Optimization of the richardson integration over fluctuations of its step sizes

Abstract In this paper, we examine the optimization of Richardson numerical integration of an arbitrary real valued function in the space of step sizes. Namely, as one of the most efficient numerical integrations of an integrable function, the Richardson method is optimized under the variations of its step sizes. Subsequently, we classify the stability domains of the Richardson integration of real valued functions. We discuss stability criteria of the Richardson integration via the sign of the fluctuation discriminant as a quintic or lower degree polynomials as a function of the step size parameter. As special cases, our proposal optimizes the trapezoidal, Romberg and other numerical integrations. Hereby, we consider the optimization of the Richardson schemes as a weighted estimation in the light of extrapolation techniques. Finally, optimal Richardson integrations are discussed towards prospective theoretical and experimental applications and their industrial counterparts.