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.