Iterative Refinement of Schur decompositions

TL;DR

This paper introduces a Newton-like iterative algorithm for refining Schur decompositions of matrices, achieving high accuracy efficiently, especially in mixed precision environments, significantly reducing computation time.

Contribution

The authors develop a novel quadratic convergence algorithm for improving approximate Schur decompositions using minimal high-precision computations.

Findings

Quadratic convergence proven for matrices with distinct eigenvalues.

Algorithm reduces quadruple precision Schur computation time by up to 20 times.

Fast practical convergence observed in numerical experiments.

Abstract

The Schur decomposition of a square matrix $A$ is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following task: Compute a (more) accurate Schur decomposition of $A$ from a given approximate Schur decomposition. This task arises, for example, in the context of parameter-dependent eigenvalue problems and mixed precision computations. We have developed a Newton-like algorithm that requires the solution of a triangular matrix equation and an approximate orthogonalization step in every iteration. We prove local quadratic convergence for matrices with mutually distinct eigenvalues and observe fast convergence in practice. In a mixed low-high precision environment, our algorithm essentially reduces to only four high-precision matrix-matrix multiplications per iteration. When refining double to quadruple precision, it often needs only 3-4 iterations, which reduces the time of computing a quadruple precision Schur decomposition by up to a factor of 10-20.