Householder orthogonalization with a non-standard inner product

TL;DR

This paper introduces new strategies for Householder orthogonalization in non-standard inner product spaces, ensuring numerical stability and broadening its applicability in linear algebra computations.

Contribution

It proposes novel algorithms and variants for Householder orthogonalization tailored to non-standard inner products, addressing a key challenge in the field.

Findings

Algorithms are numerically stable under mild assumptions

Strategies effectively handle the lack of an initial orthonormal basis

Numerical experiments confirm the approach's stability and effectiveness

Abstract

Householder orthogonalization plays an important role in numerical linear algebra. It attains perfect orthogonality regardless of the conditioning of the input. However, in the context of a non-standard inner product, it becomes difficult to apply Householder orthogonalization due to the lack of an initial orthonormal basis. We propose strategies to overcome this obstacle and discuss algorithms and variants of Householder orthogonalization with a non-standard inner product. Theoretical analysis and numerical experiments demonstrate that our approach is numerically stable under mild assumptions.