A fast algorithm for LR-2 factorization of Toeplitz matrices

In this paper a new order recursive algorithm for the efficient −1 factorization of Toeplitz matrices is described. The proposed algorithm can be seen as a fast modified Gram-Schmidt method which recursively computes the orthonormal columns i, i = 1,2, …,p, of , as well as the elements of R−1, of a Toeplitz matrix with dimensions L × p. The factor estimation requires 8Lp MADS (multiplications and divisions). Matrix −1 is subsequently estimated using 3p2 MADS. A faster algorithm, based on a mixed and −1 updating scheme, is also derived. It requires 7Lp + 3.5p2 MADS. The algorithm can be efficiently applied to batch least squares FIR filtering and system identification. When determination of the optimal filter is the desired task it can be utilized to compute the least squares filter in an order recursive way. The algorithm operates directly on the experimental data, overcoming the need for covariance estimates. An orthogonalized version of the proposed −1 algorithm is derived. Matlab code implementing the algorithm is also supplied

A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

New Structured Matrix Methods for Real and Complex Polynomial Root-finding

We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular numerical approximation of the real roots of a polynomial. Our analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page

A fast solver for linear systems with displacement structure

We describe a fast solver for linear systems with reconstructable Cauchy-like structure, which requires O(rn^2) floating point operations and O(rn) memory locations, where n is the size of the matrix and r its displacement rank. The solver is based on the application of the generalized Schur algorithm to a suitable augmented matrix, under some assumptions on the knots of the Cauchy-like matrix. It includes various pivoting strategies, already discussed in the literature, and a new algorithm, which only requires reconstructability. We have developed a software package, written in Matlab and C-MEX, which provides a robust implementation of the above method. Our package also includes solvers for Toeplitz(+Hankel)-like and Vandermonde-like linear systems, as these structures can be reduced to Cauchy-like by fast and stable transforms. Numerical experiments demonstrate the effectiveness of the software.Comment: 27 pages, 6 figure

A fast semi-direct least squares algorithm for hierarchically block separable matrices

We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data-sparse and can describe many important operators including those derived from asymptotically smooth radial kernels that are not too oscillatory. The algorithm is based on a recursive skeletonization procedure that exposes this sparsity and solves the dense least squares problem as a larger, equality-constrained, sparse one. It relies on a sparse QR factorization coupled with iterative weighted least squares methods. In essence, our scheme consists of a direct component, comprised of matrix compression and factorization, followed by an iterative component to enforce certain equality constraints. At most two iterations are typically required for problems that are not too ill-conditioned. For an $M \times N$ HBS matrix with $M \geq N$ having bounded off-diagonal block rank, the algorithm has optimal $\mathcal{O} (M + N)$ complexity. If the rank increases with the spatial dimension as is common for operators that are singular at the origin, then this becomes $\mathcal{O} (M + N)$ in 1D, $\mathcal{O} (M + N^{3/2})$ in 2D, and $\mathcal{O} (M + N^{2})$ in 3D. We illustrate the performance of the method on both over- and underdetermined systems in a variety of settings, with an emphasis on radial basis function approximation and efficient updating and downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App

Fast Algorithms for Displacement and Low-Rank Structured Matrices

This tutorial provides an introduction to the development of fast matrix algorithms based on the notions of displacement and various low-rank structures