Can someone explain to me what is the convergence criteria for newton's method? As I was trying to code my own Square root function: sqrt(double n) iteratively, can I use: absolute(estimation^2 - n) as an indicator of when the iteration should stop?
I understand that convergence is also a great stopping criteria but why can't I use: absolute(estimation^2 - n)
You can use both $|x_k-n| < \epsilon$ and $|x_{k + 1}-x_k| < \epsilon'$ as stopping criteria. Although you must be aware that there exists some sequence $(x_k)$ where for any $\epsilon > 0$, it exists $k$ such that $|x_{k + 1}-x_k| < \epsilon$ but $(x_k)$ does not converge. But $x\mapsto x^2$ is convex and you can prove Newton's method converge towards the root for convex functions.