Abstract
We present an overview of two approaches to validated, one dimensional indefinite integration. The first approach is to find an inclusion of the integrand, then integrate this inclusion to obtain an inclusion of the indefinite integral. Inclusions for the integrand may be obtained from Taylor polynomials, Tschebyscheff polynomials, or other approximating forms which have a known error term. The second approach finds an inclusion of the indefinite integral directly as a linear combination of function evaluations plus an interval-valued error term. The second approach can be applied to any quadrature formula such as Gaussian or Newton-Cotes quadrature with a known error expression. In either approach, composite formulae improve the accuracy of the inclusion.
The result of the validated indefinite integration is an algorithm which may be represented as a character string, as a subroutine in a high level programming language such a Pascal-SC or Fortran, or as a collection of data. An example is given showing the application of validated indefinite integration to constructing a validated inclusion of the error function, erf (x).
This work was supported in part by IBM Deutschland, GmbH, and in part by the National Science Foundation under Grant No. CCC-8802429. The Government has certain rights to this material.
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Corliss, G.F. (1991). Validated Anti-Derivatives. In: Meyer, K.R., Schmidt, D.S. (eds) Computer Aided Proofs in Analysis. The IMA Volumes in Mathematics and Its Applications, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9092-3_9
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DOI: https://doi.org/10.1007/978-1-4613-9092-3_9
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