Solving Nonlinear System of Equations with 10 variables

I'm trying to solve the system of equations below but I keep running into problems. I tried to solve it with fsolve, but it would always return an error saying "FSOLVE requires all values returned by functions to be of data type double." or would just run forever and I would have to force stop it. I looked into doing the Newton-Raphson method, but I didn't want to manually type 100 different partial derivatives. If anyone knows a good way to solve this, I would realy appreciate it.

clc
W=100;
s=10;
S=23;
F=13;
m=9;
eqn1= 5-(W+S*9)*X1+S*9*X2-S*20*X1^2+S*20*X2^2==0;
eqn2= W*X1-(W+S*9)*X2+S*9*X3-S*20*X2^2+S*20*X3^2==0;
eqn3= W*X2-(W+S*9)*X3+S*9*X4-S*20*X3^2+S*20*X4^2==0;
eqn4= W*X3-(W+S*9)*X4+S*9*X5-S*20*X4^2+S*20*X5^2==0;
eqn5= W*X4-(W+S*9)*X5+S*9*X6-S*20*X5^2+S*20*X6^2==0;
eqn6= W*X5-(W+S*9)*X6+S*9*X7-S*20*X6^2+S*20*X7^2==0;
eqn7= 0.039+W*X6+(-W+9*S)*X7-9*s*X8+20*S*X7^2-20*s*X8^2==0;
eqn8= W*X7-(W+9*s)*X8-9*s*X9-20*s*X8^2-20*s*X9^2==0;
eqn9= W*X8-(W+9*s)*X9-9*s*X10-20*s*X9^2-20*s*X10^2==0;
eqn10=W*X9-(W+9*s)*X10-20*s*X10^2==0;

Answer 1

[John D'Errico]

Use fsolve. You are trying to formulate this in terrm of a symbolic variables, but fsolve is a NUMERICAL solver. So you need to write this in terms of a function, that returns a vector of length 10, as a function of the vector x, also of length 10.

fsolve does not like symbolic variables. Again, they are not numbers. So fsolve will have a fit.

[Owen Jones]

How would I convert symbolic variables into a function? I'm fairly new to Matlab, so a lot of these things just don't make sense to me yet.

[Walter Roberson]

matlabFunction() and use the 'vars' option with a cell array that is a single entry that is a vector of the variable names (giving the order to use them)

Answer 2

[Torsten]

W=100;
s=10;
S=23;
%F=13;
%m=9;
fun = @(X1,X2,X3,X4,X5,X6,X7,X8,X9,X10)...
          [5-(W+S*9)*X1+S*9*X2-S*20*X1^2+S*20*X2^2
           W*X1-(W+S*9)*X2+S*9*X3-S*20*X2^2+S*20*X3^2
           W*X2-(W+S*9)*X3+S*9*X4-S*20*X3^2+S*20*X4^2
           W*X3-(W+S*9)*X4+S*9*X5-S*20*X4^2+S*20*X5^2
           W*X4-(W+S*9)*X5+S*9*X6-S*20*X5^2+S*20*X6^2
           W*X5-(W+S*9)*X6+S*9*X7-S*20*X6^2+S*20*X7^2
           0.039+W*X6+(-W+9*S)*X7-9*s*X8+20*S*X7^2-20*s*X8^2
           W*X7-(W+9*s)*X8-9*s*X9-20*s*X8^2-20*s*X9^2
           W*X8-(W+9*s)*X9-9*s*X10-20*s*X9^2-20*s*X10^2
           W*X9-(W+9*s)*X10-20*s*X10^2];
Fun = @(x)fun(x(1),x(2),x(3),x(4),x(5),x(6),x(7),x(8),x(9),x(10));
format long
sol = fsolve(Fun,ones(10,1))

Equation solved.

fsolve completed because the vector of function values is near zero
as measured by the value of the function tolerance, and
the problem appears regular as measured by the gradient.

sol = 10×1

0.022650801201351 0.010341116968466 0.004585328417275 0.001843921328684 0.000526509707922 -0.000109331562119 -0.000416860856866 -0.000182933614111 -0.000077073111109 -0.000040566527577

norm(Fun(sol))

ans = 
     7.757362175446520e-14