# A direct solver for solving systems of linear equations with banded ill-conditioned Toeplitz matrices

Document Type : Research Article

Author

School of Mathematics and Computer Science, Damghan University, Damghan, Iran

Abstract

In this paper, the banded Toeplitz matrices generated by $f(\theta)=(2(1-\cos(\theta-\tilde{\theta})))^d$ are studied. The function $f$ is a real non-negative function with a zero of order $2d$ at $\tilde{\theta}$ and the generated matrices are ill-conditioned Hermitian positive definite. We show that these banded Toeplitz matrices are similar to the banded real symmetric positive definite Toeplitz matrices that are generated by $f(\theta)=(2(1-\cos(\theta)))^d$.  A fast direct solver is proposed to compute the inverse of these real matrices. Numerical experiments show that our proposed method is faster and more stable than the stable Levinson algorithm.

Keywords