http://mypeoplepc.com/members/jon8338/math/id49.html

This is the intersection of a line with a step graph.

For 10^100 points, to save labor select 10 of them and find the curve that 
passes through them.  Solve the matrix for the coefficients of ten 9th 
degree polyomials.  Input the 10 abcissas into the independent variable of 
the equations to equal the 10 ordinates and solve for the coefficients using 
linear algebra.

I'm sure this is common practice.  While the equation doesn't have to be a 
polynomial, all functions are power series anyway.  If the character is 
mostly manifested in the first 9 terms of the series, it works better.

Jon Giffen