Graphics Programs Reference
In-Depth Information
=
e
−
x
,
a
=
=∞
=
As an illustration, let
w
(
x
)
0,
b
and
n
2. The four equations
determining
x
1
,
x
2
,
A
1
and
A
2
are
∞
e
−
x
dx
=
A
1
+
A
2
0
∞
e
−
x
x dx
=
A
1
x
1
+
A
2
x
2
0
∞
e
−
x
x
2
dx
A
1
x
1
A
2
x
2
=
+
0
∞
e
−
x
x
3
dx
A
1
x
1
A
2
x
2
=
+
0
After evaluating the integrals, we get
A
1
+
A
2
=
1
A
1
x
1
+
A
2
x
2
=
1
A
1
x
1
A
2
x
2
+
=
2
A
1
x
1
A
2
x
2
+
=
6
The solutionis
√
2
√
2
+
1
x
1
=
2
−
A
1
=
2
√
2
√
2
+
√
2
−
1
x
2
=
2
A
2
=
2
√
2
so that the quadratureformulabecomes
∞
(
√
2
1)
f
2
−
√
2
1)
f
2
+
√
2
(
√
2
1
2
√
2
e
−
x
f
(
x
)
dx
≈
+
+
−
0
Duetothenonlinearity of the equations, this approach will not work well for
large
n
. Practical methodsoffinding
x
i
and
A
i
requiresomeknowledgeoforthogo-
nal polynomials and their relationship to Gaussian quadrature. There are, however,
several “classical” Gaussian integration formulasforwhich the abscissas and weights
have been computedwith great precision and tabulated. These formulascan used
withoutknowing the theory behind them,since all one needsfor Gaussian integra-
tionare the values of
x
i
and
A
i
. If youdo not intend to ventureoutside the classical
formulas, you can skip the nexttwotopics.
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