Review of Cyclic Reduction for Parallel Solution of Hermitian Positive Definite Block-Tridiagonal Linear Systems
For researchers and practitioners solving large block-tridiagonal systems, this review consolidates known advantages of cyclic reduction, but the contribution is incremental as it does not introduce new algorithms or results.
This paper reviews cyclic reduction for solving Hermitian positive definite block-tridiagonal linear systems, highlighting its numerical stability without pivoting, suitability for parallel computation, and reduced computational cost via symmetry exploitation. The method also provides a robust check for positive definiteness.
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.