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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.

Function
Mark of
Introduction

Purpose
s01bac 7 nag_specfun_log_shifted
ln(1+x)
s10aac 1 nag_specfun_tanh
Hyperbolic tangent, tanh x
s10abc 1 nag_specfun_sinh
Hyperbolic sine, sinh x
s10acc 1 nag_specfun_cosh
Hyperbolic cosine, cosh x
s11aac 1 nag_specfun_arctanh
Inverse hyperbolic tangent, arctanh x
s11abc 1 nag_specfun_arcsinh
Inverse hyperbolic sine, arcsinh x
s11acc 1 nag_specfun_arccosh
Inverse hyperbolic cosine, arccosh x
s13aac 1 nag_specfun_integral_exp
Exponential integral E1(x)
s13acc 1 nag_specfun_integral_cos
Cosine integral Ci(x)
s13adc 1 nag_specfun_integral_sin
Sine integral Si(x)
s14aac 1 nag_specfun_gamma
Gamma function Γ(x)
s14abc 1 nag_specfun_gamma_log_real
Log gamma function ln(Γ(x))
s14acc 7 nag_specfun_polygamma
ψ(x)-lnx
s14adc 7 nag_specfun_polygamma_deriv
Scaled derivatives of ψ(x)
s14aec 6 nag_specfun_psi_deriv_real
Derivative of the psi function ψ(x)
s14afc 6 nag_specfun_psi_deriv_complex
Derivative of the psi function ψ(z)
s14agc 7 nag_specfun_gamma_log_complex
Logarithm of the gamma function lnΓ(z), complex argument
s14ahc 9 nag_specfun_gamma_log_scaled_real
Scaled log gamma function lnG(x), where G(x)=Γ(x+1)/(x/e)x
s14anc 27 nag_specfun_gamma_vector
Gamma function, vectorized Γ(x)
s14apc 27 nag_specfun_gamma_log_real_vector
Log gamma function, vectorized ln(Γ(x))
s14bac 1 nag_specfun_gamma_incomplete
Incomplete gamma functions P(a,x) and Q(a,x)
s14bnc 27 nag_specfun_gamma_incomplete_vector
Incomplete gamma functions, vectorized P(a,x) and Q(a,x)
s14cbc 23 nag_specfun_beta_log_real
Logarithm of the beta function lnB(a,b)
s14ccc 23 nag_specfun_beta_incomplete
Regularized incomplete beta function Ix(a,b) and its complement 1-Ix
s14cpc 27 nag_specfun_beta_log_real_vector
Logarithm of the beta function, vectorized lnB(a,b)
s14cqc 27 nag_specfun_beta_incomplete_vector
Regularized incomplete beta function, vectorized Ix(a,b) and its complement 1-Ix
s15abc 1 nag_specfun_cdf_normal
Cumulative Normal distribution function P(x)
s15acc 1 nag_specfun_compcdf_normal
Complement of cumulative Normal distribution function Q(x)
s15adc 1 nag_specfun_erfc_real
Complement of error function erfc(x)
s15aec 1 nag_specfun_erf_real
Error function erf(x)
s15afc 7 nag_specfun_dawson
Dawson's integral
s15agc 9 nag_specfun_erfcx_real
Scaled complement of error function, erfcx(x)
s15apc 27 nag_specfun_cdf_normal_vector
Cumulative Normal distribution function, vectorized P(x)
s15aqc 27 nag_specfun_compcdf_normal_vector
Complement of cumulative Normal distribution function, vectorized Q(x)
s15arc 27 nag_specfun_erfc_real_vector
Complement of error function, vectorized erfc(x)
s15asc 27 nag_specfun_erf_real_vector
Error function, vectorized erf(x)
s15atc 27 nag_specfun_dawson_vector
Dawson's integral, vectorized
s15auc 27 nag_specfun_erfcx_real_vector
Scaled complement of error function, vectorized erfcx(x)
s15ddc 7 nag_specfun_erfc_complex
Scaled complex complement of error function, exp(-z2)erfc(-iz)
