Adaptive, Self-Validating Numerical Quadrature
George F. Corliss, Louis B. Rall · SIAM Journal on Scientific and Statistical Computing · 1987
Integrals of a function of a single variable can be expressed as the sum of a numerical quadrature rule and a remainder term. The quadrature rule is a linear combination of function values and weights, or the integral of a Taylor polynomial, while the remainder term depends on some derivative of the integrand evaluated at an unknown point in the interval of integration. Numerical quadrature is made self validating by using interval computations to capture both the roundoff and truncation errors made when using a given formula. Necessary derivatives can be generated automatically by using well-known recurrence relations for Taylor coefficients. In order for quadrature methods of this type to be accurate (in the sense that small intervals are produced) and efficient (to obtain results of given accuracy in a reasonably short time), an accurate scalar product and an adaptive strategy are required. The necessary scalar product and support for interval arithmetic are provided in Pascal-SC (for microcomputers) and ACRITH (for IBM 370 computers). The adaptive strategy chooses the subintervals of integration and the order of the quadrature formula in each subinterval on the basis of guaranteed, rather than estimated, information about the error of numerical integration in each subinterval. The program described here implements standard Newton-Cotes, Gaussian and Taylor series formulas for numerical integration. The handling of singularities is discussed, and comparisons are given with a standard numerical integration method.