Iterations for the Unitary Sign Decomposition and the Unitary Eigendecomposition

TL;DR

This paper introduces fast, structure-preserving iterative methods for computing the sign decomposition of unitary matrices, offering improved convergence and stability over existing methods, and applies these to develop a stable spectral divide-and-conquer eigendecomposition algorithm.

Contribution

The paper develops new iterative algorithms for the sign decomposition of unitary matrices that are faster and more stable than previous methods, especially near eigenvalues at ±i.

Findings

Iterations converge faster than Padé when eigenvalues are near ±i.

Numerical evidence shows backward stability of the proposed iterations.

The method enables a stable spectral divide-and-conquer eigendecomposition for unitary matrices.

Abstract

We construct fast, structure-preserving iterations for computing the sign decomposition of a unitary matrix $A$ with no eigenvalues equal to $\pm i$. This decomposition factorizes $A$ as the product of an involutory matrix $S = \operatorname{sign}(A) = A(A^2)^{-1/2}$ times a matrix $N = (A^2)^{1/2}$ with spectrum contained in the open right half of the complex plane. Our iterations rely on a recently discovered formula for the best (in the minimax sense) unimodular rational approximant of the scalar function $\operatorname{sign}(z) = z/\sqrt{z^2}$ on subsets of the unit circle. When $A$ has eigenvalues near $\pm i$, the iterations converge significantly faster than Pad\'e iterations. Numerical evidence indicates that the iterations are backward stable, with backward errors often smaller than those obtained with direct methods. This contrasts with other iterations like the scaled Newton iteration, which suffers from numerical instabilities if $A$ has eigenvalues near $\pm i$. As an application, we use our iterations to construct a stable spectral divide-and-conquer algorithm for the unitary eigendecomposition.