Computing the complete CS decomposition

TL;DR

This paper introduces a novel algorithm to compute the complete CS decomposition of a partitioned unitary matrix, advancing beyond previous methods that only computed a reduced version, and employs a two-phase process involving bidiagonalization and simultaneous diagonalization.

Contribution

It presents the first fully specified algorithm for the complete 2-by-2 CS decomposition of a unitary matrix, improving upon prior reduced methods.

Findings

The algorithm computes the complete CS decomposition.

It is numerically stable and efficient.

It is the only fully specified algorithm for this purpose.

Abstract

An algorithm is developed to compute the complete CS decomposition (CSD) of a partitioned unitary matrix. Although the existence of the CSD has been recognized since 1977, prior algorithms compute only a reduced version (the 2-by-1 CSD) that is equivalent to two simultaneous singular value decompositions. The algorithm presented here computes the complete 2-by-2 CSD, which requires the simultaneous diagonalization of all four blocks of a unitary matrix partitioned into a 2-by-2 block structure. The algorithm appears to be the only fully specified algorithm available. The computation occurs in two phases. In the first phase, the unitary matrix is reduced to bidiagonal block form, as described by Sutton and Edelman. In the second phase, the blocks are simultaneously diagonalized using techniques from bidiagonal SVD algorithms of Golub, Kahan, and Demmel. The algorithm has a number of desirable numerical features.