An isospectral transformation between Hessenberg matrix and Hessenberg-bidiagonal matrix pencil without using subtraction

TL;DR

This paper presents a subtraction-free, eigenvalue-preserving transformation algorithm that converts a generalized eigenvalue problem involving Hessenberg and bidiagonal matrices into a standard form, enhancing numerical stability.

Contribution

It introduces a novel subtraction-free transformation method based on orthogonal polynomial theory for Hessenberg and bidiagonal matrix pencils, improving numerical stability.

Findings

The algorithm preserves eigenvalues during transformation.

It maintains sparsity of matrices, reducing computational complexity.

The method extends to Hessenberg-type matrices.

Abstract

We introduce an eigenvalue-preserving transformation algorithm from the generalized eigenvalue problem by matrix pencil of the upper and the lower bidiagonal matrices into a standard eigenvalue problem while preserving sparsity, using the theory of orthogonal polynomials. The procedure is formulated without subtraction, which causes numerical instability. Furthermore, the algorithm is discussed for the extended case where the upper bidiagonal matrix is of Hessenberg type.