3
x1*x2^2 + x1*x3^2 - 1.1*x1 + 1;
x2*x1^2 + x2*x3^2 - 1.1*x2 + 1;
x3*x1^2 + x3*x2^2 - 1.1*x3 + 1;
TITLE : A neural network modeled by an adaptive Lotka-Volterra system, n=3
ROOT COUNTS :
total degree : 27
generalized Bezout bound : 21
set structure:
{x1} {x2 x3} {x2 x3}
{x2} {x1 x3} {x1 x3}
{x3} {x1 x2} {x1 x2}
mixed volume : 21
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., Vol. 49, No. 6, 1779-1792, 1989.
Jan Verschelde and Ann Haegemans:
`The GBQ-Algorithm for constructing start systems of homotopies for polynomial
systems, SIAM J. Numer. Anal., Vol. 30, No. 2, pp 583-594, 1993.
NOTE :
The coefficients have been chosen so that full permutation symmetry
was obtained. The parameter c = 1.1.
The orbits of solutions : 3 4 1
3*(a,a,a) 4*(a,a,b) and 1*(a,b,c)
THE GENERATING SOLUTIONS :
8 3
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.09959548065397E-01 4.79766841277292E-01
x2 : 5.09959548065397E-01 4.79766841277292E-01
x3 : 5.09959548065397E-01 4.79766841277292E-01
== err : 5.911E-16 = rco : 2.745E-01 = res : 2.493E-16 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.35560902253960E+00 -1.28703758201580E-17
x2 : -6.77804511269800E-01 -5.27500584353303E-01
x3 : -6.77804511269800E-01 5.27500584353303E-01
== err : 7.955E-16 = rco : 2.529E-01 = res : 4.003E-16 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -4.44383120980212E-01 -3.01427302521399E-61
x2 : -1.29427788609688E+00 1.65298843318187E-61
x3 : -1.29427788609688E+00 1.65298843318187E-61
== err : 2.849E-15 = rco : 2.583E-01 = res : 3.886E-16 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 8.98653694263692E-01 3.48820047576431E-01
x2 : -1.65123467890611E-01 7.61734168646636E-01
x3 : 8.98653694263692E-01 3.48820047576431E-01
== err : 3.699E-16 = rco : 2.949E-01 = res : 1.665E-16 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.09959548065397E-01 -4.79766841277292E-01
x2 : 5.09959548065397E-01 -4.79766841277292E-01
x3 : 5.09959548065397E-01 -4.79766841277292E-01
== err : 5.242E-16 = rco : 2.745E-01 = res : 5.662E-17 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -5.03029502430507E-01 -2.11770212163837E-55
x2 : -5.03029502430507E-01 1.95759134039956E-54
x3 : 1.68372096585234E+00 -1.95759134039956E-54
== err : 1.716E-15 = rco : 3.764E-01 = res : 3.331E-16 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.65123467890611E-01 -7.61734168646636E-01
x2 : 8.98653694263692E-01 -3.48820047576431E-01
x3 : 8.98653694263692E-01 -3.48820047576431E-01
== err : 3.702E-16 = rco : 2.761E-01 = res : 2.791E-16 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.01991909613079E+00 -9.49556774575980E-66
x2 : -1.01991909613079E+00 -7.59645419660784E-65
x3 : -1.01991909613079E+00 -4.74778387287990E-65
== err : 3.234E-15 = rco : 2.070E-01 = res : 6.661E-16 ==
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