# Numerical analysis problem

• February 21st 2009, 08:28 AM
tedeman
Numerical analysis problem
Let A be a given positive constant and g(x)=2x-Ax^2

a. Show that if fixed-point iteration converges to a nonzero limit, then the
limit is p=1/A, so the reciprocal of a number can be found using only
multiplications and subtractions
b. Find an interval about 1/A for which fixed-point iteration converges,
provided p0 is in the interval

Can any one help with this problem
• February 22nd 2009, 12:19 PM
CaptainBlack
Quote:

Originally Posted by tedeman
Let A be a given positive constant and g(x)=2x-Ax^2

a. Show that if fixed-point iteration converges to a nonzero limit, then the
limit is p=1/A, so the reciprocal of a number can be found using only
multiplications and subtractions
b. Find an interval about 1/A for which fixed-point iteration converges,
provided p0 is in the interval

Can any one help with this problem

if the iteration $x_{n+1}=g(x_n)$ converges then it convergent to a root of:

$g(x)=x$

For the second part you need to find an interval about $1/A$ such that $g'(x) on that interval, then the contraction mapping (Banach's fixed point) theorem will guarantee convergence.

CB