Graphics Programs Reference
In-Depth Information
Gauss-Chebyshev quadrature
1
1
x
2
−
1
/
2
n
≈
n
−
f
(
x
)
dx
f
(
x
i
)
(6.31)
−
1
i
=
1
Note that all the weights areequal:
A
i
=
π/
n
. The abscissasofthenodes, which are
symmetric about
x
=
0, are givenby
−
π
(2
i
1)
x
i
=
cos
(6.32)
2
n
The truncation erroris
π
2
2
n
(2
n
)!
2
f
(2
n
)
(
c
),
E
=
−
1
<
c
<
1
(6.33)
Gauss-Laguerre quadrature
∞
n
e
−
x
f
(
x
)
dx
A
i
f
(
x
i
)
(6.34)
≈
0
i
=
1
x
i
A
i
x
i
A
i
n
=
2
n
=
5
0
.
585 786
0
.
853 554
0
.
263 560
0
.
521 756
3
.
414 214
0
.
146 447
1
.
413 403
0
.
398 667
n
=
3
3
.
596 426
(
−
1)0
.
759 424
0
.
415 775
0
.
711 093
7
.
085 810
(
−
2)0
.
361 175
2
.
294 280
0
.
278 517
12
.
640 801
(
−
4)0
.
233 670
6
.
289 945
(
−
1)0
.
103 892
n
=
6
n
=
4
0
.
222 847
0
.
458 964
.
.
.
.
0
322 548
0
603154
1
188 932
0
417 000
1
.
745 761
0
.
357 418
2
.
992 736
0
.
113 373
4
.
536 620
(
−
1)0
.
388 791
5
.
775 144
(
−
1)0
.
103 992
9
.
395 071
(
−
3)0
.
539 295
9
.
837 467
(
−
3)0
.
261 017
15
.
982 874
(
−
6)0
.
898 548
Table 6.4.
Multiply numbers by 10
k
, where
k
is given in parentheses
(
n
!)
2
(2
n
)!
f
(2
n
)
(
c
),
E
=
0
<
c
<
∞
(6.35)
Gauss-Hermite quadrature:
∞
n
e
−
x
2
f
(
x
)
dx
A
i
f
(
x
i
)
(6.36)
≈
−∞
i
=
1
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