Numerical Approximation to Integration

Simpson's Rule is meet for sinusoids, but doesn't fit general expectations

Simpson's Rule, the weight 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 ... even numbers of samples require making one end or the other fit by Newton's Rule, the weight vector [3 9 9 3]/8 , while the remainder are fitted by Simpson's Rule. But then, this suggests to us that if we try Newton on each end, and average, we should get a better fit than one-try one-end only. And this averaged weight vector is [17 59 43 49 48 48 ... 59 17]/48 - 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 mathematician expected in the middle: all points equally unit-weighted, with only the end-points weight 'pulled-back' [from the very end] because there is no information from beyond, and therefore less information as the estimate approaches the end-points ... generally of course we might choose a simpler estimate: the plain trapezoidal estimate, weighted [1 2 2 2 ... 2 1]/2 , always takes the inside of the curve, while the tangential fit, [9 28 23 24 24 ...]/24 with parabolic interpolation to the end-points, or [5 13 12 12 ...]/12 with straight interpolation to the end-points, takes the outside of the curve ... and whence we may average this with the trapezoidal fit for [21 52 47 48 48 ...]/48 ~ [.43750 1.08333 .97917 1 1 ...] parabolic to the end-point, or [11 25 24 24 ...]/24 ~ [.45833 1.04167 1 1 ...] straight to the end point.

The parabolic interpolation is Simpson-like; and the tangential fit with parabolic interpolation to the end-points, 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 mid-points (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...

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