2. (a) Briey describe how orthogonal polynomials can be used to fi nd the nodes of Gaussian quadra-ture rules for a weighted integral
∫
a
b
f
(
x
)
w
(
x
)
d
x
.
\int_{a}^bf(x)w(x)\mathrm{d}x.
∫abf(x)w(x)dx.(b) Using your described approach to find the nodes, and then the method of undetermined coefficients to find the weights, derive the Gauss quadrature rule of the form,
∫
−
1
1
f
(
x
)
(
1
−
x
2
)
1
/
2
d
x
≈
w
0
f
(
x
0
)
+
w
1
f
(
x
1
)
.
\int_{-1}^{1}f(x)(1-x^{2})^{1/2}\,\mathrm{d}x\approx w_{0}f(x_{0})+w_{1}f(x_{1}).
∫−11f(x)(1−x2)1/2dx≈w0f(x0)+w1f(x1).You may use the following facts without proof:
ϕ
0
(
x
)
=
1
,
ϕ
1
(
x
)
=
x
,
ϕ
2
(
x
)
=
x
t
,
ϕ
2
(
x
)
=
x
2
−
1
/
4.
\phi_{0}(x)=1, \phi_{1}(x)=x, \phi_{2}(x)= x t, \phi_{2}(x)=x^{2}-1/4.
ϕ0(x)=1,ϕ1(x)=x,ϕ2(x)=xt,ϕ2(x)=x2−1/4. are orthogonal polynomials w.r.t. the weight function
w
(
x
)
=
(
1
−
x
2
)
1
/
2
w(x)=(1-x^{2})^{1/2}
w(x)=(1−x2)1/2 on
(
−
1
,
1
)
(-1,1)
(−1,1), and
∫
−
1
1
(
1
−
x
2
)
1
/
2
d
x
=
π
/
2.
\int_{-1}^1(1-x^{2})^{1/2}\mathrm{d}x=\pi/2.
∫−11(1−x2)1/2dx=π/2.
Ans:
4. Let
f
∈
[
−
1
,
1
]
f\in [-1,1]
f∈[−1,1] and let
p
n
p_n
pn be the best weighted
L
2
L_2
L2 approximation to
f
f
f from polynomials of degree at most
n
n
n with respect to the weight function
w
(
x
)
=
(
1
−
x
2
)
−
1
/
2
w(x)=(1-x^{2})^{-1/2}
w(x)=(1−x2)−1/2 on
(
−
1
;
1
)
.
(-1; 1).
(−1;1).