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How Bad Are Vandermonde Matrices?

The work on the estimation of the condition numbers of Vandermonde matrices, motivated by applications to interpolation and quadrature, can be traced back at least to the 1970s. Empirical study has shown consistently that Vandermonde matrices tend to be badly ill-conditioned, with a narrow class of notable exceptions, such as the matrices of the discrete Fourier transform (hereafter referred to as DFT). So far formal support for this empirical observation, however, has been limited to the matrices defined by the real set of knots. We prove that, more generally, any Vandermonde matrix of a large size is badly ill-conditioned unless its knots are more or less equally spaced on or about the circle $C(0,1)=\{x:\,|x|=1\}$. The matrices of DFT are perfectly conditioned, being defined by a cyclic sequence of knots, equally spaced on that circle, but we prove that even a slight modification of the knots into the so called quasi-cyclic sequence on this circle defines badly ill-conditioned Vandermonde matrices. Likewise we prove that the half-size leading block of a large DFT matrix is badly ill-conditioned. (This result was motivated by an application to pre-conditioning of an input matrix for Gaussian elimination with no pivoting.) Our analysis involves the Ekkart--Young theorem, the Vandermonde-to-Cauchy transformation of matrix structure, our new inversion formula for a Cauchy matrix, and low-rank approximation of its large submatrices.