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convergence help

house2012

New member
Mar 17, 2012
2
Hey guys, I can't get his question dealing with orders of convergence at all so any help would be nice.

Q: Find the conditions on A so that the iteration $$x_{n+1}=x_n-Af(x_n)$$ will converge to a root of f if stared near the root.

I know I should look at the taylor series expansion of f about its root but I am stuck with working out.

Thanks for your help!
 

CaptainBlack

Well-known member
Jan 26, 2012
890
Hey guys, I can't get his question dealing with orders of convergence at all so any help would be nice.

Q: Find the conditions on A so that the iteration $$x_{n+1}=x_n-Af(x_n)$$ will converge to a root of f if stared near the root.

I know I should look at the taylor series expansion of f about its root but I am stuck with working out.

Thanks for your help!
Write \(x_n=x_0+\varepsilon_n\), where \(x_0\) is the root of \(f(x)\)

Then:

\[
x_{n+1}=x_0+\varepsilon_{n+1}=x_0+\varepsilon_n - A\{f(x_0)+\varepsilon_n f'(x_0)+...\}
\]

Ignoring terms or order 2 and higher in \(\varepsilon_n\) we find:

\[\varepsilon_{n+1}=\varepsilon_n(1-Af'(x_0)) \]

So convergence occurs when the initial estimate is close enough to the solution when:

\[|1-Af'(x_0)|<1\]

CB
 
Last edited:

chisigma

Well-known member
Feb 13, 2012
1,704
Hey guys, I can't get his question dealing with orders of convergence at all so any help would be nice.

Q: Find the conditions on A so that the iteration $$x_{n+1}=x_n-Af(x_n)$$ will converge to a root of f if stared near the root.

I know I should look at the taylor series expansion of f about its root but I am stuck with working out.

Thanks for your help!
In order to avoid confusion we indicate with $x^{*}$ the root of $f(*)$ and with $x_{0}$ the starting point of iterations. Other hypotheses are...

a) $A>0$...

b) $f(*)$ crosses the x axes with positive slope...

If a) and b) are satisfied, then, as explained in...

http://www.mathhelpboards.com/showthread.php?426-Difference-equation-tutorial-draft-of-part-I

... the sequence $x_{n}$ will converge to $x^{*}$ if it exists an interval $a<x<b$ which contains $x^{*}$ and $x_{0}$ ad where for any $x \ne x^{*}$ is...


$\displaystyle |A\ f(x)|<2\ |x-x^{*}|$ (1)

More precisely if is...

$\displaystyle |A\ f(x)|\le |x-x^{*}|$ (2)

... the convergence will be 'monotonic' and if is...

$\displaystyle |x-x^{*}|<|A\ f(x)|<2\ |x-x^{*}|$ (3)

... the convergence will be 'oscillating'...

Kind regards

$\chi$ $\sigma$