How to find e^x using Newton Raphson method??

Question

Is there a way to find e^x using Newton Raphson method, I found this link: Iterative refinement algorithm for computing exp(x) with arbitrary precision but I don't know how to implement it. Please help me out.

Answer 1

For a given, fixed $x\in\mathbb{R}$, you need to find a function $F(y)$ which fulfils $F(e^x)=0$. Then you can apply Newton’s method for finding the zeros of $F$. The problem lies in finding a suitable $F$. In the question you mention, the author suggests using $F(y):= \ln(y) - x$. The Newton iteration for this Function is

$$ y_{n+1} = y_n - y_n(\ln(y_n)-x) $$

The problem is that you need to be able to compute $\ln(y_n)$ to do this. I don’t see an easy way how you would use Newton’s method for computing $e^x$ without using $\ln (x)$.

If you are just looking for an algorithm to compute $e^x$ or $\ln(x)$ to very high precision, I recommend you this paper: https://cr.yp.to/bib/1976/brent-elementary.pdf, section 6.