s15drc 27 nag_specfun_erfc_complex_vector
Scaled complex complement of error function, vectorized exp(-z2)erfc(-iz)
s17acc 1 nag_specfun_bessel_y0_real
Bessel function Y0(x)
s17adc 1 nag_specfun_bessel_y1_real
Bessel function Y1(x)
s17aec 1 nag_specfun_bessel_j0_real
Bessel function J0(x)
s17afc 1 nag_specfun_bessel_j1_real
Bessel function J1(x)
s17agc 1 nag_specfun_airy_ai_real
Airy function Ai(x)
s17ahc 1 nag_specfun_airy_bi_real
Airy function Bi(x)
s17ajc 1 nag_specfun_airy_ai_deriv
Airy function Ai(x)
s17akc 1 nag_specfun_airy_bi_deriv
Airy function Bi(x)
s17alc 6 nag_specfun_bessel_zeros
Zeros of Bessel functions Jα(x), Jα(x), Yα(x) or Yα(x)
s17aqc 23 nag_specfun_bessel_y0_real_vector
Bessel function vectorized Y0(x)
s17arc 23 nag_specfun_bessel_y1_real_vector
Bessel function vectorized Y1(x)
s17asc 23 nag_specfun_bessel_j0_real_vector
Bessel function vectorized J0(x)
s17atc 23 nag_specfun_bessel_j1_real_vector
Bessel function vectorized J1(x)
s17auc 23 nag_specfun_airy_ai_real_vector
Airy function vectorized Ai(x)
s17avc 23 nag_specfun_airy_bi_real_vector
Airy function vectorized Bi(x)
s17awc 23 nag_specfun_airy_ai_deriv_vector
Derivatives of the Airy function, vectorized Ai(x)
s17axc 23 nag_specfun_airy_bi_deriv_vector
Derivatives of the Airy function, vectorized Bi(x)
s17dcc 7 nag_specfun_bessel_y_complex
Bessel functions Yν+a(z), real a0, complex z, ν=0,1,2,
s17dec 7 nag_specfun_bessel_j_complex
Bessel functions Jν+a(z), real a0, complex z, ν=0,1,2,
s17dgc 7 nag_specfun_airy_ai_complex
Airy functions Ai(z) and Ai(z), complex z
s17dhc 7 nag_specfun_airy_bi_complex
Airy functions Bi(z) and Bi(z), complex z
s17dlc 7 nag_specfun_hankel_complex
Hankel functions Hν+a(j)(z), j=1,2, real a0, complex z, ν=0,1,2,
s17gac 26.1 nag_specfun_struve_h0
Struve function of order 0, H0(x)
s17gbc 26.1 nag_specfun_struve_h1
Struve function of order 1, H1(x)
s18acc 1 nag_specfun_bessel_k0_real
Modified Bessel function K0(x)
s18adc 1 nag_specfun_bessel_k1_real
Modified Bessel function K1(x)
s18aec 1 nag_specfun_bessel_i0_real
Modified Bessel function I0(x)
s18afc 1 nag_specfun_bessel_i1_real
Modified Bessel function I1(x)
s18aqc 23 nag_specfun_bessel_k0_real_vector
Modified Bessel function vectorized K0(x)
s18arc 23 nag_specfun_bessel_k1_real_vector
Modified Bessel function vectorized K1(x)
s18asc 23 nag_specfun_bessel_i0_real_vector
Modified Bessel function vectorized I0(x)
s18atc 23 nag_specfun_bessel_i1_real_vector
Modified Bessel function vectorized I1(x)
s18ccc 2 nag_specfun_bessel_k0_scaled
Scaled modified Bessel function exK0(x)
s18cdc 2 nag_specfun_bessel_k1_scaled
Scaled modified Bessel function exK1(x)
s18cec 2 nag_specfun_bessel_i0_scaled
Scaled modified Bessel function e-|x|I0(x)
s18cfc 2 nag_specfun_bessel_i1_scaled
Scaled modified Bessel function e-|x|I1(x)
s18cqc 23 nag_specfun_bessel_k0_scaled_vector
Scaled modified Bessel function vectorized exK0(x)
s18crc 23 nag_specfun_bessel_k1_scaled_vector
Scaled modified Bessel function vectorized exK1(x)
s18csc 23 nag_specfun_bessel_i0_scaled_vector
Scaled modified Bessel function vectorized e-|x|I0(x)
