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/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/* jv.c
*
* Bessel function of noninteger order
*
*
*
* SYNOPSIS:
*
* double v, x, y, jv();
*
* y = jv( v, x );
*
*
*
* DESCRIPTION:
*
* Returns Bessel function of order v of the argument,
* where v is real. Negative x is allowed if v is an integer.
*
* Several expansions are included: the ascending power
* series, the Hankel expansion, and two transitional
* expansions for large v. If v is not too large, it
* is reduced by recurrence to a region of best accuracy.
* The transitional expansions give 12D accuracy for v > 500.
*
*
*
* ACCURACY:
* Results for integer v are indicated by *, where x and v
* both vary from -125 to +125. Otherwise,
* x ranges from 0 to 125, v ranges as indicated by "domain."
* Error criterion is absolute, except relative when |jv()| > 1.
*
* arithmetic v domain x domain # trials peak rms
* IEEE 0,125 0,125 100000 4.6e-15 2.2e-16
* IEEE -125,0 0,125 40000 5.4e-11 3.7e-13
* IEEE 0,500 0,500 20000 4.4e-15 4.0e-16
* Integer v:
* IEEE -125,125 -125,125 50000 3.5e-15* 1.9e-16*
*
*/
/*
* Cephes Math Library Release 2.8: June, 2000
* Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
*/
#pragma once
#include "../config.h"
#include "../error.h"
#include "airy.h"
#include "cbrt.h"
#include "rgamma.h"
#include "j0.h"
#include "j1.h"
#include "polevl.h"
namespace xsf {
namespace cephes {
namespace detail {
constexpr double jv_BIG = 1.44115188075855872E+17;
/* Reduce the order by backward recurrence.
* AMS55 #9.1.27 and 9.1.73.
*/
XSF_HOST_DEVICE inline double jv_recur(double *n, double x, double *newn, int cancel) {
double pkm2, pkm1, pk, qkm2, qkm1;
/* double pkp1; */
double k, ans, qk, xk, yk, r, t, kf;
constexpr double big = jv_BIG;
int nflag, ctr;
int miniter, maxiter;
/* Continued fraction for Jn(x)/Jn-1(x)
* AMS 9.1.73
*
* x -x^2 -x^2
* ------ --------- --------- ...
* 2 n + 2(n+1) + 2(n+2) +
*
* Compute it with the simplest possible algorithm.
*
* This continued fraction starts to converge when (|n| + m) > |x|.
* Hence, at least |x|-|n| iterations are necessary before convergence is
* achieved. There is a hard limit set below, m <= 30000, which is chosen
* so that no branch in `jv` requires more iterations to converge.
* The exact maximum number is (500/3.6)^2 - 500 ~ 19000
*/
maxiter = 22000;
miniter = std::abs(x) - std::abs(*n);
if (miniter < 1) {
miniter = 1;
}
if (*n < 0.0) {
nflag = 1;
} else {
nflag = 0;
}
fstart:
pkm2 = 0.0;
qkm2 = 1.0;
pkm1 = x;
qkm1 = *n + *n;
xk = -x * x;
yk = qkm1;
ans = 0.0; /* ans=0.0 ensures that t=1.0 in the first iteration */
ctr = 0;
do {
yk += 2.0;
pk = pkm1 * yk + pkm2 * xk;
qk = qkm1 * yk + qkm2 * xk;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
/* check convergence */
if (qk != 0 && ctr > miniter)
r = pk / qk;
else
r = 0.0;
if (r != 0) {
t = std::abs((ans - r) / r);
ans = r;
} else {
t = 1.0;
}
if (++ctr > maxiter) {
set_error("jv", SF_ERROR_UNDERFLOW, NULL);
goto done;
}
if (t < MACHEP) {
goto done;
}
/* renormalize coefficients */
if (std::abs(pk) > big) {
pkm2 /= big;
pkm1 /= big;
qkm2 /= big;
qkm1 /= big;
}
} while (t > MACHEP);
done:
if (ans == 0)
ans = 1.0;
/* Change n to n-1 if n < 0 and the continued fraction is small */
if (nflag > 0) {
if (std::abs(ans) < 0.125) {
nflag = -1;
*n = *n - 1.0;
goto fstart;
}
}
kf = *newn;
/* backward recurrence
* 2k
* J (x) = --- J (x) - J (x)
* k-1 x k k+1
*/
pk = 1.0;
pkm1 = 1.0 / ans;
k = *n - 1.0;
r = 2 * k;
do {
pkm2 = (pkm1 * r - pk * x) / x;
/* pkp1 = pk; */
pk = pkm1;
pkm1 = pkm2;
r -= 2.0;
/*
* t = fabs(pkp1) + fabs(pk);
* if( (k > (kf + 2.5)) && (fabs(pkm1) < 0.25*t) )
* {
* k -= 1.0;
* t = x*x;
* pkm2 = ( (r*(r+2.0)-t)*pk - r*x*pkp1 )/t;
* pkp1 = pk;
* pk = pkm1;
* pkm1 = pkm2;
* r -= 2.0;
* }
*/
k -= 1.0;
} while (k > (kf + 0.5));
/* Take the larger of the last two iterates
* on the theory that it may have less cancellation error.
