We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has built-in function certify , which proves the correctness of an isolated nonsingular solution to square system polynomial equations. The implementation rests on Krawczyk’s method. demonstrate that it dramatically outperforms earlier approaches certi...

Our concern is with the construction of a frame in L 2 (S) consisting of smooth functions based on kernels of spherical harmonics. The corresponding decomposition and reconstruction algorithms utilize discrete spherical Fourier transforms. Numerical examples connrm the theoretical expectations. x1. Introduction Traditionally, wavelets were tailored to problems on the Euclidean space IR d. Howev...

Recently, there has been an increasing interest in the study of singular and perturbed systems. In this paper we propose a collocation method for solving singularly perturbed Volterra integro-differential and Volterra integral equations. The method is based upon radial basis functions, using zeros of the shifted Legendre polynomial as the collocation points. The results of numerical experiments...

This paper proposes a direct approach for solving optimal control problems. The time domain is divided into multiple subdomains, and a Lagrange interpolating polynomial using the Legendre–Gauss– Lobatto points is used to approximate the states and controls. The state equations are enforced at the Legendre–Gauss–Lobatto nodes in a nonlinear programming implementation by partial Gauss–Lobatto qua...

A binomial identity ((1) below), which relates the famous Apéry numbers and the sums of cubes of binomial coefficients (for which Franel has established a recurrence relation almost 100 years ago), can be seen as a particular instance of a Legendre transform between sequences. A proof of this identity can be based on the more general fact that the Apéry and Franel recurrence relations themselve...

Superconvergence phenomenon of the Legendre spectral collocation method and the p-version finite element method is discussed under the one dimensional setting. For a class of functions that satisfy a regularity condition (M): ‖u‖L∞ ≤ cMk on a bounded domain, it is demonstrated, both theoretically and numerically, that the optimal convergent rate is supergeometric. Furthermore, at proper Gaussia...

The complex method of interpolation, going back to Calderón and Coifman et al., on the one hand, and the Alexander–Wermer–Slodkowski theorem on polynomial hulls with convex fibers, on the other hand, are generalized to a method of interpolation of real (finite-dimensional) Banach spaces and of convex functions. The underlying duality in this method is given by the Legendre transform. Our result...

We discuss the impact of modal filtering in Legendre spectral methods, both on accuracy and stability. For the former, we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity. Computational results confirm that less strict necessary conditions appear to be adequate. We proceed to discuss a instability mechanism in polynomial spectral method...