Effective Methods for Solving Banded Toeplitz Systems

We propose new algorithms for solving n x n banded Toeplitz systems with bandwidth m. If the function associated with the Toeplitz matrix has no zero in the unit circle, then O(nlog m + mlog ² mloglog ε⁻¹) arithmetic operations (ops) are sufficient to approximate the solution of the system up to within the error ε; otherwise the cost becomes O(nlog m +mlog² mlog n m) ops. Here $m=o(n)$ and n>log ε⁻¹. Some applications are presented. The methods can be applied to infinite and bi-infinite systems and to block matrices.