Fast Parallel Algorithms for QR and Triangular Factorization

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...