We began, 25 years ago, to take up [the conditioning of] the class of Vandermonde matrices. The original motivation came from unpleasant experiences with the computation of Gauss type quadrature rules from the moments of the underlying weight function.

- WALTER GAUTSCHI, How (Un)stable Are Vandermonde Systems? (1990)

Extreme ill-conditioning of the [Vandermonde] linear systems will eventually manifest itself as n increases by yielding an error curve which is not sufficiently levelled on the current reference ... or more seriously fails to have the correct number of sign changes.

- M. ALMACANY, C. B. DUNHAM, and J. WILLIAMS, Discrete Chebyshev Approximation by Interpolating Rationals (1984)

Overview

A Vandermonde matrix is defined in terms of scalars by

Vandermonde matrices play an important role in various problems, such as in polynomial interpolation. Suppose we wish to find the polynomial p _n( x) = a _nx ⁿ + a _{n ?1} x ^{n ?1} + ? + a ₀ that interpolates to the data ( ? _i, f _i) ⁿ _i=0, for distinct points ? _i, that is, p _n( ? _i) = f _i, i = 0: n. Then the desired coefficient vector a = [ a ₀, a ₁,..., a _n] ^T is the solution of the dual Vandermonde system

The primal system

represents a moment problem, which arises, for example, when determining the weights for a...