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

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.

💡 Research Summary

The paper presents a numerically robust variant of the classic Kogbetliantz algorithm for computing the singular value decomposition (SVD) of real 2 × 2 matrices. The authors identify two main shortcomings of the existing LAPACK routine xLASV2, which is widely used for the SVD of upper‑triangular 2 × 2 blocks: (i) loss of relative accuracy for the smaller singular value when the matrix entries are close to the limits of the floating‑point range, and (ii) insufficient orthogonality of the left singular vectors. To address these issues, the authors propose a three‑phase pipeline that can be applied to any 2 × 2 matrix, whether triangular or full.

Phase 1 – Almost exact prescaling.
The input matrix G is first examined to determine its pattern of zero entries (16 possible patterns) and a scaling exponent s is chosen so that the scaled matrix G′ = 2ˢ G has all non‑zero entries within the normal floating‑point range