Updating and downdating an upper trapezoidal sparse orthogonal factorization†

Abstract

We describe how to update and downdate an upper trapezoidal sparse orthogonal factorization, namely the sparse QR factorization of A T k , where A k is a "tall and thin" full column rank matrix formed with a subset of the columns of a fixed matrix A. In order to do that, we have adapted to rectangular matrices (with fewer columns than rows) Saunders' techniques of early 70s for square matrices, by using the static data structure of George and Heath of early 80s but allowing row downdating on it. An implicitly determined column permutation allow us to dispense with computing a new ordering after each update/downdate; it fits well into the Linpack downdating algorithm and ensures that the updated trapezoidal factor will remain sparse. We give all the necessary formulae even if the orthogonal factor is not available, and we comment on our implementation using the sparse toolbox of Matlab 5. Aims, difficulties and related workIn certain non-simplex active-set methods (also currently known as basisdeficiency-allowing simplex variations) for linear programming we need to solve a sequence of sparse compatible systems of the form:where b and c are iteration-dependent vectors, and x, y, and d are unknown vectors. Furthermore, the matrix A k ∈ R n×m k is a "tall and thin" iterationdependent full column rank matrix with m k ≤ n and rank(A k ) = m k . Such