Theorem of Sturm

The theorem of Sturm makes it possible to calculate the number of real roots distinct from a Fonction polynomial included/understood in a given interval. This theorem was established in 1829 by Charles Sturm.

Statement of the theorem

The number of real roots distinct in a interval from a polynomial at Coefficient S realities, whose has and B is not roots, is equal to the difference in the number of changes of sign of the continuation of Sturm at the boundaries of this interval.

Continuation of Sturm

The continuation of Sturm or chain of Sturm is built starting from the polynomials $P\_0=P~$ and of its Dérivée $P\_1=P^\; \backslash \; prime$

$P=x^n\; +\; \backslash \; ldots\; +\; a\_1\; X\; +\; a\_0$

$P^\; \backslash \; prime=n*x^\; \{(n-1)\}\; +\; \backslash \; ldots\; +\; a\_1$

This continuation is the sequence of intermediate results which one obtains by applying the Algorithme of Euclide to $P\_0~$ and its derivative $P\_1~$.

To obtain this continuation one calculates:

$\backslash begin\{matrix\}$

P_0&=&P_1 * Q_1 - P_2 \ \ P_1&=&P_2 * Q_2 - P_3 \ \ & \ ldots& \ \ \end{matrix}

The $P\_i$ are thus the opposites of the successive remainders of the division of the two preceding terms of the continuation. If $P~$ has only distinct roots, the last term is constant nonnull. If this term is null, $P~$ admits multiple roots, and one can in this case apply the theorem of Sturm by using the continuation $T\_0,\; T\_1,\; \backslash \; ldots,\; T\_\; \{r-2\},\; 1$ that one obtains by dividing the $P\_1,\; P\_2,\; \backslash \; ldots,\; P\_\; \{r-1\}\; \backslash \; by\; \backslash \; P\_\; \{r-1\}$.

If one notes $\backslash \; sigma\; (\backslash \; xi)\; ~$ the number of changes of sign (zero is not counted like a change of sign) in the sequence

$P\; (\backslash \; xi),\; P\_1\; (\backslash \; xi),\; P\_2\; (\backslash \; xi),\; \backslash \; ldots,\; P\_r\; (\backslash \; xi)$.

the theorem of Sturm says to us that for two $a\; real\; numbers,\; b~$,

$\backslash \; sigma\; (A)\; -\; \backslash \; sigma\; (b)~$.

One can use this theorem to calculate the number of real roots distinct by choosing in a suitable way the terminals $a~$ and $b~$, for example all the real roots of a polynomial are in the interval $M~$ with:

$M=\; \backslash \; max\; (1,\; \backslash \; sum\; |a\_i|)$.

External bonds

• On the resolution of the numerical equations, by C. Sturm mathematical and physical Sciences, Volume VI Paris 1835 p 271-318

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