MHB Find the Conditions on A for Convergence of f(x) Root

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The discussion focuses on determining the conditions on A for the iteration \(x_{n+1}=x_n-Af(x_n)\) to converge to a root of the function f when starting near that root. It is established that convergence occurs if the initial estimate is sufficiently close to the solution, specifically when \(|1-Af'(x_0)|<1\). Additional conditions include A being positive and the function f crossing the x-axis with a positive slope. The convergence behavior can be categorized as monotonic or oscillating based on the relationship between \(A f(x)\) and the distance from the root. These insights are crucial for ensuring the effectiveness of the iterative method in finding roots.
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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!
 
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house2012 said:
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:
house2012 said:
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

$\chi$ $\sigma$
 
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