*/
if (cancel) {
if ((kf >= 0.0) && (std::abs(pk) > std::abs(pkm1))) {
k += 1.0;
pkm2 = pk;
}
}
*newn = k;
return (pkm2);
}
/* Ascending power series for Jv(x).
* AMS55 #9.1.10.
*/
XSF_HOST_DEVICE inline double jv_jvs(double n, double x) {
double t, u, y, z, k;
int ex, sgngam;
z = -x * x / 4.0;
u = 1.0;
y = u;
k = 1.0;
t = 1.0;
while (t > MACHEP) {
u *= z / (k * (n + k));
y += u;
k += 1.0;
if (y != 0)
t = std::abs(u / y);
}
t = std::frexp(0.5 * x, &ex);
ex = ex * n;
if ((ex > -1023) && (ex < 1023) && (n > 0.0) && (n < (MAXGAM - 1.0))) {
t = std::pow(0.5 * x, n) * xsf::cephes::rgamma(n + 1.0);
y *= t;
} else {
t = n * std::log(0.5 * x) - lgam_sgn(n + 1.0, &sgngam);
if (y < 0) {
sgngam = -sgngam;
y = -y;
}
t += std::log(y);
if (t < -MAXLOG) {
return (0.0);
}
if (t > MAXLOG) {
set_error("Jv", SF_ERROR_OVERFLOW, NULL);
return (std::numeric_limits<double>::infinity());
}
y = sgngam * std::exp(t);
}
return (y);
}
/* Hankel's asymptotic expansion
* for large x.
* AMS55 #9.2.5.
*/
XSF_HOST_DEVICE inline double jv_hankel(double n, double x) {
double t, u, z, k, sign, conv;
double p, q, j, m, pp, qq;
int flag;
m = 4.0 * n * n;
j = 1.0;
z = 8.0 * x;
k = 1.0;
p = 1.0;
u = (m - 1.0) / z;
q = u;
sign = 1.0;
conv = 1.0;
flag = 0;
t = 1.0;
pp = 1.0e38;
qq = 1.0e38;
while (t > MACHEP) {
k += 2.0;
j += 1.0;
sign = -sign;
u *= (m - k * k) / (j * z);
p += sign * u;
k += 2.0;
j += 1.0;
u *= (m - k * k) / (j * z);
q += sign * u;
t = std::abs(u / p);
if (t < conv) {
conv = t;
qq = q;
pp = p;
flag = 1;
}
/* stop if the terms start getting larger */
if ((flag != 0) && (t > conv)) {
goto hank1;
}
}
hank1:
u = x - (0.5 * n + 0.25) * M_PI;
t = std::sqrt(2.0 / (M_PI * x)) * (pp * std::cos(u) - qq * std::sin(u));
return (t);
}
/* Asymptotic expansion for transition region,
* n large and x close to n.
* AMS55 #9.3.23.
