Computing several eigenvalues of nonlinear eigenvalue problems by selection

Abstract

Computing more than one eigenvalue for (large sparse) one-parameter polynomial and general nonlinear eigenproblems, as well as for multiparameter linear and nonlinear eigenproblems, is a much harder task than for standard eigenvalue problems. We present simple but efficient selection methods based on divided differences to do this. In contrast to locking techniques, it is not necessary to keep converged eigenvectors in the search space, so that the entire search space may be devoted to new information. The techniques are applicable to many types of matrix eigenvalue problems; standard deflation is possible only for linear one-parameter problems. The methods are easy to understand and implement. Although divided differences are well-known in the context of nonlinear eigenproblems, the proposed selection techniques are new for one-parameter problems. For multiparameter problems, we improve on and generalize our previous work. We also show how to use divided differences in the framework of homogeneous coordinates, which may be appropriate for generalized eigenvalue problems with infinite eigenvalues.While the approaches are valuable alternatives for one-parameter nonlinear eigenproblems, they seem the only option for multiparameter problems.

Orthogonal iterations on Structured Pencils

“…For the polynomial plasma drift we repeated the experiment twice by setting the stopping criteria in (2.2) to a tolerance of 1.0e − 4, and the maximum number of iterations to 450, respectively. This is a very challenging problem for any eigensolver [25] with several eigenvalues of high multiplicity and/or clustered around zero. In this case the estimated backward error is not so significant since our measure of backward stability assumes that we are at convergence and we have identified an invariant subspace.…”

“…In this case the estimated backward error is not so significant since our measure of backward stability assumes that we are at convergence and we have identified an invariant subspace. In [25] the authors proposed a variation of the Jacobi-Davidson method for computing several eigenpairs of the polynomial eigenvalue problem. For the plasma drift problem they fixed a residual threshold of 1.0e − 2 and within 200 iterations they were able to compute the approximations of the 19 eigenvalues closer to the origin.…”

Nonlinearizing Two-parameter Eigenvalue Problems

“…We consider a BVP-eigenvalue problem, which we separate into two domains in such a way that it leads to a two-parameter eigenvalue problem. Similar techniques and analysis are found in, e.g., [9], [3,Chapter 2], and [19,Experiment 4], where it is common to force the solution to have roots within the considered interval. with a wavenumber \kappa which is discontinuous in one part of the domain and smooth in another, as in Figure 5.6.…”

Orthogonal Iterations on Companion-Like Pencils

“…For the plasma_drift polynomial, the backward errors on the pencil reflect the termination of the algorithm with the adopted criteria, and the invariant subspace is found with an adequate accuracy. The plasma_drift is a very challenging problem for any eigensolver [26], due to several eigenvalues of high multiplicity and/or clustered around zero. In [26] the authors proposed a variation of the Jacobi-Davidson method for computing several eigenpairs of the polynomial eigenvalue problem.…”

“…The plasma_drift is a very challenging problem for any eigensolver [26], due to several eigenvalues of high multiplicity and/or clustered around zero. In [26] the authors proposed a variation of the Jacobi-Davidson method for computing several eigenpairs of the polynomial eigenvalue problem. For this problem, they ran their algorithm with a residual threshold of 1.0e−2 and within 200 iterations they were able to compute the approximations of the 19 eigenvalues closer to the origin.…”