s18ctc 23 nag_specfun_bessel_i1_scaled_vector
Scaled modified Bessel function vectorized e-|x|I1(x)
s18dcc 7 nag_specfun_bessel_k_complex
Modified Bessel functions Kν+a(z), real a0, complex z, ν=0,1,2,
s18dec 7 nag_specfun_bessel_i_complex
Modified Bessel functions Iν+a(z), real a0, complex z, ν=0,1,2,
s18ecc 6 nag_bessel_i_nu_scaled
Scaled modified Bessel function e-x Iν/4(x)
s18edc 6 nag_bessel_k_nu_scaled
Scaled modified Bessel function exKν/4(x)
s18eec 6 nag_bessel_i_nu
Modified Bessel function Iν/4(x)
s18efc 6 nag_bessel_k_nu
Modified Bessel function Kν/4(x)
s18egc 6 nag_bessel_k_alpha
Modified Bessel functions Kα+n(x) for real x>0, selected values of α0 and n=0,1,,N
s18ehc 6 nag_bessel_k_alpha_scaled
Scaled modified Bessel functions exKα+n(x) for real x>0, selected values of α 0 and n=0,1,,N
s18ejc 6 nag_bessel_i_alpha
Modified Bessel functions Iα+n-1(x) or Iα-n+1(x) for real x0, non-negative α<1 and n=1,2,,|N|+1
s18ekc 6 nag_bessel_j_alpha
Bessel functions Jα+n-1(x) or Jα-n+1(x) for real x0, non-negative α<1 and n=1,2,,|N|+1
s18gac 26.1 nag_specfun_struve_l0
Modified Struve function of order 0, L0(x)
s18gbc 26.1 nag_specfun_struve_l1
Modified Struve function of order 1, L1(x)
s18gcc 26.1 nag_specfun_struve_i0ml0
The function I0(x)-L0(x), where I0(x) is a modified Bessel function and L0(x) is a Struve function
s18gdc 26.1 nag_specfun_struve_i1ml1
The function I1(x)-L1(x), where I1(x) is a modified Bessel function and L1(x) is a Struve function
s18gkc 7 nag_specfun_bessel_j_seq_complex
Bessel function of the 1st kind Jα±n(z)
s19aac 1 nag_specfun_kelvin_ber
Kelvin function berx
s19abc 1 nag_specfun_kelvin_bei
Kelvin function beix
s19acc 1 nag_specfun_kelvin_ker
Kelvin function kerx
s19adc 1 nag_specfun_kelvin_kei
Kelvin function keix
s19anc 23 nag_specfun_kelvin_ber_vector
Kelvin function vectorized berx
s19apc 23 nag_specfun_kelvin_bei_vector
Kelvin function vectorized beix
s19aqc 23 nag_specfun_kelvin_ker_vector
Kelvin function vectorized kerx
s19arc 23 nag_specfun_kelvin_kei_vector
Kelvin function vectorized keix
s20acc 1 nag_specfun_fresnel_s
Fresnel integral S(x)
s20adc 1 nag_specfun_fresnel_c
Fresnel integral C(x)
s20aqc 23 nag_specfun_fresnel_s_vector
Fresnel integral vectorized S(x)
s20arc 23 nag_specfun_fresnel_c_vector
Fresnel integral vectorized C(x)
s21bac 1 nag_specfun_ellipint_symm_1_degen
Degenerate symmetrised elliptic integral of 1st kind RC(x,y)
s21bbc 1 nag_specfun_ellipint_symm_1
Symmetrised elliptic integral of 1st kind RF(x,y,z)
s21bcc 1 nag_specfun_ellipint_symm_2
Symmetrised elliptic integral of 2nd kind RD(x,y,z)
s21bdc 1 nag_specfun_ellipint_symm_3
Symmetrised elliptic integral of 3rd kind RJ(x,y,z,r)
s21bec 9 nag_specfun_ellipint_legendre_1
Elliptic integral of 1st kind, Legendre form, F(ϕm)
s21bfc 9 nag_specfun_ellipint_legendre_2
Elliptic integral of 2nd kind, Legendre form, E (ϕm)
s21bgc 9 nag_specfun_ellipint_legendre_3
Elliptic integral of 3rd kind, Legendre form, Π (n;ϕm)
s21bhc 9 nag_specfun_ellipint_complete_1
Complete elliptic integral of 1st kind, Legendre form, K (m)