*/
constexpr double jv_PF2[] = {-9.0000000000000000000e-2, 8.5714285714285714286e-2};
constexpr double jv_PF3[] = {1.3671428571428571429e-1, -5.4920634920634920635e-2, -4.4444444444444444444e-3};
constexpr double jv_PF4[] = {1.3500000000000000000e-3, -1.6036054421768707483e-1, 4.2590187590187590188e-2,
2.7330447330447330447e-3};
constexpr double jv_PG1[] = {-2.4285714285714285714e-1, 1.4285714285714285714e-2};
constexpr double jv_PG2[] = {-9.0000000000000000000e-3, 1.9396825396825396825e-1, -1.1746031746031746032e-2};
constexpr double jv_PG3[] = {1.9607142857142857143e-2, -1.5983694083694083694e-1, 6.3838383838383838384e-3};
XSF_HOST_DEVICE inline double jv_jnt(double n, double x) {
double z, zz, z3;
double cbn, n23, cbtwo;
double ai, aip, bi, bip; /* Airy functions */
double nk, fk, gk, pp, qq;
double F[5], G[4];
int k;
cbn = cbrt(n);
z = (x - n) / cbn;
cbtwo = cbrt(2.0);
/* Airy function */
zz = -cbtwo * z;
xsf::cephes::airy(zz, &ai, &aip, &bi, &bip);
/* polynomials in expansion */
zz = z * z;
z3 = zz * z;
F[0] = 1.0;
F[1] = -z / 5.0;
F[2] = xsf::cephes::polevl(z3, jv_PF2, 1) * zz;
F[3] = xsf::cephes::polevl(z3, jv_PF3, 2);
F[4] = xsf::cephes::polevl(z3, jv_PF4, 3) * z;
G[0] = 0.3 * zz;
G[1] = xsf::cephes::polevl(z3, jv_PG1, 1);
G[2] = xsf::cephes::polevl(z3, jv_PG2, 2) * z;
G[3] = xsf::cephes::polevl(z3, jv_PG3, 2) * zz;
pp = 0.0;
qq = 0.0;
nk = 1.0;
n23 = cbrt(n * n);
for (k = 0; k <= 4; k++) {
fk = F[k] * nk;
pp += fk;
if (k != 4) {
gk = G[k] * nk;
qq += gk;
}
nk /= n23;
}
fk = cbtwo * ai * pp / cbn + cbrt(4.0) * aip * qq / n;
return (fk);
}
/* Asymptotic expansion for large n.
* AMS55 #9.3.35.
*/
constexpr double jv_lambda[] = {1.0,
1.041666666666666666666667E-1,
8.355034722222222222222222E-2,
1.282265745563271604938272E-1,
2.918490264641404642489712E-1,
8.816272674437576524187671E-1,
3.321408281862767544702647E+0,
1.499576298686255465867237E+1,
7.892301301158651813848139E+1,
4.744515388682643231611949E+2,
3.207490090890661934704328E+3};
constexpr double jv_mu[] = {1.0,
-1.458333333333333333333333E-1,
-9.874131944444444444444444E-2,
-1.433120539158950617283951E-1,
-3.172272026784135480967078E-1,
-9.424291479571202491373028E-1,
-3.511203040826354261542798E+0,
-1.572726362036804512982712E+1,
-8.228143909718594444224656E+1,
-4.923553705236705240352022E+2,
-3.316218568547972508762102E+3};
constexpr double jv_P1[] = {-2.083333333333333333333333E-1, 1.250000000000000000000000E-1};
constexpr double jv_P2[] = {3.342013888888888888888889E-1, -4.010416666666666666666667E-1,
7.031250000000000000000000E-2};
constexpr double jv_P3[] = {-1.025812596450617283950617E+0, 1.846462673611111111111111E+0,
-8.912109375000000000000000E-1, 7.324218750000000000000000E-2};
constexpr double jv_P4[] = {4.669584423426247427983539E+0, -1.120700261622299382716049E+1,
8.789123535156250000000000E+0, -2.364086914062500000000000E+0,
