Perturbations de matrices

 
 
 
• Fonction eig_plot :

 

This Maple function  generates the plot in the complex plane of the numerical eigenvalues of 1) random matrix perturbations of a given Jordan structure, 2) random polynomial matrix perturbations of a given local Smith form in zero. Several options can be specified, determining the form of the perturbation (constant or polynomial, dense or sparse, with a particular valuation profile or not) and the nature of the computations (number of random trials, of digits).    This allows for an easy experimentation of Lidskii's perturbation theory for eigenvalues and of its extension to the polynomial case by Langer and Najman. Some generic behaviors can also be observed for a number of nongeneric perturbations: Ma-Edelman's sparse perturbations of one Jordan block, conforming perturbations of an arbitrary Jordan structure or of an arbitrary local Smith form.    Numerical eigenvalue computations are done with Matlab. This Maple program has been written by Claude-Pierre Jeannerod.

 
• Fonction eig_order :

 

This Maple function uses symbolic computation to follow parametric eigenvalues that are computed numerically. Consider a matrix depending on a real parameter, namely an angle between 0 and Pi. Drawing the eigenvalues that are real everywhere on [0,Pi]  as functions of the parameter, we obtain two-dimensional curves which may cross each other for certain values of the parameter (crossing points). However, the way the eigenvalues cross at such points cannot be (numerically) detected by eigensolvers. To follow numerical eigenvalues in the presence of some crossings, one may compute their Puiseux expansions in some neighborhood of such points. Comparing with the numerical data before and after each crossing point allows for the reordering of the eigenvalue functions.. Therefore, this program combines calls to the Maple function algcurves[puiseux] and calls to Matlab eigensolver eig. The motivation for this study comes from computational problems in quantum physics. This Maple program has been written by Claude-Pierre Jeannerod.

 
• Package MP (Matrix Perturbations) :

 

MP (Matrix Perturbations) is a Maple package for square matrices depending on a parameter. It relies on ISOLDE package (E. Pflugel)  and aims to compute the first-order terms of the eigenvalue expansions of such matrices.  In the case of hermitian matrices, eigenvector approximations are also provided.  The main point of this program is the use of the lazy evaluation process of ISOLDE in order to avoid the computation of the characteristic polynomial of the matrix under consideration. This Maple package has been written by Claude-Pierre Jeannerod and Eckhard Pflugel. [More information]