# Searching for numerical algorithm

I have nonlinear system of 3 equations. Here it is: $$\frac{s_2 - K_2}{ps_2^{\gamma_1} + (1 - p)s_2^{\gamma_2}} = \frac{K_1 - s_1}{ps_1^{\gamma_1} + (1 - p)s_1^{\gamma_2}} \\ \frac{s_2}{s_2 - K_2} = \frac{p\gamma_1s_2^{\gamma_1} + (1 - p)\gamma_2 s_2^{\gamma_2}}{ps_2^{\gamma_1} + (1 - p)s_2^{\gamma_2}} \\ \frac{s_1}{s_1 - K_1} = \frac{p\gamma_1s_1^{\gamma_1} + (1 - p)\gamma_2 s_1^{\gamma_2}}{ps_1^{\gamma_1} + (1 - p)s_1^{\gamma_2}}$$ Unknown variables here is $p,s_1,s_2$. At the previous topic user EMS recommend me very simple Newton Method, but here a many problems with it.

Fisrt problem what at Newton method Im need a pretty good starting point because at another way at the second iteration I may get very irreational solution (see second problem)

Second problem what there are many restrictions to my variables: $s1 \geq 0$,$s2\geq 0$ and $0<p<1$.

So at my practise at the second iteration I got an p = -45, s1 = -1.45 (f.e.) ($\gamma$ may be negative too) and my Newton realization lie down.

Mbe exist some another methods what I can use for my system?

Thx!

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maybe something like this? en.wikipedia.org/wiki/Nonlinear_programming – Bruno Apr 10 '12 at 18:22
Pls, write it as answer and if tommorow there are no more explicit answers Il mark it as correct. – 0dd_b1t Apr 11 '12 at 8:55

## 2 Answers

There exists a family of techniques for solving these types of problem, called Nonlinear Programming. The idea is that if you have a system of equations (equalities and inequalities) with nonlinear relations and constraints on the solutions that you accept, it is possible to find solutions using (sometimes expensive) methods such as linear branch and bound. For introductory information on that area, please see http://en.wikipedia.org/wiki/Nonlinear_programming.

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If you are interested only in finding the solution of this problem you could use excel's solver add-on or some other non-linear programming software like LINDO's products.

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