s Chapter Contents
s Chapter Introduction (PDF version)
NAG C Library Manual

NAG Library Chapter Introduction

s – Approximations of Special Functions

+ Contents

1  Scope of the Chapter

This chapter is concerned with the provision of some commonly occurring physical and mathematical functions.

2  Background to the Problems

The majority of the functions in this chapter approximate real-valued functions of a single real argument, and the techniques involved are described in Section 2.1. In addition the chapter contains functions for elliptic integrals (see Section 2.2), Bessel and Airy functions of a complex argument (see Section 2.3), complementary error function of a complex argument and various option pricing functions for use in financial applications.

2.1  Functions of a Single Real Argument

Most of the functions 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 xa,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
yt=r=0 n-1brtr=r=0 n-1CrTrt.
The polynomial can then be evaluated by the efficient Horner's method of nested multiplications,
yt=b0+tb1+tb2+tbn- 2+tbn- 1.
This method of evaluation results in efficient functions 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
bn-1=Cn-1 bn-2=2tbn-1+Cn-2 bj-0=2tbj+1-bj+2+Cj,  j=n-3,n-4,,0
r=0 CrTrt=12b0-b2
and this is always stable. This method is most efficiently implemented by using three variables cyclically and explicitly constructing the recursion.
That is,
α = Cn-1 β = 2tα+Cn-2 γ = 2tβ-α+Cn-3 α = 2tγ-β+Cn-4 β = 2tα-γ+Cn-5 say ​α = 2tγ-β+C2 β = 2tα-γ+C1 yt = tβ-α+12C0
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 functions 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
E fx E xfxδ ε x f x fx δ.
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 xfx/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 xfx 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 functions in this chapter.

2.2  Approximations to Elliptic Integrals

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
RF x,y,z = 12 0 dt t+x t+y t+z ,
where x,y,z0 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
RC x,y = RF x,y,y = 12 0 dt t+x t+y ,
where the argument restrictions are now x0 and y0.
This function is related to the logarithm or inverse hyperbolic functions if 0<y<x, and to the inverse circular functions if 0xy.
The symmetrised elliptic integral of the second kind is defined by
RD x,y,z = 32 0 dt t+x t+y t+z3
with z>0, x0 and y0, 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
RJ x,y,z,ρ = 32 0 dt t+x t+y t+z t+ρ
with ρ0 and x,y,z0 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=cos2ϕ, r=1-msin2ϕ, s=1-nsin2ϕ , where 0ϕ12π , we have
the classical elliptic integral of the first kind:
Fϕm = 0ϕ 1-m sin2θ -12 dθ = sinϕ RF q,r,1 ;
the classical elliptic integral of the second kind:
Eϕm = 0ϕ 1-m sin2θ 12 dθ = sinϕ RF q,r,1 -13m sin3 ϕ RD q,r,1
the classical elliptic integral of the third kind:
Πn; ϕm = 0ϕ 1-n sin2θ -1 1-m sin2θ -12 dθ = sinϕ RF q,r,1 + 13 n sin3 ϕ RJ q,r,1,s .
Also the classical complete elliptic integral of the first kind:
Km = 0 π2 1 - m sin2θ -12 dθ = RF 0,1-m,1 ;
the classical complete elliptic integral of the second kind:
Em = 0 π2 1-m sin2 θ 12 dθ = RF 0,1-m,1 - 13 m RD 0,1-m,1 .
For convenience, Chapter s contains functions to evaluate classical and symmetrised elliptic integrals.

2.3  Bessel and Airy Functions of a Complex Argument

The functions for Bessel and Airy functions of a real argument are based on Chebyshev expansions, as described in Section 2.1. The functions provided for functions of a complex argument, however, use different methods. These functions 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)).

