3. Iterative root finding Consider the following dynamical system XN+1 = g(XN). Given we know x...

  

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3. Iterative root finding Consider the following dynamical system XN+1 = g(XN). Given we know x at some time N in the program we simply compute the next value of x by assigning it the value g(x). Imagine we do this for a very long time, one might end up in a fixed point where x stops changing at all, i.e. x = g(x). That could be useful, imagine we'd like to find the roots of some function f(x), we could simply pick g(x) = x + f(x), such that the fixed point equation becomes f(x) = 0. (1) 1. If the initial input xo would have a small error e. What would the error be after N iterations? Keep only terms of O(e). 2. How is the latter related to the condition number of the function g(x)? 3. Iterative root finding Consider the following dynamical system XN+1 = g(XN). Given we know x at some time N in the program we simply compute the next value of x by assigning it the value g(x). Imagine we do this for a very long time, one might end up in a fixed point where x stops changing at all, i.e. x = g(x). That could be useful, imagine we'd like to find the roots of some function f(x), we could simply pick g(x) = x + f(x), such that the fixed point equation becomes f(x) = 0. (1) 1. If the initial input xo would have a small error e. What would the error be after N iterations? Keep only terms of O(e). 2. How is the latter related to the condition number of the function g(x)?

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1 8 xo E xo E f xo After N iterations the error will be magnified by a factor of 8N The error wi... View the full answer