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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
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.
Next: Optimization
Up: Linear algebra
Previous: Characteristic polynomial
Contents
Jean-Pierre Merlet
2012-12-20