In Numerical analysis, to solve an equation of the form $f(x)=0$ in $[a,b]$, fixed point iteration method is useful.
To this end, we can write $f(x)=0$ in the form $g(x)=x$ and try to find a fixed point of $g$.
But there are many ways to find such function $g$. I am confused which $g$ should one choose? For example, in Burden and Faires book, it is assumed that if $|g'(x)|\leq k<1$ for every $x\in(a,b)$, then for any initial guess $x_0$ the sequence $x_n=g(x_{n-1})$ will converge to the fixed point of $g$. But it seems problematic to prove that $|g'(x)|\leq k<1$ for every $k\in(a,b)$.
My question is there an alternative way to find a suitable $g$ to ensure the convergence of the fixed point iteration method? In particular, can we choose $g$ and $x_0$ the initial guess so that only the condition $|g'(x_0)|<1$ will ensure the convergence?
Thanks in advance.
Best Answer
First of all, the condition you mention (contractivity) does not ensure the convergence for all $x_0 \in [a,b]$, it only ensures convergence if $x_0$ is close enough to the fixed point. If you want to ensure convergence for all initial approximations, you should also require invariance, i.e. $g([a,b]) \subseteq [a,b]$. So, regarding your questions,
I don't really understand why you are concerned about showing that $|g'(x)|\leq k <1$. Sure, this can be hard do do analytically if $g$ is "complicated", but is not a real obstacle.
The condition $|g'(x_0)|<1$ assures nothing. Local convergence would be a consequence of $|g'(z)|<1$, where $z$ is the fixed point.