The method of simple iteration involves writing
the equation *f(x)* = 0 in a form suitable for the construction of a
sequence of approximations to some root, in a repetitive fashion.

The iteration procedure is as
follows. In some way we obtain a rough approximation *x*_{0} of
the desired root, which may then be substituted into the right-hand side to
give a new approximation, . The new
approximation is again substituted into the right-hand side to give a further
approximation , and so on until
(hopefully) a sufficiently accurate approximation to the root is obtained. This
repetitive process, based on , is called *simple
iteration*; provided that decreases as *n*
increases, the process tends to . where denotes
the root.

The method of simple itcration
will be used to find the root of the equation 3*xe ^{x}*=1 to an
accuracy of 4

One first writes

Assuming *x*_{0} =
1 and with numbers displayed to 5*D*, successive iterations

We see Ihat after eight iterations the root is 0.2576 to 4<I>D</I>. A graphical interpretation of the first three iterations is shown in Figure

FIGURE 8. Iterative method.

Whether
or not an iteration procedure converges quickly, or indeed at all, depends on
the choice of the function , as well as the starting
value *x*_{0}. For example, thc equation *x*^{2} = 3
has two real roots: . It can be rewritten in the
form

which suggests the iteration

However, if the starting value *x*_{0}
= 1 is used, successive iterations yield

so that there is no convergence!

We can examine the convergence of the iteration process

to with the help of the Taylor series (cf. STEP 5]

where is a value between
the root and the
approximation *x _{k}*. We have

Multiplying the *n* + 1
rows together and cancelling the common factors leaves

,

Consequently,

so that the absolute error can
be made as small as we please by a sufficient number of iterations, *if **of the root*.
(Note that the derivative of is such that

- What should a programmer
guard against in a computer prgram using the method of simple iteration?
- What is necessary to
ensure that the method of simple iteration does converge to a root?

- Assuming that the
initial guess is
*x*_{0}= 1, show by the method of simple iteration that one root of the equation 2*x*- 1 - 2sin*x*= 0 is 1.4973. - Use the method of simple
itteration to find to 4
*D*the root of the equation*x*+ cos*x*. - Use the method of simple
iteration to find to 3
*D*the root of the equation in Exercise**2,2**of STEP6.

http://mpec.sc.mahidol.ac.th/numer/STEP9x.HTM. 07/06/2012