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Jacobi-Davidson methods for polynomial two-parameter eigenvalue problems. (English) Zbl 06448790
Summary: We propose Jacobi-Davidson type methods for polynomial two-parameter eigenvalue problems (PMEP). Such problems can be linearized as singular two-parameter eigenvalue problems, whose matrices are of dimension \(k(k+1) n / 2\), where \(k\) is the degree of the polynomial and \(n\) is the size of the matrix coefficients in the PMEP. When \(k^2 n\) is relatively small, the problem can be solved numerically by computing the common regular part of the related pair of singular pencils. For large \(k^2 n\), computing all solutions is not feasible and iterative methods are required.
When \(k\) is large, we propose to linearize the problem first and then apply Jacobi-Davidson to the obtained singular two-parameter eigenvalue problem. The resulting method may for instance be used for computing zeros of a system of scalar bivariate polynomials close to a given target. On the other hand, when \(k\) is small, we can apply a Jacobi-Davidson type approach directly to the original matrices. The original matrices are projected onto a low-dimensional subspace, and the projected polynomial two-parameter eigenvalue problems are solved by a linearization.

MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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