# Non linear equation

• Sep 16th 2011, 10:51 PM
edustuffs5
Non linear equation
A set of non linear equations can be solved using newton raphson method. But how to solve that set of non linear equations with an inequality constraint?
• Sep 16th 2011, 10:52 PM
Prove It
Re: Non linear equation
Quote:

Originally Posted by edustuffs5
A set of non linear equations can be solved using newton raphson method. But how to solve that set of non linear equations with an inequality constraint?

• Sep 17th 2011, 04:43 AM
HallsofIvy
Re: Non linear equation
Quote:

Originally Posted by edustuffs5
A set of non linear equations can be solved using newton raphson method. But how to solve that set of non linear equations with an inequality constraint?

If there exist a solution satifying the inequality constraint, then I would recommend using Newton-Raphson, starting with an initial value that satisfies the constraint. If that inital value is well within the constraint area, Newton-Raphson should converge to a solution in that area but the boundary of such soutions can be very complex- even fractal.

If there does not exist such a solution, you would have to try to find a value that satisfies the constraint and comes "as close as possible" to satisfying the equation- Prove It's "optimization".
• Sep 17th 2011, 06:21 AM
CaptainBlack
Re: Non linear equation
Quote:

Originally Posted by edustuffs5
A set of non linear equations can be solved using newton raphson method. But how to solve that set of non linear equations with an inequality constraint?

Solve the unconstrained system using Newton-Raphson then either the constraints are satisfied at the solution in which case you are done, or not. If not there are either multiple solutions to the unconstrained problem or no solution. If there are multiple solutions you need to restart N-R at a different point outside of the basin of convergence to your previous solution/s and see where that takes you untill you find a solution satisfying the constraints.

Looking at this problem I suspect you may well have started with a constrained minimisation(maximisation) problem which you differentiated and set the derivatives to zero. If this is the case this is a $\displaystyle \bf{bad \ idea}^{(tm)}$. You should look at constrained optimisation algorithms.

CB
• Sep 18th 2011, 06:24 PM
edustuffs5
Re: Non linear equation
Quote:

Originally Posted by HallsofIvy
If there exist a solution satifying the inequality constraint, then I would recommend using Newton-Raphson, starting with an initial value that satisfies the constraint. If that inital value is well within the constraint area, Newton-Raphson should converge to a solution in that area but the boundary of such soutions can be very complex- even fractal.

If there does not exist such a solution, you would have to try to find a value that satisfies the constraint and comes "as close as possible" to satisfying the equation- Prove It's "optimization".

I implemented Newton-Raphson initialized with starting value with in constraint area, but sometime it is converging outside the desired region and sometime bouncing around the desired solution.