# convergence

• Mar 15th 2009, 09:55 PM
pandakrap
convergence
can anyone show me...
How do you prove that an iterative method is convergent??

like for example if you have \$\displaystyle x_n = x_{n-1} - f(x_{n-1})/f'(x_{n-1})\$

thank you (Rofl)
• Mar 16th 2009, 12:42 PM
CaptainBlack
Quote:

Originally Posted by pandakrap
can anyone show me...
How do you prove that an iterative method is convergent?? or like show that it converges some how

like for example if you have y(k) = k - h(k)/h'(k)

y(kn) = k(n+1)
where this method is solving h(k) =0
thank you (Rofl)

Well this is Newtom-Raphson and its convergence depends on where you start and what h(x) actually is. See the Wikipedia article.

CB
• Mar 23rd 2009, 08:22 AM
HallsofIvy
Quote:

Originally Posted by pandakrap
can anyone show me...
How do you prove that an iterative method is convergent??

like for example if you have \$\displaystyle x_n = x_{n-1} - f(x_{n-1})/f'(x_{n-1})\$

thank you (Rofl)

You can't. It isn't always. Whether it converges of not depends on the function f, the set of possible values of f (obviously, if \$\displaystyle f(x)= x^2+ 1\$ where x is required to be real, this will not converge since it must converge to a solution of f(x)= 0 and \$\displaystyle x^2+ 1= 0\$ doesn't have a real solution) and, even when f is such that it can converge, whether it will or not depends on the initial choice of \$\displaystyle x_0\$.