File size: 6,264 Bytes
7885a28 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 |
/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* kn.c
*
* Modified Bessel function, third kind, integer order
*
*
*
* SYNOPSIS:
*
* double x, y, kn();
* int n;
*
* y = kn( n, x );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of the third kind
* of order n of the argument.
*
* The range is partitioned into the two intervals [0,9.55] and
* (9.55, infinity). An ascending power series is used in the
* low range, and an asymptotic expansion in the high range.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0,30 90000 1.8e-8 3.0e-10
*
* Error is high only near the crossover point x = 9.55
* between the two expansions used.
*/
/*
* Cephes Math Library Release 2.8: June, 2000
* Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*/
/*
* Algorithm for Kn.
* n-1
* -n - (n-k-1)! 2 k
* K (x) = 0.5 (x/2) > -------- (-x /4)
* n - k!
* k=0
*
* inf. 2 k
* n n - (x /4)
* + (-1) 0.5(x/2) > {p(k+1) + p(n+k+1) - 2log(x/2)} ---------
* - k! (n+k)!
* k=0
*
* where p(m) is the psi function: p(1) = -EUL and
*
* m-1
* -
* p(m) = -EUL + > 1/k
* -
* k=1
*
* For large x,
* 2 2 2
* u-1 (u-1 )(u-3 )
* K (z) = sqrt(pi/2z) exp(-z) { 1 + ------- + ------------ + ...}
* v 1 2
* 1! (8z) 2! (8z)
* asymptotically, where
*
* 2
* u = 4 v .
*
*/
#pragma once
#include "../config.h"
#include "../error.h"
#include "const.h"
namespace xsf {
namespace cephes {
namespace detail {
constexpr int kn_MAXFAC = 31;
}
XSF_HOST_DEVICE inline double kn(int nn, double x) {
double k, kf, nk1f, nkf, zn, t, s, z0, z;
double ans, fn, pn, pk, zmn, tlg, tox;
int i, n;
if (nn < 0)
n = -nn;
else
n = nn;
if (n > detail::kn_MAXFAC) {
overf:
set_error("kn", SF_ERROR_OVERFLOW, NULL);
return (std::numeric_limits<double>::infinity());
}
if (x <= 0.0) {
if (x < 0.0) {
set_error("kn", SF_ERROR_DOMAIN, NULL);
return std::numeric_limits<double>::quiet_NaN();
} else {
set_error("kn", SF_ERROR_SINGULAR, NULL);
return std::numeric_limits<double>::infinity();
}
}
if (x > 9.55)
goto asymp;
ans = 0.0;
z0 = 0.25 * x * x;
fn = 1.0;
pn = 0.0;
zmn = 1.0;
tox = 2.0 / x;
if (n > 0) {
/* compute factorial of n and psi(n) */
pn = -detail::SCIPY_EULER;
k = 1.0;
for (i = 1; i < n; i++) {
pn += 1.0 / k;
k += 1.0;
fn *= k;
}
zmn = tox;
if (n == 1) {
ans = 1.0 / x;
} else {
nk1f = fn / n;
kf = 1.0;
s = nk1f;
z = -z0;
zn = 1.0;
for (i = 1; i < n; i++) {
nk1f = nk1f / (n - i);
kf = kf * i;
zn *= z;
t = nk1f * zn / kf;
s += t;
if ((std::numeric_limits<double>::max() - std::abs(t)) < std::abs(s)) {
goto overf;
}
if ((tox > 1.0) && ((std::numeric_limits<double>::max() / tox) < zmn)) {
goto overf;
}
zmn *= tox;
}
s *= 0.5;
t = std::abs(s);
if ((zmn > 1.0) && ((std::numeric_limits<double>::max() / zmn) < t)) {
goto overf;
}
if ((t > 1.0) && ((std::numeric_limits<double>::max() / t) < zmn)) {
goto overf;
}
ans = s * zmn;
}
}
tlg = 2.0 * log(0.5 * x);
pk = -detail::SCIPY_EULER;
if (n == 0) {
pn = pk;
t = 1.0;
} else {
pn = pn + 1.0 / n;
t = 1.0 / fn;
}
s = (pk + pn - tlg) * t;
k = 1.0;
do {
t *= z0 / (k * (k + n));
pk += 1.0 / k;
pn += 1.0 / (k + n);
s += (pk + pn - tlg) * t;
k += 1.0;
} while (fabs(t / s) > detail::MACHEP);
s = 0.5 * s / zmn;
if (n & 1) {
s = -s;
}
ans += s;
return (ans);
/* Asymptotic expansion for Kn(x) */
/* Converges to 1.4e-17 for x > 18.4 */
asymp:
if (x > detail::MAXLOG) {
set_error("kn", SF_ERROR_UNDERFLOW, NULL);
return (0.0);
}
k = n;
pn = 4.0 * k * k;
pk = 1.0;
z0 = 8.0 * x;
fn = 1.0;
t = 1.0;
s = t;
nkf = std::numeric_limits<double>::infinity();
i = 0;
do {
z = pn - pk * pk;
t = t * z / (fn * z0);
nk1f = std::abs(t);
if ((i >= n) && (nk1f > nkf)) {
goto adone;
}
nkf = nk1f;
s += t;
fn += 1.0;
pk += 2.0;
i += 1;
} while (std::abs(t / s) > detail::MACHEP);
adone:
ans = std::exp(-x) * std::sqrt(M_PI / (2.0 * x)) * s;
return (ans);
}
} // namespace cephes
} // namespace xsf
|