Graphics Programs Reference
In-Depth Information
The classicalorthogonal polynomials are also obtainable from the formulas
dx
n
1
x
2
n
(
−
1)
n
2
n
n
!
d
n
p
n
(
x
)
=
−
cos(
n
cos
−
1
x
),
n
=
>
T
n
(
x
)
0
dx
n
x
n
e
−
x
e
x
n
!
d
n
L
n
(
x
)
=
(6.20)
1)
n
e
x
2
d
n
dx
n
(
e
−
x
2
)
and their derivatives can becalculated from
H
n
(
x
)
=
(
−
x
2
)
p
n
(
x
)
(1
−
=
n
[
−
xp
n
(
x
)
+
p
n
−
1
(
x
)
]
x
2
)
T
n
(
x
)
(1
−
=
n
[
−
xT
n
(
x
)
+
nT
n
−
1
(
x
)]
xL
n
(
x
)
=
n
[
L
n
(
x
)
−
L
n
−
1
(
x
)
]
(6.21)
H
n
(
x
)
=
2
nH
n
−
1
(
x
)
Otherproperties of orthogonal polynomials thathave relevance to Gaussian in-
tegrationare:
ϕ
n
(
x
)has
n
real, distinct zeroes in the interval(
a
,
b
).
ϕ
n
(
x
)lie between the zeroes of
ϕ
n
+
1
(
x
).
The zeroes of
Any polynomial
P
n
(
x
)ofdegree
n
can beexpressedinthe form
n
P
n
(
x
)
=
c
i
ϕ
i
(
x
)
(6.22)
=
i
0
It followsfromEq. (6.22) and the orthogonalitypropertyinEq. (6.18)that
b
w
(
x
)
P
n
(
x
)
ϕ
n
+
m
(
x
)
dx
=
0,
m
≥
0
(6.23)
a
∗
Determination of Nodal Abscissas and Weights
Theorem
The nodal abscissas
x
1
,
x
2
,...,
x
n
are the zeros of the polynomial
ϕ
n
(
x
)that
belongs to the orthogonal set definedinEq. (6.18).
Proof
Westart the proof by letting
f
(
x
)
1.
Since the Gaussian integrationwith
n
nodes isexact for this polynomial, wehave
b
=
P
2
n
−
1
(
x
) be apolynomialofdegree 2
n
−
n
w
(
x
)
P
2
n
−
1
(
x
)
dx
=
A
i
P
2
n
−
1
(
x
i
)
(a)
a
i
=
1
Apolynomialofdegree 2
n
−
1 can always writteninthe form
P
2
n
−
1
(
x
)
=
Q
n
−
1
(
x
)
+
R
n
−
1
(
x
)
ϕ
n
(
x
)
(b)
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