NAG FL Interface
S (Specfun)
Approximations of Special Functions

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S (Specfun) Chapter Introduction – A description of the Chapter and an overview of the algorithms available.

Routine
Mark of
Introduction

Purpose
s01baf 14 nagf_specfun_log_shifted
ln(1+x)
s01eaf 14 nagf_specfun_exp_complex
Complex exponential, ez
s07aaf 1 nagf_specfun_tan
tanx
s09aaf 1 nagf_specfun_arcsin
arcsinx
s09abf 3 nagf_specfun_arccos
arccosx
s10aaf 3 nagf_specfun_tanh
tanhx
s10abf 4 nagf_specfun_sinh
sinhx
s10acf 4 nagf_specfun_cosh
coshx
s11aaf 4 nagf_specfun_arctanh
arctanhx
s11abf 4 nagf_specfun_arcsinh
arcsinhx
s11acf 4 nagf_specfun_arccosh
arccoshx
s13aaf 1 nagf_specfun_integral_exp
Exponential integral E1(x)
s13acf 2 nagf_specfun_integral_cos
Cosine integral Ci(x)
s13adf 5 nagf_specfun_integral_sin
Sine integral Si(x)
s14aaf 1 nagf_specfun_gamma
Gamma function
s14abf 8 nagf_specfun_gamma_log_real
Log gamma function, real argument
s14acf 14 nagf_specfun_polygamma
ψ(x)-lnx
s14adf 14 nagf_specfun_polygamma_deriv
Scaled derivatives of ψ(x)
s14aef 20 nagf_specfun_psi_deriv_real
Polygamma function ψ(n)(x) for real x
s14aff 20 nagf_specfun_psi_deriv_complex
Polygamma function ψ(n)(z) for complex z
s14agf 21 nagf_specfun_gamma_log_complex
Logarithm of the gamma function lnΓ(z), complex argument
s14ahf 23 nagf_specfun_gamma_log_scaled_real
Scaled log gamma function
s14anf 27 nagf_specfun_gamma_vector
Gamma function, vectorized Γ(x)
s14apf 27 nagf_specfun_gamma_log_real_vector
Log gamma function, vectorized ln(Γ(x))
s14baf 14 nagf_specfun_gamma_incomplete
Incomplete gamma functions P(a,x) and Q(a,x)
s14bnf 27 nagf_specfun_gamma_incomplete_vector
Incomplete gamma functions, vectorized P(a,x) and Q(a,x)
s14cbf 24 nagf_specfun_beta_log_real
Logarithm of the beta function lnB(a,b)
s14ccf 24 nagf_specfun_beta_incomplete
Regularized incomplete beta function Ix(a,b) and its complement 1-Ix
s14cpf 27 nagf_specfun_beta_log_real_vector
Logarithm of the beta function, vectorized lnB(a,b)
s14cqf 27 nagf_specfun_beta_incomplete_vector
Regularized incomplete beta function, vectorized Ix(a,b) and its complement 1-Ix
s15abf 3 nagf_specfun_cdf_normal
Cumulative Normal distribution function P(x)
s15acf 4 nagf_specfun_compcdf_normal
Complement of cumulative Normal distribution function Q(x)
s15adf 4 nagf_specfun_erfc_real
Complement of error function erfc(x)
s15aef 4 nagf_specfun_erf_real
Error function erf(x)
s15aff 7 nagf_specfun_dawson
Dawson's integral
s15agf 22 nagf_specfun_erfcx_real
Scaled complement of error function, erfcx(x)
s15apf 27 nagf_specfun_cdf_normal_vector
Cumulative Normal distribution function, vectorized P(x)
s15aqf 27 nagf_specfun_compcdf_normal_vector
Complement of cumulative Normal distribution function, vectorized Q(x)
s15arf 27 nagf_specfun_erfc_real_vector
Complement of error function, vectorized erfc(x)
s15asf 27 nagf_specfun_erf_real_vector
Error function, vectorized erf(x)
