Numerical Analysis P107 atkinson Consider the rootfinding pr

Numerical Analysis. P107 atkinson

Consider the rootfinding problem f(x)=0 with root , with f´(x)0. Convert it to the fixed-point problem x=x+cf(x)g(x) ,with c a nonzero constant. How should c be chosen to ensure rapid convergence of xn+1=xn+cf(xn) to (Provided that x0 is chosen sufficiently close to )? Apply your way of choosing c to the rootfinding problem x^35=0

Solution

Given:

   Consider the rootfinding problem f(x)=0 with root , with f´(x)0.

Convert it to the fixed-point problem x=x+cf(x_n)g(x) with c a nonzero constant.

How should c be chosen to ensure rapid convergence of x_n+1=x_n+cf(x) to (Provided that x0 is chosen sufficiently close to )? Apply your way of choosing c to the rootfinding problem x³5=0

here,

    The fixed point iteration for g(x)=x behaves very nicely

   because if the

             |g(x)| is small near the fixed point.

then it,s smallest possible value is comes to 0.

   here, if we limit ourselves to the choice of g(x) =x-f(x)in applying this scheme.

        In general we can writeg(x) =x-H(f(x)) as long as we choose H such that H(0) = 0.

Not all choices will lead to a converging method.

For convergence we must also choose H such that|g(x)|<1

   this case is differ from above. and it is related to that.