Hartley Transform Representations of Symmetric Toeplitz Matrix Inverses with Application to Fast Matrix-Vector Multiplication

Abstract

Representations for inverses of real symmetric Toeplitz matrices involving discrete Hartley transformations are presented which can be used for fast matrix-vector multiplication. In this way, multiplication of a column vector by an inverse real symmetric Toeplitz matrix can be carried out with the help of six Hartley transformations plus two for preprocessing. Besides complexity, stability issues will also be discussed. In particular, it is shown that, for positive definite Toeplitz matrices, the relative error of the computed vector can be estimated by a multiple of the condition number of the matrix.