The high relative accuracy of the HZ method

Matejaš, Josip; Hari, Vjeran

doi:10.1016/j.amc.2022.127358

Cited by 1 publication

(3 citation statements)

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“…If a real number t is known such that B(t) = A sin t+B cos t is positive definite, then the eigenvalues of a definite pair (A, B) can be computed via those of an associated definite pair (A(t), B(t)), where A(t) = A cos t − B sin t, by methods that exploit the definiteness of B(t) [1,2].…”

Section: Introductionmentioning

confidence: 99%

“…The FL method does not use a definitizing shift. If a definitizing shift θ 0 is provided, we can apply the real or complex Hari-Zimmermann method (HZ) (see [2] and the references therein) on the pair (B i , A i − θ 0 B i ) with positive definite A i − θ 0 B i . The HZ algorithm is a normalized version of the FL algorithm.…”

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confidence: 99%

“…The HZ algorithm is a normalized version of the FL algorithm. Each of these two approaches, the FL and the HZ, has its advantages and shortcomings [2].…”

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confidence: 99%

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Subspace algorithms for detecting a definite Hermitian matrix pair

Pandur

2023

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The Hermitian pair (A,B) is called definite if some real linear combination of the matrices A and B is a positive definite matrix. There are several reasons why it is important to detect whether a given matrix pair is definite. Determining whether a given matrix pair is definite is not straightforward. There exist different kinds of algorithms for that task. Most of them are not appropriate for medium-size or large-size matrix pairs. We propose subspace algorithms for medium-size or large-size matrix pairs that are based on iterative testing of small compressed Hermitian matrix pairs formed by using subspaces of small dimensions. First, we propose a new basic subspace algorithm for detecting definite matrix pairs. Furthermore, we propose a specialized algorithm and its preconditioned variant. In this preconditioned variant, we need to solve several systems of linear equations. These systems can be solved only approximately and not necessarily in every iteration step. Numerical experiments demonstrate the efficiency of our specialized algorithm.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%