{-| module : Data.Number.Flint.Arb.Hypgeom.FFI copyright : (c) 2022 Hartmut Monien license : GNU GPL, version 2 or above (see LICENSE) maintainer : hmonien@uni-bonn.de -} module Data.Number.Flint.Arb.Hypgeom.FFI ( -- * Hypergeometric functions of real variables -- * Rising factorials _arb_hypgeom_rising_coeffs_1 , _arb_hypgeom_rising_coeffs_2 , _arb_hypgeom_rising_coeffs_fmpz , arb_hypgeom_rising_ui_forward , arb_hypgeom_rising_ui_bs , arb_hypgeom_rising_ui_rs , arb_hypgeom_rising_ui_rec , arb_hypgeom_rising_ui , arb_hypgeom_rising , arb_hypgeom_rising_ui_jet_powsum , arb_hypgeom_rising_ui_jet_bs , arb_hypgeom_rising_ui_jet_rs , arb_hypgeom_rising_ui_jet -- * Gamma function , _arb_hypgeom_gamma_stirling_term_bounds , arb_hypgeom_gamma_stirling_sum_horner , arb_hypgeom_gamma_stirling_sum_improved , arb_hypgeom_gamma_stirling , arb_hypgeom_gamma_taylor , arb_hypgeom_gamma , arb_hypgeom_gamma_fmpq , arb_hypgeom_gamma_fmpz , arb_hypgeom_rgamma , arb_hypgeom_lgamma -- * Binomial coefficients , arb_hypgeom_central_bin_ui -- * Generalized hypergeometric function , arb_hypgeom_pfq -- * Confluent hypergeometric functions , arb_hypgeom_0f1 , arb_hypgeom_m , arb_hypgeom_1f1 , arb_hypgeom_1f1_integration , arb_hypgeom_u , arb_hypgeom_u_integration -- * Gauss hypergeometric function , arb_hypgeom_2f1 , arb_hypgeom_2f1_integration -- * Error functions and Fresnel integrals , arb_hypgeom_erf , _arb_hypgeom_erf_series , arb_hypgeom_erf_series , arb_hypgeom_erfc , _arb_hypgeom_erfc_series , arb_hypgeom_erfc_series , arb_hypgeom_erfi , _arb_hypgeom_erfi_series , arb_hypgeom_erfi_series , arb_hypgeom_erfinv , arb_hypgeom_erfcinv , arb_hypgeom_fresnel , _arb_hypgeom_fresnel_series , arb_hypgeom_fresnel_series -- * Incomplete gamma and beta functions , arb_hypgeom_gamma_upper , arb_hypgeom_gamma_upper_integration , _arb_hypgeom_gamma_upper_series , arb_hypgeom_gamma_upper_series , arb_hypgeom_gamma_lower , _arb_hypgeom_gamma_lower_series , arb_hypgeom_gamma_lower_series , arb_hypgeom_beta_lower , _arb_hypgeom_beta_lower_series , arb_hypgeom_beta_lower_series -- * Internal evaluation functions , _arb_hypgeom_gamma_lower_sum_rs_1 , _arb_hypgeom_gamma_upper_sum_rs_1 , _arb_hypgeom_gamma_upper_fmpq_inf_choose_N , _arb_hypgeom_gamma_upper_fmpq_inf_bsplit , _arb_hypgeom_gamma_lower_fmpq_0_choose_N , _arb_hypgeom_gamma_lower_fmpq_0_bsplit , _arb_hypgeom_gamma_upper_singular_si_choose_N , _arb_hypgeom_gamma_upper_singular_si_bsplit , _arb_gamma_upper_fmpq_step_bsplit -- * Exponential and trigonometric integrals , arb_hypgeom_expint , arb_hypgeom_ei , _arb_hypgeom_ei_series , arb_hypgeom_ei_series , _arb_hypgeom_si_asymp , _arb_hypgeom_si_1f2 , arb_hypgeom_si , _arb_hypgeom_si_series , arb_hypgeom_si_series , _arb_hypgeom_ci_asymp , _arb_hypgeom_ci_2f3 , arb_hypgeom_ci , _arb_hypgeom_ci_series , arb_hypgeom_ci_series , arb_hypgeom_shi , _arb_hypgeom_shi_series , arb_hypgeom_shi_series , arb_hypgeom_chi , _arb_hypgeom_chi_series , arb_hypgeom_chi_series , arb_hypgeom_li , _arb_hypgeom_li_series , arb_hypgeom_li_series -- * Bessel functions , arb_hypgeom_bessel_j , arb_hypgeom_bessel_y , arb_hypgeom_bessel_jy , arb_hypgeom_bessel_i , arb_hypgeom_bessel_i_scaled , arb_hypgeom_bessel_k , arb_hypgeom_bessel_k_scaled , arb_hypgeom_bessel_i_integration , arb_hypgeom_bessel_k_integration -- * Airy functions , arb_hypgeom_airy , arb_hypgeom_airy_jet , _arb_hypgeom_airy_series , arb_hypgeom_airy_series , arb_hypgeom_airy_zero -- * Coulomb wave functions , arb_hypgeom_coulomb , arb_hypgeom_coulomb_jet , _arb_hypgeom_coulomb_series , arb_hypgeom_coulomb_series -- * Orthogonal polynomials and functions , arb_hypgeom_chebyshev_t , arb_hypgeom_chebyshev_u , arb_hypgeom_jacobi_p , arb_hypgeom_gegenbauer_c , arb_hypgeom_laguerre_l , arb_hypgeom_hermite_h , arb_hypgeom_legendre_p , arb_hypgeom_legendre_q , arb_hypgeom_legendre_p_ui_deriv_bound , arb_hypgeom_legendre_p_ui_zero , arb_hypgeom_legendre_p_ui_one , arb_hypgeom_legendre_p_ui_asymp --, arb_hypgeom_legendre_p_rec , arb_hypgeom_legendre_p_ui , arb_hypgeom_legendre_p_ui_root -- * Dilogarithm , arb_hypgeom_dilog -- * Hypergeometric sums , arb_hypgeom_sum_fmpq_arb_forward , arb_hypgeom_sum_fmpq_arb_rs , arb_hypgeom_sum_fmpq_arb , arb_hypgeom_sum_fmpq_imag_arb_forward , arb_hypgeom_sum_fmpq_imag_arb_rs , arb_hypgeom_sum_fmpq_imag_arb_bs , arb_hypgeom_sum_fmpq_imag_arb ) where -- Hypergeometric functions of real variables ---------------------------------- import Foreign.Ptr import Foreign.ForeignPtr import Foreign.C.Types import Foreign.C.String import Data.Number.Flint.Fmpz import Data.Number.Flint.Fmpq import Data.Number.Flint.Arb.Types -- Rising factorials ----------------------------------------------------------- -- | /_arb_hypgeom_rising_coeffs_1/ /c/ /k/ /n/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_rising_coeffs_1" _arb_hypgeom_rising_coeffs_1 :: Ptr CULong -> CULong -> CLong -> IO () -- | /_arb_hypgeom_rising_coeffs_2/ /c/ /k/ /n/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_rising_coeffs_2" _arb_hypgeom_rising_coeffs_2 :: Ptr CULong -> CULong -> CLong -> IO () -- | /_arb_hypgeom_rising_coeffs_fmpz/ /c/ /k/ /n/ -- -- Sets /c/ to the coefficients of the rising factorial polynomial -- \((X+k)_n\). The /1/ and /2/ versions respectively compute single-word -- and double-word coefficients, without checking for overflow, while the -- /fmpz/ version allows arbitrarily large coefficients. These functions -- are mostly intended for internal use; the /fmpz/ version does not use an -- asymptotically fast algorithm. The degree /n/ must be at least 2. foreign import ccall "arb_hypgeom.h _arb_hypgeom_rising_coeffs_fmpz" _arb_hypgeom_rising_coeffs_fmpz :: Ptr CFmpz -> CULong -> CLong -> IO () -- | /arb_hypgeom_rising_ui_forward/ /res/ /x/ /n/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_rising_ui_forward" arb_hypgeom_rising_ui_forward :: Ptr CArb -> Ptr CArb -> CULong -> CLong -> IO () -- | /arb_hypgeom_rising_ui_bs/ /res/ /x/ /n/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_rising_ui_bs" arb_hypgeom_rising_ui_bs :: Ptr CArb -> Ptr CArb -> CULong -> CLong -> IO () -- | /arb_hypgeom_rising_ui_rs/ /res/ /x/ /n/ /m/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_rising_ui_rs" arb_hypgeom_rising_ui_rs :: Ptr CArb -> Ptr CArb -> CULong -> CULong -> CLong -> IO () -- | /arb_hypgeom_rising_ui_rec/ /res/ /x/ /n/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_rising_ui_rec" arb_hypgeom_rising_ui_rec :: Ptr CArb -> Ptr CArb -> CULong -> CLong -> IO () -- | /arb_hypgeom_rising_ui/ /res/ /x/ /n/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_rising_ui" arb_hypgeom_rising_ui :: Ptr CArb -> Ptr CArb -> CULong -> CLong -> IO () -- | /arb_hypgeom_rising/ /res/ /x/ /n/ /prec/ -- -- Computes the rising factorial \((x)_n\). -- -- The /forward/ version uses the forward recurrence. The /bs/ version uses -- binary splitting. The /rs/ version uses rectangular splitting. It takes -- an extra tuning parameter /m/ which can be set to zero to choose -- automatically. The /rec/ version chooses an algorithm automatically, -- avoiding use of the gamma function (so that it can be used in the -- computation of the gamma function). The default versions (/rising_ui/ -- and /rising_ui/) choose an algorithm automatically and may additionally -- fall back on the gamma function. foreign import ccall "arb_hypgeom.h arb_hypgeom_rising" arb_hypgeom_rising :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_rising_ui_jet_powsum/ /res/ /x/ /n/ /len/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_rising_ui_jet_powsum" arb_hypgeom_rising_ui_jet_powsum :: Ptr CArb -> Ptr CArb -> CULong -> CLong -> CLong -> IO () -- | /arb_hypgeom_rising_ui_jet_bs/ /res/ /x/ /n/ /len/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_rising_ui_jet_bs" arb_hypgeom_rising_ui_jet_bs :: Ptr CArb -> Ptr CArb -> CULong -> CLong -> CLong -> IO () -- | /arb_hypgeom_rising_ui_jet_rs/ /res/ /x/ /n/ /m/ /len/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_rising_ui_jet_rs" arb_hypgeom_rising_ui_jet_rs :: Ptr CArb -> Ptr CArb -> CULong -> CULong -> CLong -> CLong -> IO () -- | /arb_hypgeom_rising_ui_jet/ /res/ /x/ /n/ /len/ /prec/ -- -- Computes the jet of the rising factorial \((x)_n\), truncated to length -- /len/. In other words, constructs the polynomial -- \((X + x)_n \in \mathbb{R}[X]\), truncated if -- \(\operatorname{len} < n + 1\) (and zero-extended if -- \(\operatorname{len} > n + 1\)). -- -- The /powsum/ version computes the sequence of powers of /x/ and forms -- integral linear combinations of these. The /bs/ version uses binary -- splitting. The /rs/ version uses rectangular splitting. It takes an -- extra tuning parameter /m/ which can be set to zero to choose -- automatically. The default version chooses an algorithm automatically. foreign import ccall "arb_hypgeom.h arb_hypgeom_rising_ui_jet" arb_hypgeom_rising_ui_jet :: Ptr CArb -> Ptr CArb -> CULong -> CLong -> CLong -> IO () -- Gamma function -------------------------------------------------------------- -- | /_arb_hypgeom_gamma_stirling_term_bounds/ /bound/ /zinv/ /N/ -- -- For \(1 \le n < N\), sets /bound/ to an exponent bounding the /n/-th -- term in the Stirling series for the gamma function, given a precomputed -- upper bound for \(|z|^{-1}\). This function is intended for internal use -- and does not check for underflow or underflow in the exponents. foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_stirling_term_bounds" _arb_hypgeom_gamma_stirling_term_bounds :: Ptr CLong -> Ptr CMag -> CLong -> IO () -- | /arb_hypgeom_gamma_stirling_sum_horner/ /res/ /z/ /N/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_stirling_sum_horner" arb_hypgeom_gamma_stirling_sum_horner :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> IO () -- | /arb_hypgeom_gamma_stirling_sum_improved/ /res/ /z/ /N/ /K/ /prec/ -- -- Sets /res/ to the final sum in the Stirling series for the gamma -- function truncated before the term with index /N/, i.e. computes -- \(\sum_{n=1}^{N-1} B_{2n} / (2n(2n-1) z^{2n-1})\). The /horner/ version -- uses Horner scheme with gradual precision adjustments. The /improved/ -- version uses rectangular splitting for the low-index terms and reexpands -- the high-index terms as hypergeometric polynomials, using a splitting -- parameter /K/ (which can be set to 0 to use a default value). foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_stirling_sum_improved" arb_hypgeom_gamma_stirling_sum_improved :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_gamma_stirling/ /res/ /x/ /reciprocal/ /prec/ -- -- Sets /res/ to the gamma function of /x/ computed using the Stirling -- series together with argument reduction. If /reciprocal/ is set, the -- reciprocal gamma function is computed instead. foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_stirling" arb_hypgeom_gamma_stirling :: Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /arb_hypgeom_gamma_taylor/ /res/ /x/ /reciprocal/ /prec/ -- -- Attempts to compute the gamma function of /x/ using Taylor series -- together with argument reduction. This is only supported if /x/ and -- /prec/ are both small enough. If successful, returns 1; otherwise, does -- nothing and returns 0. If /reciprocal/ is set, the reciprocal gamma -- function is computed instead. foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_taylor" arb_hypgeom_gamma_taylor :: Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO CInt -- | /arb_hypgeom_gamma/ /res/ /x/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma" arb_hypgeom_gamma :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_gamma_fmpq/ /res/ /x/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_fmpq" arb_hypgeom_gamma_fmpq :: Ptr CArb -> Ptr CFmpq -> CLong -> IO () -- | /arb_hypgeom_gamma_fmpz/ /res/ /x/ /prec/ -- -- Sets /res/ to the gamma function of /x/ computed using a default -- algorithm choice. foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_fmpz" arb_hypgeom_gamma_fmpz :: Ptr CArb -> Ptr CFmpz -> CLong -> IO () -- | /arb_hypgeom_rgamma/ /res/ /x/ /prec/ -- -- Sets /res/ to the reciprocal gamma function of /x/ computed using a -- default algorithm choice. foreign import ccall "arb_hypgeom.h arb_hypgeom_rgamma" arb_hypgeom_rgamma :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_lgamma/ /res/ /x/ /prec/ -- -- Sets /res/ to the log-gamma function of /x/ computed using a default -- algorithm choice. foreign import ccall "arb_hypgeom.h arb_hypgeom_lgamma" arb_hypgeom_lgamma :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- Binomial coefficients ------------------------------------------------------- -- | /arb_hypgeom_central_bin_ui/ /res/ /n/ /prec/ -- -- Computes the central