Solve complex equation numerically

[enricfemi]

all i want to do is finding all the solution in a area.

i know the Newton's method, but the problem is how can i divide the area into much smaller areas that i can judge it whether include a solution and can only have one solution.

and how can judge it?

any talk will be appreciated. it's really messing me up.

 

[D H]

To start with, what problem are you trying to solve? Numerical methods is a huge topic; a reasonable introduction to the concept deserves a book and a college class. In fact, Introduction to Numerical Methods is a class at several colleges.

 

[enricfemi]

thanks D H!
i have passed the class of numerical methods. but there isnot introduce to the solution about complex problems.
the problem is i have a complex function(can't analysis), and i want to find out when it go to zero

 

[D H]

Finding a zero of a multidimensional function is a tougher (much tougher) problem than is single dimensional zero finding. There is no good general purpose technique for finding a zero of a multidimensional function.

That said, you may want to use Newton's method as extended to multiple dimensions (you will need the Jacobian) or use Broyden's method, which is an extension of the secant method to multiple dimensions. Both techniques can fail miserably.

 

[enricfemi]

more than considering the techniques, what puzzling me now, indeed, is the problem of finding zero of a multidimensional function.

i need a law, like in one dimension, can identify whether exist a zero of this function in a small area.
whether exist such a law? i can't find it out in any calculus book.

 

[D H]

Since such a "law" does not even exist for a real function of one real variable, what makes you think it will exist for a multivariate function? There is a rule for real analytic functions of one variable: If f(x) is real analytic over some interval (a,b) and f(a)*f(b) < 0, then f(x) has an odd number of zeros on (a,b). You don't know how many, and you don't know much of anything at all if f(a)*f(b) > 0.

Complex analyticity offers an even more powerful tool: the http://mathworld.wolfram.com/VariationofArgument.html" technique. If f(z) is complex function that is not identically zero and is analytic over some region R, evalate the variation of the argument around the boundary of R, divide by 2*pi, and voila! you have a count of the number of zeros in R.

All bets are off if you are working with a non-analytic complex function or a function that maps from Rⁿ to R^m, n,m>1. There is no general rule, period.

 

[enricfemi]

to tell the truth, i am not sophisticated in complex analysis. but just now i have found some books and read the part about principle of argument.
the only thing i confused is how to calculate the variation of the argument in a closed boundary, both analytic and numerically.

 

[D H]

Google is your friend.

It took me a short time to find these:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.3379"
http://cat.inist.fr/?aModele=afficheN&cpsidt=17978784"