Spectral radius

Safe calculation of the spectral radius of a square real or interval matrix may be obtained with the procedures

 
int Spectral_Radius(INTERVAL_MATRIX &AA,double eps,double *ro,int iter)
int Spectral_Radius(MATRIX &AA,double eps,double *ro,int inter)

where

• AA: the matrix
• eps: a real that will be used to increment the solutions found for the median polynomial (i.e. the polynomial whose coefficients are the mid-point of the interval coefficients of the characteristic polynomial) until the polynomial evaluation does not contain 0
• ro: the upper bound of the spectral radius
• iter: the maximal number of allowed iteration

These procedures return 1 on success, 0 on failure (in which case increasing eps may be a good option) and -1 if the matrix is not square. The procedures

 
int Spectral_Radius(INTERVAL_MATRIX &AA,double eps,double *ro)
int Spectral_Radius(MATRIX &AA,double eps,double *ro)

may also be used with a maximum number of iteration fixed to 100.

If it intended just to show that the spectral radius is larger than a given value seuil then you may use

 
int Spectral_Radius(INTERVAL_MATRIX &A,double eps,double *ro,double seuil);
int Spectral_Radius(MATRIX &A,double eps,double *ro,double seuil);
int Spectral_Radius(INTERVAL_MATRIX &A,double eps,double *ro,int iter,double seuil);
int Spectral_Radius(MATRIX &A,double eps,double *ro,int iter,double seuil);

Note that the calculation of the spectral radius may be used to check the regularity of an interval matrix. Indeed let ${\bf A}$ be an interval matrix of dimension $n$, $I_n$ the identity matrix of dimension $n$. If ${\bf A_c}$ is the mid-matrix of ${\bf A}$ we may write

$${\bf A}=[{\bf A_c}-\Delta,{\bf A_c}+\Delta]$$

Let ${\bf R}$ be an arbitrary matrix. It can be shown that if

$$\rho(\vert I_n-{\bf R}{\bf A_c}\vert+\vert{\bf R}\vert\Delta)<1$$

where $\rho$ denotes the spectral radius, then ${\bf A}$ is regular [22]: this is known as the Beeck-Ris test.