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(n log m+ m log(2) m log log epsilon(-1)) arithmetic operations (ops) are sufficient to approximate the solution of the system up to within the error epsilon; otherwise the cost becomes O(n log m + m log(2) m log n/m) ops. Here m = o(n) and n > log epsilon(-1). Some applications are presented. The methods can be applied to infinite and bi-infinite systems and to block matrices.