
744 Chapter 16 Numerical Methods
In general, the extrapolation table is an n × m lower triangular matrix T = [T
rc
],
where
T
rc
= A
c
h
2
r−1
and
A
c
(h) =
4
c−1
A
c−1
h
2
− A
c−1
(h)
4
c−1
− 1
Trapezoid Rule
An approximation for
1
b
a
f(x)dx can be computed by first approximating f(x)by
the linear function
L(x) =
x −b
a − b
f(a)+
x −a
b − a
f(b)
and using h[f(b)+ f(a)]/2 =
1
b
a
L(x) dx
.
=
1
b
a
f(x)dx. Some calculus shows that
0
b
a
f(x)dx =
f(b)+ f(a)
2
h + O(h
3
)
When f(x)>0, the approximation is the area of a trapezoid with vertices at (a,0),
(a, f(a)), (b,0), and (b, f(b)).
The integration interval [a , b] can be divided into N subinter vals over which the
integration can be composited. Define ...