Barycentric Lagrange Interpolation

Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable. It deserves to be known as the standard method of polynomial interpolation.\ud \ud Dedicated to the memory of Peter Henrici (1923-1987

Stability of barycentric interpolation formulas

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or "first barycentric" formula dating to Jacobi in 1825. This difference in stability confirms the theory published by N. J. Higham in 2004 (IMA J. Numer. Anal., v. 24) and has practical consequences for computation with rational functions

High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures

The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.Comment: 30 pages, 10 figures, 1 tabl

On the numerical stability of the second barycentric formula for trigonometric interpolation in shifted equispaced points

We consider the numerical stability of the second barycentric formula for evaluation at points in [0,2?] of trigonometric interpolants in an odd number of equispaced points in that interval. We show that, contrary to the prevailing view, which claims that this formula is always stable, it actually possesses a subtle instability that seems not to have been noticed before. This instability can be corrected by modifying the formula. We establish the forward stability of the resulting algorithm by using techniques that mimic those employed previously by Higham (2004, The numerical stability of barycentric Lagrange interpolation. IMA J. Numer. Anal., 24, 547–556) to analyse the second barycentric formula for polynomial interpolation. We show how these results can be extended to interpolation on other intervals of length-2? in many cases. Finally, we investigate the formula for an even number of points and show that, in addition to the instability that affects the odd-length formula, it possesses another instability that is more difficult to correct