Adaptive, Self-Validating Numerical Quadrature.
Technical summary rept.,
WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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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 computation to capture both the roundoff and truncation errors made when using a given rule. 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 and efficient 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 used in each subinterval on the basis of guaranteed, rather than estimated, information about the error of the numerical integration in each subinterval. The program described implements standard Newton-Cotes, Gaussian, and Taylor series methods for numerical intergration. Ways to handle singularities are discussed, and comparisons are given with a standard numerical integration method. Author
- Theoretical Mathematics