s15atf 27 nagf_specfun_dawson_vector
Dawson's integral, vectorized
s15auf 27 nagf_specfun_erfcx_real_vector
Scaled complement of error function, vectorized erfcx(x)
s15ddf 14 nagf_specfun_erfc_complex
Scaled complex complement of error function, exp(-z2)erfc(-iz)
s15drf 27 nagf_specfun_erfc_complex_vector
Scaled complex complement of error function, vectorized exp(-z2)erfc(-iz)
s17acf 1 nagf_specfun_bessel_y0_real
Bessel function Y0(x)
s17adf 1 nagf_specfun_bessel_y1_real
Bessel function Y1(x)
s17aef 5 nagf_specfun_bessel_j0_real
Bessel function J0(x)
s17aff 5 nagf_specfun_bessel_j1_real
Bessel function J1(x)
s17agf 8 nagf_specfun_airy_ai_real
Airy function Ai(x)
s17ahf 8 nagf_specfun_airy_bi_real
Airy function Bi(x)
s17ajf 8 nagf_specfun_airy_ai_deriv
Airy function Ai(x)
s17akf 8 nagf_specfun_airy_bi_deriv
Airy function Bi(x)
s17alf 20 nagf_specfun_bessel_zeros
Zeros of Bessel functions Jα(x), Jα(x), Yα(x) or Yα(x)
s17aqf 24 nagf_specfun_bessel_y0_real_vector
Bessel function vectorized Y0(x)
s17arf 24 nagf_specfun_bessel_y1_real_vector
Bessel function vectorized Y1(x)
s17asf 24 nagf_specfun_bessel_j0_real_vector
Bessel function vectorized J0(x)
s17atf 24 nagf_specfun_bessel_j1_real_vector
Bessel function vectorized J1(x)
s17auf 24 nagf_specfun_airy_ai_real_vector
Airy function vectorized Ai(x)
s17avf 24 nagf_specfun_airy_bi_real_vector
Airy function vectorized Bi(x)
s17awf 24 nagf_specfun_airy_ai_deriv_vector
Derivatives of the Airy function, vectorized Ai(x)
s17axf 24 nagf_specfun_airy_bi_deriv_vector
Derivatives of the Airy function, vectorized Bi(x)
s17dcf 13 nagf_specfun_bessel_y_complex
Bessel functions Yν+a(z), real a0, complex z, ν=0,1,2,
s17def 13 nagf_specfun_bessel_j_complex
Bessel functions Jν+a(z), real a0, complex z, ν=0,1,2,
s17dgf 13 nagf_specfun_airy_ai_complex
Airy functions Ai(z) and Ai(z), complex z
s17dhf 13 nagf_specfun_airy_bi_complex
Airy functions Bi(z) and Bi(z), complex z
s17dlf 13 nagf_specfun_hankel_complex
Hankel functions Hν+a(j)(z), j=1,2, real a0, complex z, ν=0,1,2,
s17gaf 26.1 nagf_specfun_struve_h0
Struve function of order 0, H0(x)
s17gbf 26.1 nagf_specfun_struve_h1
Struve function of order 1, H1(x)
s18acf 1 nagf_specfun_bessel_k0_real
Modified Bessel function K0(x)
s18adf 1 nagf_specfun_bessel_k1_real
Modified Bessel function K1(x)
s18aef 5 nagf_specfun_bessel_i0_real
Modified Bessel function I0(x)
s18aff 5 nagf_specfun_bessel_i1_real
Modified Bessel function I1(x)
s18aqf 24 nagf_specfun_bessel_k0_real_vector
Modified Bessel function vectorized K0(x)
s18arf 24 nagf_specfun_bessel_k1_real_vector
Modified Bessel function vectorized K1(x)
s18asf 24 nagf_specfun_bessel_i0_real_vector
Modified Bessel function vectorized I0(x)
s18atf 24 nagf_specfun_bessel_i1_real_vector
Modified Bessel function vectorized I1(x)
s18ccf 10 nagf_specfun_bessel_k0_scaled
Scaled modified Bessel function exK0(x)
s18cdf 10 nagf_specfun_bessel_k1_scaled
Scaled modified Bessel function exK1(x)
