Interpolation numerical calculus for analytic functions by using algebraic polynomials

Abstract The finite difference method is a powerful method for numerical analysis of equilibrium and steady states of physical phenomena. In the traditional finite difference method, second-order accuracy differences have been used. From an engineering point of view, this approach is often regarded as having sufficient accuracy. However, much research has been done on increasing the accuracy of this numerical analysis. Virtual error zero (VE0)calculations are defined as calculations that can perform numerical calculations with exact solutions for 15 significant digits under double-precision calculations. VE0 calculation is considered to be the ultimate goal of high-accuracy numerical analysis. This paper is written from the viewpoint that such numerical calculations have inherent value. VE0 calculations are always obtained as long as we deal with functions having no singularities in the computational domain for ordinary differential equations as boundary value problems. Furthermore, VE0 calculation is possible even for ordinary differential equations as initial value problems. In both numerical analyses, algebraic polynomials commonly play an important role. This paper comprehensively examines the important role that algebraic polynomials play in increasing the accuracy of numerical calculations in various fields of numerical analysis, such as numerical differentiation, numerical integration, and numerical analysis of integral and integrodifferential equations.