The Newton-Raphson method is an iterative method for finding a root of a function, and it is self-correcting in the sense that any error in the initial input is reduced with each iteration so that it will still converge to the correct answer.
Is there a method to compute pi to higher and higher precision that has the same self-correcting property but also does not use transcendental functions? I have only been able to find methods that use transcendental functions, like applying Newton-Raphson to $x - \tan(x)$ or iterating the formula $x = x + \sin(x)$ over and over.
I was given this one $$a_n=a_{n-1}+\sin (a_{n-1}) \qquad \text{with} \qquad a_0=3$$
$$\left( \begin{array}{cc} n & a_n \\ 0 & \color{red}{3.}0000000000000000000000000000000000000000000000000 \\ 1 & \color{red}{3.141}1200080598672221007448028081102798469332642523 \\ 2 & \color{red}{3.1415926535}721955587348885681408797467429928172150 \\ 3 & \color{red}{3.14159265358979323846264338327950}9759271524572210 \\ 4 &\color{red}{ 3.1415926535897932384626433832795028841971693993751} \\ \end{array} \right)$$