Generalized Quadrature Formula for Convex Functions

A generalized quadrature formula was introduced to estimate the numerical solutions of the integral of convex functions. The formula was composed of the difference between two parts of the areas of the trapezoid and the weighted convex patch. The range of weight coefficient of the convex patch was from 0 to 1 evaluated by the Hadamard´s inequality. Tuning the weighted value of the convex patch could be considered as using different order curve to approximate the integrand. The classical trapezoid and Simpson quadrature formulas could also be reformulated and composed of the difference between the areas of the trapezoid and the patch weighted by 0 and 2/3, respectively. Thus those classical quadrature formulas were generalized. The numerical experiments were performed to compare the computing performance of our proposed equation using different weight values. In addition, the generalized formula looks much simple and understandable compared with the classical.