The problem is not simple and there are book on the solution equilibrium. On book is written by smith and missen. I can suggest you these papers for an efficient equilibrium algorithm.

The reference for the smith book is

Chemical Reaction Equilibrium Analysis: Theory and Algorithms, W.R. Smith and R.W. Missen

This is my second paper

If I understand the question correctly you have an equation of the form f(x) = 0. Most solution techniques would involve an inverse of the Jacobian which for many problems can be singular or very ill-conditioned that solution will diverge. 

Some newly developed algorithms can help overcome the singularity or the ill-conditioning of the Jacobian. They were also shown to be less sensitive to initial guess as Newton's method.

I attached a couple of papers that may help

Every deterministic numerical solver (such as fsolve) will be sensitive to the initial guess. But you can easily transform your problem to the optimization problem and try to use some heuristic methods, such as Particle Swarm Optimzation or Simulated Annealing. There are plenty of nice MATLAB submission with those algorithms.

Thanks, Gianpiero. I will see whether the method in your paper can be used in my problem.

Hi, Tarek. Yes, what I am doing is finding the root of f(x) = 0. Thank you very much for your valuable information  and papers.

Hi, Jan. Yes, it is a good method to transform the problem to the optimization problem. I will try this way.  Thank you very much for the suggestions.

Yes, Prof. Yan. The analysis of the problem is very important. An iteration method developed by myself based on the equilibrium equation performs much better in accuracy and efficiency than the solver in Matlab, but it is only effective for some families of the solutions, not all of them.

Numerical integration to the equations of motion of a satellite (spacecraft) in Sun-Earth system, for example, are not suitable if the trajectories passing near one of the primaries. That is because the radius vector becomes very small or close to zero. In this case, it is better to use one of the “regularization” methods that transforms the equations of motion to the harmonic oscillations. The position vector r will disappear from the denominator of the new system of equations. You can see, for example,  the book:  “Linear and Regular Celestial Mechanics”, by Stiefel and Scheifele.

Thanks, Prof. Saad. I will look at the “regularization” method.

Dear Yue Wang,

I am new in ResearchGate, but I try to help you here.

I have developed the so-called Multi-Dimensional Bisection Method, which is robust, and has quite high efficiency.

"

This robust technique is able to find the submanifolds of the roots of a system of non-linear equations in any high dimension and co-dimension.

The MDBM can be used for the determination of multiple boundary curves and it can even find the closed curves of stable and unstable islands in the stability chart automatically.

"

Theoretically it work is any high parameter dimension, but due to some memory consumption problem it works for m=1,2,3,4,(5),((6)) parameter dimension. The number of equation (n) is also arbitrary, but the best case if n=(m),m-1,m-2.

The matlab version is available for free, with many example. See my web page:

http://www.mm.bme.hu/~bdaniel/

Best Regards,

Daniel Bachrathy

Article Bisection method in higher dimensions and the efficiency number

Dear @Daniel_Bachrathy, have you ever use your nonlinear solver on the solution of differential equations with an implicit method?. I'm just wondering that is it really better than newton-raphson method while solving nonlinear algebraic system of equtions appearing in implicit methods?

sir i myself always remain in problem but help non

I have found fsolve to be pretty reliable, though it can get trapped in a local minimum that is not a solution. You might check out homotopy methods, which "guarantee" convergence to a solution (though not necessarily in the time you have available). They don't get trapped in a local minimum, but they only converge to the nearest solution. To find other solutions, you have to try other starting points.

For example:

Chein-Shan Liu, Weichung Yeih, Chung-Lun Kuo, and Satya N. Atluri. A scalar homotopy method for solving an over/under-determined system of non-linear algebraic equations. Computer Modeling in Engineering and Sciences, 53(1):47--71, July 2009.

Cheng-Yu Ku, Weichung Yeih, and Chein-Shan Liu. Solving non-linear algebraic equations by a scalar Newton-homotopy continuation method. Intl J. Nonlinear Sciences & Numerical Simulation, 11(6):435--450, 2010.

Tarek A. Elgohary. Novel Computationaal and Analytic Techniques for Nonlinear Systems Applied to Structural and Celestial Mechanics. PhD thesis, Texas A & M University, May 2011.

Weichung Yeih, Cheng-Yu Ku, Chein-Shan Liu, and I-Yao Chan. A scalar homotopy method with optimal hybrid search directions for solving nonlinear algebraic equations. Computer Modeling in Engineering and Sciences, 90(4):255--282, 2013.

I will try the homotopy method. Thank you very much, Professor Van Zandt.

You could refer to this source code using matlab. It solves similar equations using many methods (load control method - displacement control method - arc length method - work control method)

Code Solving Nonlinear Equations Using Matlab

Code Solving Nonlinear Equations Using Matlab

Khaled Megahed Khaled, thanks a lot!

Yue Wang - thanks for the complement, but the title should be just "Dr."

For the homotopy method, I suggest you check out these packages: PHCpack, PHClab, and Hom4PS-3.

In the MDBM method, the system (implicit) no-linear equation can be solved and it provides the detected (multiple) solutions.

If the initial grid is not fine enough, then there is a chance to miss some solution if the number of equation is equal to the number of parameter.

However, if an additional variable (parameter is presented), this chance is greatly reduced.

p.s.:

for the free mdbm matlab code see: https://github.com/bachrathyd/MDBM-Matlab

or

https://www.mathworks.com/matlabcentral/fileexchange/69414-multi-dimensional-bisection-method

I can cooperate with you if you wish, send the equations. There is a new method called piecewise analytic method (PAM).

Piecewise analytic method (PAM) is a new technique used for solving nonlinear differential equation. The advantages of PAM are( these points are proofed):

1. It gives a general analytic form that can be used in differentiation and integration.

2. It can solve highly non-linear differential equation.

3. The accuracy and error can be controlled according to our needs very easily.

4. It can solve problems which other famous techniques can’t solve.

5. In some cases, it gives the exact solution.

The Method still needs too much work.

If anybody wishes to cooperate with me, he is welcome.

[email protected]

Preprint Piecewise Analytic Method VS Runge-Kutta Method (Comparative Study)

Preprint Solving Nonlinear 2 nd Order Differential Equations Using Pi...

I have a problem involved an external simulator, it depends on integer values and also real ones, but generates a lot of local maximums to be find (1-10 in a variable taking the other fixed). Every simulation takes from 1 to 60 seconds (as better is the performance, more time is needed)

Exists a method that throw some random multidimensional vector?

The library should help to use computer clusters.