Most of the methods provided for functions of a single real argument have been based on truncated Chebyshev expansions. This method of approximation was adopted as a compromise between the conflicting requirements of efficiency and ease of implementation on many different machine ranges. For details of the reasons behind this choice and the production and testing procedures followed in constructing this chapter see Schonfelder (1976).

Basically, if the function to be approximated is fx, then for x∈a,b an approximation of the form is used (∑' denotes, according to the usual convention, a summation in which the first term is halved), where gx is some suitable auxiliary function which extracts any singularities, asymptotes and, if possible, zeros of the function in the range in question and t=tx is a mapping of the general range a,b to the specific range [-1,+1] required by the Chebyshev polynomials, Trt. For a detailed description of the properties of the Chebyshev polynomials see Clenshaw (1962) and Fox and Parker (1968).

The essential property of these polynomials for the purposes of function approximation is that Tnt oscillates between ±1 and it takes its extreme values n+1 times in the interval [-1,+1]. Therefore, provided the coefficients Cr decrease in magnitude sufficiently rapidly the error made by truncating the Chebyshev expansion after n terms is approximately given by That is, the error oscillates between ±Cn and takes its extreme value n+1 times in the interval in question. Now this is just the condition that the approximation be a minimax representation, one which minimizes the maximum error. By suitable choice of the interval, [a,b], the auxiliary function, gx, and the mapping of the independent variable, tx, it is almost always possible to obtain a Chebyshev expansion with rapid convergence and hence truncations that provide near minimax polynomial approximations to the required function. The difference between the true minimax polynomial and the truncated Chebyshev expansion is seldom sufficiently great enough to be of significance.

The evaluation of the Chebyshev expansions follows one of two methods. The first and most efficient, and hence the most commonly used, works with the equivalent simple polynomial. The second method, which is used on the few occasions when the first method proves to be unstable, is based directly on the truncated Chebyshev series, and uses backward recursion to evaluate the sum. For the first method, a suitably truncated Chebyshev expansion (truncation is chosen so that the error is less than the machine precision) is converted to the equivalent simple polynomial. That is, we evaluate the set of coefficients br such that The polynomial can then be evaluated by the efficient Horner's method of nested multiplications, This method of evaluation results in efficient methods but for some expansions there is considerable loss of accuracy due to cancellation effects. In these cases the second method is used. It is well known that if then and this is always stable. This method is most efficiently implemented by using three variables cyclically and explicitly constructing the recursion.

That is, The auxiliary functions used are normally functions compounded of simple polynomial (usually linear) factors extracting zeros, and the primary compiler-provided functions, sin, cos, ln, exp, sqrt, which extract singularities and/or asymptotes or in some cases basic oscillatory behaviour, leaving a smooth well-behaved function to be approximated by the Chebyshev expansion which can therefore be rapidly convergent.

The mappings of [a,b] to [-1,+1] used range from simple linear mappings to the case when b is infinite, and considerable improvement in convergence can be obtained by use of a bilinear form of mapping. Another common form of mapping is used when the function is even; that is, it involves only even powers in its expansion. In this case an approximation over the whole interval [-a,a] can be provided using a mapping t=2 x/a 2-1. This embodies the evenness property but the expansion in t involves all powers and hence removes the necessity of working with an expansion with half its coefficients zero.

For many of the methods an analysis of the error in principle is given, namely, if E and ∇ are the absolute errors in function and argument and ε and δ are the corresponding relative errors, then If we ignore errors that arise in the argument of the function by propagation of data errors, etc., and consider only those errors that result from the fact that a real number is being represented in the computer in floating-point form with finite precision, then δ is bounded and this bound is independent of the magnitude of x. For example, on an 11-digit machine (This of course implies that the absolute error ∇=xδ is also bounded but the bound is now dependent on x.) However, because of this the last two relations above are probably of more interest. If possible the relative error propagation is discussed; that is, the behaviour of the error amplification factor xf′x/fx is described, but in some cases, such as near zeros of the function which cannot be extracted explicitly, absolute error in the result is the quantity of significance and here the factor xf′x is described. In general, testing of the functions has shown that their error behaviour follows fairly well these theoretical error behaviours. In regions where the error amplification factors are less than or of the order of one, the errors are slightly larger than the above predictions. The errors are here limited largely by the finite precision of arithmetic in the machine, but ε is normally no more than a few times greater than the bound on δ. In regions where the amplification factors are large, of order ten or greater, the theoretical analysis gives a good measure of the accuracy obtainable.

It should be noted that the definitions and notations used for the functions in this chapter are all taken from Abramowitz and Stegun (1972). You are strongly recommended to consult this book for details before using the methods in this chapter.

Four functions provided here are symmetrised variants of the classical (Legendre) elliptic integrals. These alternative definitions have been suggested by Carlson (1965), Carlson (1977b) and Carlson (1977a) and he also developed the basic algorithms used in this chapter.

The symmetrised elliptic integral of the first kind is represented by where x,y,z≥0 and at most one may be equal to zero.

The normalization factor, 12 , is chosen so as to make If any two of the variables are equal, RF degenerates into the second function where the argument restrictions are now x≥0 and y≠0.

This function is related to the logarithm or inverse hyperbolic functions if 0<y<x, and to the inverse circular functions if 0≤x≤y.

The symmetrised elliptic integral of the second kind is defined by with z>0, x≥0 and y≥0, but only one of x or y may be zero.

The function is a degenerate special case of the symmetrised elliptic integral of the third kind with ρ≠0 and x,y,z≥0 with at most one equality holding. Thus RDx,y,z=RJx,y,z,z. The normalization of both these functions is chosen so that The algorithms used for all these functions are based on duplication theorems. These allow a recursion system to be established which constructs a new set of arguments from the old using a combination of arithmetic and geometric means. The value of the function at the original arguments can then be simply related to the value at the new arguments. These recursive reductions are used until the arguments differ from the mean by an amount small enough for a Taylor series about the mean to give sufficient accuracy when retaining terms of order less than six. Each step of the recurrences reduces the difference from the mean by a factor of four, and as the truncation error is of order six, the truncation error goes like 4096-n, where n is the number of iterations.

The above forms can be related to the more traditional canonical forms (see Section 17.2 of Abramowitz and Stegun (1972)), as follows.

If we write q=cos2⁡ϕ, r=1-m.sin2⁡ϕ, s=1-n.sin2⁡ϕ , where 0≤ϕ≤12π , we have

the classical elliptic integral of the first kind: the classical elliptic integral of the second kind: the classical elliptic integral of the third kind: Also the classical complete elliptic integral of the first kind: the classical complete elliptic integral of the second kind:

For convenience, S class contains methods to evaluate classical and symmetrised elliptic integrals.

The methods for Bessel and Airy functions of a real argument are based on Chebyshev expansions, as described in [Functions of a Single Real Argument]. The methods provided for functions of a complex argument, however, use different methods. These methods relate all functions to the modified Bessel functions Iνz and Kνz computed in the right-half complex plane, including their analytic continuations. Iν and Kν are computed by different methods according to the values of z and ν. The methods include power series, asymptotic expansions and Wronskian evaluations. The relations between functions are based on well known formulae (see Abramowitz and Stegun (1972)).

The method s21ba for RC is largely included as an auxiliary to the other methods for elliptic integrals. This integral essentially calculates elementary functions, e.g., In general this method of calculating these elementary functions is not recommended as there are usually much more efficient specific methods available in the Library. However, s21ba may be used, for example, to compute ln⁡x/x-1 when x is close to 1, without the loss of significant figures that occurs when ln⁡x and x-1 are computed separately.