Is there any criteria or a way to take a good initial guess for solving non-linear system of equations using Newton Method.Or one shoud try different guess randomly again and again until one get the required result.Working with non linear system of equations and using Newton method.I will be thankful to any guidance in this regard
Hi, another approach is to use homotooy methods ie to start with an initial equation with a known solution as f1 (x)=x -x0 =0 and to build a family of equations depending on a parameter lambda:
G (x)=lambda*f (x) + (1-lambda)*f1 (x)=0
The above equation is solved numerically starting with lambda=0 and x=x0 until lamba=1is reached.
NR method is guaranteed to converge if the initial guess x0 is close enough, but it is hard to make a clear statement about what we mean by "close enough" because this is highly problem specific. If you're lucky, Newton-Raphson might still work even if this initial guess is quite far from the actual solution. If you're unlucky, you can try another guess. There are limited ways to find an initial guess.
1) A sketch of the graph of f(x) can help you decide on an appropriate initial guess x0 for a particular problem.
2) It sometimes helps if you can isolate the roots in intervals. If you can find a
Sir Luis, The basic idea of Newton method is now clear to me and I think I will get the solution.
And Sir Vikash you gave me some more useful ideas.I will also work on that.
I m thankful to both of you.
And Sir Richard I will also try your idea of Homotopy method for my problem.To me it looks that Newton Method is more useful but will see now what gives better result.
Dear Shakeel
A VERY NICE METHOD TO KNOW IF THE GUESS IS CONVERGE OR NOT, IS THE BASINS OF ATTRACTION OF THE ROOTS OF THE NONLINEAR EQUATION (SYSTEM OF NONLINEAR EQUATION).
One can plot the basins of attraction of the roots, and make a region around it, and then plot the basins of attractions of each root, and assign a special color for all points that are converge to special root.
indeed, you can assign all points which are diverge by a black color.
in the attachment you will find an example of what I say, it is the graph of the basins of attraction of the roots of f(x)=x^5-1, which have 5 roots.
the points which have the same color, means that they will converge to the same root, and the black points means that these points will diverge (those points together are called "Julia set").
I didn't make to find quickly more pictures like the one presented above by Obadah Said Solaiman. But the conclusion is simple and obvious: there are cases, not very sophisticated ones, when "basins of retractions" happen to be arbitrarily close to the solution, interspersed with basins of attraction. Consequently, there is no simple, always working answer for your question. Personally I'm more and more inclined to use interval methods (see https://en.wikipedia.org/wiki/Interval_arithmetic ), which may but need not to be linked with Newton's method.
First few steps of bisection method give you a suitable starting point.
This has been discussed before. Please see the link below.
https://www.researchgate.net/post/How_we_take_suitable_initial_values_in_Newton_Raphson_based_function_libraries/1
You can use a quasi-Newton method (Broyden, ...) first because it has a larger convergence domain. Then switch to the true Newton-Raphson method for quadratic convergence rate near the root.
Extreme example : Newton's method always diverges for the function x -> arctan(x), whereas the bisection method converges.
There are enough answers reported. Aren't there.
If Dr. Shakeel Khan is not satisfied yet. I would like to ask him to inform others.
Sir Jafar Biazer, You are right, there are enough answers and I am almost done for my problem.All the researchers above gave their valuable ideas and directions to me.I have now many options to see this problem.
I am thankful to all of them and to you, Sir.
Dear Dr.S.A.Khan,
I'm glad you are satisfied, and have many options.
Wish you all the best
if you know the solution of the nonlinear system then you can choose the initial guess very close to it. But if you don't know the solution, then you search for its neighborhood and choose the initial guess within it.
You can use some globalization strategy, such that, trust region.
Now, the ball of convergence of Newton's method is already established, depends on the solution
Choose the initial guess within the neighborhood of the solution since the method has local convergence property
you can use the w4 method it s the extension of the NR method, but this one you are able to choose initial conditions that are not close enough to the solution.
This article may help you "The w4 method a new multi-dimensional root-finding scheme for a non linear systems of equations".
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