2.4  Option Pricing Functions

The option pricing functions evaluate the closed form solutions or approximations to the equations that define mathematical models for the prices of selected financial option contracts. These solutions can be viewed as special functions determined by the underlying equations. The terminology associated with these functions arises from their setting in financial markets and is briefly outlined below. See Joshi (2003) for a comprehensive introduction to this subject. An option is a contract which gives the holder the right, but not the obligation, to buy (if it is a call) or sell (if it is a put) a particular asset, S. A European option can be exercised only at the specified expiry time, T, while an American option can be exercised at any time up to T. For Asian options the average underlying price over a pre-set time period determines the payoff.
The asset is bought (if a call) or sold (if a put) at a pre-specified strike price X. Thus, an option contract has a payoff to the holder of maxST-X,0 for a call or maxX-ST,0, for a put, which depends on whether the asset price at the time of exercise is above (call) or below (put) the strike, X. If at any moment in time a contract is currently showing a theoretical profit then it is deemed ‘in-the-money’; otherwise it is deemed ‘out-of-the-money’.
The option contract itself therefore has a value and, in many cases, can be traded in markets. Mathematical models, such as the Black–Scholes model, give theoretical prices for particular option contracts using a number of assumptions about the behaviour of financial markets. Typically, the price, St, of the underlying asset at time t, is modelled as the solution of a stochastic differential equation for the return, dSt/St, on the asset price over a time interval, dt,
dSt St = μ dt + σdWt ,
where dWt is a Brownian motion. The drift, μ, defines the trend in the movements of S, while the volatility, σ, measures the risk and may be taken to be the standard deviation of the returns on the asset price. In addition the model requires a riskless money market account or bond with value, Bt, at time t and risk-free rate, r, such that
dBt = rBt dt .
This leads to the determination of the Black–Scholes option price, P, by a martingale method or via the derivation of the Black–Scholes partial differential equation,
P t + P S rS+ 12 2P S2 σ2 S2 - rP=0 .
For this case a closed form solution exists which is evaluated by nag_bsm_price (s30aac).
A number of different option types where the solution exists in closed form or as a closed form approximation are presented in this chapter. See Haug (2007) for an extensive listing of option pricing formulae.

3  Recommendations on Choice and Use of Available Functions

3.1  Elliptic Integrals

IMPORTANT ADVICE: users who encounter elliptic integrals in the course of their work are strongly recommended to look at transforming their analysis directly to one of the Carlson forms, rather than to the traditional canonical Legendre forms. In general, the extra symmetry of the Carlson forms is likely to simplify the analysis, and these symmetric forms are much more stable to calculate.
The function nag_elliptic_integral_rc (s21bac) for RC is largely included as an auxiliary to the other functions for elliptic integrals. This integral essentially calculates elementary functions, e.g.,
lnx =x-1RC 1+x2 2,x ,  x>0; arcsinx =xRC1-x2,1,x1; arcsinhx =xRC1+x2,1,etc.
In general this method of calculating these elementary functions is not recommended as there are usually much more efficient specific functions available in the Library. However, nag_elliptic_integral_rc (s21bac) may be used, for example, to compute lnx/x-1 when x is close to 1, without the loss of significant figures that occurs when lnx and x-1 are computed separately.

3.2  Bessel and Airy Functions

For computing the Bessel functions Jνx, Yνx, Iνx and Kνx where x is real and ν=0​ or ​1, special functions are provided, which are much faster than the more general functions that allow a complex argument and arbitrary real ν0. Similarly, special functions are provided for computing the Airy functions and their derivatives Aix, Bix, Aix, Bix for a real argument which are much faster than the functions for complex arguments.

