| Simpson's Rule is meet for sinusoids, but doesn't fit general expectations |
Simpson's Rule, the weighting vector [1 4 2 4 2 ... 4 2 4 1]/3 , applies to equally spaced samples of fairly smooth functions, to estimate its total area across the included domain between the first and last points... But Simpson's Rule only applies to odd, numbers of points, but more than one, whereas even numbers of samples, but more than two, require making one end or the other fit by Newton's Rule, the weighting vector [3 9 9 3]/8 , while the remainder are fitted by Simpson's Rule... But then, this suggests to us that if we use Newton on each, end, both ways, and average, we should get a better fit than by one-try, one-end-only, and this averaged weighting vector is [17 59 43 49 48 48 48 ... 17]/48 ~ [0.354 1.229 0.896 1.021 1 1 1 ... 0.354]—which is marvelously not-the-ordinary textbook conclusion looking more like a sinusoidal fit because that is better for sinusoids as Simpson did—and is just what the statistical-data-mathematician expected in the middle: all points equally unit-weighted, with only the endpoint weights 'pulled-back' [from the very ends] because there is no information from beyond, and, therefore, less information as the estimate approaches, the endpoints... generally of course we might choose a simpler estimate: the plain trapezoidal estimate, weighted [1 2 2 2 ... 1]/2 , always takes the inside-chords of a curve, while the tangential fit, [9 28 23 24 24 24 ... 9]/24 mildly simplified of the averaged result found already, with parabolic interpolation to the endpoints, or [5 13 12 12 12 ... 5]/12 with straight interpolation to the endpoints, takes the outside of a curve ... and whence we may average this with the trapezoidal fit for [21 52 47 48 48 48 ... 21]/48 ~ [0.438 1.083 0.979 1 1 1 ... 0.438] parabolic to the endpoint, or [11 25 24 24 24 ... 11]/24 ~ [0.458 1.042 1 1 1 ... 0.458] straight to the end point.
(Note the weighting does not work for one, odd, point, [1]/1, because it spans zero-width; and-yet for two points, is linearly exact [1 1]/2...)
The parabolic interpolation is Simpson-like; and the tangential fit with parabolic interpolation to the endpoints, is itself close to our prior suggested average - we might arbitrarily pick a simple weight of [18 57 45 48 48 ...]/48 = [3/8 19/16 15/16 1 1 ...] = [0.375 1.1875 .9375 1.0 1.0 ...].
The other proposed fits of higher order might be similarly improved by transposing and sliding them left and right, and finding their average: expectably [... 1 1 1 ...] for the midpoints (inner-points). Only the endpoints contain any special information, and most of that is not usable - a paradox of maybe have maybe not - but knowledge of the type of curve is useful: and Simpson's may help with sino-elliptic curve fitting.
OTHER RULES AND METHODS:
[2014/11/5] Another numerical integration method, for tracing paths through vector fields aka ordinary differential equations, the "Runge-Kutta" approximation, in its case, tends one-sided rather than bracketing the result, and should be instead a balance between its dy/dx version and its multiplicative/derivative inverse dx/dy, (a hint of the "Newton-Raphson" method for numerically estimating square-roots where the verse √n and inverse n/√n, bracket the successive-approximation thereto)... The "Runge-Kutta" method should probably also step units along the hypotenuse d͢s = d͢x + d͢y especially where the dy/dx or dx/dy slope gets steep, or in applications with slope-reversal instabilities, but that is already categorically "modified Runge-Kutta" whereas the importance of dx/dy is contemporarily less known...