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// kbessel.cc: implementation of K-Bessel function for arbitrary real
//////////////////////////////////////////////////////////////////////////
//
// Copyright 1990-2023 John Cremona
//
// This file is part of the eclib package.
//
// eclib is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by the
// Free Software Foundation; either version 2 of the License, or (at your
// option) any later version.
//
// eclib is distributed in the hope that it will be useful, but WITHOUT
// ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
// FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
// for more details.
//
// You should have received a copy of the GNU General Public License
// along with eclib; if not, write to the Free Software Foundation,
// Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
//
//////////////////////////////////////////////////////////////////////////
// parameter nu. Adapted from the PARI function kbessel.
#include <iostream> // for debugging only
#include <cmath>
#include <eclib/templates.h>
#include <eclib/kbessel.h>
#ifndef M_LN2
#define M_LN2 0.69314718055994530942
#endif
#ifndef M_PI
#define M_PI 3.14159265358979323846
#endif
#define LOG2 M_LN2
#define PI M_PI
#define expo(x) ((x)==0?-1000:(long)floor(log(fabs((x)))/LOG2))
#define lbin -64 // = bit-precision of doubles (or should be)
double kbessel(double nu, double gx, int debug)
{
if(debug) {
cout << "\nCalled kbessel("<<nu<<","<<gx<<").\n";
}
double x=gx, nu2=-4*nu*nu, y;
long n=(long)(32*LOG2+PI*nu/2);
long n2=(n<<1);
if (x<n)
{
if(debug) {
cout << "In the x<n case.\n";
}
double
zf=sqrt(PI/n2),
zz=1.0/(n2<<2),
s=1.0,
t=0.0;
long k2=2*n2+1, ex;
for (long k=n2;k>0;--k)
{
k2-=2;
double p1=k2*k2+nu2;
double ak=-p1*zz/k;
s=1+ak*s;
t=k2+ak*t;
}
t/=2.0;
double u=s*zf;
double
v=-(t*zf+u*nu)/n2,
q=n2,
r=x+x;
if(debug) {
cout << "Finished k loop. lbin = "<<lbin<<endl;
cout << "t,u,v,q,r = "<<t<<", "<<u<<", "<<v<<", "<<q<<", "<<r<<endl;
}
do
{
double p1=5.0/q;
if (expo(p1)>= -1) p1=0.5;
double p2=1.0-r/q;
if (p1>p2) p1=p2;
double
c=-p1,
d=1,
e=u,
f=v;
long k=1;
if(debug) {
cout << "...outer loop: p1 = "<<p1<<", expo(p1) = "<<expo(p1)<<endl;
}
do
{
double w=(((k-0.5)*u)+((q-k)*v));
w+=nu*(u-(v+v));
u=q*v/k;
v=w/k;
d*=c;
e+=d*u;
f+=(p1=(d*v));
k++;
ex=expo(p1)-expo(f);
if(debug) {
cout << "......inner loop: f = "<<f<<", expo(f) = "<<expo(f)<<endl;
cout << " p1= "<<p1<<", expo(p1) = "<<expo(p1)<<", ex = "<<ex<<endl;
}
}
while(ex>lbin);
if(debug) {
cout << "...finished inner loop -- ex = "<<ex<<endl;
}
p1=u;u=e;e=p1;p1=v;v=f;f=p1;
q*=(1+c);
p1=q-r; ex=expo(p1);
if(debug) {
cout << "...outer loop -- ex = "<<ex<<endl;
cout << "u,q,r,p1 = " << u<<", "<<q<<", "<<r<<", "<<r<< endl;
}
}
while(ex>lbin);
y=u*pow((x/n),nu);
}
else // x>=n
{
if(debug) {
cout << "In the x>=n case.\n";
}
double p2=2*x;
double
zf=sqrt(PI/p2),
zz=1.0/(4*p2),
s=1.0;
long k2=2*n2+1;
for (long k=n2;k>0;--k)
{
k2-=2;
double p1=k2*k2+nu2;
double ak=(p1*zz)/k;
s=1.0-(ak*s);
}
y=s*zf;
}
y/=exp(x);
if(debug) {
cout << "kbessel returns " << y << endl;
}
return y;
}
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