Theoretical analysis of the extended cyclic reduction algorithm

TL;DR

This paper provides a theoretical analysis of the extended cyclic reduction algorithm, focusing on the zeros of matrix polynomials and the algorithm's error behavior when solving block-tridiagonal systems.

Contribution

It establishes the relationship between zeros of matrix polynomials and eigenvalues of submatrices, and analyzes the forward error of the algorithm.

Findings

Zeros of matrix polynomial are eigenvalues of principal submatrices.

Theoretical results on the zeros of $B_{i}^{(r)}$ are derived.

Forward error analysis of the algorithm is conducted.

Abstract

The extended cyclic reduction algorithm developed by Swarztrauber in 1974 was used to solve the block-tridiagonal linear system. The paper fills in the gap of theoretical results concerning the zeros of matrix polynomial $B_{i}^{(r)}$ with respect to a tridiagonal matrix which are computed by Newton's method in the extended cyclic reduction algorithm. Meanwhile, the forward error analysis of the extended cyclic reduction algorithm for solving the block-tridiagonal system is studied. To achieve the two aims, the critical point is to find out that the zeros of matrix polynomial $B_{i}^{(r)}$ are eigenvalues of a principal submatrix of the coefficient matrix.