Sturm series

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, the Sturm series[1] associated with a pair of polynomials is named after Jacques Charles François Sturm.


Further information: 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.

Sturm series associated to a characteristic polynomial[edit]

Let us see now Sturm series p_0,p_1,\dots,p_k associated to a characteristic polynomial P in the variable \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 \{1,\dots,k\} are rational functions in \mathbb{R}(Z) with the coordinate set Z. The series begins with two polynomials obtained by dividing P(\imath \mu) by \imath ^k where \imath represents the imaginary unit equal to \sqrt{-1} and separate real and imaginary parts:

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

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})\mu which provides the condition c_{i,0}\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}} 
c_{i-1,0} & c_{i-1,j+1} \\
c_{i,0} & c_{i,j+1}

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 h<k.


  1. ^ (French) C. F. Sturm. Résolution des équations algébriques. Bulletin de Férussac. 11:419–425. 1829.