It's often good to exploit special properties of the function under consideration.

In this case, if your processor has a fast *sqrt*, then
remove the singularity at *x=1* by approximating the
following function instead.

Using a Taylor series is almost always wrong. (It's one of those ideas taught in freshman classes because they (that is, the ideas, not the freshmen) are simple, not because they're appropriate when solving applications. Another example is the linked list.) Nevertheless, here is the Taylor series, expanded about the origin, and the error plot.

It's almost always better to expand a Taylor series about the
center of the interval, *x=0.5* in this case.

The Taylor series is designed to fit the first *n* derivatives
at one point. However, we're interested in approximating a function
over some interval. In this case,a Chebyshev approximation is far
better than a Taylor approximation, since it is a polynomial
approximation that comes closer to minimizing the maximum error over a
given interval. (However, contrary to a common belief, it doesn't
exactly do this.) Again, use symmetry and make the interval [0,1].
Here is the 6th degree Chebyshev, in both a Chebyshev basis and a
power basis.

or

The Pade approximation is a formal transformation of the Taylor
series into a rational expression. Altho it does not add any
information, and calculating it does not refer back to the
*arcsin*, yet it is often a better approximation then the
Taylor series it was derived from.

Here is the Pade approximation, with 7 d.f., derived from the Taylor series centered about the origin. When counting the number of degrees of freedom, note that one leading coefficient can be normalized to be one (altho Maple doesn't automatically do this).

The Pade approximation centered about *x=0.5* is this.

Altho it is accurate over most of the interval, it is useless
because of the singularity at *x=.9626070222*. This happens sometimes
with rational approximations.

The Chebyshev-Pade approximation is just the formal transformation of the Chebyshev approximation to a, rational, quotient.

or

The minimax polynomial approximation is this.

The minimax is always the best polynomial approximation. Perhaps the reason that it is not used more is that it is more difficult to compute.

Finally, rational minimax approximations would be expected to be
even better than polynomials, especially when the function has
non-polynomiallike things, such as singularities. However, in most
cases, Maple and Mathematica fail to compute them, with the error
message *Error, (in numapprox/remez) error curve fails to oscillate
sufficiently; try different degrees*. These cases failed here:
[3,3], [4,2], [2,4], [1,5]. Here is the [5,1] quotient. Note that
the error curve is not equally oscillating. This is probably caused
by insufficient precision during the computation. It could be fixed
by increasing Digits, but is left here as a warning of what can
happen, and of why it's useful to plot the error curves.

Here is the [4,2] quotient.

Finally, here is the [2,4] quotient.

Another test, set Digits:=30, then tried to compute every rational minimax approximation with total degree up to 19. The following cases succeeded; all others failed.

[0,1], [1,0], [0,2], [1,1], [2,0], [0,3], [1,2], [3,0], [1,3], [4,0], [1,4], [5,0], [1,5], [6,0], [n,0] for n=7 to 9. The failures of the hi-degree polynomial minimax approximations may be worth further study.

Email: wrfATecse.rpi.edu

http://wrfranklin.org/

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ECSE Dept., 6026 JEC, Rensselaer Polytechnic Inst, Troy NY, 12180 USA

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