Solving Toeplitz Systems

Abstract

An n×n Toeplitz matrix is a square matrix in which element a, _ij = a _{i- j} . An n×n circulant matrix is a square matrix in which element a, _ij = a _{((i- j) )}. A circulant matrix is a Toeplitz matrix. A Toeplitz system of equations is given by the matrix-vector equation Af = g. The computational task of solving the Toeplitz system of equations is the task of computing the vector f when given the vector g and the elements of the Toeplitz matrix on the left. One way to solve for f is to compute the matrix inverse. But to compute the matrix inverse for very large n may be impractical, both because of the amount of computation and because of problems of precision.