An indefinite LOBPCG type of algorithm for detecting a definite Hermitian matrix pair
For researchers working on eigenvalue problems with indefinite matrix pairs, this provides a faster detection method.
The paper proposes a subspace algorithm for detecting definiteness of a Hermitian matrix pair (A,B) with indefinite B, and a specialized variant with parameter m. Numerical experiments show the algorithm detects definiteness much faster than other methods.
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.