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Iterative Methods

  1. Nov 14, 2008 #1
    http://img142.imageshack.us/img142/6899/asdaps7.jpg

    I cant 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
     
  2. jcsd
  3. Nov 15, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Well, it says "this question is based on theory only". Okay, what theory do you know?
     
  4. Nov 15, 2008 #3
    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.
     
  5. Nov 15, 2008 #4

    Mark44

    Staff: Mentor

    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?
     
  6. Nov 16, 2008 #5
    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.
     
  7. Nov 17, 2008 #6
    Anyone? I really can't get this and it seems like I should know it fundamentally to understand the method better.
     
  8. Nov 17, 2008 #7
    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.
     
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