Solution of system of non-linear equations

[JulieK]

1. Is there a general condition for the existence and uniqueness of solution of a system of simultaneous non-linear equations similar to the determinant test for a system of linear equations.

2. What are the solution methods (theoretical and numerical) for solving a system of simultaneous non-linear equations.

 

[WWGD]

Only idea I can think of is that if your equations describe manifolds, then the dimension of the intersection is the sum of the codimensions of the manifolds, i.e., in R^m , if the solution to f(x_1,x_2,..,x_n) and g(x_1,..,x_k) are respectively an n-manifold and a k-manifold, then the intersection (which is not necessarily a manifold) will have dimension m-n-k. If your equations are of the type R[x_1,..,x_n] , i.e., if they are varieties, then you can use results from Algebraic Geometry, like Bezout's theorem : http://en.wikipedia.org/wiki/Bezout's_theorem.

 

[Stephen Tashi]

2. What are the solution methods (theoretical and numerical) for solving a system of simultaneous non-linear equations.

Are you using "non-linear" to mean equations involving polynomials? Or are you asking about any kind of non-linear equation?

 

[JulieK]

Any kind not only polynomials.

 

[Stephen Tashi]

Any kind not only polynomials.

For such a large class of equations, there is no general test like the determinant test.

 

[Chestermiller]

What are the solution methods (theoretical and numerical) for solving a system of simultaneous non-linear equations.

For numerically solving sets of coupled non-linear algebraic equations, Newton-Raphson (and modifications thereof) are often very effective.

Chet

 

[JulieK]

But if there is nothing that guarantees the existence and uniqueness of solution, how can we verify that Newton-Raphson or any other method will pick the required solution (i.e. the solution that we are looking for in a certain physical setting)?

 

[Chestermiller]

But if there is nothing that guarantees the existence and uniqueness of solution, how can we verify that Newton-Raphson or any other method will pick the required solution (i.e. the solution that we are looking for in a certain physical setting)?

If it is a physical problem, and the model equations are formulated correctly, then there should exist a solution. As far as obtaining the required solution to a set of non-linear algebraic equations for a physical problem, there is no set recipe. The trick is to get an initial guess that is close enough to the required solution for Newton-Raphson (or other method, such as successive substitution) to converge. The method used for getting a good initial guess depends on the specific problem. But it is mostly a matter of playing with the equations, and having some experience. Do you have a specific problem in mind that you would like to lay on the table?

Chet