https://github.com/janverschelde/PHCpack
Tip revision: 1b7c69d37cd919359b5166b1c97d94c065360905 authored by Jan Verschelde on 17 November 2021, 01:50:30 UTC
loaded multipliers for the normalization of complex vectors in octo double precision into local variables
loaded multipliers for the normalization of complex vectors in octo double precision into local variables
Tip revision: 1b7c69d
butcher
7
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;
TITLE : Butcher's problem
ROOT COUNTS :
total degree : 4608
4-homogeneous Bezout number : 1361
with partition : {{z y t }{u v }{w }{x }}
multi-homogeneous Bezout number 1209,
with the following degree structure :
The partition for equation 1 : {{z y t }{u v }{w }}
The partition for equation 2 : {{z y t }{u v }{w }}
The partition for equation 3 : {{z t }{v }{w }{x }}
The partition for equation 4 : {{z y t }{u v }{w }}
The partition for equation 5 : {{z t }{u }{v }{w }{x }}
The partition for equation 6 : {{z t }{v }{w }{x }}
The partition for equation 7 : {{t }{w }}
generalized Bezout number : 605
based on the set structure :
{z y t w }{u v w }
{z y t w }{u v w }{u v w }
{z t w }{v w }{w x }
{z y t w }{u v w }{u v w }{u v w }
{z t w }{u w }{v w }{w x }
{z t w }{v w }{v w }{w x }
{t w }{w }{w }{w }
mixed volume: 24
REFERENCES :
The example has been retrieved from the POSSO test suite,
available by anonymous ftp from the site gauss.dm.unipi.it,
from the directory pub/posso.
See also
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.
DESCRIPTION :
There are 5 regular solutions and two singular solutions
The two singular solutions belong to a manifold of solutions:
t=-1=w, z=0=y, with u and v arbitrary complex numbers.
There are 3 regular real solutions.
THE SOLUTIONS :
7 7
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : -2.29379273840329E-02 7.44087924070508E-36
u : 8.16496580927762E-01 -1.29774576330781E-35
y : -4.58758547680744E-02 -6.84137712178571E-36
v : 4.08248290463845E-01 -1.10731567847459E-35
t : -1.00000000000000E+00 3.37954625861408E-37
w : -9.08248290463859E-01 -3.28550671054829E-36
x : 8.16496580927830E-01 -6.65800000305744E-35
== err : 4.965E-14 = rco : 1.736E-04 = res : 3.331E-16 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : 2.88808493551858E-01 -1.92592994438724E-34
u : 7.21933058546343E-01 9.62964972193618E-34
y : -2.44033884709223E-01 -9.14816723583937E-34
v : -6.24774425776102E-01 1.73333694994851E-33
t : 1.27806694145366E+00 -1.17963209093718E-33
w : 2.78066941453658E-01 -5.05556610401649E-34
x : -1.13792449427108E+00 -5.87408633038107E-33
== err : 2.785E-15 = rco : 1.273E-02 = res : 8.327E-17 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : -2.27062072615966E-01 -3.85185988877447E-34
u : -8.16496580927726E-01 1.92592994438724E-34
y : -4.54124145231932E-01 3.61111864572607E-34
v : -4.08248290463863E-01 1.38426214752833E-34
t : -1.00000000000000E+00 -2.46383615932351E-35
w : -9.17517095361370E-02 1.55165254308542E-36
x : -8.16496580927726E-01 0.00000000000000E+00
== err : 8.898E-16 = rco : 2.386E-03 = res : 2.776E-17 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : 7.32450810630072E-16 1.37145635748911E-15
u : 4.17630643926077E-01 -7.58875284416709E-01
y : 1.67709576191771E-15 -4.57212901658091E-15
v : 4.94521940247853E-01 -3.05129784662510E-02
t : -1.00000000000000E+00 4.68824600419586E-16
w : -1.00000000000000E+00 3.75059625315153E-15
x : -1.50630009976240E+00 -2.13582124544133E+00
== err : 0.000E+00 = rco : 2.854E-17 = res : 0.000E+00 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : 1.27097621050230E-16 3.03259676112193E-17
u : 2.34958226863350E+00 -1.87549206427436E+00
y : -1.91328685223823E-15 -3.04962129204523E-16
v : 1.73977586284498E+00 4.84530770879628E-01
t : -1.00000000000000E+00 -3.24271157948272E-15
w : -9.99999999999994E-01 -2.68584216774777E-15
x : 5.62376352154760E+00 -2.71542574259875E+00
== err : 0.000E+00 = rco : 8.236E-19 = res : 0.000E+00 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : 3.59196576269340E-01 -1.71748996563448E-01
u : 1.38903347072683E+00 -3.85150602548912E-01
y : 2.35528602162567E-01 9.24893027822689E-02
v : 1.22905387955472E+00 3.00007066016267E-01
t : 6.10966529273171E-01 3.85150602548917E-01
w : -3.89033470726829E-01 3.85150602548913E-01
x : 3.87782471854638E-01 -2.22654061728370E-01
== err : 4.135E-15 = rco : 1.198E-03 = res : 2.776E-16 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z : 3.59196576269339E-01 1.71748996563450E-01
u : 1.38903347072683E+00 3.85150602548910E-01
y : 2.35528602162569E-01 -9.24893027822688E-02
v : 1.22905387955472E+00 -3.00007066016269E-01
t : 6.10966529273174E-01 -3.85150602548915E-01
w : -3.89033470726829E-01 -3.85150602548913E-01
x : 3.87782471854640E-01 2.22654061728369E-01
== err : 5.151E-15 = rco : 1.198E-03 = res : 4.965E-16 ==