s21bjc 9 nag_specfun_ellipint_complete_2
Complete elliptic integral of 2nd kind, Legendre form, E (m)
s21cac 7 nag_specfun_jacellip_real
Jacobian elliptic functions sn, cn and dn of real argument
s21cbc 6 nag_specfun_jacellip_complex
Jacobian elliptic functions sn, cn and dn of complex argument
s21ccc 6 nag_specfun_jactheta_real
Jacobian theta functions with real arguments
s21dac 6 nag_specfun_ellipint_general_2
Elliptic integrals of the second kind with complex arguments
s22aac 6 nag_specfun_legendre_p
Legendre and associated Legendre functions of the first kind with real arguments
s22bac 24 nag_specfun_hyperg_confl_real
Real confluent hypergeometric function F 1 1 (a;b;x)
s22bbc 24 nag_specfun_hyperg_confl_real_scaled
Real confluent hypergeometric function F 1 1 (a;b;x) in scaled form
s22bec 24 nag_specfun_hyperg_gauss_real
Real Gauss hypergeometric function F1 2 (a,b;c;x)
s22bfc 24 nag_specfun_hyperg_gauss_real_scaled
Real Gauss hypergeometric function F 1 2 (a,b;c;x) in scaled form
s22cac 27 nag_specfun_mathieu_ang_periodic_real
Calculates values of real periodic angular Mathieu functions
s30aac 9 nag_specfun_opt_bsm_price
Black–Scholes–Merton option pricing formula
s30abc 9 nag_specfun_opt_bsm_greeks
Black–Scholes–Merton option pricing formula with Greeks
s30acc 27.1 nag_specfun_opt_imp_vol
Black–Scholes–Merton implied volatility
s30bac 9 nag_specfun_opt_lookback_fls_price
Floating-strike lookback option pricing formula in the Black-Scholes-Merton model
s30bbc 9 nag_specfun_opt_lookback_fls_greeks
Floating-strike lookback option pricing formula with Greeks in the Black-Scholes-Merton model
s30cac 9 nag_specfun_opt_binary_con_price
Binary option, cash-or-nothing pricing formula
s30cbc 9 nag_specfun_opt_binary_con_greeks
Binary option, cash-or-nothing pricing formula with Greeks
s30ccc 9 nag_specfun_opt_binary_aon_price
Binary option, asset-or-nothing pricing formula
s30cdc 9 nag_specfun_opt_binary_aon_greeks
Binary option, asset-or-nothing pricing formula with Greeks
s30fac 9 nag_specfun_opt_barrier_std_price
Standard barrier option pricing formula
s30jac 9 nag_specfun_opt_jumpdiff_merton_price
Jump-diffusion, Merton's model, option pricing formula
s30jbc 9 nag_specfun_opt_jumpdiff_merton_greeks
Jump-diffusion, Merton's model, option pricing formula with Greeks
s30nac 9 nag_specfun_opt_heston_price
Heston's model option pricing formula
s30nbc 23 nag_specfun_opt_heston_greeks
Heston's model option pricing formula with Greeks
s30ncc 24 nag_specfun_opt_heston_term
Heston's model option pricing with term structure
s30ndc 28.5 (Experimental) nag_specfun_opt_heston_more_greeks
Heston's model option pricing formula with Greeks, sensitivities of model parameters and negative rates
s30qcc 9 nag_specfun_opt_amer_bs_price
American option, Bjerksund and Stensland pricing formula
s30sac 9 nag_specfun_opt_asian_geom_price
Asian option, geometric continuous average rate pricing formula
s30sbc 9 nag_specfun_opt_asian_geom_greeks
Asian option, geometric continuous average rate pricing formula with Greeks