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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.

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  • $\begingroup$ There are many infinite sums and infinite products for $\pi$ where the error is reduced each time you include a new term. $\endgroup$
    – Gerry Myerson
    Commented Nov 5, 2023 at 6:40
  • $\begingroup$ Take your pick. $\endgroup$
    – J.G.
    Commented Nov 5, 2023 at 10:08
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    $\begingroup$ @noumenon28 Do you consider the following quadratically convergent iteration based on the AGM (from Borwein & Borwein, Pi and the AGM, Wiley 1987) as an iteration of the desired type? On p. 46: $x_{0}:=\sqrt{2}$, $\pi_{0}:=2+\sqrt{2}$, and $y_{1}:=2^{1/4}$. Define $x_{n+1}:=\frac{1}{2}\left(\sqrt{x_{n}}+\frac{1}{\sqrt{x_{n}}}\right)$, $y_{n+1}:=\frac{y_{n}\sqrt{x_{n}}+1/\sqrt{x_{n}}}{y_{n}+1}$, $\pi_{n}:=\pi_{n-1}\frac{x_{n}+1}{y_{n}+1}$. The $\pi_{n}$ converge monotonically to $\pi$. $\endgroup$
    – njuffa
    Commented Nov 6, 2023 at 4:47
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    $\begingroup$ If you make your question precise, then I think it would require pi to be an algebraic number and it is not. $\endgroup$
    – Somos
    Commented Nov 6, 2023 at 11:29

1 Answer 1

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I was given this one $$a_n=a_{n-1}+\sin (a_{n-1}) \qquad \text{with} \qquad a_0=3$$

$$\left( \begin{array}{cc} n & a_n \\ 0 & \color{red}{3.}0000000000000000000000000000000000000000000000000 \\ 1 & \color{red}{3.141}1200080598672221007448028081102798469332642523 \\ 2 & \color{red}{3.1415926535}721955587348885681408797467429928172150 \\ 3 & \color{red}{3.14159265358979323846264338327950}9759271524572210 \\ 4 &\color{red}{ 3.1415926535897932384626433832795028841971693993751} \\ \end{array} \right)$$

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  • $\begingroup$ Note that the iteration of $x \mapsto x + \sin(x)$ is already mentioned in the question, and the author explicitly asks for methods which do “not use transcendental functions.” $\endgroup$
    – Martin R
    Commented Nov 16, 2023 at 15:40
  • $\begingroup$ I also doubt that the method is “self-correcting.” If some $a_n > \pi$ due to rounding errors, then the next itererates increase further towards $2\pi$. $\endgroup$
    – Martin R
    Commented Nov 16, 2023 at 15:48
  • $\begingroup$ @MartinR. I do not agree. Use Darboux theorem and you avoid all problems. $\endgroup$
    – Claude Leibovici
    Commented Nov 17, 2023 at 7:59
  • $\begingroup$ I am not sure what you mean. $(a_n)$ is strictly increasing except at the points $k \pi$, so any rounding error which leads to some $a_n > \pi$ will not be “corrected.” Perhaps it depends what “self-correcting” exactly means. – In any case, my first comment still applies. $\endgroup$
    – Martin R
    Commented Nov 17, 2023 at 8:07

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