Numerical Integration

Abstract

Physical simulations often involve the calculation of definite integrals over complicated functions, for instance, the Coulombic interaction between two electrons. Integration is also the elementary step in solving equations of motion. In general a definite integral can be approximated numerically as the weighted average over a finite number of function values, a so-called integral rule. Newton–Cotes rules use equidistant sample points and can be combined with the Romberg extrapolation method to provide higher order approximations. Optimization of the sample point positions increases the accuracy. The Gaussian method is discussed in detail. A computer experiment investigates the efficiency of the Romberg integration method.