Sturm series
Encyclopedia
In mathematics, the Sturm series associated with a pair of polynomial
s is named after Jacques Charles François Sturm.
and 
two univariate polynomials. Suppose that they do not have a common root and the degree of 
is greater than the degree of 
. The Sturm series is constructed by:
This is almost the same algorithm as Euclid's
but the remainder
has negative sign.
associated to a characteristic polynomial
in the variable 
:
where
for 
in 
are rational functions in 
with the coordinate set 
. The series begins with two polynomials obtained by dividing 
by 
where 
represents the imaginary unit equal to 
and separate real and imaginary parts:
The remaining terms are defined with the above relation. Due to the special structure of these polynomials, they can be written in the form:
In these notations, the quotient
is equal to 
which provides the condition 
. Moreover, the polynomial 
replaced in the above relation gives the following recursive formulas for computation of the coefficients 
.
If
for some 
, the quotient 
is a higher degree polynomial and the sequence 
stops at 
with 
.
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s is named after Jacques Charles François Sturm.
Definition
Let and two univariate polynomials. Suppose that they do not have a common root and the degree of is greater than the degree of . The Sturm series is constructed by:This is almost the same algorithm as Euclid's
Euclidean algorithm
In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
but the remainder
has negative sign.Sturm series associated to a characteristic polynomial
Let us see now Sturm series associated to a characteristic polynomialCharacteristic polynomial
In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....
in the variable :where
for in are rational functions in with the coordinate set . The series begins with two polynomials obtained by dividing by where represents the imaginary unit equal to and separate real and imaginary parts:The remaining terms are defined with the above relation. Due to the special structure of these polynomials, they can be written in the form:
In these notations, the quotient
is equal to which provides the condition . Moreover, the polynomial replaced in the above relation gives the following recursive formulas for computation of the coefficients .If
for some , the quotient is a higher degree polynomial and the sequence stops at with .