Scaling Bini's Algorithm for Fast Inversion of Triangular Toeplitz Matrices

In this paper, motivated by Lin, Ching and Ng [Theoretical Computer Science, 315:511523 (2004)], a scaling version of Bini's algorithm [SIAM J. Comput., 13:268-276 (1984)] for an approximate inversion of a triangular Toeplitz matrix is proposed. The scaling algorithm. introduces a new scale parameter and is mathematically equivalent to the original Bini's. Its computational cost is about two fast Fourier transforms of n-vectors (FFTs(n)), equal to that of Bini's. We also improve the accuracy of the proposed approach by embedding the n-by-n triangular Toeplitz matrix into an (n + n(0))-by-(n + n(0)) triangular (banded) Toeplitz matrix, where n(0) is a positive integer. The complexity of the resulting revised scaling Bini's algorithm is about two FFTs(2n). Several numerical examples are given to illustrate the effectiveness and stability of the proposed methods.