Automatic quadrature of functions of the form \(g(| f(x)|)\)
This paper provides a description for an interesting and useful modification to adaptive integration methods when applied to integrands of the form \(g(|f(x)|)\), for smooth functions \(f(x)\) and \(g(x)\). The key idea is to use signs of the \(f(x)\) values that are computed during the adaptivity and integral estimation process, for standard adaptive integration algorithms, to detect the presence of zeros within the integration interval. When a sign change is found, a little extra work is used to accurately locate a zero of the integrand, and these zeros are used for interval subdivision points as the algorithm continues. Test results from a battery of examples are used by the authors to illustrate how software using this approach can be more efficient than standard software.
scientific article; zbMATH DE number 885560