03 Mar 2026

Iteration Method

Fixed-point iteration, convergence condition, error reduction, and practical numerical stopping rules.

msc semester-i numerical-methods iteration

An iteration method starts with an approximate value and improves it repeatedly. Many numerical methods have the form

\[x_{n+1}=g(x_n).\]

If the sequence $x_0,x_1,x_2,\ldots$ approaches a fixed value $\alpha$, then

\[\alpha=g(\alpha).\]

Fixed-point form

To solve $f(x)=0$, rewrite it as

\[x=g(x).\]

Then choose an initial guess $x_0$ and calculate

\[x_1=g(x_0),\qquad x_2=g(x_1),\qquad x_3=g(x_2),\ldots\]

Convergence condition

A useful local condition for convergence is

\[|g'(\alpha)|<1.\]

If $

g’(\alpha)

>1$, the iteration usually moves away from the fixed point.

Example pattern

For the equation

\[x=\cos x,\]

one may use

\[x_{n+1}=\cos x_n.\]

Starting from a reasonable value, the sequence approaches the solution.

Error estimate

The difference between two successive approximations,

\[|x_{n+1}-x_n|,\]

is often used as a practical indication of convergence.

Key points

• The same equation may be rearranged into many iteration formulas.
• Not every rearrangement converges.
• A numerical result should be tested by substitution into the original equation.