Reliable numerical evaluation of eigenvalues involved in polynomial systems solving
Résumé
In order to improve the performances of symbolic algorithms for solving large polynomial systems, certified numerical computations may be performed. We present here how to control the validity of the numerical evaluation of eigenvalues involved in polynomial systems solving. The accuracy of the computed eigenvalues can be estimated from the eigenvalue condition number. Round-off errors due to the use of floating-point arithmetic for eigenvalues computation can be estimated using Discrete Stochastic Arithmetic implemented in the CADNA library. Guaranteed inclusions of the eigenvalues can be computed using the INTLAB toolbox. These numerical validation tools have been compared, taking into account the exact eigenvalues computed using symbolic algorithms. In order to improve the numerical quality of the results, experiments have also been carried out using extended precision arithmetic and the MPFR multiple precision library.