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Routh

The Routh algorithm allows to calculate the number of roots with positive real part of a polynomial being given the coefficients of the polynomial. It is implemented as:

int Routh(int Degree,double *Coeff)
that returns the number of roots with positive real part or -1 if it has not been possible to compute the Routh table (because the first element of a row of the Routh table is close to 0).

A similar algorithm allows to deal with polynomial with interval coefficients:

INTERVAL Routh(int Degree,INTERVAL_VECTOR &Coeff)
This algorithm returns in its interval:
-1,-1
: the Routh table cannot be computed
a,a+1
: there is at least a roots with positive real part but the exact number of roots with positive real part cannot be calculated
a,a
: there is exactly a roots with positive real part
A similar procedure may be used when the coefficients are functions of parameters:

INTERVAL Routh(int Degree,INTERVAL_VECTOR (* TheCoeff)(int,int,INTERVAL_VECTOR &),
INTERVAL_VECTOR &Input)
where:
• TheCoeff: a procedure that allow to calculate the coefficients being given the range for the parameters. It returns a Degree+1 interval vector. The two first integers l1, l2 of the procedure allows one to specify which coefficients are calculated. For example if V=TheCoeff(1,5,Input), then V(1..5) will be the first 5 coefficients of the polynomial and if V=TheCoeff(1,Degree+1,Input), then all the coefficients of the polynomial will be available in V
• Input: the intervals for the parameter
If the derivatives of the coefficients with respect to the parameters are known an even better procedure will be:

INTERVAL Routh(int Degree,INTERVAL_VECTOR (*TheCoeff)(int,int,INTERVAL_VECTOR &),
INTERVAL_MATRIX (* TheCoeffG)(int,int,INTERVAL_VECTOR &),
INTERVAL_VECTOR &Input)
where TheCoeffG is a procedure that allow to compute the derivatives of the coefficients with respect to the parameters (see note 2.4.2.2). This procedure allows to a certain amount to take into account the dependency between the coefficients.

Note also that the procedure Routh of ALIAS-Maple allows an even better calculation of the Routh table when dealing with parametric polynomials as the elements of the Routh table are calculated symbolically.    Next: Weyl filter Up: Parametric polynomials and eigenvalues Previous: Implementation   Contents
Jean-Pierre Merlet 2012-12-20