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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* j0.c
*
* Bessel function of order zero
*
*
*
* SYNOPSIS:
*
* double x, y, j0();
*
* y = j0( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of order zero of the argument.
*
* The domain is divided into the intervals [0, 5] and
* (5, infinity). In the first interval the following rational
* approximation is used:
*
*
* 2 2
* (w - r ) (w - r ) P (w) / Q (w)
* 1 2 3 8
*
* 2
* where w = x and the two r's are zeros of the function.
*
* In the second interval, the Hankel asymptotic expansion
* is employed with two rational functions of degree 6/6
* and 7/7.
*
*
*
* ACCURACY:
*
* Absolute error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 60000 4.2e-16 1.1e-16
*
*/
/* y0.c
*
* Bessel function of the second kind, order zero
*
*
*
* SYNOPSIS:
*
* double x, y, y0();
*
* y = y0( x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of the second kind, of order
* zero, of the argument.
*
* The domain is divided into the intervals [0, 5] and
* (5, infinity). In the first interval a rational approximation
* R(x) is employed to compute
* y0(x) = R(x) + 2 * log(x) * j0(x) / M_PI.
* Thus a call to j0() is required.
*
* In the second interval, the Hankel asymptotic expansion
* is employed with two rational functions of degree 6/6
* and 7/7.
*
*
*
* ACCURACY:
*
* Absolute error, when y0(x) < 1; else relative error:
*
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 1.3e-15 1.6e-16
*
*/
/*
* Cephes Math Library Release 2.8: June, 2000
* Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*/
/* Note: all coefficients satisfy the relative error criterion
* except YP, YQ which are designed for absolute error. */
#pragma once
#include "../config.h"
#include "../error.h"
#include "const.h"
#include "polevl.h"
namespace xsf {
namespace cephes {
namespace detail {
constexpr double j0_PP[7] = {
7.96936729297347051624E-4, 8.28352392107440799803E-2, 1.23953371646414299388E0, 5.44725003058768775090E0,
8.74716500199817011941E0, 5.30324038235394892183E0, 9.99999999999999997821E-1,
};
constexpr double j0_PQ[7] = {
9.24408810558863637013E-4, 8.56288474354474431428E-2, 1.25352743901058953537E0, 5.47097740330417105182E0,
8.76190883237069594232E0, 5.30605288235394617618E0, 1.00000000000000000218E0,
};
constexpr double j0_QP[8] = {
-1.13663838898469149931E-2, -1.28252718670509318512E0, -1.95539544257735972385E1, -9.32060152123768231369E1,
-1.77681167980488050595E2, -1.47077505154951170175E2, -5.14105326766599330220E1, -6.05014350600728481186E0,
};
constexpr double j0_QQ[7] = {
/* 1.00000000000000000000E0, */
6.43178256118178023184E1, 8.56430025976980587198E2, 3.88240183605401609683E3, 7.24046774195652478189E3,
5.93072701187316984827E3, 2.06209331660327847417E3, 2.42005740240291393179E2,
};
constexpr double j0_YP[8] = {
1.55924367855235737965E4, -1.46639295903971606143E7, 5.43526477051876500413E9,
-9.82136065717911466409E11, 8.75906394395366999549E13, -3.46628303384729719441E15,
4.42733268572569800351E16, -1.84950800436986690637E16,
};
constexpr double j0_YQ[7] = {
/* 1.00000000000000000000E0, */
1.04128353664259848412E3, 6.26107330137134956842E5, 2.68919633393814121987E8, 8.64002487103935000337E10,
2.02979612750105546709E13, 3.17157752842975028269E15, 2.50596256172653059228E17,
};
/* 5.783185962946784521175995758455807035071 */
constexpr double j0_DR1 = 5.78318596294678452118E0;
/* 30.47126234366208639907816317502275584842 */
constexpr double j0_DR2 = 3.04712623436620863991E1;
constexpr double j0_RP[4] = {
-4.79443220978201773821E9,
1.95617491946556577543E12,
-2.49248344360967716204E14,
9.70862251047306323952E15,
};
constexpr double j0_RQ[8] = {
/* 1.00000000000000000000E0, */
4.99563147152651017219E2, 1.73785401676374683123E5, 4.84409658339962045305E7, 1.11855537045356834862E10,
2.11277520115489217587E12, 3.10518229857422583814E14, 3.18121955943204943306E16, 1.71086294081043136091E18,
};
} // namespace detail
XSF_HOST_DEVICE inline double j0(double x) {
double w, z, p, q, xn;
if (x < 0) {
x = -x;
}
if (x <= 5.0) {
z = x * x;
if (x < 1.0e-5) {
return (1.0 - z / 4.0);
}
p = (z - detail::j0_DR1) * (z - detail::j0_DR2);
p = p * polevl(z, detail::j0_RP, 3) / p1evl(z, detail::j0_RQ, 8);
return (p);
}
w = 5.0 / x;
q = 25.0 / (x * x);
p = polevl(q, detail::j0_PP, 6) / polevl(q, detail::j0_PQ, 6);
q = polevl(q, detail::j0_QP, 7) / p1evl(q, detail::j0_QQ, 7);
xn = x - M_PI_4;
p = p * std::cos(xn) - w * q * std::sin(xn);
return (p * detail::SQRT2OPI / std::sqrt(x));
}
/* y0() 2 */
/* Bessel function of second kind, order zero */
/* Rational approximation coefficients YP[], YQ[] are used here.
* The function computed is y0(x) - 2 * log(x) * j0(x) / M_PI,
* whose value at x = 0 is 2 * ( log(0.5) + EUL ) / M_PI
* = 0.073804295108687225.
*/
XSF_HOST_DEVICE inline double y0(double x) {
double w, z, p, q, xn;
if (x <= 5.0) {
if (x == 0.0) {
set_error("y0", SF_ERROR_SINGULAR, NULL);
return -std::numeric_limits<double>::infinity();
} else if (x < 0.0) {
set_error("y0", SF_ERROR_DOMAIN, NULL);
return std::numeric_limits<double>::quiet_NaN();
}
z = x * x;
w = polevl(z, detail::j0_YP, 7) / p1evl(z, detail::j0_YQ, 7);
w += M_2_PI * std::log(x) * j0(x);
return (w);
}
w = 5.0 / x;
z = 25.0 / (x * x);
p = polevl(z, detail::j0_PP, 6) / polevl(z, detail::j0_PQ, 6);
q = polevl(z, detail::j0_QP, 7) / p1evl(z, detail::j0_QQ, 7);
xn = x - M_PI_4;
p = p * std::sin(xn) + w * q * std::cos(xn);
return (p * detail::SQRT2OPI / std::sqrt(x));
}
} // namespace cephes
} // namespace xsf
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