My problem is:

An iterative method to find $n$-th root of a positive number $a$ is given by $x_{k+1}=\frac{1}{2} \left[x_k +\frac{a}{x_k^{n-1}}\right]$

Find the value of $n$ for which this iterative method fails to converge.

I tried to use $|g'(x)|<1$ but could not get it .

Please help


I hope you found that for $$g(x)=\frac12(x+a/x^{n-1})$$ the derivative is $$g'(x)=\frac12(1-(n-1)a/x^n).$$

To have a useful numerical method this needs to be contractive at least in the solution of the problem. There $$ g'(\sqrt[n]a)=1-n/2 $$ which has to fall inside the interval $(-1,1)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.