Hello, For a function, I want to calculate the range where one root exists. For example, f(x) has a root in subrange [a1,b1], another root in subrange[a2,b2] in a total range of [X,Y]. Is any such algorithm available to calculate the ranges [a1,b1],[a2,b2]? Thanks in advance.

I remember asking about this when I took numerical analysis. It really does seem like an important part of the problem.

Unfortunately, even if f is differentiable on (a,b), it can have infinitely many roots.

What you might try is making some more constraining assumptions about your function. Something like differentiable on [a,b] with the derivative bounded by a constant. That enables you to calculate neighborhoods around a non-root point where you know there can't be any roots. I'm guessing you can slowly fill up the interval that way, finding most roots in the process. Even still, imagine the function f(x)=0. How would you know it doesn't have a little bump somewhere in your interval?

I wish I knew of a paper about this. I would imagine someone has 'solved' this problem before.

I found the method "bracketing method" in the following link

http://civilwares.free.fr/27102005/Applied%20Numerical%20Methods%20with%20MATLAB%20for%20Engineers%20and%20Scientists/cha92657_ch05.pdf

It works fine for my purpose. Thanks to all.

If the function f is an analytic function some powerful tools from complex analysis can be used. Specially Rouche's theorem can be used to determine the number of roots inside some closed contour. If there is only one root inside the contour, even the position of the root can be determined by a numerical integration.
More information and some samples can be found in chapter 3 of the book Complex Analysis by Freitag and Busam .
Unfortunately there aren't such elegant theorems available for non analytic function's. But for continuous (and smooth enough) functions one can take many samples and use the fact that if f(a) and f(b) have different signs then there exists at least one root in the interval (a,b) .