Hybrid Triple Quadrature Rule Blending Some Gauss-Type Rules with the classical or the Derivative-Based Newton-Cotes-Type Rules.

Abstract

Hybrid numerical quadrature rules are widespread techniques for approximate computations of definite integrals. Such hybrid rules combine as many quadrature rules as long as they possess the same degree of precision. The revenue is a new mixed rule with a higher degree of precision than its constituted rules at least by two. Moreover, such mixed rules are quite simple and handy, because they do not involve any extra evaluations of the integrand. That is by relying on the same number of quadrature points of the constituted rules, the acquired hybrid rule performs more efficiently than its ingredients rules. In this paper; a triple hybrid quadrature rule has been constructed for the numerical integration of real definite integrals that do not possess a closed-form anti-derivative.  At First, a dual hybrid rule was produced by blending Milne’s rule of Newton-Cotes type with the anti-Gaussian quadrature rule to prevail a dual rule of a degree of precision equal to five. Then the acquired dual rule is recombined with the composite derivative-based and mid-Point Newton–Cotes formula producing a hybrid triple rule of degree of precision equal to seven. The accomplished approach is satisfactory and efficient in the approximate evaluation of definite real integrals as confirmed analytically by the error analysis and numerically by some verification examples. To promote the degree of precision of the proposed triple approach, the numerical computations have been implemented in an adaptive environment.