Actual source code: ex41.c
slepc-3.17.0 2022-03-31
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*/