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

In this paper, the banded Toeplitz matrices generated by f(theta) = (2(1-cos(theta-(theta) over tilde)))(d) are studied. The function f is a real non-negative function with a zero of order 2d at (theta) over tilde 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.