80 4. numerical integration and differentiation 3 ...facstaff.cbu.edu/wschrein/media/m329...
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80 4. NUMERICAL INTEGRATION AND DIFFERENTIATION
3. Composite Quadrature Rules
There are two primary problems with Newton-Coates methods. The first is thatthey are unsuitable for large intervals since high degree formulas are requiredand the coe�cients of the formulas are hard to find. The second problem isthat they are based on interpolating polynomials and high degree polynomialsoscillate over large intervals.
A solution to these problems is to use a piecewise approach with low-orderNewton-Coates formulas.
Composite Simpson’s Rule
If f 2 C4[a, b], then a number ⇠ in (x0, x2) exists withZ x2
x0
f(x) dx =h
3
f(x0) + 4f(x1) + f(x2)
�� h5
90f (4)(⇠)
where h =x2 � x0
2.
Divide [a, b] into n intervals, n even, h =b� a
n, xj = a + jh. Then apply
Simpson’s Rule to successive pairs of intervals.
1 4 11 4 1
1 4 11 4 1
Thus the composite weight pattern is
1� 4� 2� 4� · · ·� 2� 4� 1.