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In-Depth Information
It turns out that we can solve (
4.23
) analytically only in very few cases. In other
cases, we need to compute the solution by some sort of numerical scheme. The
purpose of this chapter is to develop numerical methods for equations that cannot
be solved directly by analytical expressions.
Before we start developing methods, let us just consider (
4.2
) once more. Since
u
nC1
u
n
D
t g.
u
nC1
/
(4.29)
and t is a small number, we know that
u
nC1
is close to
u
n
, provided that g is
bounded. This turns out to be important when we want to construct numerical meth-
ods for solving the equation. The reason for this is that we will develop iterative
methods. Such methods depend on a good initial guess of the solution. We have just
seen that, for finding
u
nC1
, the previous value
u
n
is a good initial guess.
In the rest of this chapter we study methods for solving equations of the form
f.x/
D
0;
(4.30)
where f is a nonlinear function. We will also study systems of such equations later.
Solutions of (
4.30
) are called zeros, or roots. Motivated by the discussion above, we
will assume that we know a value x
0
close to x
,where
f.x
/
D
0:
(4.31)
Furthermore, we assume that f has no other zeros in a small region around x
.
4.1
The Bisection Method
Consider the function
f.x/
D
2
C
x
e
x
(4.32)
for x ranging from 0 to 3, see the graph in Fig.
4.1
.
We want to compute x
D
x
such that
f.x
/
D
0:
(4.33)
Let
x
0
D
0
(4.34)
and
x
1
D
3:
(4.35)