binomial coefficient \({2n \choose n}\). foreign import ccall "arb_hypgeom.h arb_hypgeom_central_bin_ui" arb_hypgeom_central_bin_ui :: Ptr CArb -> CULong -> CLong -> IO () -- Generalized hypergeometric function ----------------------------------------- -- | /arb_hypgeom_pfq/ /res/ /a/ /p/ /b/ /q/ /z/ /regularized/ /prec/ -- -- Computes the generalized hypergeometric function \({}_pF_{q}(z)\), or -- the regularized version if /regularized/ is set. foreign import ccall "arb_hypgeom.h arb_hypgeom_pfq" arb_hypgeom_pfq :: Ptr CArb -> Ptr CArb -> CLong -> Ptr CArb -> CLong -> Ptr CArb -> CInt -> CLong -> IO () -- Confluent hypergeometric functions ------------------------------------------ -- | /arb_hypgeom_0f1/ /res/ /a/ /z/ /regularized/ /prec/ -- -- Computes the confluent hypergeometric limit function \({}_0F_1(a,z)\), -- or \(\frac{1}{\Gamma(a)} {}_0F_1(a,z)\) if /regularized/ is set. foreign import ccall "arb_hypgeom.h arb_hypgeom_0f1" arb_hypgeom_0f1 :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /arb_hypgeom_m/ /res/ /a/ /b/ /z/ /regularized/ /prec/ -- -- Computes the confluent hypergeometric function -- \(M(a,b,z) = {}_1F_1(a,b,z)\), or -- \(\mathbf{M}(a,b,z) = \frac{1}{\Gamma(b)} {}_1F_1(a,b,z)\) if -- /regularized/ is set. foreign import ccall "arb_hypgeom.h arb_hypgeom_m" arb_hypgeom_m :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /arb_hypgeom_1f1/ /res/ /a/ /b/ /z/ /regularized/ /prec/ -- -- Alias for @arb_hypgeom_m@. foreign import ccall "arb_hypgeom.h arb_hypgeom_1f1" arb_hypgeom_1f1 :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /arb_hypgeom_1f1_integration/ /res/ /a/ /b/ /z/ /regularized/ /prec/ -- -- Computes the confluent hypergeometric function using numerical -- integration of the representation -- -- \[`\] -- \[{}_1F_1(a,b,z) = \frac{\Gamma(b)}{\Gamma(a) \Gamma(b-a)} \int_0^1 e^{zt} t^{a-1} (1-t)^{b-a-1} dt.\] -- -- This algorithm can be useful if the parameters are large. This will -- currently only return a finite enclosure if \(a \ge 1\) and -- \(b - a \ge 1\). foreign import ccall "arb_hypgeom.h arb_hypgeom_1f1_integration" arb_hypgeom_1f1_integration :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /arb_hypgeom_u/ /res/ /a/ /b/ /z/ /prec/ -- -- Computes the confluent hypergeometric function \(U(a,b,z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_u" arb_hypgeom_u :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_u_integration/ /res/ /a/ /b/ /z/ /regularized/ /prec/ -- -- Computes the confluent hypergeometric function \(U(a,b,z)\) using -- numerical integration of the representation -- -- \[`\] -- \[U(a,b,z) = \frac{1}{\Gamma(a)} \int_0^{\infty} e^{-zt} t^{a-1} (1+t)^{b-a-1} dt.\] -- -- This algorithm can be useful if the parameters are large. This will -- currently only return a finite enclosure if \(a \ge 1\) and \(z > 0\). foreign import ccall "arb_hypgeom.h arb_hypgeom_u_integration" arb_hypgeom_u_integration :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- Gauss hypergeometric function ----------------------------------------------- -- | /arb_hypgeom_2f1/ /res/ /a/ /b/ /c/ /z/ /regularized/ /prec/ -- -- Computes the Gauss hypergeometric function \({}_2F_1(a,b,c,z)\), or -- \(\mathbf{F}(a,b,c,z) = \frac{1}{\Gamma(c)} {}_2F_1(a,b,c,z)\) if -- /regularized/ is set. -- -- Additional evaluation flags can be passed via the /regularized/ -- argument; see @acb_hypgeom_2f1@ for documentation. foreign import ccall "arb_hypgeom.h arb_hypgeom_2f1" arb_hypgeom_2f1 :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /arb_hypgeom_2f1_integration/ /res/ /a/ /b/ /z/ /regularized/ /prec/ -- -- Computes the Gauss hypergeometric function using numerical integration -- of the representation -- -- \[`\] -- \[{}_2F_1(a,b,c,z) = \frac{\Gamma(a)}{\Gamma(b) \Gamma(c-b)} \int_0^1 t^{b-1} (1-t)^{c-b-1} (1-zt)^{-a} dt.\] -- -- This algorithm can be useful if the parameters are large. This will -- currently only return a finite enclosure if \(b \ge 1\) and -- \(c - b \ge 1\) and \(z < 1\), possibly with /a/ and /b/ exchanged. foreign import ccall "arb_hypgeom.h arb_hypgeom_2f1_integration" arb_hypgeom_2f1_integration :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- Error functions and Fresnel integrals --------------------------------------- -- | /arb_hypgeom_erf/ /res/ /z/ /prec/ -- -- Computes the error function \(\operatorname{erf}(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_erf" arb_hypgeom_erf :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /_arb_hypgeom_erf_series/ /res/ /z/ /zlen/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_erf_series" _arb_hypgeom_erf_series :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_erf_series/ /res/ /z/ /len/ /prec/ -- -- Computes the error function of the power series /z/, truncated to length -- /len/. foreign import ccall "arb_hypgeom.h arb_hypgeom_erf_series" arb_hypgeom_erf_series :: Ptr CArbPoly -> Ptr CArbPoly -> CLong -> CLong -> IO () -- | /arb_hypgeom_erfc/ /res/ /z/ /prec/ -- -- Computes the complementary error function -- \(\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)\). This function -- avoids catastrophic cancellation for large positive /z/. foreign import ccall "arb_hypgeom.h arb_hypgeom_erfc" arb_hypgeom_erfc :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /_arb_hypgeom_erfc_series/ /res/ /z/ /zlen/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_erfc_series" _arb_hypgeom_erfc_series :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_erfc_series/ /res/ /z/ /len/ /prec/ -- -- Computes the complementary error function of the power series /z/, -- truncated to length /len/. foreign import ccall "arb_hypgeom.h arb_hypgeom_erfc_series" arb_hypgeom_erfc_series :: Ptr CArbPoly -> Ptr CArbPoly -> CLong -> CLong -> IO () -- | /arb_hypgeom_erfi/ /res/ /z/ /prec/ -- -- Computes the imaginary error function -- \(\operatorname{erfi}(z) = -i\operatorname{erf}(iz)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_erfi" arb_hypgeom_erfi :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /_arb_hypgeom_erfi_series/ /res/ /z/ /zlen/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_erfi_series" _arb_hypgeom_erfi_series :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_erfi_series/ /res/ /z/ /len/ /prec/ -- -- Computes the imaginary error function of the power series /z/, truncated -- to length /len/. foreign import ccall "arb_hypgeom.h arb_hypgeom_erfi_series" arb_hypgeom_erfi_series :: Ptr CArbPoly -> Ptr CArbPoly -> CLong -> CLong -> IO () -- | /arb_hypgeom_erfinv/ /res/ /z/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_erfinv" arb_hypgeom_erfinv :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_erfcinv/ /res/ /z/ /prec/ -- -- Computes the inverse error function \(\operatorname{erf}^{-1}(z)\) or -- inverse complementary error function \(\operatorname{erfc}^{-1}(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_erfcinv" arb_hypgeom_erfcinv :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_fresnel/ /res1/ /res2/ /z/ /normalized/ /prec/ -- -- Sets /res1/ to the Fresnel sine integral \(S(z)\) and /res2/ to the -- Fresnel cosine integral \(C(z)\). Optionally, just a single function can -- be computed by passing /NULL/ as the other output variable. The -- definition \(S(z) = \int_0^z \sin(t^2) dt\) is used if /normalized/ is -- 0, and \(S(z) = \int_0^z \sin(\tfrac{1}{2} \pi t^2) dt\) is used if -- /normalized/ is 1 (the latter is the Abramowitz & Stegun convention). -- \(C(z)\) is defined analogously. foreign import ccall "arb_hypgeom.h arb_hypgeom_fresnel" arb_hypgeom_fresnel :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /_arb_hypgeom_fresnel_series/ /res1/ /res2/ /z/ /zlen/ /normalized/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_fresnel_series" _arb_hypgeom_fresnel_series :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_fresnel_series/ /res1/ /res2/ /z/ /normalized/ /len/ /prec/ -- -- Sets /res1/ to the Fresnel sine integral and /res2/ to the Fresnel -- cosine integral of the power series /z/, truncated to length /len/. -- Optionally, just a single function can be computed by passing /NULL/ as -- the other output variable. foreign import ccall "arb_hypgeom.h arb_hypgeom_fresnel_series" arb_hypgeom_fresnel_series :: Ptr CArbPoly -> Ptr CArbPoly -> Ptr CArbPoly -> CInt -> CLong -> CLong -> IO () -- Incomplete gamma and beta functions ----------------------------------------- -- | /arb_hypgeom_gamma_upper/ /res/ /s/ /z/ /regularized/ /prec/ -- -- If /regularized/ is 0, computes the upper incomplete gamma function -- \(\Gamma(s,z)\). -- -- If /regularized/ is 1, computes the regularized upper incomplete gamma -- function \(Q(s,z) = \Gamma(s,z) / \Gamma(s)\). -- -- If /regularized/ is 2, computes the generalized exponential integral -- \(z^{-s} \Gamma(s,z) = E_{1-s}(z)\) instead (this option is mainly -- intended for internal use; @arb_hypgeom_expint@ is the intended -- interface for computing the exponential integral). foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_upper" arb_hypgeom_gamma_upper :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /arb_hypgeom_gamma_upper_integration/ /res/ /s/ /z/ /regularized/ /prec/ -- -- Computes the upper incomplete gamma function using numerical -- integration. foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_upper_integration" arb_hypgeom_gamma_upper_integration :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /_arb_hypgeom_gamma_upper_series/ /res/ /s/ /z/ /zlen/ /regularized/ /n/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_upper_series" _arb_hypgeom_gamma_upper_series :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_gamma_upper_series/ /res/ /s/ /z/ /regularized/ /n/ /prec/ -- -- Sets /res/ to an upper incomplete gamma function where /s/ is a constant -- and /z/ is a power series, truncated to length /n/. The /regularized/ -- argument has the same interpretation as in @arb_hypgeom_gamma_upper@. foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_upper_series" arb_hypgeom_gamma_upper_series :: Ptr CArbPoly -> Ptr CArb -> Ptr CArbPoly -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_gamma_lower/ /res/ /s/ /z/ /regularized/ /prec/ -- -- If /regularized/ is 0, computes the lower incomplete gamma function -- \(\gamma(s,z) = \frac{z^s}{s} {}_1F_1(s, s+1, -z)\). -- -- If /regularized/ is 1, computes the regularized lower incomplete gamma -- function \(P(s,z) = \gamma(s,z) / \Gamma(s)\). -- -- If /regularized/ is 2, computes a further regularized lower incomplete -- gamma function \(\gamma^{*}(s,z) = z^{-s} P(s,z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_lower" arb_hypgeom_gamma_lower :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /_arb_hypgeom_gamma_lower_series/ /res/ /s/ /z/ /zlen/ /regularized/ /n/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_lower_series" _arb_hypgeom_gamma_lower_series :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_gamma_lower_series/ /res/ /s/ /z/ /regularized/ /n/ /prec/ -- -- Sets /res/ to an lower incomplete gamma function where /s/ is a constant -- and /z/ is a power series, truncated to length /n/. The /regularized/ -- argument has the same interpretation as in @arb_hypgeom_gamma_lower@. foreign import ccall "arb_hypgeom.h arb_hypgeom_gamma_lower_series" arb_hypgeom_gamma_lower_series :: Ptr CArbPoly -> Ptr CArb -> Ptr CArbPoly -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_beta_lower/ /res/ /a/ /b/ /z/ /regularized/ /prec/ -- -- Computes the (lower) incomplete beta function, defined by -- \(B(a,b;z) = \int_0^z t^{a-1} (1-t)^{b-1}\), optionally the regularized -- incomplete beta function \(I(a,b;z) = B(a,b;z) / B(a,b;1)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_beta_lower" arb_hypgeom_beta_lower :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /_arb_hypgeom_beta_lower_series/ /res/ /a/ /b/ /z/ /zlen/ /regularized/ /n/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_beta_lower_series" _arb_hypgeom_beta_lower_series :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_beta_lower_series/ /res/ /a/ /b/ /z/ /regularized/ /n/ /prec/ -- -- Sets /res/ to the lower incomplete beta function \(B(a,b;z)\) -- (optionally the regularized version \(I(a,b;z)\)) where /a/ and /b/ are -- constants and /z/ is a power series, truncating the result to length -- /n/. The underscore method requires positive lengths and does not -- support aliasing. foreign import ccall "arb_hypgeom.h arb_hypgeom_beta_lower_series" arb_hypgeom_beta_lower_series :: Ptr CArbPoly -> Ptr CArb -> Ptr CArb -> Ptr CArbPoly -> CInt -> CLong -> CLong -> IO () -- Internal evaluation functions ----------------------------------------------- -- | /_arb_hypgeom_gamma_lower_sum_rs_1/ /res/ /p/ /q/ /z/ /N/ /prec/ -- -- Computes \(\sum_{k=0}^{N-1} z^k / (a)_k\) where \(a = p/q\) using -- rectangular splitting. It is assumed that \(p + qN\) fits in a limb. foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_lower_sum_rs_1" _arb_hypgeom_gamma_lower_sum_rs_1 :: Ptr CArb -> CULong -> CULong -> Ptr CArb -> CLong -> CLong -> IO () -- | /_arb_hypgeom_gamma_upper_sum_rs_1/ /res/ /p/ /q/ /z/ /N/ /prec/ -- -- Computes \(\sum_{k=0}^{N-1} (a)_k / z^k\) where \(a = p/q\) using -- rectangular splitting. It is assumed that \(p + qN\) fits in a limb. foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_upper_sum_rs_1" _arb_hypgeom_gamma_upper_sum_rs_1 :: Ptr CArb -> CULong -> CULong -> Ptr CArb -> CLong -> CLong -> IO () -- | /_arb_hypgeom_gamma_upper_fmpq_inf_choose_N/ /err/ /a/ /z/ /abs_tol/ -- -- Returns number of terms /N/ and sets /err/ to the truncation error for -- evaluating \(\Gamma(a,z)\) using the asymptotic series at infinity, -- targeting an absolute tolerance of /abs_tol/. The error may be set to -- /err/ if the tolerance cannot be achieved. Assumes that /z/ is positive. foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_upper_fmpq_inf_choose_N" _arb_hypgeom_gamma_upper_fmpq_inf_choose_N :: Ptr CMag -> Ptr CFmpq -> Ptr CArb -> Ptr CMag -> IO CLong -- | /_arb_hypgeom_gamma_upper_fmpq_inf_bsplit/ /res/ /a/ /z/ /N/ /prec/ -- -- Sets /res/ to the approximation of \(\Gamma(a,z)\) obtained by -- truncating the asymptotic series at infinity before term /N/. The -- truncation error bound has to be added separately. foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_upper_fmpq_inf_bsplit" _arb_hypgeom_gamma_upper_fmpq_inf_bsplit :: Ptr CArb -> Ptr CFmpq -> Ptr CArb -> CLong -> CLong -> IO () -- | /_arb_hypgeom_gamma_lower_fmpq_0_choose_N/ /err/ /a/ /z/ /abs_tol/ -- -- Returns number of terms /N/ and sets /err/ to the truncation error for -- evaluating \(\gamma(a,z)\) using the Taylor series at zero, targeting an -- absolute tolerance of /abs_tol/. Assumes that /z/ is positive. foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_lower_fmpq_0_choose_N" _arb_hypgeom_gamma_lower_fmpq_0_choose_N :: Ptr CMag -> Ptr CFmpq -> Ptr CArb -> Ptr CMag -> IO CLong -- | /_arb_hypgeom_gamma_lower_fmpq_0_bsplit/ /res/ /a/ /z/ /N/ /prec/ -- -- Sets /res/ to the approximation of \(\gamma(a,z)\) obtained by -- truncating the Taylor series at zero before term /N/. The truncation -- error bound has to be added separately. foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_lower_fmpq_0_bsplit" _arb_hypgeom_gamma_lower_fmpq_0_bsplit :: Ptr CArb -> Ptr CFmpq -> Ptr CArb -> CLong -> CLong -> IO () -- | /_arb_hypgeom_gamma_upper_singular_si_choose_N/ /err/ /n/ /z/ /abs_tol/ -- -- Returns number of terms /N/ and sets /err/ to the truncation error for -- evaluating \(\Gamma(-n,z)\) using the Taylor series at zero, targeting -- an absolute tolerance of /abs_tol/. foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_upper_singular_si_choose_N" _arb_hypgeom_gamma_upper_singular_si_choose_N :: Ptr CMag -> CLong -> Ptr CArb -> Ptr CMag -> IO CLong -- | /_arb_hypgeom_gamma_upper_singular_si_bsplit/ /res/ /n/ /z/ /N/ /prec/ -- -- Sets /res/ to the approximation of \(\Gamma(-n,z)\) obtained by -- truncating the Taylor series at zero before term /N/. The truncation -- error bound has to be added separately. foreign import ccall "arb_hypgeom.h _arb_hypgeom_gamma_upper_singular_si_bsplit" _arb_hypgeom_gamma_upper_singular_si_bsplit :: Ptr CArb -> CLong -> Ptr CArb -> CLong -> CLong -> IO () -- | /_arb_gamma_upper_fmpq_step_bsplit/ /Gz1/ /a/ /z0/ /z1/ /Gz0/ /expmz0/ /abs_tol/ /prec/ -- -- Given /Gz0/ and /expmz0/ representing the values \(\Gamma(a,z_0)\) and -- \(\exp(-z_0)\), computes \(\Gamma(a,z_1)\) using the Taylor series at -- \(z_0\) evaluated using binary splitting, targeting an absolute error of -- /abs_tol/. Assumes that \(z_0\) and \(z_1\) are positive. foreign import ccall "arb_hypgeom.h _arb_gamma_upper_fmpq_step_bsplit" _arb_gamma_upper_fmpq_step_bsplit :: Ptr CArb -> Ptr CFmpq -> Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CMag -> CLong -> IO () -- Exponential and trigonometric integrals ------------------------------------- -- | /arb_hypgeom_expint/ /res/ /s/ /z/ /prec/ -- -- Computes the generalized exponential integral \(E_s(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_expint" arb_hypgeom_expint :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_ei/ /res/ /z/ /prec/ -- -- Computes the exponential integral \(\operatorname{Ei}(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_ei" arb_hypgeom_ei :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /_arb_hypgeom_ei_series/ /res/ /z/ /zlen/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_ei_series" _arb_hypgeom_ei_series :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_ei_series/ /res/ /z/ /len/ /prec/ -- -- Computes the exponential integral of the power series /z/, truncated to -- length /len/. foreign import ccall "arb_hypgeom.h arb_hypgeom_ei_series" arb_hypgeom_ei_series :: Ptr CArbPoly -> Ptr CArbPoly -> CLong -> CLong -> IO () -- | /_arb_hypgeom_si_asymp/ /res/ /z/ /N/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_si_asymp" _arb_hypgeom_si_asymp :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> IO () -- | /_arb_hypgeom_si_1f2/ /res/ /z/ /N/ /wp/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_si_1f2" _arb_hypgeom_si_1f2 :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_si/ /res/ /z/ /prec/ -- -- Computes the sine integral \(\operatorname{Si}(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_si" arb_hypgeom_si :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /_arb_hypgeom_si_series/ /res/ /z/ /zlen/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_si_series" _arb_hypgeom_si_series :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_si_series/ /res/ /z/ /len/ /prec/ -- -- Computes the sine integral of the power series /z/, truncated to length -- /len/. foreign import ccall "arb_hypgeom.h arb_hypgeom_si_series" arb_hypgeom_si_series :: Ptr CArbPoly -> Ptr CArbPoly -> CLong -> CLong -> IO () -- | /_arb_hypgeom_ci_asymp/ /res/ /z/ /N/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_ci_asymp" _arb_hypgeom_ci_asymp :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> IO () -- | /_arb_hypgeom_ci_2f3/ /res/ /z/ /N/ /wp/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_ci_2f3" _arb_hypgeom_ci_2f3 :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_ci/ /res/ /z/ /prec/ -- -- Computes the cosine integral \(\operatorname{Ci}(z)\). The result is -- indeterminate if \(z < 0\) since the value of the function would be -- complex. foreign import ccall "arb_hypgeom.h arb_hypgeom_ci" arb_hypgeom_ci :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /_arb_hypgeom_ci_series/ /res/ /z/ /zlen/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_ci_series" _arb_hypgeom_ci_series :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_ci_series/ /res/ /z/ /len/ /prec/ -- -- Computes the cosine integral of the power series /z/, truncated to -- length /len/. foreign import ccall "arb_hypgeom.h arb_hypgeom_ci_series" arb_hypgeom_ci_series :: Ptr CArbPoly -> Ptr CArbPoly -> CLong -> CLong -> IO () -- | /arb_hypgeom_shi/ /res/ /z/ /prec/ -- -- Computes the hyperbolic sine integral -- \(\operatorname{Shi}(z) = -i \operatorname{Si}(iz)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_shi" arb_hypgeom_shi :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /_arb_hypgeom_shi_series/ /res/ /z/ /zlen/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_shi_series" _arb_hypgeom_shi_series :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_shi_series/ /res/ /z/ /len/ /prec/ -- -- Computes the hyperbolic sine integral of the power series /z/, truncated -- to length /len/. foreign import ccall "arb_hypgeom.h arb_hypgeom_shi_series" arb_hypgeom_shi_series :: Ptr CArbPoly -> Ptr CArbPoly -> CLong -> CLong -> IO () -- | /arb_hypgeom_chi/ /res/ /z/ /prec/ -- -- Computes the hyperbolic cosine integral \(\operatorname{Chi}(z)\). The -- result is indeterminate if \(z < 0\) since the value of the function -- would be complex. foreign import ccall "arb_hypgeom.h arb_hypgeom_chi" arb_hypgeom_chi :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /_arb_hypgeom_chi_series/ /res/ /z/ /zlen/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_chi_series" _arb_hypgeom_chi_series :: Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_chi_series/ /res/ /z/ /len/ /prec/ -- -- Computes the hyperbolic cosine integral of the power series /z/, -- truncated to length /len/. foreign import ccall "arb_hypgeom.h arb_hypgeom_chi_series" arb_hypgeom_chi_series :: Ptr CArbPoly -> Ptr CArbPoly -> CLong -> CLong -> IO () -- | /arb_hypgeom_li/ /res/ /z/ /offset/ /prec/ -- -- If /offset/ is zero, computes the logarithmic integral -- \(\operatorname{li}(z) = \operatorname{Ei}(\log(z))\). -- -- If /offset/ is nonzero, computes the offset logarithmic integral -- \(\operatorname{Li}(z) = \operatorname{li}(z) - \operatorname{li}(2)\). -- -- The result is indeterminate if \(z < 0\) since the value of the function -- would be complex. foreign import ccall "arb_hypgeom.h arb_hypgeom_li" arb_hypgeom_li :: Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /_arb_hypgeom_li_series/ /res/ /z/ /zlen/ /offset/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_li_series" _arb_hypgeom_li_series :: Ptr CArb -> Ptr CArb -> CLong -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_li_series/ /res/ /z/ /offset/ /len/ /prec/ -- -- Computes the logarithmic integral (optionally the offset version) of the -- power series /z/, truncated to length /len/. foreign import ccall "arb_hypgeom.h arb_hypgeom_li_series" arb_hypgeom_li_series :: Ptr CArbPoly -> Ptr CArbPoly -> CInt -> CLong -> CLong -> IO () -- Bessel functions ------------------------------------------------------------ -- | /arb_hypgeom_bessel_j/ /res/ /nu/ /z/ /prec/ -- -- Computes the Bessel function of the first kind \(J_{\nu}(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_bessel_j" arb_hypgeom_bessel_j :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_bessel_y/ /res/ /nu/ /z/ /prec/ -- -- Computes the Bessel function of the second kind \(Y_{\nu}(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_bessel_y" arb_hypgeom_bessel_y :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_bessel_jy/ /res1/ /res2/ /nu/ /z/ /prec/ -- -- Sets /res1/ to \(J_{\nu}(z)\) and /res2/ to \(Y_{\nu}(z)\), computed -- simultaneously. foreign import ccall "arb_hypgeom.h arb_hypgeom_bessel_jy" arb_hypgeom_bessel_jy :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_bessel_i/ /res/ /nu/ /z/ /prec/ -- -- Computes the modified Bessel function of the first kind -- \(I_{\nu}(z) = z^{\nu} (iz)^{-\nu} J_{\nu}(iz)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_bessel_i" arb_hypgeom_bessel_i :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_bessel_i_scaled/ /res/ /nu/ /z/ /prec/ -- -- Computes the function \(e^{-z} I_{\nu}(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_bessel_i_scaled" arb_hypgeom_bessel_i_scaled :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_bessel_k/ /res/ /nu/ /z/ /prec/ -- -- Computes the modified Bessel function of the second kind \(K_{\nu}(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_bessel_k" arb_hypgeom_bessel_k :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_bessel_k_scaled/ /res/ /nu/ /z/ /prec/ -- -- Computes the function \(e^{z} K_{\nu}(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_bessel_k_scaled" arb_hypgeom_bessel_k_scaled :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_bessel_i_integration/ /res/ /nu/ /z/ /scaled/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_bessel_i_integration" arb_hypgeom_bessel_i_integration :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /arb_hypgeom_bessel_k_integration/ /res/ /nu/ /z/ /scaled/ /prec/ -- -- Computes the modified Bessel functions using numerical integration. foreign import ccall "arb_hypgeom.h arb_hypgeom_bessel_k_integration" arb_hypgeom_bessel_k_integration :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- Airy functions -------------------------------------------------------------- -- | /arb_hypgeom_airy/ /ai/ /ai_prime/ /bi/ /bi_prime/ /z/ /prec/ -- -- Computes the Airy functions -- \((\operatorname{Ai}(z), \operatorname{Ai}'(z), \operatorname{Bi}(z), \operatorname{Bi}'(z))\) -- simultaneously. Any of the four function values can be omitted by -- passing /NULL/ for the unwanted output variables, speeding up the -- evaluation. foreign import ccall "arb_hypgeom.h arb_hypgeom_airy" arb_hypgeom_airy :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_airy_jet/ /ai/ /bi/ /z/ /len/ /prec/ -- -- Writes to /ai/ and /bi/ the respective Taylor expansions of the Airy -- functions at the point /z/, truncated to length /len/. Either of the -- outputs can be /NULL/ to avoid computing that function. The variable /z/ -- is not allowed to be aliased with the outputs. To simplify the -- implementation, this method does not compute the series expansions of -- the primed versions directly; these are easily obtained by computing one -- extra coefficient and differentiating the output with -- @_arb_poly_derivative@. foreign import ccall "arb_hypgeom.h arb_hypgeom_airy_jet" arb_hypgeom_airy_jet :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> CLong -> IO () -- | /_arb_hypgeom_airy_series/ /ai/ /ai_prime/ /bi/ /bi_prime/ /z/ /zlen/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_airy_series" _arb_hypgeom_airy_series :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_airy_series/ /ai/ /ai_prime/ /bi/ /bi_prime/ /z/ /len/ /prec/ -- -- Computes the Airy functions evaluated at the power series /z/, truncated -- to length /len/. As with the other Airy methods, any of the outputs can -- be /NULL/. foreign import ccall "arb_hypgeom.h arb_hypgeom_airy_series" arb_hypgeom_airy_series :: Ptr CArbPoly -> Ptr CArbPoly -> Ptr CArbPoly -> Ptr CArbPoly -> Ptr CArbPoly -> CLong -> CLong -> IO () -- | /arb_hypgeom_airy_zero/ /a/ /a_prime/ /b/ /b_prime/ /n/ /prec/ -- -- Computes the /n/-th real zero \(a_n\), \(a'_n\), \(b_n\), or \(b'_n\) -- for the respective Airy function or Airy function derivative. Any -- combination of the four output variables can be /NULL/. The zeros are -- indexed by increasing magnitude, starting with \(n = 1\) to follow the -- convention in the literature. An index /n/ that is not positive is -- invalid input. The implementation uses asymptotic expansions for the -- zeros < [PS1991]> together with the interval Newton method for -- refinement. foreign import ccall "arb_hypgeom.h arb_hypgeom_airy_zero" arb_hypgeom_airy_zero :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CFmpz -> CLong -> IO () -- Coulomb wave functions ------------------------------------------------------ -- | /arb_hypgeom_coulomb/ /F/ /G/ /l/ /eta/ /z/ /prec/ -- -- Writes to /F/, /G/ the values of the respective Coulomb wave functions -- \(F_{\ell}(\eta,z)\) and \(G_{\ell}(\eta,z)\). Either of the outputs can -- be /NULL/. foreign import ccall "arb_hypgeom.h arb_hypgeom_coulomb" arb_hypgeom_coulomb :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_coulomb_jet/ /F/ /G/ /l/ /eta/ /z/ /len/ /prec/ -- -- Writes to /F/, /G/ the respective Taylor expansions of the Coulomb wave -- functions at the point /z/, truncated to length /len/. Either of the -- outputs can be /NULL/. foreign import ccall "arb_hypgeom.h arb_hypgeom_coulomb_jet" arb_hypgeom_coulomb_jet :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> CLong -> IO () -- | /_arb_hypgeom_coulomb_series/ /F/ /G/ /l/ /eta/ /z/ /zlen/ /len/ /prec/ foreign import ccall "arb_hypgeom.h _arb_hypgeom_coulomb_series" _arb_hypgeom_coulomb_series :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> CLong -> CLong -> IO () -- | /arb_hypgeom_coulomb_series/ /F/ /G/ /l/ /eta/ /z/ /len/ /prec/ -- -- Computes the Coulomb wave functions evaluated at the power series /z/, -- truncated to length /len/. Either of the outputs can be /NULL/. foreign import ccall "arb_hypgeom.h arb_hypgeom_coulomb_series" arb_hypgeom_coulomb_series :: Ptr CArbPoly -> Ptr CArbPoly -> Ptr CArb -> Ptr CArb -> Ptr CArbPoly -> CLong -> CLong -> IO () -- Orthogonal polynomials and functions ---------------------------------------- -- | /arb_hypgeom_chebyshev_t/ /res/ /nu/ /z/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_chebyshev_t" arb_hypgeom_chebyshev_t :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_chebyshev_u/ /res/ /nu/ /z/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_chebyshev_u" arb_hypgeom_chebyshev_u :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_jacobi_p/ /res/ /n/ /a/ /b/ /z/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_jacobi_p" arb_hypgeom_jacobi_p :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_gegenbauer_c/ /res/ /n/ /m/ /z/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_gegenbauer_c" arb_hypgeom_gegenbauer_c :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_laguerre_l/ /res/ /n/ /m/ /z/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_laguerre_l" arb_hypgeom_laguerre_l :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_hermite_h/ /res/ /nu/ /z/ /prec/ -- -- Computes Chebyshev, Jacobi, Gegenbauer, Laguerre or Hermite polynomials, -- or their extensions to non-integer orders. foreign import ccall "arb_hypgeom.h arb_hypgeom_hermite_h" arb_hypgeom_hermite_h :: Ptr CArb -> Ptr CArb -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_legendre_p/ /res/ /n/ /m/ /z/ /type/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_legendre_p" arb_hypgeom_legendre_p :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /arb_hypgeom_legendre_q/ /res/ /n/ /m/ /z/ /type/ /prec/ -- -- Computes Legendre functions of the first and second kind. See -- @acb_hypgeom_legendre_p@ and @acb_hypgeom_legendre_q@ for definitions. foreign import ccall "arb_hypgeom.h arb_hypgeom_legendre_q" arb_hypgeom_legendre_q :: Ptr CArb -> Ptr CArb -> Ptr CArb -> Ptr CArb -> CInt -> CLong -> IO () -- | /arb_hypgeom_legendre_p_ui_deriv_bound/ /dp/ /dp2/ /n/ /x/ /x2sub1/ -- -- Sets /dp/ to an upper bound for \(P'_n(x)\) and /dp2/ to an upper bound -- for \(P''_n(x)\) given /x/ assumed to represent a real number with -- \(|x| \le 1\). The variable /x2sub1/ must contain the precomputed value -- \(1-x^2\) (or \(x^2-1\)). This method is used internally to bound the -- propagated error for Legendre polynomials. foreign import ccall "arb_hypgeom.h arb_hypgeom_legendre_p_ui_deriv_bound" arb_hypgeom_legendre_p_ui_deriv_bound :: Ptr CMag -> Ptr CMag -> CULong -> Ptr CArb -> Ptr CArb -> IO () -- | /arb_hypgeom_legendre_p_ui_zero/ /res/ /res_prime/ /n/ /x/ /K/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_legendre_p_ui_zero" arb_hypgeom_legendre_p_ui_zero :: Ptr CArb -> Ptr CArb -> CULong -> Ptr CArb -> CLong -> CLong -> IO () -- | /arb_hypgeom_legendre_p_ui_one/ /res/ /res_prime/ /n/ /x/ /K/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_legendre_p_ui_one" arb_hypgeom_legendre_p_ui_one :: Ptr CArb -> Ptr CArb -> CULong -> Ptr CArb -> CLong -> CLong -> IO () -- | /arb_hypgeom_legendre_p_ui_asymp/ /res/ /res_prime/ /n/ /x/ /K/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_legendre_p_ui_asymp" arb_hypgeom_legendre_p_ui_asymp :: Ptr CArb -> Ptr CArb -> CULong -> Ptr CArb -> CLong -> CLong -> IO () -- -- | /arb_hypgeom_legendre_p_rec/ /res/ /res_prime/ /n/ /x/ /prec/ -- foreign import ccall "arb_hypgeom.h arb_hypgeom_legendre_p_rec" -- arb_hypgeom_legendre_p_rec :: Ptr CArb -> Ptr CArb -> CULong -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_legendre_p_ui/ /res/ /res_prime/ /n/ /x/ /prec/ -- -- Evaluates the ordinary Legendre polynomial \(P_n(x)\). If /res_prime/ is -- non-NULL, simultaneously evaluates the derivative \(P'_n(x)\). -- -- The overall algorithm is described in < [JM2018]>. -- -- The versions /zero/, /one/ respectively use the hypergeometric series -- expansions at \(x = 0\) and \(x = 1\) while the /asymp/ version uses an -- asymptotic series on \((-1,1)\) intended for large /n/. The parameter -- /K/ specifies the exact number of expansion terms to use (if the series -- expansion truncated at this point does not give the exact polynomial, an -- error bound is computed automatically). The asymptotic expansion with -- error bounds is given in < [Bog2012]>. The /rec/ version uses the -- forward recurrence implemented using fixed-point arithmetic; it is only -- intended for the interval \((-1,1)\), moderate /n/ and modest precision. -- -- The default version attempts to choose the best algorithm automatically. -- It also estimates the amount of cancellation in the hypergeometric -- series and increases the working precision to compensate, bounding the -- propagated error using derivative bounds. foreign import ccall "arb_hypgeom.h arb_hypgeom_legendre_p_ui" arb_hypgeom_legendre_p_ui :: Ptr CArb -> Ptr CArb -> CULong -> Ptr CArb -> CLong -> IO () -- | /arb_hypgeom_legendre_p_ui_root/ /res/ /weight/ /n/ /k/ /prec/ -- -- Sets /res/ to the /k/-th root of the Legendre polynomial \(P_n(x)\). We -- index the roots in decreasing order -- -- \[`\] -- \[1 > x_0 > x_1 > \ldots > x_{n-1} > -1\] -- -- (which corresponds to ordering the roots of \(P_n(\cos(\theta))\) in -- order of increasing \(\theta\)). If /weight/ is non-NULL, it is set to -- the weight corresponding to the node \(x_k\) for Gaussian quadrature on -- \([-1,1]\). Note that only \(\lceil n / 2 \rceil\) roots need to be -- computed, since the remaining roots are given by \(x_k = -x_{n-1-k}\). -- -- We compute an enclosing interval using an asymptotic approximation -- followed by some number of Newton iterations, using the error bounds -- given in < [Pet1999]>. If very high precision is requested, the root is -- subsequently refined using interval Newton steps with doubling working -- precision. foreign import ccall "arb_hypgeom.h arb_hypgeom_legendre_p_ui_root" arb_hypgeom_legendre_p_ui_root :: Ptr CArb -> Ptr CArb -> CULong -> CULong -> CLong -> IO () -- Dilogarithm ----------------------------------------------------------------- -- | /arb_hypgeom_dilog/ /res/ /z/ /prec/ -- -- Computes the dilogarithm \(\operatorname{Li}_2(z)\). foreign import ccall "arb_hypgeom.h arb_hypgeom_dilog" arb_hypgeom_dilog :: Ptr CArb -> Ptr CArb -> CLong -> IO () -- Hypergeometric sums --------------------------------------------------------- -- | /arb_hypgeom_sum_fmpq_arb_forward/ /res/ /a/ /alen/ /b/ /blen/ /z/ /reciprocal/ /N/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_sum_fmpq_arb_forward" arb_hypgeom_sum_fmpq_arb_forward :: Ptr CArb -> Ptr CFmpq -> CLong -> Ptr CFmpq -> CLong -> Ptr CArb -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_sum_fmpq_arb_rs/ /res/ /a/ /alen/ /b/ /blen/ /z/ /reciprocal/ /N/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_sum_fmpq_arb_rs" arb_hypgeom_sum_fmpq_arb_rs :: Ptr CArb -> Ptr CFmpq -> CLong -> Ptr CFmpq -> CLong -> Ptr CArb -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_sum_fmpq_arb/ /res/ /a/ /alen/ /b/ /blen/ /z/ /reciprocal/ /N/ /prec/ -- -- Sets /res/ to the finite hypergeometric sum -- \(\sum_{n=0}^{N-1} (\textbf{a})_n z^n / (\textbf{b})_n\) where -- \(\textbf{x}_n = (x_1)_n (x_2)_n \cdots\), given vectors of rational -- parameters /a/ (of length /alen/) and /b/ (of length /blen/). If -- /reciprocal/ is set, replace \(z\) by \(1 / z\). The /forward/ version -- uses the forward recurrence, optimized by delaying divisions, the /rs/ -- version uses rectangular splitting, and the default version uses an -- automatic algorithm choice. foreign import ccall "arb_hypgeom.h arb_hypgeom_sum_fmpq_arb" arb_hypgeom_sum_fmpq_arb :: Ptr CArb -> Ptr CFmpq -> CLong -> Ptr CFmpq -> CLong -> Ptr CArb -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_sum_fmpq_imag_arb_forward/ /res1/ /res2/ /a/ /alen/ /b/ /blen/ /z/ /reciprocal/ /N/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_sum_fmpq_imag_arb_forward" arb_hypgeom_sum_fmpq_imag_arb_forward :: Ptr CArb -> Ptr CArb -> Ptr CFmpq -> CLong -> Ptr CFmpq -> CLong -> Ptr CArb -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_sum_fmpq_imag_arb_rs/ /res1/ /res2/ /a/ /alen/ /b/ /blen/ /z/ /reciprocal/ /N/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_sum_fmpq_imag_arb_rs" arb_hypgeom_sum_fmpq_imag_arb_rs :: Ptr CArb -> Ptr CArb -> Ptr CFmpq -> CLong -> Ptr CFmpq -> CLong -> Ptr CArb -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_sum_fmpq_imag_arb_bs/ /res1/ /res2/ /a/ /alen/ /b/ /blen/ /z/ /reciprocal/ /N/ /prec/ foreign import ccall "arb_hypgeom.h arb_hypgeom_sum_fmpq_imag_arb_bs" arb_hypgeom_sum_fmpq_imag_arb_bs :: Ptr CArb -> Ptr CArb -> Ptr CFmpq -> CLong -> Ptr CFmpq -> CLong -> Ptr CArb -> CInt -> CLong -> CLong -> IO () -- | /arb_hypgeom_sum_fmpq_imag_arb/ /res1/ /res2/ /a/ /alen/ /b/ /blen/ /z/ /reciprocal/ /N/ /prec/ -- -- Sets /res1/ and /res2/ to the real and imaginary part of the finite -- hypergeometric sum -- \(\sum_{n=0}^{N-1} (\textbf{a})_n (i z)^n / (\textbf{b})_n\). If -- /reciprocal/ is set, replace \(z\) by \(1 / z\). foreign import ccall "arb_hypgeom.h arb_hypgeom_sum_fmpq_imag_arb" arb_hypgeom_sum_fmpq_imag_arb :: Ptr CArb -> Ptr CArb -> Ptr CFmpq -> CLong -> Ptr CFmpq -> CLong -> Ptr CArb -> CInt -> CLong -> CLong -> IO ()