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
butcher8
8
b1 + b2 + b3 - (a+b);
b2*c2 + b3*c3 - (1/2 + 1/2*b + b**2 - a*b);
b2*c2**2 + b3*c3**2 - (a*(1/3+b**2) - 4/3*b - b**2 - b**3);
b3*a32*c2 - (a*(1/6 + 1/2*b + b**2) - 2/3*b - b**2 - b**3);
b2*c2**3 + b3*c3**3 - (1/4 + 1/4*b + 5/2*b**2 + 3/2*b**3 + b**4 - a*(b+b**3));
b3*c3*a32*c2 - (1/8 + 3/8*b + 7/4*b**2 + 3/2*b**3 + b**4
- a*(1/2*b + 1/2*b**2 + b**3));
b3*a32*c2**2 - (1/12 + 1/12*b + 7/6*b**2 + 3/2*b**3 + b**4
- a*(2/3*b + b**2 + b**3));
1/24 + 7/24*b + 13/12*b**2 + 3/2*b**3 + b**4 - a*(1/3*b + b**2 + b**3);
TITLE : 8-variable version of Butcher's problem
ROOT COUNTS :
total degree : 4608
5-homogeneous Bezout number : 1361
with partition : {b1 }{b2 b3 a }{b }{c2 c3 }{a32 }
general linear-product Bezout number : 605
based on the set structure :
{ b1 b2 b3 a b }
{ b2 b3 a b }{ b c2 c3 }
{ b2 b3 a b }{ b c2 c3 }{ b c2 c3 }
{ b3 a b }{ b c2 }{ b a32 }
{ b2 b3 a b }{ b c2 c3 }{ b c2 c3 }{ b c2 c3 }
{ b3 a b }{ b c2 }{ b c3 }{ b a32 }
{ b3 a b }{ b c2 }{ b c2 }{ b a32 }
{ a b }{ b }{ b }{ b }
mixed volume : 24
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.
NOTE :
The system has 5 regular solutions. Two paths converged to highly
singular solutions, which indicates that the system probably has an
positive dimensional solutions component.
THE SOLUTIONS :
7 8
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
b1 : -4.10565491688240E-01 3.76158192263132E-37
b2 : -4.54124145231932E-01 -7.52316384526264E-37
b3 : -2.27062072615965E-01 5.17217514361807E-37
a : -1.00000000000000E+00 5.87747175411144E-38
b : -9.17517095361370E-02 6.39175053259619E-38
c2 : -4.08248290463863E-01 -3.52648305246686E-38
c3 : -8.16496580927726E-01 4.70197740328915E-37
a32 : -8.16496580927727E-01 1.41059322098675E-36
== err : 5.051E-15 = rco : 2.063E-03 = res : 1.422E-16 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
b1 : -3.72792119885564E-01 8.49560898879008E-01
b2 : 2.35528602162567E-01 9.24893027822684E-02
b3 : 3.59196576269339E-01 -1.71748996563447E-01
a : 6.10966529273170E-01 3.85150602548915E-01
b : -3.89033470726828E-01 3.85150602548914E-01
c2 : 1.22905387955472E+00 3.00007066016266E-01
c3 : 1.38903347072683E+00 -3.85150602548913E-01
a32 : 3.87782471854639E-01 -2.22654061728370E-01
== err : 1.629E-14 = rco : 1.346E-03 = res : 1.180E-15 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
b1 : -3.72792119885562E-01 -8.49560898879006E-01
b2 : 2.35528602162566E-01 -9.24893027822658E-02
b3 : 3.59196576269336E-01 1.71748996563446E-01
a : 6.10966529273169E-01 -3.85150602548912E-01
b : -3.89033470726828E-01 -3.85150602548915E-01
c2 : 1.22905387955472E+00 -3.00007066016267E-01
c3 : 1.38903347072683E+00 3.85150602548917E-01
a32 : 3.87782471854639E-01 2.22654061728372E-01
== err : 6.667E-15 = rco : 1.346E-03 = res : 1.228E-15 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
b1 : -2.00000000000000E+00 7.85417544673964E-15
b2 : -2.21811329325582E-15 -4.68716554016273E-15
b3 : 7.44660099998350E-16 1.32692288000702E-15
a : -1.00000000000000E+00 4.99325865176077E-16
b : -1.00000000000000E+00 3.99460692140786E-15
c2 : 6.10228849392220E-01 7.80261520828316E-01
c3 : -5.75622432602162E-01 -8.16657458049619E-01
a32 : -4.18737481532424E+00 -1.05540120319375E+00
== err : 6.808E+00 = rco : 7.493E-18 = res : 4.678E-14 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
b1 : 1.51135927406468E+00 -5.58329856879419E-46
b2 : -2.44033884709223E-01 2.24426707177021E-46
b3 : 2.88808493551859E-01 3.28429327576129E-46
a : 1.27806694145366E+00 6.36331822178750E-47
b : 2.78066941453657E-01 -6.91070043441438E-47
c2 : -6.24774425776101E-01 1.53267019535527E-46
c3 : 7.21933058546342E-01 -2.66164600889821E-46
a32 : -1.13792449427108E+00 4.26958125848968E-46
== err : 4.383E-15 = rco : 8.828E-03 = res : 7.078E-16 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
b1 : -1.83943450831177E+00 -1.24157286016722E-39
b2 : -4.58758547680644E-02 1.03270091626882E-39
b3 : -2.29379273840380E-02 -4.18192303305384E-40
a : -9.99999999999999E-01 7.74974102710197E-41
b : -9.08248290463868E-01 -7.04550447087089E-40
c2 : 4.08248290463805E-01 -2.26037289566993E-39
c3 : 8.16496580927691E-01 -3.95897084218590E-39
a32 : 8.16496580927679E-01 8.36563972814201E-40
== err : 7.525E-13 = rco : 1.468E-04 = res : 1.060E-15 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
b1 : -1.99999999999998E+00 -1.77453270253141E-14
b2 : -2.29308802652977E-14 1.60405121178499E-14
b3 : 2.07541705350947E-17 7.73078577002929E-16
a : -1.00000000000000E+00 -1.03526258940137E-16
b : -9.99999999999999E-01 -8.28210071521105E-16
c2 : 3.87642666200723E-02 2.48399890452917E-01
c3 : -5.33762129425664E-01 -4.01820103923349E-01
a32 : -3.94790779697364E+00 2.03513755526150E+01
== err : 2.515E+01 = rco : 2.347E-19 = res : 2.591E-14 ==