Block RLS using row householder reflections

Abstract

Householder reflections applied from the left are generally used to zero a contiguous sequence of entries in a column of a matrix A. Our purpose in this paper is to introduce new row Householder and row hyperbolic Householder reflections which are also applied from the left, but now zero a contiguous sequence of entries in a row of A. We then show how these row Householder reflections can be used to design efficient sliding-data-window recursive least-squares (RLS) covariance algorithms, which are based upon rank- k modifications to the inverse Cholesky factor R -1 of the covariance matrix. The algorithms are rich in matrix-matrix BLAS-3 computations, making them efficient on vector and parallel architectures. Preliminary numerical experiments are reported, comparing these row Householder-based rank- k modification schemes with k applications of the classical updating and downdating covariance schemes which use Givens and hyperbolic rotations.