Graphics Programs Reference
In-Depth Information
Thisequation can besatisfied for arbitrary
Q
n
−
1
onlyif
b
A
i
−
w
(
x
)
i
(
x
)
dx
=
0,
i
=
1
,
2
,...,
n
a
which isequivalenttoEq. (6.24).
ϕ
n
(
x
)
belonging to an orthogonal set by one of the methods discussedinChapter 4. Once
the zeros areknown, the weights
A
i
,
i
It is not difficult to compute the zeros
x
i
,
i
=
1
,
2
,...,
n
of apolynomial
n
couldbefound fromEq. (6.24).
However the following formulas(givenwithout proof ) areeasier to compute
=
1
,
2
,...,
2
Gauss-Legendre
A
i
=
)
p
n
(
x
i
)
2
x
i
(1
−
1
x
i
L
n
(
x
i
)
2
Gauss-Laguerre
A
i
=
(6.25)
2
n
+
1
n
!
√
π
H
n
(
x
i
)
2
Gauss-Hermite
A
i
=
Abscissas and Weights for Gaussian Quadratures
Welist heresomeclassicalGaussian integration formulas. The tables of nodal abscis-
sas and weights, covering
n
2 to 6,have beenrounded off to six decimal places.
These tables shouldbe adequate for hand computation, but in programming you
may needmore precision oralarger number of nodes. In thatcase you should consult
otherreferences,
12
oruse a subroutinetocompute the abscissas and weights within
the integrationprogram.
13
The truncation errorinGaussian quadrature
=
b
n
E
=
w
(
x
)
f
(
x
)
dx
−
A
i
f
(
x
i
)
a
i
=
1
=
K
(
n
)
f
(2
n
)
(
c
), where
a
<
<
has the form
E
b
(the valueof
c
is unknown; onlyits
bounds are given). The expression for
K
(
n
) dependson the particular quadrature
being used. If the derivatives of
f
(
x
)can beevaluated, the error formulas are useful
is estimating the errorbounds.
c
12
Handbook of Mathematical Functions
,M. Abramowitz and I.A. Stegun, Dover Publications(1965);
A.H.Stroud and D. Secrest,
Gaussian Quadrature Formulas
,Prentice-Hall (1966).
13
Severalsuch subroutines arelistedin
Numerical Recipes in Fortran 90
, W. H. Press et al., Cambridge
University Press (1996).
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