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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
##### Software:
Chebfun; Chebfun2; Matlab; MultiParEig; PHClab; PHCpack; rootsb
Full Text:
##### References:
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