On condition numbers in the cyclic reduction processes of a tridiagonal matrix

Abstract

The cyclic reduction method is a direct method for solving tridiagonal linear systems. At the first step of this method, a tridiagonal coefficient matrix is transformed into a pentadiagonal form. In this article, we prove that the condition number for eigenvalues of some classes of coefficient matrices always decreases after the first step of the cyclic reduction method.

Keywords:

• cyclic reduction method
• linear systems
• matrix condition number
• tridiagonal
• positive definite

2000 AMS Subject Classifications:

• 11D04
• 65F05
• 65F15

Acknowledgements

This study is partially supported by Grant for Scientific Research No.19656025 of Japan Society for the Promotion of Science.