Accurate SVDs of polynomial Vandermonde matrices involving orthonormal polynomials

Abstract

We present a new O( n 3) algorithm for computing the SVD of an n × n polynomial Vandermonde matrix V P = [ P i−1 ( x j )] to high relative accuracy in O( n 3) time. The P i are orthonormal polynomials, deg P i = i, and x j are complex nodes. The small singular values of V P can be arbitrarily smaller than the largest ones, so that traditional algorithms typically compute them with no relative accuracy at all. We show that the singular values, even the tiniest ones, are usually well-conditioned functions of the data x j , justifying this computation. We also explain how this theory can be extended to other polynomial Vandermonde matrices, involving polynomials that are not orthonormal or even orthogonal.