Fast Parallel Algorithms for QR and Triangular Factorization

Abstract

We present a family of new fast algorithms for QR factorization of certain structured matrices, including rectangular Toeplitz matrices and a variety of other Toeplitz-like matrices. It possesses a very regular structure, and appears to be very convenient for parallel implementation. Moreover it is shown that the same architecture can be used for either triangular factorization or QR factorization. Our approach separates the conceptual and implementational aspects of the problem. Our analysis reveals a variety of algorithmic implementations of the basic procedure, all with potentially different numerical properties that need further examination.Our approach is based on the observations that the matrix ${\bf R}$ in ${\bf A} = {\bf {QR}}$ is the Cholesky factor of ${\bf A}^T {\bf A}$ and that fast algorithms for Cholesky factorization of positive definite Toeplitz and certain “close-to-Toeplitz” matrices are readily obtained via a so-called displacement representation of matrices. In this paper we show that if ${\bf A}$ is Toeplitz or is expressed in suitable displacement form then ${\bf A}^T {\bf A}$ also has such a displacement representation, so that ${\bf R}$ can be obtained by a suitable fast Cholesky algorithm. The matrix ${\bf Q}$ is then found by extending a result, also presented in this paper, that ${\bf R}$ and ${\bf R}^{ - 1} $ can be found simultaneously by appropriate circular and hyperbolic rotations applied to a matrix (with Toeplitz blocks) constructed from the elements of ${\bf A}$.