Solution of Equations  1
15  Iterative Methods for the Solution of
Nonlinear Algebraic Equations in One
Variable
Motivation
For any given
y
, it is easy to find an
x
such that
y=35*x
.
For more complicated
equations, the solution is not so simple.
Consider finding
x
such that
)
13
,
(
10
*
4
(
8288
.
3
18
x
i
abs
β
+
=
where
β
(x,13)
is a Bessel function of the first kind. (The answer is 42).
In this section, we discuss certain simple iterative methods for the solution of nonlinear
algebraic equations of the form
f
(
x
) = 0, where
f
is a continuous function (i.e., it does not
exhibit "jumps" for any values of
x
).
They can easily be modified to an equation of the
form
f(x)=y
(where y is a known number) by simply solving
f(x)y=0
.
The answer is
called the
root
of the equation.
In the iterative methods discussed below, we solve equations by starting with a guess for
x
(which could be a very bad guess).
Then we calculate what
f(x)
would be for that
guess.
We see how far off we were (i.e. by how much
f(x)
doesn’t equal zero), adjust our
guess and try again.
Eventually, we reason, we should get to a pretty good guess which
will be our answer.
Iterative Methods
Convergence
All iterative methods start with an initial guess,
x
0
, and produce a sequence {
x
n
} =
x
1
,
x
2
,
... , which, hopefully, converges to a true value of
x
, called
x’
which solves the equation..
We say that the sequence {
x
n
}, which converges to x’ , converges with order
α
when the
error e
n
=x’x
n
is such that
C
e
e
n
n
n
=
+
∞
→
α
1
lim
C is a constant which is known as the
asymptotic error constant
.
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 Spring '08
 Patzek
 Calculus, Tol, Bisection Method

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