Sturm series explained

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

Definition

Sturm series associated to a characteristic polynomial

Let us see now Sturm series

p_0,p_1,...,p_k

associated to a characteristic polynomial

in the variable

P(λ)=a₀λ^k+a₁λ^k-1+ … +a_k-1λ+a_k

where

a_i

for

\{1,...,k\}

are rational functions in

R(Z)

with the coordinate set

. The series begins with two polynomials obtained by dividing

P(\imath\mu)

\imath^k

where

\imath

represents the imaginary unit equal to

\sqrt{-1}

and separate real and imaginary parts:

\begin{align} p_0(\mu)&:=\Re\left(

	P(\imath\mu)
	\imath^k

\right)=a₀\mu^k-a₂\mu^k-2+a₄\mu^k-4\pm … \\ p_1(\mu)&:=-\Im\left(

	P(\imath\mu)
	\imath^k

\right)=a₁\mu^k-1-a₃\mu^k-3+a₅\mu^k-5\pm … \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+ …

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 ≠ 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

	c_i-1,0
	c_i,0

-c_i-1,j+1=

	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}.

c_i,0=0

for some

, the quotient

q_i

is a higher degree polynomial and the sequence

p_i

stops at

p_h

with

h<k