On polynomial interpolation in the monomial basis

TL;DR

The paper analyzes polynomial interpolation in the monomial basis over simply connected compact domains in the complex plane and shows that, when the Vandermonde condition number satisfies ${\kappa}(V^{(N)}) \lesssim \frac{1}{u}$, the monomial basis achieves backward-error-controlled accuracy comparable to well-conditioned bases. It derives a practical framework and a priori error bounds, notably ${\lVert F-\widehat{P}_N \rVert}_{\infty} \lesssim {\lVert F-P_N \rVert}_{\infty} + u{\lVert a^{(N)} \rVert}_2$, and establishes tight bounds on ${\lVert (V^{(N)})^{-1} \rVert}_2$ and ${\lVert a^{(N)} \rVert}_2$ via Lebesgue constants and domain geometry. The authors propose a robust, piecewise monomial interpolation strategy that performs well in practice, supported by extensive numerical experiments on intervals and general regions, and demonstrate useful applications to oscillatory/singular integrals and root finding. The work also provides a new general bound on Vandermonde conditioning, unifying prior results and guiding when the monomial basis is advantageous for interpolation and related computations.

Abstract

In this paper, we show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. This leads to a practical algorithm for piecewise polynomial interpolation over general regions in the complex plane using the monomial basis. Our analysis also yields a new upper bound for the condition number of an arbitrary Vandermonde matrix, which generalizes several previous results.

Figures (17)

• Figure 1: Polynomial interpolation of $\cos(2x+1)$ in the monomial basis over $[-1,1]$. The $x$-axis label $N$ denotes the order of approximation. The polynomial $P_N$ denotes the interpolating polynomial approximated using the Barycentric interpolation formula. The polynomial $\widehat{P}_N$ denotes the computed monomial expansion. The $L^\infty$ error is estimated by comparing the approximated function values at $10000$ equidistant points over $[-1,1]$ with the true function values.
• Figure 2: Polynomial interpolation of more complicated functions in the monomial basis, over $[-1,1]$.
• Figure 3: Polynomial interpolation of functions with a singularity near the approximation domain, in the monomial basis, over $[-1,1]$.
• Figure 4: Polynomial interpolation of non-smooth functions in the monomial basis, over $[-1,1]$.
• Figure 5: Polynomial interpolation error, monomial approximation error and $u\cdot {\lVert a^{(N)} \rVert} _2$, for $\Omega=[-1,1]$. These functions are the ones that appear in Section \ref{['sec:intro']}.

Theorems & Definitions (15)

• Lemma 2.1
• Theorem 2.2
• Remark 2.1
• Remark 2.2
• Definition 2.1
• Remark 2.3
• Definition 2.2
• Lemma 2.3
• Theorem 2.4
• Remark 2.4