Non-uniform UE-spline quasi-interpolants and their application to the numerical solution of integral equations

A construction of Marsden’s identity for UE-splines is developed and a complete proof is given. With the help of this identity, a new non-uniform quasi-interpolant that repro-duces the spaces of polynomial, trigonometric and hyperbolic functions are defined. Effi-cient quadrature rules based on integrating these quasi-interpolation schemes are derived and analyzed. Then, a quadrature formula associated with non-uniform quasi-interpolation along with Nyström’s method is used to numericallysolve Hammerstein and Fredholm integral equations. Numerical results that illustrate the effectiveness of these rules are pre-sented.Universidad de Granada / CBU

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