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evaluate.Another advantageofGaussian quadrature is its ability to handle integrable
singularities, enabling ustoevaluate expressionssuch as
1
g
(
x
)
√
1
x
2
dx
provided that
g
(
x
) is awell-behaved function.
−
0
6.2
Newton-Cotes Formulas
f
(
x
)
P
n-
1
(
x
)
h
Figure 6.1.
Polynomial approximation of
f
(
x
).
x
x
1
x
2
x
3
x
4
x
n
-1
x
n
a
b
Consider the definite integral
b
f
(
x
)
dx
(6.1)
a
We divide the rangeofintegration (
a
,
b
) into
n
−
1 equal intervals of length
h
=
−
/
−
(
b
a
)
(
n
1)each, as shown in Fig. 6.1, and denote the abscissas of the resulting
nodes by
x
1
,
x
2
,...,
x
n
. Next we approximate
f
(
x
) byapolynomialofdegree
n
−
1 that
intersects all the nodes. Lagrange'sform of this polynomial, Eq. (3.1a), is
n
P
n
−
1
(
x
)
=
f
(
x
i
)
i
(
x
)
i
=
1
where
i
(
x
) are the cardinalfunctions definedinEq. (3.1b). Therefore, an approxima-
tion to the integral in Eq. (6.1) is
f
(
x
i
)
b
a
i
(
x
)
dx
b
n
n
I
=
P
n
−
1
(
x
)
dx
=
=
A
i
f
(
x
i
)
(6.2a)
a
i
=
1
i
=
1
where
b
a
i
(
x
)
dx
,
A
i
=
i
=
1
,
2
,...,
n
(6.2b)
Equations(6.2) are the
Newton-Cotes formulas
.Classicalexamples of these formu-
las are the
trapezoidal rule
(
n
=
2),
Simpson's rule
(
n
=
3) and
Simpson's 3
/
8 rule
(
n
4). The most important of these is the trapezoidal rule. It can becombinedwith
Richardson extrapolationinto an efficient algorithm known as
Romberg integration
,
which makes the other classical rules somewhat redundant.
=
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