1.121520996093750000000000E-1};
constexpr double jv_P5[] = {-2.8212072558200244877E1, 8.4636217674600734632E1, -9.1818241543240017361E1,
4.2534998745388454861E1, -7.3687943594796316964E0, 2.27108001708984375E-1};
constexpr double jv_P6[] = {2.1257013003921712286E2, -7.6525246814118164230E2, 1.0599904525279998779E3,
-6.9957962737613254123E2, 2.1819051174421159048E2, -2.6491430486951555525E1,
5.7250142097473144531E-1};
constexpr double jv_P7[] = {-1.9194576623184069963E3, 8.0617221817373093845E3, -1.3586550006434137439E4,
1.1655393336864533248E4, -5.3056469786134031084E3, 1.2009029132163524628E3,
-1.0809091978839465550E2, 1.7277275025844573975E0};
XSF_HOST_DEVICE inline double jv_jnx(double n, double x) {
double zeta, sqz, zz, zp, np;
double cbn, n23, t, z, sz;
double pp, qq, z32i, zzi;
double ak, bk, akl, bkl;
int sign, doa, dob, nflg, k, s, tk, tkp1, m;
double u[8];
double ai, aip, bi, bip;
/* Test for x very close to n. Use expansion for transition region if so. */
cbn = cbrt(n);
z = (x - n) / cbn;
if (std::abs(z) <= 0.7) {
return (jv_jnt(n, x));
}
z = x / n;
zz = 1.0 - z * z;
if (zz == 0.0) {
return (0.0);
}
if (zz > 0.0) {
sz = std::sqrt(zz);
t = 1.5 * (std::log((1.0 + sz) / z) - sz); /* zeta ** 3/2 */
zeta = cbrt(t * t);
nflg = 1;
} else {
sz = std::sqrt(-zz);
t = 1.5 * (sz - std::acos(1.0 / z));
zeta = -cbrt(t * t);
nflg = -1;
}
z32i = std::abs(1.0 / t);
sqz = cbrt(t);
/* Airy function */
n23 = cbrt(n * n);
t = n23 * zeta;
xsf::cephes::airy(t, &ai, &aip, &bi, &bip);
/* polynomials in expansion */
u[0] = 1.0;
zzi = 1.0 / zz;
u[1] = xsf::cephes::polevl(zzi, jv_P1, 1) / sz;
u[2] = xsf::cephes::polevl(zzi, jv_P2, 2) / zz;
u[3] = xsf::cephes::polevl(zzi, jv_P3, 3) / (sz * zz);
pp = zz * zz;
u[4] = xsf::cephes::polevl(zzi, jv_P4, 4) / pp;
u[5] = xsf::cephes::polevl(zzi, jv_P5, 5) / (pp * sz);
pp *= zz;
u[6] = xsf::cephes::polevl(zzi, jv_P6, 6) / pp;
u[7] = xsf::cephes::polevl(zzi, jv_P7, 7) / (pp * sz);
pp = 0.0;
qq = 0.0;
np = 1.0;
/* flags to stop when terms get larger */
doa = 1;
dob = 1;
akl = std::numeric_limits<double>::infinity();
bkl = std::numeric_limits<double>::infinity();
for (k = 0; k <= 3; k++) {
tk = 2 * k;
tkp1 = tk + 1;
zp = 1.0;
ak = 0.0;
bk = 0.0;
for (s = 0; s <= tk; s++) {
if (doa) {
if ((s & 3) > 1)
sign = nflg;
else
sign = 1;
ak += sign * jv_mu[s] * zp * u[tk - s];
}
if (dob) {
m = tkp1 - s;
if (((m + 1) & 3) > 1)
sign = nflg;
else
sign = 1;
bk += sign * jv_lambda[s] * zp * u[m];
}
zp *= z32i;
}
if (doa) {
ak *= np;
t = std::abs(ak);
if (t < akl) {
akl = t;
pp += ak;
} else
doa = 0;
}
if (dob) {
bk += jv_lambda[tkp1] * zp * u[0];
bk *= -np / sqz;
t = std::abs(bk);
if (t < bkl) {
bkl = t;
qq += bk;
} else
dob = 0;
}
if (np < MACHEP)
break;
np /= n * n;
}
/* normalizing factor ( 4*zeta/(1 - z**2) )**1/4 */
t = 4.0 * zeta / zz;
