The QS-Householder Sliding Window Bi-SVD Subspace Tracker

Abstract

A fast algorithm for computing the sliding window bi-SVD subspace tracker is introduced. This algorithm produces, in each time step, a dominant rank- <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> SVD subspace approximant of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> times <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> rectangular sliding window data matrix. The method is based on the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">QS</i> (orthonormal-square) decomposition. It uses two row-Householder transformations for updating and one nonorthogonal Householder transformation for downdating in each time step. The resulting algorithm is long-term stable and shows excellent numerical and structural properties, as known from pure Householder-type algorithms. The dominant complexity is <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4Lr</i> +3 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Nr</i> multiplications per time update, which is also the lower bound in dominant complexity for an algorithm of this kind. A completely self-contained algorithm summary is provided and a Fortran subroutine of the algorithm is available for download from http://webuser.hs-furtwangen.de/~strobach/qsh-bisvd.for.