Fast Algorithms for Structured Matrices with Arbitrary Rank Profile

Abstract

Triangular factorization, solution to linear equations, inversion, computation of rank profile and inertia (in the Hermitian case) etc. of general $n \times n$ matrices require O($n\sp3$) operations. For certain structured matrices including Toeplitz and Hankel matrices the computational complexity is known to be O($n\sp2$) or better. Fast (i.e. O($n\sp2$)) algorithms for these structured matrices have been actively studied for over twenty five years. However almost all the authors have assumed that the underlying matrices are strongly regular i.e. every principal submatrix is nonsingular. Although some fast algorithms have recently been developed for certain problems involving some of these structured matrices which may have one or more zero minors, several other problems remain unresolved; also a unified approach towards these problems is lacking. In this dissertation, we obtain several new results through a unified approach to the problems mentioned earlier. First we derive fast (i.e. O($n\sp2$)) procedures for computing a triangular factorization which is a LDU factorization where L is lower triangular (resp. U is upper triangular) with unit diagonal entries and D is a block diagonal matrix with possibly varying block sizes. For the matrices we study the diagonal blocks are also structured. A particular application of our result is a fast method of computing rank profile and inertia, leading to alternative proofs of certain results due to Iohvidov (1974) on rank profile and inertia of Hankel and Toeplitz matrices. Next using the results on modified triangular factorization we extend the Schur complement based approach of Chun (1989) for inversion of strongly regular Toeplitz and Hankel matrices to Hermitian Toeplitz, Quasi-Toeplitz, Hankel, and Quasi-Hankel matrices with arbitrary rank profile. We apply our O($n\sp2$) procedures for computing inertia to derive fast procedures for computing root distribution of polynomials. In particular it has been possible to derive the first general recursive procedure for determining the root-distribution of a polynomial with respect to the unit circle; the classical Schur-Cohn test for this problem fails in the presence of certain singularities, which correspond to a Quasi-Toeplitz Bezoutian matrix being non-strongly-regular.