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.

/pdf/barycentric-lagrange-interpolation-4t9rmsagmj.pdf

Barycentric Lagrange Interpolation

A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.

/pdf/a-matlab-differentiation-matrix-suite-1wlicky0xm.pdf

A MATLAB differentiation matrix suite

We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision).

Spectral Deferred Correction Methods for Ordinary Differential Equations

The Lagrange representation of the interpolating polynomial can be rewritten in two more computationally attractive forms: a modified Lagrange form and a barycentric form. We give an error analysis of the evaluation of the interpolating polynomial using these two forms. The modified Lagrange formula is shown to be backward stable. The barycentric formula has a less favourable error analysis, but is forward stable for any set of interpolating points with a small Lebesgue constant. Therefore the barycentric formula can be significantly less accurate than the modified Lagrange formula only for a poor choice of interpolating points. This analysis provides further weight to the argument of Berrut and Trefethen that barycentric Lagrange interpolation should be the polynomial interpolation method of choice.

/pdf/the-numerical-stability-of-barycentric-lagrange-3jvujnubo6.pdf

The numerical stability of barycentric Lagrange interpolation

The black-box fast multipole method

For functions tabulated at Chebyshev nodes on an interval, spectral interpolation and indefinite integration can be performed stably and efficiently via the fast Fourier transform. For many other sets of nodes (such as the Gaussian nodes on an interval) the classical interpolation and indefinite integration schemes are stable but slow, requiring $O(N^2 )$ arithmetic operations with N the number of nodes in the discretization of the interval. In this paper, a group of algorithms is presented for the efficient evaluation of Lagrange polynomial interpolants at multiple points on the line and for the rapid indefinite integration and differentiation of functions tabulated at nodes other than Chebyshev. The interpolation scheme requires $O(N \cdot \log ({1 / \varepsilon }))$arithmetic operations, and $O(N \cdot \log N + N \cdot \log ({1 / \varepsilon }))$ operations are required for the integration and differentiation schemes, where $\varepsilon $ is the precision of computations and N is the number of nodes (t...

Fast algorithms for polynomial interpolation, integration, and differentiation

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Fast algorithms for polynomial interpolation, integration, and differentiation