Fast Qr Factorization of Low-rank Changes of Vandermonde-like Matrices

VANDERMONDE-LIKE MATRICES LUCA GEMIGNANI Abstract. This paper is concerned with the solution of linear systems with coe cient matrices which are Vandermonde-like matrices modi ed by adding low-rank corrections. Hereafter we refer to these matrices as to modi ed Vandermonde-like matrices. The solution of modi ed Vandermondelike linear systems arises in the approximation theory both when we use Remez algorithms for nding minimax approximations and when we consider least squares problems with constraints. Our approach relies on the computation of an inverse QR factorization. More speci cally, we show that some classical orthogonalization schemes for m n, m n, Vandermonde-like matrices can be extended to compute e ciently an inverse QR factorization of modi ed Vandermonde-like matrices. The resulting algorithm has a cost of O(mn) arithmetical operations. Moreover it requires O(m) storage since the matrices Q and R are not to be stored.