What are some good online solvers of huge systems of non-linear equations?

Question

I'm wondering cause I want to solve a non-linear system of 8 equations. https://quickmath.com/ worked for 6 equations, but for 8 it says "internal error". Can anyone help? This is quite urgent. I want all of the solutions. Thanks!$$\left\{\begin{matrix} s+w=0 \\t+ws+x=s \\wv+ty+sz+ux=0 \\wu+v+tx+sy+z=t \\tz+uy+vx=u \\uz+vy=0 \\vz=v \\wt+u+sx+y=0 \end{matrix}\right.$$

Answer 1

Not a full answer, but Wolfram Cloud solves this system and obtains 45 solutions. It refuses to display them all though.

You can sign up and check out yourself.

Though I wonder where and why would you need all 45 solutions.

Answer 2

For rather large and complicated systems of equations, Excel may suit your purposes.

As it would turn out, there is a solver plugin in Excel which allows one to attempt to optimize systems of inequalities. For linear systems, the Simplex LP method is recommended. For smooth problems, the GRG Nonlinear method is recommended (which is your case). If it's really bad you can try the evolutionary solver, but this is usually not recommendable.

To solve systems of equations, you can attempt to approximate them by rewriting an equation such as $t+ws+x=s$ into $e_1=(t+ws+x-s)^2$ as the error squared. You can then set the objective function to be the sum of all of these errors.

To get multiple solutions, you can save each set of solutions and then add in the constraint $\sum_x(x-\bar x)^2\ge\epsilon$ for each variable $x$ and the previous solution $\bar x$ and some desired $\epsilon$. This can be repeated for each set of solutions until the objective function (i.e. the errors) are larger than you'd like.