Arithmetical enhancements of the Kogbetliantz method for the SVD of order two

TL;DR

This paper introduces an improved Kogbetliantz method for 2x2 SVD that enhances accuracy and stability through prescaling, refined triangularization, and a new SVD computation approach, tested on various matrices.

Contribution

It presents a novel, arithmetically enhanced Kogbetliantz method for 2x2 SVD with improved accuracy and robustness over existing routines like xLASV2.

Findings

High relative accuracy for singular values within a wide safe range.

Effective heuristic improves numerical orthogonality of singular vectors.

Method outperforms xLASV2 on upper triangular matrices in accuracy.

Abstract

An enhanced Kogbetliantz method for the singular value decomposition (SVD) of general matrices of order two is proposed. The method consists of three phases: an almost exact prescaling, that can be beneficial to the LAPACK's xLASV2 routine for the SVD of upper triangular 2x2 matrices as well, a highly relatively accurate triangularization in the absence of underflows, and an alternative procedure for computing the SVD of triangular matrices, that employs the correctly rounded hypot function. A heuristic for improving numerical orthogonality of the left singular vectors is also presented and tested on a wide spectrum of random input matrices. On upper triangular matrices under test, the proposed method, unlike xLASV2, finds both singular values with high relative accuracy as long as the input elements are within a safe range that is almost as wide as the entire normal range. On general matrices of order two, the method's safe range for which the smaller singular values remain accurate is of about half the width of the normal range.