MATCOM Special Issue MAMERN VI-2015: 6th International Conference on Approximation Methods and Numerical Modeling in Environment and Natural Resources

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A recursive construction of Hermite spline interpolants and applications

AbstractLet fk be the Hermite spline interpolant of class Ck and degree 2k+1 to a real function f which is defined by its values and derivatives up to order k at some knots of an interval [a,b]. We present a quite simple recursive method for the construction of fk. We show that if at the step k, the values of the kth derivative of f are known, then fk can be obtained as a sum of fk-1 and of a particular spline gk-1 of class Ck-1 and degree 2k+1. Beyond the simplicity of the evaluation of gk-1, we prove that it has other interesting properties. We also give some applications of this method in numerical approximation

On two families of near-best spline quasi-interpolants on non-uniform partitions of the real line

The univariate spline quasi-interpolants (abbr. QIs) studied in this paper are approximation operators using B-spline expansions with coefficients which are linear combinations of discrete or weighted mean values of the function to be approximated. When working with nonuniform partitions, the main challenge is to find QIs which have both good approximation orders and uniform norms which are bounded independently of the given partition. Near-best QIs are obtained by minimizing an upper bound of the infinity norm of QIs depending on a certain number of free parameters, thus reducing this norm. This paper is devoted to the study of two families of near-best QIs of approximation order 3

Product integration methods based on discrete spline quasi-interpolants and application to weakly singular integral equations

AbstractQuadrature formulae are established for product integration rules based on discrete spline quasi-interpolants on a bounded interval. The integrand considered may have algebraic or logarithmic singularities. These formulae are then applied to the numerical solution of integral equations with weakly singular kernels

A fourth order method for finding a simple root of univariate function

In this paper, we describe an iterative method for approximating a simple zero $z$ of a real defined function. This method is a essentially based on the idea to extend Newton's method to be the inverse quadratic interpolation. We prove that for a sufficiently smooth function $f$ in a neighborhood of $z$ the order of the convergence is quartic. Using Mathematica with its high precision compatibility, we present some numerical examples to confirm the theoretical results and to compare our method with the others given in the literature