Iterative Methods: Faster Convergence Analysis

[Firepanda]

http://img142.imageshack.us/img142/6899/asdaps7.jpg

I can't see how to do this at all, I can see how the methods come about easily enough, and of course find the root if needed and then show which converged faster. But I have nothing in my notes to hint me on how I can find which converges the fastest without working anything out..

Any ideas?

Thanks

 

[HallsofIvy]

Well, it says "this question is based on theory only". Okay, what theory do you know?

 

[Firepanda]

Well, it says "this question is based on theory only". Okay, what theory do you know?

I know how to use the x0 close to the root to find an xn as an approximation the the root, also some notes on errors, but nothing on rate of convergence to the root.

 

[Mark44]

http://img142.imageshack.us/img142/6899/asdaps7.jpg

I can't see how to do this at all, I can see how the methods come about easily enough, and of course find the root if needed and then show which converged faster. But I have nothing in my notes to hint me on how I can find which converges the fastest without working anything out..

Any ideas?

Thanks

You said you don't have anything in your notes about this. Is there something in your text (assuming you're using a textbook) that discusses convergence rates?

 

[Firepanda]

You said you don't have anything in your notes about this. Is there something in your text (assuming you're using a textbook) that discusses convergence rates?

Basically in my notes I have a page titled 'Rate of convergence' which is all basically just a proof using the taylor expansion to show that Newtons method is a quadratic convergence.

I've just recently done more looking into it and everywhere says this under all google searches so I assume this is the only bit of theory I need to know?

In that case I'll take a wild stab in the dark and say the 1st equation has a faster rate as it seems liek the x^3 term would have a bigger impact on the next term than simply diving by x.

 

[Firepanda]

Anyone? I really can't get this and it seems like I should know it fundamentally to understand the method better.

 

[TheoMcCloskey]

Look-up "Fixed point iteration". In particular, what can said about the convergence rate realtive to the magnitude of the derivative of the iteration formula near the root. Compare the magnitude of the derivatives of these two functions over the domain.