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

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