Existential Proof of a Unique Solution to a Set of Non-Linear Equations

[natski]

Dear all,

I have a set of 5 non-linear equations with highly complicated and long forms for which I wish to find the unique solution. I was going to tackle this problem with Broyden's method since the derivatives cannot be easily found.

However, even if I get a solution from this, this is not the same as proofing that only one unique solution exists.

What method would readers recommend I employ to attempt to prove whether there is a unique solution or not to this problem? I was thinking along the lines of covariance matrices...

Natski

 

[MrJB]

Assume that two distinct solutions exist, then derive a contradiction.
It's difficult to say further without more information about your problem.

 

[mjames1]

Have you tried assuming that there are two solutions (just label as s_1 and s_2). Then apply them through the equations and find a contradiction to the assumption.

 

[nasu]

Dear all,

I have a set of 5 non-linear equations with highly complicated and long forms for which I wish to find the unique solution. I was going to tackle this problem with Broyden's method since the derivatives cannot be easily found.

What makes you think that the solution is (should be) unique, to start with?
A set of nonlinear algebraic equations (it seems to me you are not talking about differential equations) can have more than one solution.
For example, a set of two quadratic equations with two unknowns can have two distinct solutions.