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the particular example we considered above, we found that c could be chosen to be
0:5129,cf.(1.20). The aim of this project is to show that this result is fairly general
and holds for all functions f having a bounded second-order derivative. In this
project, we will use some results from Calculus. They are listed below, see page 20.
We start by recalling that the problem at hand is to compute an approximation to
the definite integral
Z
b
f.x/dx;
(1.27)
a
and we first consider the trapezoidal method using one single interval. We recall that
this method was derived using a linear function given by
y.x/
D
f.a/
C
f.b/
f.a/
b
a
.x
a/;
(1.28)
which approximates the function f ,cf.Fig.
1.2
. Note, in particular, that y.a/
D
f.a/ and y.b/
D
f.b/. By integrating this linear function, we get the trapezoidal
approximation given by
9
T.f;a;b/
D
Z
b
a
y.x/dx
D
.b
a/
1
2
.f .a/
C
f.b//:
(1.29)
(a) The error of the trapezoidal method is defined by
E
D
ˇ
f.x/dx
T.f;a;b/
ˇ
Z
b
:
a
Show that
ˇ
ˇ
Z
b
E
D
.f .x/
y.x//dx
:
a
(b) Use (1.34) below to show that
E
Z
b
a
j
f.x/
y.x/
j
dx:
(c) Use (1.35) below to show that
E
.b
a/ max
axb
j
f.x/
y.x/
j
:
Furthermore, if ˇ
D
1, the convergence is said to be
linear
.Ifˇ
D
2, the convergence is
quadratic
,ifˇ
3, the convergence is
cubic
, and so on.
Now this is probably a bit more than you would like to know, but, nevertheless; if the error is
ch
and c depends on h and goes to zero as h goes to zero, the convergence is said to be
superlinear
.
9
We add the explicit dependence of T on f , a,andb for future use.
D
