Graphics Programs Reference
In-Depth Information
Gauss-Legendre quadrature
1
n
f
(
ξ
)
d
ξ
≈
A
i
f
(
ξ
i
)
(6.26)
−
1
i
=
1
±
ξ
i
A
i
±
ξ
i
A
i
=
=
n
2
n
5
0
.
577 350
1
.
000 000
0
.
000 000
0
.
568 889
n
=
3
0
.
538 469
0
.
478 629
0
.
000 000
0
.
888 889
0
.
906180
0
.
236 927
0
.
774 597
0
.
555 556
n
=
6
n
=
4
0
.
238 619
0
.
467 914
0
.
339 981
0
.
652 145
0
.
661 209
0
.
360 762
0
.
861 136
0
.
347 855
0
.
932 470
0
.
171 324
Table 6.3
This is the most oftenused Gaussian integration formula. The nodes are arranged
symmetricallyabout
0, and the weights associatedwith a symmetric pair of nodes
areequal.For example, for
n
ξ
=
=
2wehave
ξ
1
=−
ξ
2
and
A
1
=
A
2
. The truncation error
in Eq. (6.26) is
2
2
n
+
1
(
n
!)
4
(2
n
f
(
2
n
)
(
c
),
E
=
−
1
<
c
<
1
(6.27)
1) [(2
n
)!]
3
+
To apply Gauss-Legendrequadrature to the integral
a
f
(
x
)
dx
, we must first map
1).Wecan accomplish this
the integration range(
a
,
b
) into the “standard” range(
−
1
,
by the transformation
b
+
a
b
−
a
x
=
+
ξ
(6.28)
2
2
Now
dx
=
d
ξ
(
b
−
a
)
/
2, and the quadrature becomes
b
n
b
−
a
f
(
x
)
dx
A
i
f
(
x
i
)
(6.29)
≈
2
a
i
=
1
where the abscissas
x
i
must becomputed fromEq. (6.28). The truncation error
here is
a
)
2
n
+
1
(
n
!)
4
(
b
−
f
(2
n
)
(
c
)
E
=
,
a
<
c
<
b
(6.30)
1)
[
(2
n
)!
]
3
(2
n
+
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