The GerschgorinConsistency procedure

Any ALIAS maple procedure involved in the calculation on eigenvalues of a matrix A may use the Gerschgorin circles methods that states that all the eigenvalues of a matrix are enclosed in the union of a set of circles (in the complex plane) whose center and radii are calculated as functions of the coefficients of the matrix.

The purpose of the GerschgorinConsistency procedure is to generate the C++ code for a simplification procedure that may be used by the ALIAS-Maple procedures doing calculation on the eigenvalues of a square matrix. The syntax is:

 
GerschgorinConsistency(Func,Vars,Gradient,n,procname)

where:

• Func: list of constraints (equation or inequality), the last element must be the matrix.
• Vars: list of parameters name including as first element an auxiliary name that will be the name of the unknown in the characteristic polynomial name
• Gradient: a flag that indicates if the derivatives of the matrix coefficients with respect to the unknowns may be used (1) or not (0)
• n: an integer, the order of the method, see below
• procname: the name of the simplification procedure. The name of the created file will be procname.C

Here we try to improve the bounds given by the Gerschgorin method using the fact that for any diagonal matrix D with positive components the eigenvalues of $DAD^{-1}$ are the same than the eigenvalues of A. The method is not able to determine D such that the Gerschgorin circles are minimal (i.e. give the best bounds) and only try a set of at most $n^2$ different D where n is the order of the method.

To check only if the eigenvalues lie in the interval provided by the C++ procedure without modifying this interval use a negative n