Mathematics of Computation · 1967 · 48 citations · 1 references
Natural SciencesComputer ScienceMathematicsAlgorithmsComputational MechanicsComputational PhysicsIntegral EquationsNumerical AlgorithmsNumerical AnalysisNumerical IntegrationNumerical Methods For Differential EquationsTime IntegrationAccuracyNumerical Integration FundamentalsNumerical Methods For Integral EquationsOscillatory Integral EvaluationQuadrature RulesQuadrature TheorySimpson's RuleTrapezoidal Approximation
which retains uniform accuracy even when co is so large that many oscillations of the integrand occur within a given element bt of the range of integration. The original Filon formula [1] was derived on the assumption that f(t), rather than the complete integrand, may be approximated stepwise by parabolas, so that it may be called a 'Filon-Simpson' rule. More sophisticated 'Filon' rules have appeared (e.g. [2], and the references quoted in [2]), but in fact with fast computers it is more useful to go in the other direction, towards the least sophisticated integration formula of all. The ordinary trapezoidal rule gives as an approximation
1