A small collection of nonlinear systems of equations
These are taken from the COPRIN collection
  Bronstein : This is a system of three equations in three unknowns with a total of 4 different solutions: 
 
        f(1) = x(1)**2+x(2)**2+x(3)**2-36.d0
        f(2) = x(1)+x(2)-x(3)
        f(3) = x(1)*x(2)+x(3)**2 - 1.d0
The solutions are obtained from 

        289 - 106x22 + 4x24 = 0 
        x1 = -17/(2 x2 )
        x3 = x1 + x2
Proposal for an initial guess: 1,1,2 .
  COCONUT ''CHEM'' . x(1),..,x(5) are concentrations and the system has one positive solution,
0.3114098139D-02 0.3459797242D+02 0.6504173202D-01
0.8593780389D+00 0.3695185962D-01
but there are more non physical solutions. 

      xh1=x(1)
      xh2=x(2)
      xh3=x(3)
      xh4=x(4)
      xh5=x(5)
c******** some constants
      y(1) = 10.d0
      y(5) = 0.193d0
      y(6) = 0.002597d0/dsqrt(40.d0)
      y(7) = 0.003448d0/dsqrt(40.d0)
      y(8) = 0.00001799d0/40.d0
      y(9) = 0.0002155d0/dsqrt(40.d0)
      y(10)= 0.00003846d0/40.d0
c******** the equations      
      f(1)=3*xh5 - xh1*(xh2 + 1.d0)
      f(2)=xh2*(2*xh1 + xh3**2 + y(8) + 2.d0*y(10)*xh2
     *  + y(7)*xh3 + y(9)*xh4)
     *  +xh1 - y(1)*xh5
      f(3)=xh3*(2.d0*xh2*xh3 + 2.d0*y(5)*xh3 + y(6) + y(7)*xh2) -
     *    8.d0*xh5
      f(4)=xh4*(y(9)*xh2 + 2.d0*xh4) - 4.d0*y(1)*xh5
      f(5)=xh2*(xh1+y(10)*xh2+xh3**2+y(8)+y(7)*xh3+y(9)*xh4)+xh1
     *  +y(5)*xh3**2+xh4**2-1.d0+y(6)*xh3
Proposal for an initial guess 1,20,1,1,1 .
  COCONUT ''I5''. This system has only one solution in the unit box [-1,1]10 but 30 solutions in [-100,100]10 
 
      f(1)= x(1)  - 0.18324757d0*(x(4)*x(3)*x(9))**3
     +         + x(3)**4*x(9)**7 - 0.25428722d0
      f(2)= x(2)  - 0.16271449d0*(x(1)*x(10)*x(6))**3
     +         + x(10)**4*x(6)**7 - 0.37842197d0
      f(3)= x(3)  - 0.16955071d0*(x(1)*x(2)*x(10))**3
     +         + x(2)**4*x(10)**7 - 0.27162577d0
      f(4)= x(4)  - 0.15585316d0*(x(7)*x(1)*x(6))**3
     +         + x(1)**4*x(6)**7 - 0.19807914d0
      f(5)= x(5)  - 0.19950920d0*(x(7)*x(6)*x(3))**3
     +          + x(6)**4*x(3)**7- 0.44166728d0
      f(6)= x(6)  - 0.18922793d0*(x(8)*x(5)*x(10))**3
     +         + x(5)**4*x(10)**7 - 0.14654113d0
      f(7)= x(7)  - 0.21180486d0*(x(2)*x(5)*x(8))**3
     +         + x(5)**4*x(8)**7 - 0.42937161d0
      f(8)= x(8)  - 0.17081208d0*(x(1)*x(7)*x(6))**3
     +         + x(7)**4*x(6)**7 - 0.07056438d0
      f(9)= x(9)  - 0.19612740d0*(x(10)*x(6)*x(8))**3
     +         + x(6)**4*x(8)**7 - 0.34504906d0
      f(10)= x(10) - 0.21466544d0*(x(4)*x(8)*x(1))**3
     +           + x(8)**4*x(1)**7 - 0.42651102d0

Proposal for an initial guess: 1,2,1,2,1,2,1,2,1,2
  ''BULLARD''. This system has two solutions given by the intersection of the two curves (x1, x2=1/(10000x1) and (x1, x2=-log(1.001-exp(-x1)), one in the first and the second in the third quadrant. 
 
        f(1)=1.0d4*x(1)*x(2)-1.d0
        f(2)=dexp(-x(1))+dexp(-x(2))-1.001d0

Proposal for an initial guess: 1,0.0001 .
  ''BRANIN'' with an unique solution 

        f(1)=2.d0*dsin(0.4d0*pi*x(1))*dsin(0.4d0*pi*x(3))-x(2)
        f(2)=2.5d0-x(3)-x(1)+0.1d0*x(2)*dsin(2.d0*pi*x(3))
        f(3)=1.d0+0.1d0*x(2)*dsin(2.d0*pi*x(1))-x(3)

Proposal for an initial guess: 1,1,1 .
  COPRIN mixed algebraic trigonometric with an unique solution

        f(1)=-1.d0-x(1)+x(2)+x(3)+2.d0*dsin(x(2)-x(3))
        f(2)=-1.d0-x(1)+x(2)-x(3)+2.d0*dsin(x(1)-x(3))
        f(3)=-1.d0+x(1)+x(2)-x(3)+2.d0*dsin(x(1)-x(2))

Proposal for an initial guess: 1,-1,1 .
 
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01.08.2013