Solution of system of non-linear equations

1. Jun 15, 2014

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.

2. Jun 15, 2014

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.

3. Jun 16, 2014

Stephen Tashi

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

4. Jun 16, 2014

JulieK

Any kind not only polynomials.

5. Jun 16, 2014

Stephen Tashi

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

6. Jun 16, 2014

Staff: Mentor

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

Chet

7. Jun 19, 2014

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)?

8. Jun 19, 2014

Staff: Mentor

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