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/* */
/* Copyright 2010-2011 by Ullrich Koethe */
/* */
/* This file is part of the VIGRA computer vision library. */
/* The VIGRA Website is */
/* http://hci.iwr.uni-heidelberg.de/vigra/ */
/* Please direct questions, bug reports, and contributions to */
/* ullrich.koethe@iwr.uni-heidelberg.de or */
/* vigra@informatik.uni-hamburg.de */
/* */
/* Permission is hereby granted, free of charge, to any person */
/* obtaining a copy of this software and associated documentation */
/* files (the "Software"), to deal in the Software without */
/* restriction, including without limitation the rights to use, */
/* copy, modify, merge, publish, distribute, sublicense, and/or */
/* sell copies of the Software, and to permit persons to whom the */
/* Software is furnished to do so, subject to the following */
/* conditions: */
/* */
/* The above copyright notice and this permission notice shall be */
/* included in all copies or substantial portions of the */
/* Software. */
/* */
/* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND */
/* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES */
/* OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND */
/* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT */
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/* OTHER DEALINGS IN THE SOFTWARE. */
/* */
/************************************************************************/
#ifndef VIGRA_BESSEL_HXX
#define VIGRA_BESSEL_HXX
#include "mathutil.hxx"
#ifdef HasBoostMath
#include <boost/math/special_functions/bessel.hpp>
#endif
namespace vigra {
/** \addtogroup MathFunctions
*/
//@{
namespace detail {
template <class REAL>
int msta1(REAL x, int mp)
{
double a0,f0,f1,f;
int i,n0,n1,nn;
a0 = abs(x);
n0 = (int)(1.1*a0)+1;
f0 = 0.5*std::log10(6.28*n0) - n0*std::log10(1.36*a0/n0)-mp;
n1 = n0+5;
f1 = 0.5*std::log10(6.28*n1) - n1*std::log10(1.36*a0/n1)-mp;
for(i=0;i<20;i++)
{
nn = int(n1-(n1-n0)/(1.0-f0/f1));
f = 0.5*std::log10(6.28*nn) - nn*std::log10(1.36*a0/nn)-mp;
if(abs(nn-n1) < 1)
break;
n0 = n1;
f0 = f1;
n1 = nn;
f1 = f;
}
return nn;
}
template <class REAL>
int msta2(REAL x, int n, int mp)
{
double a0,ejn,hmp,f0,f1,f,obj;
int i,n0,n1,nn;
a0 = abs(x);
hmp = 0.5*mp;
ejn = 0.5*std::log10(6.28*n) - n*std::log10(1.36*a0/n);
if (ejn <= hmp)
{
obj = mp;
n0 = (int)(1.1*a0);
if (n0 < 1)
n0 = 1;
}
else
{
obj = hmp+ejn;
n0 = n;
}
f0 = 0.5*std::log10(6.28*n0) - n0*std::log10(1.36*a0/n0)-obj;
n1 = n0+5;
f1 = 0.5*std::log10(6.28*n1) - n1*std::log10(1.36*a0/n1)-obj;
for (i=0;i<20;i++)
{
nn = int(n1-(n1-n0)/(1.0-f0/f1));
f = 0.5*std::log10(6.28*nn) - nn*std::log10(1.36*a0/nn)-obj;
if (abs(nn-n1) < 1)
break;
n0 = n1;
f0 = f1;
n1 = nn;
f1 = f;
}
return nn+10;
}
//
// INPUT:
// double x -- argument of Bessel function of 1st and 2nd kind.
// int n -- order
//
// OUPUT:
//
// int nm -- highest order actually computed (nm <= n)
// double jn[] -- Bessel function of 1st kind, orders from 0 to nm
// double yn[] -- Bessel function of 2nd kind, orders from 0 to nm
//
// Computes Bessel functions of all order up to 'n' using recurrence
// relations. If 'nm' < 'n' only 'nm' orders are returned.
