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arXiv · 2606.04259

An indefinite LOBPCG type of algorithm for detecting a definite Hermitian matrix pair

Marija Miloloža Pandur
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Abstract

A Hermitian matrix pair $(A,B)$ is called definite if some real linear combination of the matrices $A$ and $B$ is a positive definite matrix. Detection of the definiteness is not straightforward. We propose a basic subspace algorithm for detecting a large definite matrix pair $(A,B)$ with indefinite $B$. The proposed subspace algorithm is based on iterative testing of small projected Hermitian matrix pairs formed by using subspaces of small dimensions. Furthermore, we propose a specialized algorithm with parameter $m$, and its preconditioned variant. In the specialized algorithm with $m=3$ we choose the subspaces like in the indefinite locally optimal block preconditioned conjugate gradient (LOBPCG) method. Numerical experiments demonstrate the efficiency of our specialized algorithm, applied on medium-sized pairs, as well as, on large and banded pairs. Our algorithm very quickly detects (in)definiteness; much faster than some other algorithms.

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Marija Miloloža Pandur. 2026-06-02. An indefinite LOBPCG type of algorithm for detecting a definite Hermitian matrix pair. https://doi.org/10.1016/j.laa.2026.05.020

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