Actual source code: ex41.c

slepc-3.16.1 2021-11-17
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  1: /*
  2:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  3:    SLEPc - Scalable Library for Eigenvalue Problem Computations
  4:    Copyright (c) 2002-2021, 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*/