I suggest check for any discontinuities in range and start iteration further away from local minima.

To overcome local minima, you could try to use simulated annealing or one of its many variants, see, e.g. the article linked below.

Please explain in more detail how you use an iterative procedure to check convergence, why you think of local minima, and what the original problem is.

http://www.sciencedirect.com/science/article/pii/089571779390204C

Respected Professor, I was just trying to solve one problem in electromagnetic. I got THREE coupled system of transcendental equations and each equation has the basic form:

(S1*S2 - S5*S6)*(S3*S4 - S5*S6) = 0 where S1 to S6, all are transcendental equations. I wanted to use bisection method to solve this problem (No Newton's method where initial guess is required and I do not know the guess for higher order root). Final convergence is obtained using iterative procedure (accuracy = 1e-12).

I can easily obtain the solution for HEM12 or HEM33 or HEM2,13 etc modes. In case of HEM11 and HEM22 modes, solutions regions are repeating but no solution is obtained. If one solution is satisfied, other one does not. But the same code is able to solve for other modes. I think, solutions for HEM11 and HEM22 modes got stuck in LOCAL minima. How can I overcome this problem in iteration procedure?

Hm, this sounds more like a periodic orbit. Do you use a one dimensional bisection method  for each of the three coupled equations, say taking one step of bisection for the first equation, then one for second, then for the third, and then for the first again, and so on? Or are you subdividing the whole parameter space into boxes that are becoming smaller in each "bisection" step? In the first case, you could try  a few steps for the first, then a number of steps for the third, then some for the second, ... Of course, this is just an experimental type of approach, with no guarantee for success. Perhaps you can learn something about the nature of your problem by monitoring each of the factors of the three equation. Looking for a zero of a product means that one of the factors has to vanish. - Did you try to  reduce the interval sizes? - In Newton-type methods, you can enforce slower but steady convergence by reducing the step by a damping in some cases. Perhaps you can use a sort of damping in your algorithm somehow. Just some ideas that might, or might not help.

Respected Prof. Herbert, The solution procedure is similar to your 1st case. As per your recommendation I will also try to use  "Newton-type methods" for system of non-linear equations. I've basic idea but can you suggest some reference (including book)s to learn the Newton type methods in detail. A good 'BOOK' always give better understanding the solution procedure (i.e. region of convergence, error, iteration no, improvement  etc etc). I'm waiting for your kind reply.

Respected Prof. Herbert, In my electromagnetic problem, each equation is in the form of multiplication of TWO factor. At solution point, one factor = 0. Considering 1st factor of 1st equation  = 0 at (x,y,z) solution point. Now remaining TWO equations have FOUR factors. At solution point I have to use trial-error method to select corresponding TWO other factors. As this method will be trial-error based method, I do not want to go for it.

Respected Prof. Leigh, I took long time to investigate 'discontinuity'. I used 32 bit OS and MATLAB 2010a. Bisection method is stopped if relation accuracy < 1e-12. In my (electromagnetic) problem, all equations are transcendental equations and function of tan() as given below:

Equation 1: function of tan(x) and 'yz' term comes as multiplication factor of tan(x). Therefore, for fixed value of 'y' and 'z', equation 1 will give solution for 'x' only. Similarly, equation 2 and 3 are function tan(y) and tan(z) respectively, they will give the solution for 'y' and 'z' only respectively. 'ZERO' crossing point is a toot. When solution found, I have plotted all equations and it is found that they are crossing ZERO. I did not found tan(pi/2) point as a solution.

This is my procedure. Can you please guide me, how to check 'discontinuity' ? I'm waiting for your kind reply, Prof. Leigh

tan (THETA ) is discontinuous at THETA =PI/2, 3PI/2  ...etc . you would have to work in 3D domains excluding these points.

Respected Prof. Leigh, I've checked these already. In case of 2D or 3D problem, I can check graphically. In higher order problem, 'graphical checking procedure' may not be possible. In that case, how can I check discontinuity?

In code, I can use if-else statement to check the argument. I will do it. Is there any other procedure? I'm waiting for your kind reply, Prof. Leigh.