By William C. Lyons
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If the function f(x) is approximated by parabolas, Simpson’s rule is obtained, by which (the number of panels n being even) 1 (f 2 a k I = Sn = 1 Dx f0 + 4 3 n−1 n−2 fi + fn + E fi + 2 i=1 i odd i=2 i even where E is the dominant error term involving the fourth derivative of f, so that it is impractical to attempt to provide error correction by approximating this term. Instead, Simpson’s rule with end correction (sixth order rather than fourth order) may be applied where n−2 1 1 fi Dx 14 (f0 + fn ) + I = Sn = 15 2 i=2 i even n−1 +16 fi + Dx[f (a) − f (b)] i=1 i odd The original Simpson’s formula without end correction may be generalized in a similar way as the trapezoidal formula for n = 2k panels, using Dxk = (b − a)/2k and increasing k until sufficient accuracy is achieved, where 1 Sk = Dxk fa +4 3 n−1 n−2 f(a +iDxk )+2 i=1 f(a +iDxk )+fb i=2 i even i odd For the next higher level of integration algorithm, f(x) over segments of [a, b] can be approximated by a cubic, and if this kth order result is Ck , then Cote’s rule can be given as Ck = Sk + (Sk − Sk−1 )/15 and the next higher degree approximation as Dk = Ck + (Ck − Ck−1 )/63 The limit suggested by the sequence Tk → Sk → Ck → Dk → .
I+m−1) −1 starting with Ri = yi and returning an estimate of error, calculated by C and D in a manner analogous with Neville’s algorithm for polynomial approximation. In a high-order polynomial, the highly inflected character of the function can more accurately be reported by the cubic spline function. Given a series of xi (i = 0, 1, . . , n) and corresponding f(xi ), consider that for two arbitrary and adjacent points xi and xi+j , the cubic fitting these points is Fi (x) = a0 + ai x + a2 x2 + a3 x 3 (xi ≤ x ≤ xi+1 ) The approximating cubic spline function g(x) for the region (x0 ≤ x ≤ xn ) is constructed by matching the first and second derivatives (slope and curvature) of Fi (x) to those of Fi−1 (x), with special treatment (outlined below) at the end points, so that g(x) is the set of cubics Fi (x), i = 0, 1, 2, .
7 NUMERICAL METHODS See References 1–14 for additional information. 1 Expansion in Series If the value of a function f(x) can be expressed in the region close to x = a, and if all derivatives of f(x) near a exist and are finite, then by the infinite power series f(x) = f(a) + (x − a)f (a) + + (x − a)2 f (a) + . . 2! (x − a)n n f (a) + . . n! and f(x) is analytic near x = a. The preceding power series is called the Taylor series expansion of f(x) near x = a. If for some value of x as [x − a] is increased, the series is no longer convergent, then that value of x is outside the radius of convergence of the series.