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.
bini's algorithm inverse toeplitz matrix fast fourier transform triangular matrix