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/*************************************************************************
*
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
*
* Copyright 2000, 2010 Oracle and/or its affiliates.
*
* OpenOffice.org - a multi-platform office productivity suite
*
* This file is part of OpenOffice.org.
*
* OpenOffice.org is free software: you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License version 3
* only, as published by the Free Software Foundation.
*
* OpenOffice.org 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 Lesser General Public License version 3 for more details
* (a copy is included in the LICENSE file that accompanied this code).
*
* You should have received a copy of the GNU Lesser General Public License
* version 3 along with OpenOffice.org. If not, see
* <http://www.openoffice.org/license.html>
* for a copy of the LGPLv3 License.
*
************************************************************************/
#include "bessel.hxx"
#include "analysishelper.hxx"
#include <rtl/math.hxx>
using ::com::sun::star::lang::IllegalArgumentException;
namespace sca {
namespace analysis {
// ============================================================================
const double f_PI = 3.1415926535897932385;
const double f_2_PI = 2.0 * f_PI;
const double f_PI_DIV_2 = f_PI / 2.0;
const double f_PI_DIV_4 = f_PI / 4.0;
const double f_2_DIV_PI = 2.0 / f_PI;
const double THRESHOLD = 30.0; // Threshold for usage of approximation formula.
const double MAXEPSILON = 1e-10; // Maximum epsilon for end of iteration.
const sal_Int32 MAXITER = 100; // Maximum number of iterations.
// ============================================================================
// BESSEL J
// ============================================================================
/* The BESSEL function, first kind, unmodified:
inf (-1)^k (x/2)^(n+2k)
J_n(x) = SUM TERM(n,k) with TERM(n,k) := ---------------------
k=0 k! (n+k)!
Approximation for the BESSEL function, first kind, unmodified, for great x:
J_n(x) ~ sqrt( 2 / (PI x) ) cos( x - n PI/2 - PI/4 ) for x>=0.
*/
// ----------------------------------------------------------------------------
double BesselJ( double x, sal_Int32 n ) throw( IllegalArgumentException )
{
if( n < 0 )
throw IllegalArgumentException();
double fResult = 0.0;
if( fabs( x ) <= THRESHOLD )
{
/* Start the iteration without TERM(n,0), which is set here.
TERM(n,0) = (x/2)^n / n!
*/
double fTerm = pow( x / 2.0, (double)n ) / Fak( n );
sal_Int32 nK = 1; // Start the iteration with k=1.
fResult = fTerm; // Start result with TERM(n,0).
const double fSqrX = x * x / -4.0;
do
{
/* Calculation of TERM(n,k) from TERM(n,k-1):
(-1)^k (x/2)^(n+2k)
TERM(n,k) = ---------------------
k! (n+k)!
(-1)(-1)^(k-1) (x/2)^2 (x/2)^(n+2(k-1))
= -----------------------------------------
k (k-1)! (n+k) (n+k-1)!
-(x/2)^2 (-1)^(k-1) (x/2)^(n+2(k-1))
= ---------- * -----------------------------
k(n+k) (k-1)! (n+k-1)!
-(x^2/4)
= ---------- TERM(n,k-1)
k(n+k)
*/
fTerm *= fSqrX; // defined above as -(x^2/4)
fTerm /= (nK * (nK + n));
fResult += fTerm;
}
while( (fabs( fTerm ) > MAXEPSILON) && (++nK < MAXITER) );
}
else
{
/* Approximation for the BESSEL function, first kind, unmodified:
J_n(x) ~ sqrt( 2 / (PI x) ) cos( x - n PI/2 - PI/4 ) for x>=0.
The BESSEL function J_n with n IN {0,2,4,...} is axially symmetric at
x=0, means J_n(x) = J_n(-x). Therefore the approximation for x<0 is:
J_n(x) = J_n(|x|) for x<0 and n IN {0,2,4,...}.
The BESSEL function J_n with n IN {1,3,5,...} is point-symmetric at
x=0, means J_n(x) = -J_n(-x). Therefore the approximation for x<0 is:
J_n(x) = -J_n(|x|) for x<0 and n IN {1,3,5,...}.
*/
double fXAbs = fabs( x );
fResult = sqrt( f_2_DIV_PI / fXAbs ) * cos( fXAbs - n * f_PI_DIV_2 - f_PI_DIV_4 );
if( (n & 1) && (x < 0.0) )
fResult = -fResult;
}
return fResult;
}
// ============================================================================
// BESSEL I
// ============================================================================
/* The BESSEL function, first kind, modified:
inf (x/2)^(n+2k)
I_n(x) = SUM TERM(n,k) with TERM(n,k) := --------------
k=0 k! (n+k)!
