butcher -- Butcher's problem

Synopsis

Usage:

butcher(kk)
Inputs:
- kk, a ring, the coefficient ring
Outputs:
- a list, a list of the polynomials in this example

Description

This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.

There were 368 solutions found in 82.589 seconds (with a Bezout bound of 4608).

References:

W. Boege, R. Gebauer, and H. Kredel: "Some examples for solving systems of algebraic equations by calculating Groebner bases", J. Symbolic Computation, 2:83-98, 1986.

C. Butcher: "An application of the Runge-Kutta space". BIT, 24, pages 425–440, 1984.

This example is from the POSSO test suite available by anonymous ftp from the site gauss.dm.unipi.it, from the directory pub/posso.

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

i1 : butcher(QQ)

                         2   1    1     2      2      2    3    2   1    4  
o1 = {z*u + y*v + t*w - w  - -w - -, z*u  + y*v  - t*w  + w  + w  - -t + -w,
                             2    2                                 3    3  
     ------------------------------------------------------------------------
          2    3           1       2   1    2      3      3      3    4   3 3
     - t*w  + w  + z*v*x - -t*w + w  - -t + -w, z*u  + y*v  + t*w  - w  - -w 
                           2           6    3                             2  
     ------------------------------------------------------------------------
             5 2   1    1     3    4             1   2   3 3   1      7 2  
     + t*w - -w  - -w - -, t*w  - w  + z*u*v*x + -t*w  - -w  + -t*w - -w  -
             2     4    4                        2       2     2      4    
     ------------------------------------------------------------------------
     3    1     3    4      2       2   3 3   2      7 2    1     1       3  
     -w - -, t*w  - w  + z*v x + t*w  - -w  + -t*w - -w  - --w - --, - t*w  +
     8    8                             2     3      6     12    12          
     ------------------------------------------------------------------------
      4      2   3 3   1      13 2    7     1
     w  - t*w  + -w  - -t*w + --w  + --w + --}
                 2     3      12     24    24

o1 : List

Ways to use butcher :

butcher(Ring)

For the programmer

The object butcher is a method function.