High-Performance Algorithms for Computing the Sign Function of Triangular Matrices

TL;DR

This paper introduces new recursive and cache-efficient algorithms for computing the sign function of triangular matrices, significantly improving speed and efficiency over previous methods through optimized recursive and block techniques.

Contribution

The paper presents novel recursive and cache-efficient algorithms for the matrix sign function, leveraging Higham's stabilization and optimized computational kernels.

Findings

Recursive algorithms are asymptotically optimal in cache usage.

Blocked and recursive algorithms outperform previous methods.

Performance depends on matrix inertia, aligning with theoretical analysis.

Abstract

Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks in algorithms for computing the sign of arbitrary square real or complex matrices. We present novel recursive and cache efficient algorithms that are based on Higham's stabilized specialization of Parlett's substitution algorithm for computing the sign of a triangular matrix. We show that the new recursive algorithms are asymptotically optimal in terms of the number of cache misses that they generate. One of the novel algorithms that we present performs more arithmetic than the non-recursive version, but this allows it to benefit from calling highly-optimized matrix-multiplication routines; the other performs the same number of operations as the non-recursive version, but it uses custom computational kernels instead. We present implementations of both, as well as a cache-efficient implementation of a block version of Parlett's algorithm. Our experiments show that the blocked and recursive versions are much faster than the previous algorithms, and that the inertia strongly influences their relative performance, as predicted by our analysis.