Actual source code: ex5.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.
8:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
9: */

11: static char help[] = "Eigenvalue problem associated with a Markov model of a random walk on a triangular grid. "
12:   "It is a standard nonsymmetric eigenproblem with real eigenvalues and the rightmost eigenvalue is known to be 1.\n"
13:   "This example illustrates how the user can set the initial vector.\n\n"
14:   "The command line options are:\n"
15:   "  -m <m>, where <m> = number of grid subdivisions in each dimension.\n\n";

17: #include <slepceps.h>

19: /*
20:    User-defined routines
21: */
22: PetscErrorCode MatMarkovModel(PetscInt m,Mat A);

24: int main(int argc,char **argv)
25: {
26:   Vec            v0;              /* initial vector */
27:   Mat            A;               /* operator matrix */
28:   EPS            eps;             /* eigenproblem solver context */
29:   EPSType        type;
30:   PetscInt       N,m=15,nev;
31:   PetscMPIInt    rank;
32:   PetscBool      terse;

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

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

41:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
42:      Compute the operator matrix that defines the eigensystem, Ax=kx
43:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

45:   MatCreate(PETSC_COMM_WORLD,&A);
46:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);
47:   MatSetFromOptions(A);
48:   MatSetUp(A);
49:   MatMarkovModel(m,A);

51:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
52:                 Create the eigensolver and set various options
53:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

55:   /*
56:      Create eigensolver context
57:   */
58:   EPSCreate(PETSC_COMM_WORLD,&eps);

60:   /*
61:      Set operators. In this case, it is a standard eigenvalue problem
62:   */
63:   EPSSetOperators(eps,A,NULL);
64:   EPSSetProblemType(eps,EPS_NHEP);

66:   /*
67:      Set solver parameters at runtime
68:   */
69:   EPSSetFromOptions(eps);

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

86:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
87:                       Solve the eigensystem
88:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

90:   EPSSolve(eps);

92:   /*
93:      Optional: Get some information from the solver and display it
94:   */
95:   EPSGetType(eps,&type);
96:   PetscPrintf(PETSC_COMM_WORLD," Solution method: %s\n\n",type);
97:   EPSGetDimensions(eps,&nev,NULL,NULL);
98:   PetscPrintf(PETSC_COMM_WORLD," Number of requested eigenvalues: %D\n",nev);

100:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
101:                     Display solution and clean up
102:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

104:   /* show detailed info unless -terse option is given by user */
105:   PetscOptionsHasName(NULL,NULL,"-terse",&terse);
106:   if (terse) {
107:     EPSErrorView(eps,EPS_ERROR_RELATIVE,NULL);
108:   } else {
109:     PetscViewerPushFormat(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_ASCII_INFO_DETAIL);
110:     EPSReasonView(eps,PETSC_VIEWER_STDOUT_WORLD);
111:     EPSErrorView(eps,EPS_ERROR_RELATIVE,PETSC_VIEWER_STDOUT_WORLD);
112:     PetscViewerPopFormat(PETSC_VIEWER_STDOUT_WORLD);
113:   }
114:   EPSDestroy(&eps);
115:   MatDestroy(&A);
116:   VecDestroy(&v0);
117:   SlepcFinalize();
118:   return ierr;
119: }

121: /*
122:     Matrix generator for a Markov model of a random walk on a triangular grid.

124:     This subroutine generates a test matrix that models a random walk on a
125:     triangular grid. This test example was used by G. W. Stewart ["{SRRIT} - a
126:     FORTRAN subroutine to calculate the dominant invariant subspaces of a real
127:     matrix", Tech. report. TR-514, University of Maryland (1978).] and in a few
128:     papers on eigenvalue problems by Y. Saad [see e.g. LAA, vol. 34, pp. 269-295
129:     (1980) ]. These matrices provide reasonably easy test problems for eigenvalue
130:     algorithms. The transpose of the matrix  is stochastic and so it is known
131:     that one is an exact eigenvalue. One seeks the eigenvector of the transpose
132:     associated with the eigenvalue unity. The problem is to calculate the steady
133:     state probability distribution of the system, which is the eigevector
134:     associated with the eigenvalue one and scaled in such a way that the sum all
135:     the components is equal to one.

137:     Note: the code will actually compute the transpose of the stochastic matrix
138:     that contains the transition probabilities.
139: */
140: PetscErrorCode MatMarkovModel(PetscInt m,Mat A)
141: {
142:   const PetscReal cst = 0.5/(PetscReal)(m-1);
143:   PetscReal       pd,pu;
144:   PetscInt        Istart,Iend,i,j,jmax,ix=0;
145:   PetscErrorCode  ierr;

148:   MatGetOwnershipRange(A,&Istart,&Iend);
149:   for (i=1;i<=m;i++) {
150:     jmax = m-i+1;
151:     for (j=1;j<=jmax;j++) {
152:       ix = ix + 1;
153:       if (ix-1<Istart || ix>Iend) continue;  /* compute only owned rows */
154:       if (j!=jmax) {
155:         pd = cst*(PetscReal)(i+j-1);
156:         /* north */
157:         if (i==1) {
158:           MatSetValue(A,ix-1,ix,2*pd,INSERT_VALUES);
159:         } else {
160:           MatSetValue(A,ix-1,ix,pd,INSERT_VALUES);
161:         }
162:         /* east */
163:         if (j==1) {
164:           MatSetValue(A,ix-1,ix+jmax-1,2*pd,INSERT_VALUES);
165:         } else {
166:           MatSetValue(A,ix-1,ix+jmax-1,pd,INSERT_VALUES);
167:         }
168:       }
169:       /* south */
170:       pu = 0.5 - cst*(PetscReal)(i+j-3);
171:       if (j>1) {
172:         MatSetValue(A,ix-1,ix-2,pu,INSERT_VALUES);
173:       }
174:       /* west */
175:       if (i>1) {
176:         MatSetValue(A,ix-1,ix-jmax-2,pu,INSERT_VALUES);
177:       }
178:     }
179:   }
180:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
181:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
182:   return(0);
183: }

185: /*TEST

187:    test:
188:       suffix: 1
189:       args: -eps_largest_real -eps_nev 4 -terse

191: TEST*/
```