$\pi$ polynomials whose real zeros approximate $\pi$

Question

Let's have the following polynomials
$$x^4+105x^2-1134=0,$$ $$x^6+126x^4+10395x^2-115830=0,$$ $$3x^8+550x^6+45045x^4+3378375x^2-38288250=0$$ The positive real zeros of these equations are good approximations of $\pi$. Does anyone know how to formulate the next polynomial so its real positive zeros give a better aproxmation of $\pi$?

Answer 1

How do you measure the quality of these approximations? Here are the errors in the roots of your polynomials: $$ \begin{eqnarray} \text{deg}&=&4; \qquad \rho=3.1419530007425911; \qquad\epsilon=3.6\times10^{-4}\\ \text{deg}&=&6;\qquad \rho=3.1415990271727633; \qquad \epsilon=6.4\times10^{-6}\\ \text{deg}&=&8; \qquad\rho=3.1415927638681944; \qquad \epsilon=1.1\times10^{-7}. \end{eqnarray} $$ I would argue that these are fairly inefficient... you've used $24$ digits of coefficients to get only $7$ correct digits of $\pi$, for instance, in the degree-$8$ example. It would be more efficient to just use $1000000000x - 3141592653=0$!

There is a terrific compilation of approximations for $\pi$, including polynomial-root approximations, over at The Contest Center's Pi Competition. My favorite is $$6x^6-4x^5+5x^4+2x^3-2x^2+3x-5083=0,$$ which has just ten digits of coefficients and leads to the fourteen-digit approximation $$ \rho=3.1415926535898031685143792;\qquad\epsilon=1.0\times10^{-14}. $$ The best approximation of any kind on the page is $$ \frac{\log\left((5!\times5336)^3 + 4! + 6!\right)}{\sqrt{163}}, $$ which is correct to thirty decimal places.

Answer 2

From ancient relation, $\frac{\pi}{(\phi+1)}= \frac{6}{5}$ I had such approximation (not very good, but ancient) $25x^2 - 90x + 36$ error $3\times10^{-3}$