//
// code has been adapted from C.R. Bond's implementation
// see http://www.crbond.com/math.htm
//
template <class REAL>
void bessjyn(int n, REAL x,int &nm, double *jn, double *yn)
{
double t1,t2,f,f1,f2,bj0,bj1,bjk,by0,by1,cu,s0,su,sv;
double ec,bs,byk,p0,p1,q0,q1;
double a[] = {
-0.7031250000000000e-1,
0.1121520996093750,
-0.5725014209747314,
6.074042001273483};
double b[] = {
0.7324218750000000e-1,
-0.2271080017089844,
1.727727502584457,
-2.438052969955606e1};
double a1[] = {
0.1171875,
-0.1441955566406250,
0.6765925884246826,
-6.883914268109947};
double b1[] = {
-0.1025390625,
0.2775764465332031,
-1.993531733751297,
2.724882731126854e1};
int i,k,m;
nm = n;
if (x < 1e-15)
{
for (i=0;i<=n;i++)
{
jn[i] = 0.0;
yn[i] = -1e308;
}
jn[0] = 1.0;
return;
}
if (x <= 300.0 || n > (int)(0.9*x))
{
if (n == 0)
nm = 1;
m = msta1(x,200);
if (m < nm)
nm = m;
else
m = msta2(x,nm,15);
bs = 0.0;
su = 0.0;
sv = 0.0;
f2 = 0.0;
f1 = 1.0e-100;
for (k = m;k>=0;k--)
{
f = 2.0*(k+1.0)/x*f1 - f2;
if (k <= nm)
jn[k] = f;
if ((k == 2*(int)(k/2)) && (k != 0))
{
bs += 2.0*f;
su += (-1)*((k & 2)-1)*f/(double)k;
}
else if (k > 1)
{
sv += (-1)*((k & 2)-1)*(double)k*f/(k*k-1.0);
}
f2 = f1;
f1 = f;
}
s0 = bs+f;
for (k=0;k<=nm;k++)
{
jn[k] /= s0;
}
ec = std::log(0.5*x) + M_EULER_GAMMA;
by0 = M_2_PI*(ec*jn[0]-4.0*su/s0);
yn[0] = by0;
by1 = M_2_PI*((ec-1.0)*jn[1]-jn[0]/x-4.0*sv/s0);
yn[1] = by1;
}
else
{
t1 = x-M_PI_4;
p0 = 1.0;
q0 = -0.125/x;
for (k=0;k<4;k++)
{
p0 += a[k]*std::pow(x,-2*k-2);
q0 += b[k]*std::pow(x,-2*k-3);
}
cu = std::sqrt(M_2_PI/x);
bj0 = cu*(p0*std::cos(t1)-q0*std::sin(t1));
by0 = cu*(p0*std::sin(t1)+q0*std::cos(t1));
jn[0] = bj0;
yn[0] = by0;
t2 = x-0.75*M_PI;
p1 = 1.0;
q1 = 0.375/x;
for (k=0;k<4;k++)
{
p1 += a1[k]*std::pow(x,-2*k-2);
q1 += b1[k]*std::pow(x,-2*k-3);
}
bj1 = cu*(p1*std::cos(t2)-q1*std::sin(t2));
by1 = cu*(p1*std::sin(t2)+q1*std::cos(t2));
jn[1] = bj1;
yn[1] = by1;
for (k=2;k<=nm;k++)
{
bjk = 2.0*(k-1.0)*bj1/x-bj0;
jn[k] = bjk;
bj0 = bj1;
bj1 = bjk;
}
}
for (k=2;k<=nm;k++)
{
byk = 2.0*(k-1.0)*by1/x-by0;
yn[k] = byk;
by0 = by1;
by1 = byk;
}
}
} // namespace detail
/** \brief Bessel function of the first kind.
Computes the value of BesselJ of integer order <tt>n</tt> and argument <tt>x</tt>.
Negative <tt>x</tt> are unsupported and will result in a <tt>std::domain_error</tt>.
This function wraps a number of existing implementations and falls back to
a rather slow algorithm if none of them is available. In particular,
it uses boost::math when <tt>HasBoostMath</tt> is \#defined, or native
implementations on gcc and MSVC otherwise.
<b>\#include</b> \<vigra/bessel.hxx\><br>
Namespace: vigra
*/
inline double besselJ(int n, double x)
{
if(x < 0.0)
throw std::domain_error("besselJ(n, x): x cannot be negative");
if(x < 1e-15)
return n == 0 ? 1.0 : 0.0;
#if defined(HasBoostMath)
return boost::math::cyl_bessel_j((double)n, x);
#elif defined(__GNUC__)
return ::jn(n, x);
#elif defined(_MSC_VER)
return _jn(n, x);
#else
int an = abs(n), nr = n, s = an+2;
ArrayVector<double> t(2*s);
detail::bessjyn(an, x, nr, &t[0], &t[s]);
if(n < 0 && odd(an))
return -t[an];
else
return t[an];
#endif
}
/** \brief Bessel function of the second kind.
Computes the value of BesselY of integer order <tt>n</tt> and argument <tt>x</tt>.
Negative <tt>x</tt> are unsupported and will result in a <tt>std::domain_error</tt>.
This function wraps a number of existing implementations and falls back to
a rather slow algorithm if none of them is available. In particular,
it uses boost::math when <tt>HasBoostMath</tt> is \#defined, or native
implementations on gcc and MSVC otherwise.
<b>\#include</b> \<vigra/bessel.hxx\><br>
Namespace: vigra
*/
inline double besselY(int n, double x)
{
if(x < 0.0)
throw std::domain_error("besselY(n, x): x cannot be negative");
if(x == 0.0 )
return -std::numeric_limits<double>::infinity();
#if defined(HasBoostMath)
return boost::math::cyl_neumann((double)n, x);
#elif defined(__GNUC__)
return ::yn(n, x);
#elif defined(_MSC_VER)
return _yn(n, x);
#else
int an = abs(n), nr = n, s = an+2;
ArrayVector<double> t(2*s);
detail::bessjyn(an, x, nr, &t[0], &t[s]);
if(n < 0.0 && odd(n))
return -t[an+s];
else
return t[an+s];
#endif
}
//@}
} // namespace vigra
#endif // VIGRA_BESSEL_HXX
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