4  Functionality Index

Airy function, 
    Ai, real argument nag_airy_ai (s17agc)
    Ai or Ai ′ , complex argument, optionally scaled nag_complex_airy_ai (s17dgc)
    Ai ′ , real argument nag_airy_ai_deriv (s17ajc)
    Bi, real argument nag_airy_bi (s17ahc)
    Bi or Bi ′ , complex argument, optionally scaled nag_complex_airy_bi (s17dhc)
    Bi ′ , real argument nag_airy_bi_deriv (s17akc)
    vectorized Ai, real argument nag_airy_ai_vector (s17auc)
    vectorized Ai ′ , real argument nag_airy_ai_deriv_vector (s17awc)
    vectorized Bi, real argument nag_airy_bi_vector (s17avc)
    vectorized Bi ′ , real argument nag_airy_bi_deriv_vector (s17axc)
    inverse hyperbolic cosine nag_arccosh (s11acc)
    inverse hyperbolic sine nag_arcsinh (s11abc)
    inverse hyperbolic tangent nag_arctanh (s11aac)
Bessel function, 
    I0, real argument nag_bessel_i0 (s18aec)
    I1, real argument nag_bessel_i1 (s18afc)
    Iα + n − 1(x) or Iα − n + 1(x), real argument nag_bessel_i_alpha (s18ejc)
    Iν, complex argument, optionally scaled nag_complex_bessel_i (s18dec)
    Iν / 4(x), real argument nag_bessel_i_nu (s18eec)
    J0, real argument nag_bessel_j0 (s17aec)
    J1, real argument nag_bessel_j1 (s17afc)
    Jα  +  n − 1(x) or Jα − n + 1(x), real argument nag_bessel_j_alpha (s18ekc)
    Jα ± n(z), complex argument nag_complex_bessel_j_seq (s18gkc)
    Jν, complex argument, optionally scaled nag_complex_bessel_j (s17dec)
    K0, real argument nag_bessel_k0 (s18acc)
    K1, real argument nag_bessel_k1 (s18adc)
    Kα + n(x), real argument nag_bessel_k_alpha (s18egc)
    Kν, complex argument, optionally scaled nag_complex_bessel_k (s18dcc)
    Kν / 4(x), real argument nag_bessel_k_nu (s18efc)
    vectorized I0, real argument nag_bessel_i0_vector (s18asc)
    vectorized I1, real argument nag_bessel_i1_vector (s18atc)
    vectorized J0, real argument nag_bessel_j0_vector (s17asc)
    vectorized J1, real argument nag_bessel_j1_vector (s17atc)
    vectorized K0, real argument nag_bessel_k0_vector (s18aqc)
    vectorized K1, real argument nag_bessel_k1_vector (s18arc)
    vectorized Y0, real argument nag_bessel_y0_vector (s17aqc)
    vectorized Y1, real argument nag_bessel_y1_vector (s17arc)
    Y0, real argument nag_bessel_y0 (s17acc)
    Y1, real argument nag_bessel_y1 (s17adc)
    Yν, complex argument, optionally scaled nag_complex_bessel_y (s17dcc)
Beta function 
    incomplete nag_incomplete_beta (s14ccc)
Complement of the Cumulative Normal distribution nag_cumul_normal_complem (s15acc)
Complement of the Error function, 
    complex argument, scaled nag_complex_erfc (s15ddc)
    real argument nag_erfc (s15adc)
    real argument, scaled nag_erfcx (s15agc)
    hyperbolic nag_cosh (s10acc)
Cosine Integral nag_cos_integral (s13acc)
Cumulative Normal distribution function nag_cumul_normal (s15abc)
Dawson's Integral nag_dawson (s15afc)
Digamma function, scaled nag_polygamma_deriv (s14adc)
Elliptic functions, Jacobian, sn, cn, dn 
    complex argument nag_jacobian_elliptic (s21cbc)
    real argument nag_real_jacobian_elliptic (s21cac)
Elliptic integral, 
        of 2nd kind, F(z , k ′  , a , b) nag_general_elliptic_integral_f  (s21dac)
    Legendre form, 
        complete of 1st kind, K(m) nag_elliptic_integral_complete_K (s21bhc)
        complete of 2nd kind, E (m) nag_elliptic_integral_complete_E (s21bjc)
        of 1st kind, F(ϕ | m) nag_elliptic_integral_F (s21bec)
        of 2nd kind, E (ϕ ∣ m) nag_elliptic_integral_E (s21bfc)
        of 3rd kind, Π (n ; ϕ ∣ m) nag_elliptic_integral_pi (s21bgc)
        degenerate of 1st kind, RC nag_elliptic_integral_rc (s21bac)
        of 1st kind, RF nag_elliptic_integral_rf (s21bbc)
        of 2nd kind, RD nag_elliptic_integral_rd (s21bcc)
        of 3rd kind, RJ nag_elliptic_integral_rj (s21bdc)
    real argument nag_erf (s15aec)
    complex argument, scaled nag_complex_erfc (s15ddc)
    real argument nag_erfc (s15adc)
    real argument nag_erfcx (s15agc)
Exponential Integral nag_exp_integral (s13aac)
Fresnel integral, 
    C nag_fresnel_c (s20adc)
    S nag_fresnel_s (s20acc)
    vectorized C nag_fresnel_c_vector (s20arc)
    vectorized S nag_fresnel_s_vector (s20aqc)
Gamma function nag_gamma (s14aac)
Gamma function, 
    incomplete nag_incomplete_gamma (s14bac)
Generalized factorial function nag_gamma (s14aac)
Hankel function Hν(1) or Hν(2), 
