Review of Cyclic Reduction for Parallel Solution of Hermitian Positive Definite Block-Tridiagonal Linear Systems

TL;DR

This paper reviews a numerically stable, parallel-friendly cyclic reduction method tailored for Hermitian positive definite banded linear systems, highlighting its advantages and robustness in determining positive definiteness.

Contribution

It provides a comprehensive review of a specialized cyclic reduction method optimized for Hermitian positive definite systems, emphasizing its stability and parallel efficiency.

Findings

Method is numerically stable without pivoting

Suitable for parallel computations

Robust in determining positive definiteness

Abstract

Cyclic reduction is a method for the solution of (block-)tridiagonal linear systems. In this note we review the method tailored to hermitian positive definite banded linear systems. The reviewed method has the following advantages: It is numerically stable without pivoting. It is suitable for parallel computations. In the presented form, it uses fewer computations by exploiting symmetry. Like Cholesky, the reviewed method breaks down when the matrix is not positive definite, offering a robust way for determining positive definiteness.