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export {"butcher"}
butcher = method()
butcher(Ring) := kk -> (
(x,y,z,t,u,v,w) := (symbol x, symbol y, symbol z,
symbol t, symbol u, symbol v, symbol w);
R := kk[z,y,t,u,v,w,x];
{ z*u+y*v+t*w-w^2-1/2*w-1/2,
z*u^2+y*v^2-t*w^2+w^3+w^2-1/3*t+4/3*w,
x*z*v-t*w^2+w^3-1/2*t*w+w^2-1/6*t+2/3*w,
z*u^3+y*v^3+t*w^3-w^4-3/2*w^3+t*w-5/2*w^2-1/4*w-1/4,
x*z*u*v+t*w^3-w^4+1/2*t*w^2-3/2*w^3+1/2*t*w-7/4*w^2-3/8*w-1/8,
x*z*v^2+t*w^3-w^4+t*w^2-3/2*w^3+2/3*t*w-7/6*w^2-1/12*w-1/12,
-t*w^3+w^4-t*w^2+3/2*w^3-1/3*t*w+13/12*w^2+7/24*w+1/24
}
)
beginDocumentation()
doc ///
Key
butcher
(butcher,Ring)
Headline
Butcher's problem
Usage
butcher(kk)
Inputs
kk:Ring
the coefficient ring
Outputs
:List
a list of the polynomials in this example
Description
Text
This system was solved in May 2020, using @TO 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
Example
butcher(QQ)
///
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