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.

 

[CellCoree]

I understand your desire to prove the uniqueness of the solution to your set of non-linear equations. This is an important aspect in mathematical modeling and problem-solving. In order to prove the uniqueness of a solution, there are a few methods that can be used, such as the Jacobian matrix, the Hessian matrix, and the covariance matrix.

The Jacobian matrix is a powerful tool in analyzing the behavior of non-linear systems. It can be used to determine the number of solutions to a set of equations and to identify the conditions under which a unique solution exists. By analyzing the eigenvalues of the Jacobian matrix, you can determine if there is a single unique solution or multiple solutions to your set of equations.

Another approach is to use the Hessian matrix, which is a matrix of second-order partial derivatives. This matrix can provide information about the curvature of the solution space and can help determine if there is a unique solution or multiple solutions.

Lastly, you mentioned the use of covariance matrices. This method can also be effective in proving the uniqueness of a solution. By analyzing the covariance matrix, you can determine if there is any correlation between the variables in your equations and if this correlation leads to a unique solution.

In conclusion, I would recommend using a combination of these methods to prove the uniqueness of your solution. It is important to thoroughly analyze your equations and understand the behavior of the system in order to confidently prove the existence of a unique solution.