Fixed point or Banach's iteration

It is a Numerical Method used to find the solution to an equation (when the analytical solution is difficult or impossible to find).

It assumes that a function $f$ has a fixed point i.e. $f(x)=x$

Solution Step

1. Write the given function in terms of $x$, given $x^2+x=1$ Search $g$

$$x=g(x)=\frac{1}{x+1}$$

1. Apply the function repeatedly on a guess value till convergence

$$x_{n+1}=g(x_n)$$

where $x_n$ starts as some guess value.