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Let's have the following polynomials
$x^4+105x^2-1134=0$,$x^6+126^x4+10395x^2-115830=0$, $3x^8+550x^6+45045x^4+3378375x^2-38288250=0$ 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 $\pi$?

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How did you obtain these polynomials in the first place? –  J. M. Dec 28 '11 at 23:36
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Why not the Taylor series of $\cos(x/2)$? –  Rahul Dec 29 '11 at 2:13
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With Rahul's suggestion the degree 10 polynomial is $1-\frac{x^2}{2^2\cdot 2!}+\frac{x^4}{2^4\cdot 4!}-\frac{x^6}{2^6\cdot 6!}+\frac{x^8}{2^8\cdot 8!}-\frac{x^{10}}{2^{10}\cdot 10!}$, which has a zero at around $3.14159172406$, with error around $9.3\times 10^{-7}$. At least this gives a certain method of giving better and better approximations. –  Jonas Meyer Dec 29 '11 at 2:40
    
From the equations (x+y)^n+(y-x)^n=z^n we create different series from the coefficients. In different arrangements of these series we formulate the coefficients of the presented equations. For the time being that is all I have to say because my work on this is not complete. –  Vassilis Parassidis Dec 29 '11 at 4:01
    
I can't believe nobody has said $\ell(x)=x-\pi$... –  Jp McCarthy Jan 24 '13 at 16:16
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2 Answers 2

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.

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I quite like $86 x^4-610 x^3+1199 x^2-445 x+101$ and $16 x^6-32 x^5-61 x^4-53 x^3+162 x^2+147 x-65$ myself... –  J. M. Dec 29 '11 at 2:00
    
More broadly, this leads to the concept of norms on polynomials, for instance the polynomial's height (the largest of its coefficients in absolute value) or its $l_2$-norm (the square root of the sums of the squares of the coefficients). Of course, the OP's polynomials are fairly inefficient by either of these norms. –  Steven Stadnicki Jan 24 '13 at 16:52
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From ancient relation, $\frac{\pi}{(\phi+1)}= \frac{6}{5}$ I had such approximation (not very good, but ancient) $25x^2 - 90x + 36$ error $3\times10^{-3}$

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