Fiber product homotopy method for multiparameter eigenvalue problems

TL;DR

The paper introduces the fiber product homotopy method, a novel approach for solving multiparameter eigenvalue problems that is more efficient, accurate, and better suited for singular problems than existing methods.

Contribution

It develops a new homotopy method that requires deforming fewer equations, is provably convergent for all eigenpairs, especially effective for singular problems, and outperforms existing methods in speed and accuracy.

Findings

Requires deformation of O(1) linear equations, unlike existing methods.

Achieves backward errors around 10^{-16} without extended precision.

Significantly faster and more accurate than previous methods, especially on larger and singular problems.

Abstract

We develop a new homotopy method for solving multiparameter eigenvalue problems (MEPs) called the fiber product homotopy method. For a $k$-parameter eigenvalue problem with matrices of sizes $n_1,\dots ,n_k = O(n)$, fiber product homotopy method requires deformation of $O(1)$ linear equations, while existing homotopy methods for MEPs require $O(n)$ nonlinear equations. We show that the fiber product homotopy method theoretically finds all eigenpairs of an MEP with probability one. It is especially well-suited for dimension-deficient singular MEPs, a weakness of all other existing methods, as the fiber product homotopy method is provably convergent with probability one for such problems as well, a fact borne out by numerical experiments. More generally, our numerical experiments indicate that the fiber product homotopy method significantly outperforms the standard Delta method in terms of accuracy, with consistent backward errors on the order of $10^{-16}$, even for dimension-deficient singular problems, and without any use of extended precision. In terms of speed, it significantly outperforms previous homotopy-based methods on all problems and outperforms the Delta method on larger problems, and is also highly parallelizable. We show that the fiber product MEP that we solve in the fiber product homotopy method, although mathematically equivalent to a standard MEP, is typically a much better conditioned problem.