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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* k0.c
*
* Modified Bessel function, third kind, order zero
*
*
*
* SYNOPSIS:
*
* double x, y, k0();
*
* y = k0( x );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of the third kind
* of order zero of the argument.
*
* The range is partitioned into the two intervals [0,8] and
* (8, infinity). Chebyshev polynomial expansions are employed
* in each interval.
*
*
*
* ACCURACY:
*
* Tested at 2000 random points between 0 and 8. Peak absolute
* error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 1.2e-15 1.6e-16
*
* ERROR MESSAGES:
*
* message condition value returned
* K0 domain x <= 0 INFINITY
*
*/
/* k0e()
*
* Modified Bessel function, third kind, order zero,
* exponentially scaled
*
*
*
* SYNOPSIS:
*
* double x, y, k0e();
*
* y = k0e( x );
*
*
*
* DESCRIPTION:
*
* Returns exponentially scaled modified Bessel function
* of the third kind of order zero of the argument.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 1.4e-15 1.4e-16
* See k0().
*
*/
/*
* Cephes Math Library Release 2.8: June, 2000
* Copyright 1984, 1987, 2000 by Stephen L. Moshier
*/
#pragma once
#include "../config.h"
#include "../error.h"
#include "chbevl.h"
#include "i0.h"
namespace xsf {
namespace cephes {
namespace detail {
/* Chebyshev coefficients for K0(x) + log(x/2) I0(x)
* in the interval [0,2]. The odd order coefficients are all
* zero; only the even order coefficients are listed.
*
* lim(x->0){ K0(x) + log(x/2) I0(x) } = -EUL.
*/
constexpr double k0_A[] = {1.37446543561352307156E-16, 4.25981614279661018399E-14, 1.03496952576338420167E-11,
1.90451637722020886025E-9, 2.53479107902614945675E-7, 2.28621210311945178607E-5,
1.26461541144692592338E-3, 3.59799365153615016266E-2, 3.44289899924628486886E-1,
-5.35327393233902768720E-1};
/* Chebyshev coefficients for exp(x) sqrt(x) K0(x)
* in the inverted interval [2,infinity].
*
* lim(x->inf){ exp(x) sqrt(x) K0(x) } = sqrt(pi/2).
*/
constexpr double k0_B[] = {
5.30043377268626276149E-18, -1.64758043015242134646E-17, 5.21039150503902756861E-17,
-1.67823109680541210385E-16, 5.51205597852431940784E-16, -1.84859337734377901440E-15,
6.34007647740507060557E-15, -2.22751332699166985548E-14, 8.03289077536357521100E-14,
-2.98009692317273043925E-13, 1.14034058820847496303E-12, -4.51459788337394416547E-12,
1.85594911495471785253E-11, -7.95748924447710747776E-11, 3.57739728140030116597E-10,
-1.69753450938905987466E-9, 8.57403401741422608519E-9, -4.66048989768794782956E-8,
2.76681363944501510342E-7, -1.83175552271911948767E-6, 1.39498137188764993662E-5,
-1.28495495816278026384E-4, 1.56988388573005337491E-3, -3.14481013119645005427E-2,
2.44030308206595545468E0};
} // namespace detail
XSF_HOST_DEVICE inline double k0(double x) {
double y, z;
if (x == 0.0) {
set_error("k0", SF_ERROR_SINGULAR, NULL);
return std::numeric_limits<double>::infinity();
} else if (x < 0.0) {
set_error("k0", SF_ERROR_DOMAIN, NULL);
return std::numeric_limits<double>::quiet_NaN();
}
if (x <= 2.0) {
y = x * x - 2.0;
y = chbevl(y, detail::k0_A, 10) - std::log(0.5 * x) * i0(x);
return (y);
}
z = 8.0 / x - 2.0;
y = std::exp(-x) * chbevl(z, detail::k0_B, 25) / std::sqrt(x);
return (y);
}
XSF_HOST_DEVICE double inline k0e(double x) {
double y;
if (x == 0.0) {
set_error("k0e", SF_ERROR_SINGULAR, NULL);
return std::numeric_limits<double>::infinity();
} else if (x < 0.0) {
set_error("k0e", SF_ERROR_DOMAIN, NULL);
return std::numeric_limits<double>::quiet_NaN();
}
if (x <= 2.0) {
y = x * x - 2.0;
y = chbevl(y, detail::k0_A, 10) - std::log(0.5 * x) * i0(x);
return (y * exp(x));
}
y = chbevl(8.0 / x - 2.0, detail::k0_B, 25) / std::sqrt(x);
return (y);
}
} // namespace cephes
} // namespace xsf
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