Converge Cubically?

[DeadxBunny]

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]

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!

 

[DeadxBunny]

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]

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]

Expand x around x = r. Put
\(\displaystyle \L x = r + \epsilon\)

Insert this in the iteraton formula:

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

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