geneig -- generalized eigenvalue problem

Synopsis

Usage:

geneig(kk)
Inputs:
- kk, a ring, the coefficient ring
Outputs:
- a list, of the polynomials in the system

Description

This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.

There were 10 solutions found in 1.447 seconds (with a Bezout bound of 243).

Reference: "Homotopy method for general lambda-matrix problems" by M. Chu, T.Y. Li and T. Sauer (pages 528-536).

See also: http://homepages.math.uic.edu/~jan/Demo/geneig.html.

i1 : geneig(QQ)

             2       2      2      2       2                                
o1 = {- 10x x  + 2x x  - x x  + x x  + 3x x  + x x  + 2x x  + x x  + 2x x  +
           1 6     2 6    3 6    4 6     5 6    1 6     2 6    3 6     4 6  
     ------------------------------------------------------------------------
                                             2        2       2       2  
     x x  + 10x  + 2x  - x  + 2x  - 2x , 2x x  - 11x x  + 2x x  - 2x x  +
      5 6      1     2    3     4     5    1 6      2 6     3 6     4 6  
     ------------------------------------------------------------------------
        2                                                                    
     x x  + 2x x  + x x  + 2x x  + x x  + 3x x  + 2x  + 9x  + 3x  - x  - 2x ,
      5 6     1 6    2 6     3 6    4 6     5 6     1     2     3    4     5 
     ------------------------------------------------------------------------
          2       2        2      2      2                                 
     - x x  + 2x x  - 12x x  - x x  + x x  + x x  + 2x x  - 2x x  - 2x x  -
        1 6     2 6      3 6    4 6    5 6    1 6     2 6     4 6     5 6  
     ------------------------------------------------------------------------
                                    2       2      2        2       2        
     x  + 3x  + 10x  + 2x  - x , x x  - 2x x  - x x  - 10x x  + 2x x  + 2x x 
      1     2      3     4    5   1 6     2 6    3 6      4 6     5 6     1 6
     ------------------------------------------------------------------------
                                                                      2  
     + x x  - 2x x  + 2x x  + 3x x  + 2x  - x  + 2x  + 12x  + x , 3x x  +
        2 6     3 6     4 6     5 6     1    2     3      4    5    1 6  
     ------------------------------------------------------------------------
        2      2       2        2
     x x  + x x  + 2x x  - 11x x  + x x  + 3x x  - 2x x  + 3x x  + 3x x  -
      2 6    3 6     4 6      5 6    1 6     2 6     3 6     4 6     5 6  
     ------------------------------------------------------------------------
     2x  - 2x  - x  + x  + 10x , x  + x  + x  + x  + x  - 1}
       1     2    3    4      5   1    2    3    4    5

o1 : List

Ways to use geneig :

geneig(Ring)

For the programmer

The object geneig is a method function.