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where T is defined in (1.29)and
D
a
C
ih
D
a
C
i
b
a
n
x
i
:
(i) Define
E
n
D
ˇ
ˇ
Z
b
f.x/dx
T
n
;
a
and show that
ˇ
f.x/dx
T.f;x
i 1
;x
i
/
ˇ
Z
x
i
n
X
E
n
:
x
i 1
i D1
(j) Show that
n
X
x
i 1
/
3
:
E
n
M
.x
i
i D1
(k) Use the fact that nh
D
b
a to show that
E
n
M.b
a/h
2
:
(1.32)
(l) In comparison with (1.32), there exists a sharper error estimate of the form:
ˇ
f
00
.x/
ˇ
:
h
2
12
.b
a/ max
E
n
(1.33)
a6x6b
The above error estimate is obtained by using a refined representation of the
error. You will find the argument in the topic of Conte and de Boor [10]. Discuss
the quality of the estimate (1.33) in light of the experiments of Sect.
1.4
.
Useful Results from Calculus
In the project above, you will need some results from calculus.
-
An Integral Inequality.
Let g
D
g.x/ be a bounded function defined on the
interval Œa; b. Then,
ˇ
g.x/dx
ˇ
Z
b
Z
b
a
j
g.x/
j
dx:
(1.34)
a
-
Another Integral Inequality.
Let g
D
g.x/ be a bounded function defined on the
interval Œa; b. Then,
