Please can you send me how can I use the Newton Raphson method to solve a non linear system

@ Aravind

Given the strength of NR method, I do not know of any other method which can converge faster. NR has a convergence of 2, i.e., quadratic. Also NR has the inherent problem of diverging! You can try Muller's method which has a convergence of 1.84, besides it can yield real as well as complex solutions! However, coding Muller's method for your system of equations could be tricky.

You can also strengthen NR approach using line Searches and backtracking. Or use Multidimensional secant methods: Broyden’s method. See, Chapter 9 of Numerical recipes by Press, Teukolsky, Vetterling and Flannery.

However I am wondering, why you are not using standard approaches of matrix algebra for solving such problems? There are plethora of algorithms available if you transform your problem in the matrix framework.

@ Abderrahim,

You can find it in any standard book on Numerical analysis. Or, simply google it.

Thank you Vikash

@ hahahah. What an irony! Abderrahim says "thanks" and on the contrary, my reply gets downvoted!

@ Pandey,

Thank you. How can I incorporate the range of values to look for a solution, that is, a variable x can take only solutions in [a,b] which is a finite subset of Real line. How I impose this condition in my formulation?

@ Aravind, Well the preliminary guess and the range to look for should be intuitively obtained from the "Physics" of the problem. This is why I had included the option of Muller's method, which requires three guesses to begin with. When I use it, I try to give far away points as guesses. And then it tries to interpolate a quadratic polynomial through the points and then fine tunes itself in the successive steps to reach to the solution.

May the Iterative method for two nonlinear equations has the best efficient for find the approximate solutions.

You can see chapter 10 in  the book 

' Numerical Analysis by Richard Burden and J. Douglas Faires'

 8th Ed. or 9th Ed.

good luck

In my opinion the Barzilai and Borwein method is very effective for solving a nonlinear system of equations. This method is convergent for very complicate elliptic operators when the initial guesses are faraway from the approximate solution. See for example

T. D. Todorov, Nonlocal problem for a general second-order elliptic operator, Computers & Mathematics with Applications, vol. 69, issue 5, 2015, pp. 411-422.

Most of the Newtonian methods need the initial guesses to be very close to the exact solution. Otherwise they are divergent. On the contrary the Barzilai and Borwein method slightly depends on the initial guesses.

This is a good question. I think that you can use Runge-Kutta method for non stiff equations through Ode45 solver of Matlab bakage

Memory problems?   WIth 12x12 matrices?  Unbelievable.  I think the origin of the problem is connected with high powers, thus causing purely numerical troubles, either underflows or overflows.