t = sqrt(sqrt(t));
t *= ai * pp / cbrt(n) + aip * qq / (n23 * n);
return (t);
}
} // namespace detail
XSF_HOST_DEVICE inline double jv(double n, double x) {
double k, q, t, y, an;
int i, sign, nint;
nint = 0; /* Flag for integer n */
sign = 1; /* Flag for sign inversion */
an = std::abs(n);
y = std::floor(an);
if (y == an) {
nint = 1;
i = an - 16384.0 * std::floor(an / 16384.0);
if (n < 0.0) {
if (i & 1)
sign = -sign;
n = an;
}
if (x < 0.0) {
if (i & 1)
sign = -sign;
x = -x;
}
if (n == 0.0)
return (j0(x));
if (n == 1.0)
return (sign * j1(x));
}
if ((x < 0.0) && (y != an)) {
set_error("Jv", SF_ERROR_DOMAIN, NULL);
y = std::numeric_limits<double>::quiet_NaN();
goto done;
}
if (x == 0 && n < 0 && !nint) {
set_error("Jv", SF_ERROR_OVERFLOW, NULL);
return std::numeric_limits<double>::infinity() * rgamma(n + 1);
}
y = std::abs(x);
if (y * y < std::abs(n + 1) * detail::MACHEP) {
return std::pow(0.5 * x, n) * rgamma(n + 1);
}
k = 3.6 * std::sqrt(y);
t = 3.6 * std::sqrt(an);
if ((y < t) && (an > 21.0)) {
return (sign * detail::jv_jvs(n, x));
}
if ((an < k) && (y > 21.0))
return (sign * detail::jv_hankel(n, x));
if (an < 500.0) {
/* Note: if x is too large, the continued fraction will fail; but then the
* Hankel expansion can be used. */
if (nint != 0) {
k = 0.0;
q = detail::jv_recur(&n, x, &k, 1);
if (k == 0.0) {
y = j0(x) / q;
goto done;
}
if (k == 1.0) {
y = j1(x) / q;
goto done;
}
}
if (an > 2.0 * y)
goto rlarger;
if ((n >= 0.0) && (n < 20.0) && (y > 6.0) && (y < 20.0)) {
/* Recur backwards from a larger value of n */
rlarger:
k = n;
y = y + an + 1.0;
if (y < 30.0)
y = 30.0;
y = n + std::floor(y - n);
q = detail::jv_recur(&y, x, &k, 0);
y = detail::jv_jvs(y, x) * q;
goto done;
}
if (k <= 30.0) {
k = 2.0;
} else if (k < 90.0) {
k = (3 * k) / 4;
}
if (an > (k + 3.0)) {
if (n < 0.0) {
k = -k;
}
q = n - std::floor(n);
k = std::floor(k) + q;
if (n > 0.0) {
q = detail::jv_recur(&n, x, &k, 1);
} else {
t = k;
k = n;
q = detail::jv_recur(&t, x, &k, 1);
k = t;
}
if (q == 0.0) {
y = 0.0;
goto done;
}
} else {
k = n;
q = 1.0;
}
/* boundary between convergence of
* power series and Hankel expansion
*/
y = std::abs(k);
if (y < 26.0)
t = (0.0083 * y + 0.09) * y + 12.9;
else
t = 0.9 * y;
if (x > t)
y = detail::jv_hankel(k, x);
else
y = detail::jv_jvs(k, x);
if (n > 0.0)
y /= q;
else
y *= q;
}
else {
/* For large n, use the uniform expansion or the transitional expansion.
* But if x is of the order of n**2, these may blow up, whereas the
* Hankel expansion will then work.
*/
if (n < 0.0) {
set_error("jv", SF_ERROR_LOSS, NULL);
y = std::numeric_limits<double>::quiet_NaN();
goto done;
}
t = x / n;
t /= n;
if (t > 0.3)
y = detail::jv_hankel(n, x);
else
y = detail::jv_jnx(n, x);
}
done:
return (sign * y);
}
} // namespace cephes
} // namespace xsf
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