s18cef 10 nagf_specfun_bessel_i0_scaled
Scaled modified Bessel function e-|x|I0(x)
s18cff 10 nagf_specfun_bessel_i1_scaled
Scaled modified Bessel function e-|x|I1(x)
s18cqf 24 nagf_specfun_bessel_k0_scaled_vector
Scaled modified Bessel function vectorized exK0(x)
s18crf 24 nagf_specfun_bessel_k1_scaled_vector
Scaled modified Bessel function vectorized exK1(x)
s18csf 24 nagf_specfun_bessel_i0_scaled_vector
Scaled modified Bessel function vectorized e-|x|I0(x)
s18ctf 24 nagf_specfun_bessel_i1_scaled_vector
Scaled modified Bessel function vectorized e-|x|I1(x)
s18dcf 13 nagf_specfun_bessel_k_complex
Modified Bessel functions Kν+a(z), real a0, complex z, ν=0,1,2,
s18def 13 nagf_specfun_bessel_i_complex
Modified Bessel functions Iν+a(z), real a0, complex z, ν=0,1,2,
s18gaf 26.1 nagf_specfun_struve_l0
Modified Struve function of order 0, L0(x)
s18gbf 26.1 nagf_specfun_struve_l1
Modified Struve function of order 1, L1(x)
s18gcf 26.1 nagf_specfun_struve_i0ml0
The function I0(x)-L0(x), where I0(x) is a modified Bessel function and L0(x) is a Struve function
s18gdf 26.1 nagf_specfun_struve_i1ml1
The function I1(x)-L1(x), where I1(x) is a modified Bessel function and L1(x) is a Struve function
s18gkf 21 nagf_specfun_bessel_j_seq_complex
Bessel function of the 1st kind Jα±n(z)
s19aaf 11 nagf_specfun_kelvin_ber
Kelvin function berx
s19abf 11 nagf_specfun_kelvin_bei
Kelvin function beix
s19acf 11 nagf_specfun_kelvin_ker
Kelvin function kerx
s19adf 11 nagf_specfun_kelvin_kei
Kelvin function keix
s19anf 24 nagf_specfun_kelvin_ber_vector
Kelvin function vectorized berx
s19apf 24 nagf_specfun_kelvin_bei_vector
Kelvin function vectorized beix
s19aqf 24 nagf_specfun_kelvin_ker_vector
Kelvin function vectorized kerx
s19arf 24 nagf_specfun_kelvin_kei_vector
Kelvin function vectorized keix
s20acf 5 nagf_specfun_fresnel_s
Fresnel integral S(x)
s20adf 5 nagf_specfun_fresnel_c
Fresnel integral C(x)
s20aqf 24 nagf_specfun_fresnel_s_vector
Fresnel integral vectorized S(x)
s20arf 24 nagf_specfun_fresnel_c_vector
Fresnel integral vectorized C(x)
s21baf 8 nagf_specfun_ellipint_symm_1_degen
Degenerate symmetrised elliptic integral of 1st kind RC(x,y)
s21bbf 8 nagf_specfun_ellipint_symm_1
Symmetrised elliptic integral of 1st kind RF(x,y,z)
s21bcf 8 nagf_specfun_ellipint_symm_2
Symmetrised elliptic integral of 2nd kind RD(x,y,z)
s21bdf 8 nagf_specfun_ellipint_symm_3
Symmetrised elliptic integral of 3rd kind RJ(x,y,z,r)
s21bef 22 nagf_specfun_ellipint_legendre_1
Elliptic integral of 1st kind, Legendre form, F(ϕm)
s21bff 22 nagf_specfun_ellipint_legendre_2
Elliptic integral of 2nd kind, Legendre form, E (ϕm)
s21bgf 22 nagf_specfun_ellipint_legendre_3
Elliptic integral of 3rd kind, Legendre form, Π (n;ϕm)
s21bhf 22 nagf_specfun_ellipint_complete_1
Complete elliptic integral of 1st kind, Legendre form, K (m)
s21bjf 22 nagf_specfun_ellipint_complete_2
Complete elliptic integral of 2nd kind, Legendre form, E (m)
s21caf 15 nagf_specfun_jacellip_real
