Approximating (x^2+a^2)^(1/4) w/ Error < 1e-6

[dirk_mec1]

I want to have the following function approximated (in elementary functions) within a error of 1e-6:

(x^2+a^2)^{1/4}\ \forall x\ \in [-1,1],\ a\in \mathbb{R}If I use Legendre polynomials I have to use a lot to get convergence (since if I use an estimate of the error the error decreases with n^3/2 with n the number of polynomials). So I suppose that I can use taylor polynomials, right? But taylored around 0 the polynomials tends to have a large error moving away from zero. So my idea is to use several taylor polynomials for certain intervals. Is this an idea or do I have to do something else?

 

[Stephen Tashi]

From the point of view of computer programming, the most efficient approximations of commonly encountered functions are approximations by rational functions ( ratios of polynomials). For example, if you look at the famous book by Abramawitz and Stegun, you find that sort of approximation.