# On choices of formulations of computing the generalized singular value decomposition of a large matrix pair

## TL;DR

This paper compares two formulations for computing the GSVD of large matrices, analyzing their numerical stability and accuracy in finite precision arithmetic to guide better computational choices.

## Contribution

It provides a detailed perturbation analysis of the two formulations and offers criteria for selecting the more accurate approach in finite precision computations.

## Key findings

- One formulation is more numerically stable than the other.
- Perturbation bounds help determine the preferable formulation.
- Numerical experiments confirm the theoretical analysis.

## Abstract

For the computation of the generalized singular value decomposition (GSVD) of a large matrix pair $(A,B)$ of full column rank, the GSVD is commonly formulated as two mathematically equivalent generalized eigenvalue problems, so that a generalized eigensolver can be applied to one of them and the desired GSVD components are then recovered from the computed generalized eigenpairs. Our concern in this paper is, in finite precision arithmetic, which generalized eigenvalue formulation is numerically preferable to compute the desired GSVD components more accurately. We make a detailed perturbation analysis on the two formulations and show how to make a suitable choice between them. Numerical experiments illustrate the results obtained.