Two-point generalized Hermite interpolation: Double-weight function and functional recursion methods for solving nonlinear equations

Based on the two-point Hermite interpolation technique, the paper proposes a two-point generalized Hermite interpolation and its inversion in terms of weight functions. We prove that upon combining fourth-order optimal iterative scheme to the double Newton's method (DNM), we can yield a generalized Hermite interpolation formula to relate the first-order derivatives at two points, and the converse is also true. Resorted on the DNM and the derived formula for the generalized inverse Hermite interpolation, some new third-order iterative schemes of quadrature type are constructed. Then, the fourth-order optimal iterative schemes are devised by using a double-weight function. A functional recursion formula is developed which can generate a sequence of two-point generalized Hermite interpolations for any two given weight functions with certain constraints; hence, a more general class of fourth-order optimal iterative schemes is developed from the functional recursion formula. The constructions of fourth-order optimal iterative schemes by using the techniques of double-weight function and the recursion formula obtained from a single weight function are appeared in the literature at the first time. The novelties involve deriving a two-point generalized Hermite interpolation and its inversion in terms of weight functions subjected to two conditions and through the recursion formula, relating the DNM to the third-order iterative schemes by the inverse Hermite interpolation, formulating a functional recursion formula, deriving a broad class fourth-order optimal iterative schemes through double-weight functions rather than the previous technique with a single-weight function, and finding that the new double-weight function and the newly developed fourth-order optimal iterative schemes are inclusive being convergent faster and competitive to other iterative schemes. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.