This paper is concerned with the computation of solutions to the problem of best near-interpolation by parametric curves to constraints of the form f(ti) ∈ Ki for general sets Ki, including derivative constraints when ti has multiplicity greater than one. The minimizers are spline curves that we represent as smoothing splines with weights that are determined from the Lagrange multipliers corres...

A five-point one-step block method based on the Newton backward divided difference formulae for the solution of first-order delay differential equations is derived. The proposed block method will approximate the solutions of initial value problems at five points simultaneously using variable step size. The approximation of the delay term is calculated using Neville's interpolation. The block me...

For the numerical solution of time-dependent partial differential equations, a class of meshfree exponential integrators is proposed. These methods are of particular interest in situations where the solution of the differential equation concentrates on a small part of the computational domain which may vary in time. For the space discretization, radial basis functions with compact support are s...

We consider the extension of the method of Gauss-Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the coefficient matrix is a series with matrix coefficients, and provide a characterization for when the matrix series is regular based on the algebraic variety of an ...

In the Laurent expansion ζ(s, a) = 1 s− 1 + ∞ ∑ k=0 (−1)γk(a) k! (s− 1) , 0 < a ≤ 1, of the Riemann-Hurwitz zeta function, the coefficients γk(a) are known as Stieltjes, or generalized Euler, constants. [When a = 1, ζ(s, 1) = ζ(s) (the Riemann zeta function), and γk(1) = γk.] We present a new approach to high-precision approximation of γk(a). Plots of our results reveal much structure in the gr...

In this paper, we consider a fully discrete interpolated coefficient mixed finite element method for semilinear time fractional reaction–diffusion equations. The classic L1 scheme based on graded meshes and new triangulation is used the temporal spatial discretization, respectively. interpolation technique to deal with term, nonlinear system solved by Newton-like iterative method. Stability con...

Use of a new notion of multivariate divided difference leads to a short proof of a formula by Sauer and Xu for the error in interpolation, by polynomials of total degree ≤ n in d variables, at a ‘correct’ point set. It is the purpose of this note to give a short proof of a remarkable formula for the error in polynomial interpolation given in [3]. In [3], Sauer and Xu consider interpolation from...

We describe new fast algorithms for evaluation and interpolation on the “novel” polynomial basis over finite fields of characteristic two introduced by Lin et al. (2014). Fast are also described converting between their monomial basis, as well to from Newton associated with points algorithms. Combining yields a truncated additive Fourier transform (FFT) inverse FFT which improve upon some previ...