Fast inversion of Hankel and Toeplitz matrices

Abstract

Many superfast methods that for the inversion of an IZ X n Toeplitz matrix require O(n log*n) arithmetic operations and linear storage have been proposed (see, e.g., [2,6,7,9]). Since Hankel matrices can be obtained from Toeplitz ones by a suitable permutation of rows or columns, the Toeplitz matrix inversion algorithms apply to the Hankel matrices with merely a change of notation. Most of them usually work under the condition of strong nonsingularity of the matrix (i.e., all principal leading minors are nonzero). On the contrary, the algorithm of Brent, Gustavson and Yun [7] works for arbitrary nonsingular Toeplitz matrices and always yields the solution if it exists. It relies on suitable relations between the coefficients of the polynomials generated by the Extended Euclidean algorithm and the entries of the inverse of a Toeplitz matrix. Computations involve PadC approximants and continued fractions. Meanwhile, many other results about the links between Hankel forms and the more interesting aspects of nonlinear numerical analysis have been