How to compute Dottie number accurately? Dottie number is root of this equation : $cos \alpha = \alpha$, $\alpha \approx 0.73908513321516064165531208767\dots$.
I wonder how can I compute it ? I have tried to do it with an approximating formula:
$\alpha = \frac{5\pi^2}{\alpha^2 + \pi^2} - 4$
I have solved this equation and i got $\alpha \approx 0.738305\dots$. So , how can i compute it accurately ? Can i use taylor series, etc. ?

Taylor series of order 2 gives a simple quadratic in $\alpha$:
$$\alpha=1-\alpha^2/2\implies \alpha=0.\color{red}{73}2..$$
Of order 4 gives a bi-quadratic (there's a formula to solve roots of a polynomial of degree less than 5) in $\alpha^2$:
$$\alpha=1-\alpha^2/2+\alpha^4/4\implies 0.\color{red}{739}2..$$
Fairly accurate for practical purposes wherein the correct value is $0.739085...$
