How to find the roots of a function whose analytical form is not known, rather the function is available as a tabulated set of values?

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To find the roots of a function, we can generally use bisection method or Newton's method. For a function f(x), this is possible only when we have an analytical expression for the x-dependence of f(x).

I am trying to find the roots of such a function where I don't know the exact form of the function, rather I have a tabulated data for the values of f(x) for each values of x in a particular range of x. I am writing my program in C and I am using a for-loop to calculate f(x) for each value of x by solving a non-linear equation using bisection method and tabulating the data. Now I need to find the roots of the function f(x).

Can anyone help me with any suitable method or algorithm for the problem?

Thanks in advance!

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1 Answers

You know from where the sign changes that a root has to be between two points.

Take several nearby points, put a polynomial through them, and then solve for the root of that polynomial using Newton's method.

From your description it looks like you should be able to calculate your function at this new point. If so, then I would suggest that you calculate the value at this point, add the two nearest neighbors, calculate a parabola and solve for the root of that. If your function is smooth and has a non-zero derivative at the root, this step will make your estimate of the root several orders of magnitude more accurate.

(You can repeat again for even more accuracy. But the increased accuracy at this point may be on par with the numerical errors in your estimate of the value of the function.)

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