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05-04-2007, 09:15 PM
Question :
Consider an iteration function of the form F(x) = x + f(x)g(x), where f(r) = 0 and f'(r) != 0. Find the precise conditions on the function g so that the method of functional iteration will converge cubically to r if started near r.

Any help with this problem would be greatly appreciated! Thanks!

pka
05-04-2007, 09:58 PM
Find the precise conditions on the function g so that the method of functional iteration will converge cubically to r if started near r.
What does that mean?
Give the exact definition!

05-05-2007, 03:33 AM
I don't know what cubic convergence is really, but a few methods of functional iteration are:

Newton's method:
F(x) = x - f(x)/f'(x)

Steffensen's method:
F(x) = x - [f(x)]^2/[f(x + f(x)) - f(x)]

Thanks!

stapel
05-05-2007, 12:47 PM
I don't know what cubic convergence is really
If the tutors need the definition being used by your textbook and/or class, but you don't know what this is, then I'm afraid there may be little we can do. Sorry.

Eliz.

Count Iblis
05-05-2007, 06:54 PM
Expand x around x = r. Put
\L x = r + \epsilon

Insert this in the iteraton formula:

\L F(r+\epsilon) = r+\epsilon + f(r+\epsilon)g(r+\epsilon)

Work out the series expansion in terms of derivatives of f and g and demand that the linear and quadratic terms in \L\epsilon vanish.