Sturm series

{{See|Sturm chain}}

Let $p\_0$ and $p\_1$ two univariate polynomials. Suppose that they do not have a common root and the degree of $p\_0$ is greater than the degree of $p\_1$. The Sturm series is constructed by:

:$$

p_i := p_{i+1} q_{i+1} - p_{i+2} \text{ for } i \geq 0.

This is almost the same algorithm as Euclid's but the remainder $p\_\{i+2\}$ has negative sign.

Let us see now Sturm series $p\_0,p\_1,\backslash dots,p\_k$ associated to a characteristic polynomial $P$ in the variable $\backslash lambda$:

:$$

P(\lambda)= a_0 \lambda^k + a_1 \lambda^{k-1} + \cdots + a_{k-1} \lambda + a_k

where $a\_i$ for $i$ in $\backslash \{1,\backslash dots,k\backslash \}$ are rational functions in $\backslash mathbb\{R\}(Z)$ with the coordinate set $Z$. The series begins with two polynomials obtained by dividing $P(\backslash imath\; \backslash mu)$ by $\backslash imath\; ^k$ where $\backslash imath$ represents the imaginary unit equal to $\backslash sqrt\{-1\}$ and separate real and imaginary parts:

:$$

\begin{align}

p_0(\mu) & := \Re \left(\frac{P(\imath \mu)}{\imath^k}\right ) = a_0 \mu^k - a_2 \mu^{k-2} + a_4 \mu^{k-4} \pm \cdots \\

p_1(\mu) & := -\Im \left( \frac{P(\imath \mu)}{\imath^k}\right)= a_1 \mu^{k-1} - a_3 \mu^{k-3} + a_5 \mu^{k-5} \pm \cdots

\end{align}

The remaining terms are defined with the above relation. Due to the special structure of these polynomials, they can be written in the form:

:$$

p_i(\mu)= c_{i,0} \mu^{k-i} + c_{i,1} \mu^{k-i-2} + c_{i,2} \mu^{k-i-4}+\cdots

In these notations, the quotient $q\_i$ is equal to $(c\_\{i-1,0\}/c\_\{i,0\})\backslash mu$ which provides the condition $c\_\{i,0\}\backslash neq\; 0$. Moreover, the polynomial $p\_i$ replaced in the above relation gives the following recursive formulas for computation of the coefficients $c\_\{i,j\}$.

:$$

c_{i+1,j}= c_{i,j+1} \frac{c_{i-1,0}}{c_{i,0}}-c_{i-1,j+1} = \frac{1}{c_{i,0}}

\det

\begin{pmatrix}

c_{i-1,0} & c_{i-1,j+1} \\

c_{i,0} & c_{i,j+1}

\end{pmatrix}.

If $c\_\{i,0\}=0$ for some $i$, the quotient $q\_i$ is a higher degree polynomial and the sequence $p\_i$ stops at $p\_h$ with

{{Reflist|refs=

{{in lang|fr}} C. F. Sturm. Résolution des équations algébriques. Bulletin de Férussac. 11:419–425. 1829.

}}

Category:Series (mathematics)