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The formula in http://de.wikipedia.org/wiki/Diskussion:Nullstelle#References gives upper and lower bounds x1 and x2 for the roots of a polynomial all of whose roots are real. Where can I find a proof ? Unfortunately, Wikipedia does not give any sources. |
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Actually I happened to write a short section on wikipedia about the classic bounds here. To locate the zeros of a polynomial you can also use the Rouché theorem. Also, ymay find interesting material e.g. in the book by Pólya and Szegő. rmk 1: note that the above bounds of course hold for complex roots as well. rmk 2: the spirit of the square root formula given in the link is: assume you have a real monic polynomial $p$ of degree $n$, and you just know $a_{n-1}$ and $a_{n-2}$. These numbers put a constraint (Nebenbedingung) on the size of the roots. Remember that $-a_ {n-1}= -\sum_k z_k$ and $a_{n-2}=\sum_ {j < k} z_ j z_ k$ are elementary symmetric functions of the roots $z_k:=x_k+ i y_k$. In particular, for the real parts of the roots we have $$\sum_{k=1}^n x_k =-a_{n-1}$$ and $$\sum_{k=1}^n x_k^2 \leq a_{n-1}^2-2a_{n-2}$$ Within these constraints, the maximum, resp. the minimum, of a real root is found maximizing, resp. minimizing $x_1;$ the method of Lagrange multipliers should easily yield (I didn't check) to (a generalization of) the given result. |
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