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 Post subject: Proof of Fixed Point MethodsPosted: Sat, 17 Mar 2012 23:02:42 UTC
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Joined: Fri, 21 Oct 2011 16:04:51 UTC
Posts: 11
Let
phi(x) = 0.5(x+(x/c))
and consider the iterative process
x_n+1 = phi(x_n), n = 0,1,2,...
where x0 is a given non-zero real number. Prove that, for any positive number c, the sequence {x_n} converges and find limit as n->infinity by calculating the fixed point of phi. Determine the order of convergence of the iterative method.

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 Post subject: Re: Proof of Fixed Point MethodsPosted: Sun, 18 Mar 2012 01:19:06 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12172
Location: Austin, TX
KBrosUL wrote:
Let
phi(x) = 0.5(x+(x/c))
and consider the iterative process
x_n+1 = phi(x_n), n = 0,1,2,...
where x0 is a given non-zero real number. Prove that, for any positive number c, the sequence {x_n} converges and find limit as n->infinity by calculating the fixed point of phi. Determine the order of convergence of the iterative method.

This is false. , so , unless or this does not converge

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 Post subject: Re: Proof of Fixed Point MethodsPosted: Sun, 18 Mar 2012 12:29:31 UTC
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Joined: Fri, 21 Oct 2011 16:04:51 UTC
Posts: 11
Sorry my mistake, the function given is actually:

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 Post subject: Re: Proof of Fixed Point MethodsPosted: Sun, 18 Mar 2012 15:42:32 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6068
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
KBrosUL wrote:
Sorry my mistake, the function given is actually:

In which case, note that satisfies , so the result follows from two easy estimates (first to make sure sufficiently close will converge, second to make sure all starting values will eventually get to the sufficiently close required by the first) that I'll leave as exercise.

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