Overview
Research introduces an indefinite LOBPCG-type algorithm designed to detect definite Hermitian matrix pairs. The algorithm specifically addresses the challenge of identifying definiteness in cases where the $B$ matrix within the pair $(A,B)$ is indefinite.
Research Context
A Hermitian matrix pair $(A,B)$ is classified as definite if a real linear combination of matrices $A$ and $B$ results in a positive definite matrix. The identification of such definiteness is described as non-straightforward.
Approach
The proposed methodology includes a basic subspace algorithm and a specialized algorithm with an associated parameter, $m$, along with its preconditioned variant. The basic subspace algorithm is designed for detecting large definite matrix pairs $(A,B)$, specifically those with an indefinite $B$. This algorithm operates by iteratively testing smaller projected Hermitian matrix pairs. These smaller pairs are constructed using subspaces of reduced dimensions.
The specialized algorithm, particularly when $m=3$, selects subspaces in a manner consistent with the indefinite locally optimal block preconditioned conjugate gradient (LOBPCG) method.
Findings
Numerical experiments were conducted to evaluate the efficiency of the specialized algorithm. These experiments covered medium-sized matrix pairs, as well as large and banded pairs. The results indicated that the specialized algorithm effectively detects (in)definiteness. The algorithm demonstrated speed, identifying (in)definiteness faster than some other algorithms.
Why This Matters
The development of an efficient algorithm to detect the definiteness of Hermitian matrix pairs, particularly those with indefinite $B$ matrices, addresses a computational challenge. The observed speed of the proposed specialized algorithm over existing methods could offer practical advantages in relevant computational tasks.