4
x1*x2^2 + x1*x3^2 + x1*x4^2 - 1.1*x1 + 1;
x2*x1^2 + x2*x3^2 + x2*x4^2 - 1.1*x2 + 1;
x3*x1^2 + x3*x2^2 + x3*x4^2 - 1.1*x3 + 1;
x4*x1^2 + x4*x2^2 + x4*x3^2 - 1.1*x4 + 1;
TITLE : A neural network modeled by an adaptive Lotka-Volterra system, n=4
ROOT COUNTS :
total degree : 81
generalized Bezout bound : 73
with symmetric supporting set structure:
{x1 }{x2 x3 x4}{x2 x3 x4}
{x2 }{x1 x3 x4}{x1 x3 x4}
{x3 }{x1 x2 x4}{x1 x2 x4}
{x4 }{x1 x2 x3}{x1 x2 x3}
mixed volume : 73
REFERENCES :
Karin Gatermann:
"Symbolic solution of polynomial equation systems with symmetry",
Proceedings of ISSAC-90, pp 112-119, ACM New York, 1990.
V. W. Noonburg:
"A neural network modeled by an adaptive Lotka-Volterra system",
SIAM J. Appl. Math (1988).
SYMMETRY :
The coefficients have been chosen so that full permutation symmetry
was obtained. The parameter c = 1.1.
------------------------------------------------
| orbit information of list of solutions |
------------------------------------------------
| TYPE | NB <> | NB GEN | NB SOLS |
------------------------------------------------
| 1 2 2 2 | 2 | 4 | 16 |
| 1 1 2 3 | 3 | 3 | 36 |
| 1 1 2 2 | 2 | 3 | 18 |
| 1 1 1 1 | 1 | 3 | 3 |
------------------------------------------------
| Total number of generating solutions : 13 |
| Total number of solutions generated : 73 |
------------------------------------------------
THE GENERATING SOLUTIONS :
13 4
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.33372590962199E-01 -4.43586227384486E-01
x2 : 4.33372590962199E-01 -4.43586227384486E-01
x3 : 4.33372590962199E-01 -4.43586227384486E-01
x4 : 4.33372590962199E-01 -4.43586227384486E-01
== err : 4.169E-16 = rco : 2.776E-01 = res : 3.608E-16 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.01991909613079E+00 6.06272901205497E-17
x2 : 1.01991909613079E+00 6.06272901205497E-17
x3 : -5.09959548065397E-01 -8.48770395714556E-01
x4 : -5.09959548065397E-01 8.48770395714556E-01
== err : 3.103E-15 = rco : 3.389E-01 = res : 2.269E-16 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -2.14773924418920E-01 -6.26558193923668E-01
x2 : -2.14773924418920E-01 -6.26558193923668E-01
x3 : 1.06192282524925E+00 -3.99591785628118E-01
x4 : 1.06192282524925E+00 -3.99591785628118E-01
== err : 3.638E-15 = rco : 2.138E-01 = res : 2.831E-16 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -4.90152757143433E-01 2.95549546086774E-64
x2 : -1.20414504451724E+00 -1.37685732313517E-64
x3 : -4.90152757143432E-01 4.36796116304951E-64
x4 : -1.20414504451724E+00 -4.55787251796470E-64
== err : 4.082E-15 = rco : 1.132E-01 = res : 3.331E-16 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -6.72576917532508E-01 -3.25008188747211E-48
x2 : -9.28586522969462E-01 1.36845553156720E-48
x3 : -9.28586522969462E-01 5.98699295060652E-49
x4 : -9.28586522969462E-01 9.62195295633190E-49
== err : 2.778E-16 = rco : 6.393E-02 = res : 2.220E-16 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.35061583139308E+00 1.50901402336195E-01
x2 : -5.09959548065397E-01 4.79766841277292E-01
x3 : -5.09959548065397E-01 4.79766841277292E-01
x4 : -8.40656283327683E-01 -6.30668243613487E-01
== err : 9.546E-16 = rco : 1.640E-01 = res : 2.220E-16 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.35061583139308E+00 -1.50901402336195E-01
x2 : -8.40656283327683E-01 6.30668243613487E-01
x3 : -5.09959548065397E-01 -4.79766841277292E-01
x4 : -5.09959548065397E-01 -4.79766841277292E-01
== err : 1.027E-15 = rco : 2.148E-01 = res : 4.965E-16 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -9.17395351588499E-02 -1.00626933473987E+00
x2 : 6.75139752372438E-01 -2.43302218087769E-01
x3 : 6.75139752372438E-01 -2.43302218087769E-01
x4 : 6.75139752372438E-01 -2.43302218087769E-01
== err : 1.124E-15 = rco : 2.482E-01 = res : 3.377E-16 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -8.66745181924397E-01 -3.26265223399926E-53
x2 : -8.66745181924398E-01 -3.91518268079912E-54
x3 : -8.66745181924397E-01 -3.91518268079912E-53
x4 : -8.66745181924398E-01 7.96087145095820E-53
== err : 5.989E-16 = rco : 6.026E-02 = res : 3.331E-16 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.75139752372438E-01 2.43302218087769E-01
x2 : 6.75139752372438E-01 2.43302218087769E-01
x3 : 6.75139752372438E-01 2.43302218087769E-01
x4 : -9.17395351588496E-02 1.00626933473987E+00
== err : 1.026E-15 = rco : 1.762E-01 = res : 5.495E-16 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.06192282524925E+00 3.99591785628118E-01
x2 : 1.06192282524925E+00 3.99591785628118E-01
x3 : -2.14773924418920E-01 6.26558193923668E-01
x4 : -2.14773924418919E-01 6.26558193923668E-01
== err : 3.602E-15 = rco : 2.457E-01 = res : 3.331E-16 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.76514689694112E+00 -8.92131470234173E-57
x2 : -4.21692981775415E-01 5.22534146851444E-56
x3 : -4.21692981775415E-01 -2.35777602847603E-56
x4 : -4.21692981775415E-01 -1.68867742580040E-56
== err : 3.707E-15 = rco : 4.190E-01 = res : 2.220E-16 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.33372590962199E-01 4.43586227384486E-01
x2 : 4.33372590962199E-01 4.43586227384486E-01
x3 : 4.33372590962199E-01 4.43586227384486E-01
x4 : 4.33372590962199E-01 4.43586227384486E-01
== err : 4.169E-16 = rco : 2.776E-01 = res : 3.608E-16 ==