Information Technology Reference
In-Depth Information
which is a polynomial of order
n
.
Note that
χ(ξ)
is equal to zero at
ξ
.
Recall that at the
i
integration points,
p
(ξ
)
=
F
(ξ
).
At intermediate points, the difference between
F
(ξ )
and
i
i
p
(ξ )
can be expressed in terms of
χ(ξ).
Let
F
(ξ )
be written as
2
F
(ξ )
=
p
(ξ )
+
χ(ξ )(β
+
β
ξ
+
β
ξ
+···
)
(6.118)
0
1
2
where
β
i
,i
=
1
,
2
,
...
are appropriately chosen constants that can be used to eliminate the
gaps between
F
(ξ )
and
p
(ξ )
at the intermediate points. Integrate
F
(ξ )
to obtain
1
1
j
1
−
∞
j
d
F
(ξ )
d
ξ
=
p
(ξ )
d
ξ
+
0
β
1
χ(ξ)ξ
ξ
−
1
−
1
j
=
Split the final quantity into two parts
j
1
−
j
1
−
j
1
−
∞
n
−
1
∞
j
d
j
d
j
d
0
β
1
χ(ξ)ξ
ξ
=
0
β
1
χ(ξ)ξ
ξ
+
n
β
1
χ(ξ)ξ
ξ
j
=
j
=
j
=
and truncate the last part of the expansion. This gives the quadrature
1
1
j
1
−
n
−
1
j
d
F
(ξ )
d
ξ
≈
p
(ξ )
d
ξ
+
0
β
1
χ(ξ)ξ
ξ
(6.119)
−
1
−
1
j
=
The first integral on the right-hand side involves a polynomial of order
n
−
1, and the
Thus, the integral
1
−
−
.
(ξ )
ξ
second integral a polynomial of order 2
n
1
1
F
d
is approximated
−
.
by integrating a polynomial of order 2
n
To improve the approximation of the integral
on the left-hand side of Eq. (6.119) by the first integral on the right-hand side, the integration
points are selected to make the second integral on the right-hand side of Eq. (6.119) vanish.
Therefore, set
1
1
j
d
1
χ(ξ)ξ
ξ
=
0
j
=
0
,
...
,n
−
1
(6.120)
−
This gives a set of simultaneous nonlinear equations of order
n
for the unknown
ξ
i
,i
=
0
,
...
,n
−
1
.
Return to Eq. (6.119) to obtain
1
1
1
n
n
W
(
n
)
i
F
(ξ )
d
ξ
≈
p
(ξ )
d
ξ
=
F
(ξ
i
)
N
i
(ξ )
d
ξ
=
F
(ξ
i
)
(6.121)
−
1
−
1
−
1
i
=
1
i
=
1
1
−
(ξ )
=
i
=
1
N
i
,W
(
n
)
i
where
p
(ξ )
F
(ξ
)
=
1
N
i
(ξ )
d
ξ
, and
N
i
(ξ )
is given in Eq. (6.64). The
i
error of this quadrature is
1
F
(
2
n
+
1
)
(
r
)
2
n
R
n
=
!
(
x
)
dx
(6.122)
(
2
n
+
1
)
−
1
n
,r,
and
F
(
2
n
+
1
)
having the same meaning as in Eq. (6.115).
The integration points
with
i
and weighting coefficients
W
(
n
)
ξ
are given in Table 6.7 for various
i
n
Note that they are symmetrically distributed. This will be illustrated in the following
example.
The solutions (Gauss integration points) of Eq. (6.120) are equal to the roots of a Legendre
6
polynomial
P
n
.
(ξ )
of order
n
[Davis and Rabinivitz, 1975], where
−
−
2
k
1
k
1
(ξ )
=
(ξ )
=
ξ
(ξ )
=
ξ
(ξ )
−
(ξ )
≤
≤
P
0
1
,
1
,
k
P
k
−
1
P
k
−
2
2
k
n
(6.123)
k
k
6
Adrian Marie Legendre (1752-1833) was a timid Frenchman whose recognition as a mathematician was sup-
pressed by his colleague Laplace. He authored a variety of treatises on geometry, calculus, and the theory of
numbers. The method of least squares appeared in 1806 in his
Nouvelles methodes
. In 1812 Laplace provided a
theoretical basis for the least squares method. Legendre was best known for his work on elliptic integrals.
Search WWH ::
Custom Search