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

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 be an interval matrix of dimension , the identity matrix of dimension . If is the mid-matrix of we may write

Let be an arbitrary matrix. It can be shown that if

where denotes the spectral radius, then is regular [22]: this is known as the Beeck-Ris test.