1924–25 Beşiktaş J.K. season

From formulasearchengine
Jump to navigation Jump to search

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

Definition

Template:See Let p0 and p1 two univariate polynomials. Suppose that they do not have a common root and the degree of p0 is greater than the degree of p1. The Sturm series is constructed by:

pi:=pi+1qi+1pi+2 for i0.

This is almost the same algorithm as Euclid's but the remainder pi+2 has negative sign.

Sturm series associated to a characteristic polynomial

Let us see now Sturm series p0,p1,,pk associated to a characteristic polynomial P in the variable λ:

P(λ)=a0λk+a1λk1++ak1λ+ak

where ai for i in {1,,k} are rational functions in (Z) with the coordinate set Z. The series begins with two polynomials obtained by dividing P(ıμ) by ık where ı represents the imaginary unit equal to 1 and separate real and imaginary parts:

p0(μ):=(P(ıμ)ık)=a0μka2μk2+a4μk4±p1(μ):=(P(ıμ)ık)=a1μk1a3μk3+a5μk5±

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

pi(μ)=ci,0μki+ci,1μki2+ci,2μki4+

In these notations, the quotient qi is equal to (ci1,0/ci,0)μ which provides the condition ci,00. Moreover, the polynomial pi replaced in the above relation gives the following recursive formulas for computation of the coefficients ci,j.

ci+1,j=ci,j+1ci1,0ci,0ci1,j+1=1ci,0det(ci1,0ci1,j+1ci,0ci,j+1).

If ci,0=0 for some i, the quotient qi is a higher degree polynomial and the sequence pi stops at ph with h<k.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  1. Cite error: Invalid <ref> tag; no text was provided for refs named Sturm1829