHINE_EPSILON) {
262:       VecAXPY(u,-kr,xi);
263:       VecAXPY(u,-ki,xr);
264:     }
265:     VecNorm(u,NORM_2,&ni);
266:     *norm = SlepcAbsEigenvalue(nr,ni);
267:   }
268: #endif
269:   PetscFunctionReturn(0);
270: }

272: /*TEST

274:    testset:
275:       args: -st_type sinvert -eps_target 1.1 -eps_nev 4
276:       filter: grep -v method | sed -e "s/[+-]0\.0*i//g" | sed -e "s/[0-9]\.[0-9]*e[+-]\([0-9]*\)/removed/g"
277:       requires: !single
278:       output_file: output/ex41_1.out
279:       test:
280:          suffix: 1
281:          args: -eps_type {{power krylovschur}}
282:       test:
283:          suffix: 1_balance
284:          args: -eps_balance {{oneside twoside}} -eps_ncv 18 -eps_krylovschur_locking 0

286: TEST*/