    complex argument, optionally scaled nag_complex_hankel (s17dlc)
Jacobian theta functions θk(x , q), 
    real argument nag_jacobian_theta (s21ccc)
Kelvin function, 
    vectorized bei x nag_kelvin_bei_vector (s19apc)
    vectorized ber x nag_kelvin_ber_vector (s19anc)
    vectorized kei x nag_kelvin_kei_vector (s19arc)
    vectorized ker x nag_kelvin_ker_vector (s19aqc)
    bei x nag_kelvin_bei (s19abc)
    ber x nag_kelvin_ber (s19aac)
    kei x nag_kelvin_kei (s19adc)
    ker x nag_kelvin_ker (s19acc)
Legendre functions of 1st kind Pnm(x), Pnm(x) nag_legendre_p (s22aac)
Logarithm of 1 + x nag_shifted_log (s01bac)
Logarithm of beta function, 
    real nag_log_beta (s14cbc)
Logarithm of gamma function, 
    complex nag_complex_log_gamma (s14agc)
    real nag_log_gamma (s14abc)
    real, scaled nag_scaled_log_gamma (s14ahc)
Option Pricing, 
    American option: Bjerksund and Stensland option price nag_amer_bs_price (s30qcc)
    Asian option: geometric continuous average rate price nag_asian_geom_price (s30sac)
    Asian option: geometric continuous average rate price with Greeks nag_asian_geom_greeks (s30sbc)
    binary asset-or-nothing option price nag_binary_aon_price (s30ccc)
    binary asset-or-nothing option price with Greeks nag_binary_aon_greeks (s30cdc)
    binary cash-or-nothing option price nag_binary_con_price (s30cac)
    binary cash-or-nothing option price with Greeks nag_binary_con_greeks (s30cbc)
    Black–Scholes–Merton option price nag_bsm_price (s30aac)
    Black–Scholes–Merton option price with Greeks nag_bsm_greeks (s30abc)
    European option, option prices, using Merton jump-diffusion model nag_jumpdiff_merton_price (s30jac)
    European option, option price with Greeks, using Merton jump-diffusion model nag_jumpdiff_merton_greeks (s30jbc)
    floating-strike lookback option price nag_lookback_fls_price (s30bac)
    floating-strike lookback option price with Greeks nag_lookback_fls_greeks (s30bbc)
    Heston's model option price nag_heston_price (s30nac)
    Heston's model option price with Greeks nag_heston_greeks (s30nbc)
    standard barrier option price nag_barrier_std_price (s30fac)
Polygamma function, 
    ψ(n)(x), real x nag_real_polygamma (s14aec)
    ψ(n)(z), complex z nag_complex_polygamma (s14afc)
Psi function nag_polygamma_fun (s14acc)
Psi function derivatives, scaled nag_polygamma_deriv (s14adc)
Scaled modified Bessel function(s), 
    e − (x)I0(x), real argument nag_bessel_i0_scaled (s18cec)
    e − (x)I1(x), real argument nag_bessel_i1_scaled (s18cfc)
    e − x Iν / 4(x), real argument nag_bessel_i_nu_scaled (s18ecc)
    ex K0 (x), real argument nag_bessel_k0_scaled (s18ccc)
    ex K1 (x), real argument nag_bessel_k1_scaled (s18cdc)
    exKα + n(x), real argument nag_bessel_k_alpha_scaled (s18ehc)
    exKν / 4(x), real argument nag_bessel_k_nu_scaled (s18edc)
    vectorized e − (x)I0(x), real argument nag_bessel_i0_scaled_vector (s18csc)
    vectorized e − (x)I1(x), real argument nag_bessel_i1_scaled_vector (s18ctc)
    vectorized ex K0 (x), real argument nag_bessel_k0_scaled_vector (s18cqc)
    vectorized ex K1 (x), real argument nag_bessel_k1_scaled_vector (s18crc)
    hyperbolic nag_sinh (s10abc)
Sine Integral nag_sin_integral (s13adc)
    hyperbolic nag_tanh (s10aac)
Trigamma function, scaled nag_polygamma_deriv (s14adc)
Zeros of Bessel functions Jα(x), Jα ′ (x), Yα(x), Yα ′ (x) nag_bessel_zeros (s17alc)

5  Functions Withdrawn or Scheduled for Withdrawal


6  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Carlson B C (1965) On computing elliptic integrals and functions J. Math. Phys. 44 36–51
Carlson B C (1977a) Special Functions of Applied Mathematics Academic Press
Carlson B C (1977b) Elliptic integrals of the first kind SIAM J. Math. Anal. 8 231–242
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO
Fox L and Parker I B (1968) Chebyshev Polynomials in Numerical Analysis Oxford University Press
Haug E G (2007) The Complete Guide to Option Pricing Formulas (2nd Edition) McGraw-Hill
Joshi M S (2003) The Concepts and Practice of Mathematical Finance Cambridge University Press
Schonfelder J L (1976) The production of special function routines for a multi-machine library Softw. Pract. Exper. 6(1)

s Chapter Contents
s Chapter Introduction (PDF version)
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012