Can anyone guide me that which algorithm or Method is behind Mathematica Command " NSolve"
All methods based on Groebner bases are exponential in complexity. Polyhedral homotopy might be better in theory and is often better in practice, in terms of speed at least. Though there is an initialization step that is not cheap.
I do not know of any solver in Mathematica which would be faster than NSolve and not require an initial guess. Maybe try FindMinimum or NMinimize on the sum of squares is about all that comes to mind.
Are these methods polynomial or exponential in complexity?
It might be using a sparse homotopy method (in version 10). Or a method based on numeric Groebner bases and eigensystem computation. Depending on the specifics either one could be slow.
Thanks once again,
Can you please suggest any other similar command in mathematica which dont require any initial guess.
its being a long time i am facing problem in using Mathematica Command "NSolve". The Problem is that i have a system of 10 non-linear equations with 10 variables.
Upto a system of 8 equations i am getting my results, but for a system of 10 non-linear equations with 10 variables, i kept running my program but dont get my results even after many many hours long time. So i just wanted to know which method or algorithm is behind "NSolve" so that i may figure out anything.
There are different methods based on type of input (single equation, linear system, polynomial system, domain specification, etc.).