Actual source code: ex41.c

slepc-3.11.2 2019-07-30
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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*/