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.