University of Guilan Journal of Mathematical Modeling 2345-394X 10 4 2022 12 01 A direct solver for solving systems of linear equations with banded ill-conditioned Toeplitz matrices 453 461 5729 10.22124/jmm.2022.22278.1965 EN Nasser Akhoundi School of Mathematics and Computer Science, Damghan University, Damghan, Iran Journal Article 2022 05 11 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. https://jmm.guilan.ac.ir/article_5729_9e8c907b3f56d3cb7dc728d9fd6fe89c.pdf