Information Technology Reference
In-Depth Information
r.t
C
t/
r.t/
t
r
0
.t /
D
C
O.t/:
If we apply this observation in the case of t
D
t
n
; we have
r.t
nC1
/
r.t
n
/
t
r
0
.t
n
/
D
C
O.t/:
Thus, it follows from (2.22) that it is reasonable
11
to require that
r
nC1
r
n
t
D
f.t
n
/;
and then we have
r
nC1
D
r
n
C
tf .t
n
/:
Since r
0
is known by the initial condition, we can compute
r
1
D
r
0
C
tf .t
0
/:
And since, by now, r
1
is known, we can compute
r
2
D
r
1
C
tf .t
1
/
D
r
0
C
t .f .t
0
/
C
f.t
1
// ;
and so on. We get
D
r
0
C
t
N
X
nD0
r
N
f.t
n
/;
(2.23)
which is the Riemann sum
12
approximation of the integral.
Example 2.1.
Let us, just for the purpose of illustration, consider the case of
f.t/
D
t
2
and r.0/
D
0.Then
11
Why is this reasonable? The idea is that r
0
.t / can be approximated by a finite difference, i.e.,
r.t
n
C
1
/
r.t
n
/
r
0
.t
n
/
:
t
Now since
r
0
.t
n
/
D
f.t
n
/;
we can
define
the numbers
f
r
n
g
by requiring that
r
n
C
1
r
n
D
f.t
n
/:
t
There is nothing mysterious about this; it is simply a reasonable way of putting up a condition that
is sufficient to compute the numbers
f
r
n
g
:
12
Consult your calculus topic.
