Accurate SVDs of polynomial Vandermonde matrices involving orthonormal polynomials

We present a new O(n3) algorithm for computing the SVD of an n × n polynomial Vandermonde matrix VP = [Pi−1(xj)] to high relative accuracy in O(n3) time. The Pi are orthonormal polynomials, deg Pi = i, and xj are complex nodes. The small singular values of VP 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 xj, 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.