Jacobian elliptic functions sn, cn and dn of real argument
s21cbf 20 nagf_specfun_jacellip_complex
Jacobian elliptic functions sn, cn and dn of complex argument
s21ccf 20 nagf_specfun_jactheta_real
Jacobian theta functions θk(x,q) of real argument
s21daf 20 nagf_specfun_ellipint_general_2
General elliptic integral of 2nd kind F(z,k,a,b) of complex argument
s22aaf 20 nagf_specfun_legendre_p
Legendre functions of 1st kind Pnm(x) or Pnm¯(x)
s22baf 24 nagf_specfun_hyperg_confl_real
Real confluent hypergeometric function F 1 1 (a;b;x)
s22bbf 24 nagf_specfun_hyperg_confl_real_scaled
Real confluent hypergeometric function F 1 1 (a;b;x) in scaled form
s22bef 25 nagf_specfun_hyperg_gauss_real
Real Gauss hypergeometric function F1 2 (a,b;c;x)
s22bff 25 nagf_specfun_hyperg_gauss_real_scaled
Real Gauss hypergeometric function F 1 2 (a,b;c;x) in scaled form
s22caf 27 nagf_specfun_mathieu_ang_periodic_real
Calculates values of real periodic angular Mathieu functions
s30aaf 22 nagf_specfun_opt_bsm_price
Black–Scholes–Merton option pricing formula
s30abf 22 nagf_specfun_opt_bsm_greeks
Black–Scholes–Merton option pricing formula with Greeks
s30acf 27.1 nagf_specfun_opt_imp_vol
Black–Scholes–Merton implied volatility
s30baf 22 nagf_specfun_opt_lookback_fls_price
Floating-strike lookback option pricing formula in the Black-Scholes-Merton model
s30bbf 22 nagf_specfun_opt_lookback_fls_greeks
Floating-strike lookback option pricing formula with Greeks in the Black-Scholes-Merton model
s30caf 22 nagf_specfun_opt_binary_con_price
Binary option, cash-or-nothing pricing formula
s30cbf 22 nagf_specfun_opt_binary_con_greeks
Binary option, cash-or-nothing pricing formula with Greeks
s30ccf 22 nagf_specfun_opt_binary_aon_price
Binary option, asset-or-nothing pricing formula
s30cdf 22 nagf_specfun_opt_binary_aon_greeks
Binary option, asset-or-nothing pricing formula with Greeks
s30faf 22 nagf_specfun_opt_barrier_std_price
Standard barrier option pricing formula
s30jaf 22 nagf_specfun_opt_jumpdiff_merton_price
Jump-diffusion, Merton's model, option pricing formula
s30jbf 22 nagf_specfun_opt_jumpdiff_merton_greeks
Jump-diffusion, Merton's model, option pricing formula with Greeks
s30naf 22 nagf_specfun_opt_heston_price
Heston's model option pricing formula
s30nbf 23 nagf_specfun_opt_heston_greeks
Heston's model option pricing formula with Greeks
s30ncf 25 nagf_specfun_opt_heston_term
Heston's model option pricing with term structure
s30ndf 28.5 (Experimental) nagf_specfun_opt_heston_more_greeks
Heston's model option pricing formula with Greeks, sensitivities of model parameters and negative rates
s30qcf 22 nagf_specfun_opt_amer_bs_price
American option, Bjerksund and Stensland pricing formula
s30saf 22 nagf_specfun_opt_asian_geom_price
Asian option, geometric continuous average rate pricing formula
s30sbf 22 nagf_specfun_opt_asian_geom_greeks
Asian option, geometric continuous average rate pricing formula with Greeks