### A Simple "Filon-Trapezoidal" Rule

E. O. Tuck
1967 Mathematics of Computation
Filon's quadrature is a formula for the approximate evaluation of Fourier integrals such as (1) F(o) = f dte^fit), J-T which retains uniform accuracy even when u is so large that many oscillations of the integrand occur within a given element oí of the range of integration. The original Filon formula [1] was derived on the assumption that/(f), rather than the complete integrand, may be approximated stepwise by parabolas, so that it may be called a 'Filon-Simpson' rule. More sophisticated
more » ... ophisticated '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 The simplicity of the weights makes this the most desirable formula to use when the number 2JV 4-1 of given data values f(nôt) is large; for instance the trapezoidal rule is invariably used in power spectral analyses [3] . However, formula (2) cannot be used unless wSt <5C 1, since the whole integrand is supposed to vary linearly over an element St. But it is an exceedingly simple task to derive a modification to (2) by assuming only that/(£) itself varies linearly over the element 5t. The analysis is similar to that used to derive the Filon-Simpson rule, and will not be given here. The resulting integration formula is of the same form as (2), but the weights are now functions of wot, namely 10-N = ( 1 + iuôt -e"" ' ) /to b% , (A) iv" = ((sin|coS¿)/2^02, n ^ ±JV, wN = ( 1 -iu8t -e~lu )/oi 5t. Note that the new weights tend to the trapezoidal-rule values (3) as wSt -> 0. A particular case of interest is when the range of integration 2T is infinite. Suppose we define (5) FTRAF = St E e^'finbt) as the ordinary trapezoidal-rule approximation for this case. Then the Filon-