Approximation for the BESSEL function, first kind, modified, for great x:
I_n(x) ~ e^x / sqrt( 2 PI x ) for x>=0.
*/
// ----------------------------------------------------------------------------
double BesselI( double x, sal_Int32 n ) throw( IllegalArgumentException )
{
if( n < 0 )
throw IllegalArgumentException();
double fResult = 0.0;
if( fabs( x ) <= THRESHOLD )
{
/* Start the iteration without TERM(n,0), which is set here.
TERM(n,0) = (x/2)^n / n!
*/
double fTerm = pow( x / 2.0, (double)n ) / Fak( n );
sal_Int32 nK = 1; // Start the iteration with k=1.
fResult = fTerm; // Start result with TERM(n,0).
const double fSqrX = x * x / 4.0;
do
{
/* Calculation of TERM(n,k) from TERM(n,k-1):
(x/2)^(n+2k)
TERM(n,k) = --------------
k! (n+k)!
(x/2)^2 (x/2)^(n+2(k-1))
= --------------------------
k (k-1)! (n+k) (n+k-1)!
(x/2)^2 (x/2)^(n+2(k-1))
= --------- * ------------------
k(n+k) (k-1)! (n+k-1)!
x^2/4
= -------- TERM(n,k-1)
k(n+k)
*/
fTerm *= fSqrX; // defined above as x^2/4
fTerm /= (nK * (nK + n));
fResult += fTerm;
}
while( (fabs( fTerm ) > MAXEPSILON) && (++nK < MAXITER) );
}
else
{
/* Approximation for the BESSEL function, first kind, modified:
I_n(x) ~ e^x / sqrt( 2 PI x ) for x>=0.
The BESSEL function I_n with n IN {0,2,4,...} is axially symmetric at
x=0, means I_n(x) = I_n(-x). Therefore the approximation for x<0 is:
I_n(x) = I_n(|x|) for x<0 and n IN {0,2,4,...}.
The BESSEL function I_n with n IN {1,3,5,...} is point-symmetric at
x=0, means I_n(x) = -I_n(-x). Therefore the approximation for x<0 is:
I_n(x) = -I_n(|x|) for x<0 and n IN {1,3,5,...}.
*/
double fXAbs = fabs( x );
fResult = exp( fXAbs ) / sqrt( f_2_PI * fXAbs );
if( (n & 1) && (x < 0.0) )
fResult = -fResult;
}
return fResult;
}
// ============================================================================
double Besselk0( double fNum ) throw( IllegalArgumentException )
{
double fRet;
if( fNum <= 2.0 )
{
double fNum2 = fNum * 0.5;
double y = fNum2 * fNum2;
fRet = -log( fNum2 ) * BesselI( fNum, 0 ) +
( -0.57721566 + y * ( 0.42278420 + y * ( 0.23069756 + y * ( 0.3488590e-1 +
y * ( 0.262698e-2 + y * ( 0.10750e-3 + y * 0.74e-5 ) ) ) ) ) );
}
else
{
double y = 2.0 / fNum;
fRet = exp( -fNum ) / sqrt( fNum ) * ( 1.25331414 + y * ( -0.7832358e-1 +
y * ( 0.2189568e-1 + y * ( -0.1062446e-1 + y * ( 0.587872e-2 +
y * ( -0.251540e-2 + y * 0.53208e-3 ) ) ) ) ) );
}
return fRet;
}
double Besselk1( double fNum ) throw( IllegalArgumentException )
{
double fRet;
if( fNum <= 2.0 )
{
double fNum2 = fNum * 0.5;
double y = fNum2 * fNum2;
fRet = log( fNum2 ) * BesselI( fNum, 1 ) +
( 1.0 + y * ( 0.15443144 + y * ( -0.67278579 + y * ( -0.18156897 + y * ( -0.1919402e-1 +
y * ( -0.110404e-2 + y * ( -0.4686e-4 ) ) ) ) ) ) )
/ fNum;
}
else
{
double y = 2.0 / fNum;
fRet = exp( -fNum ) / sqrt( fNum ) * ( 1.25331414 + y * ( 0.23498619 +
y * ( -0.3655620e-1 + y * ( 0.1504268e-1 + y * ( -0.780353e-2 +
y * ( 0.325614e-2 + y * ( -0.68245e-3 ) ) ) ) ) ) );
}
return fRet;
}
double BesselK( double fNum, sal_Int32 nOrder ) throw( IllegalArgumentException )
{
switch( nOrder )
{
case 0: return Besselk0( fNum );
case 1: return Besselk1( fNum );
default:
{
double fBkp;
double fTox = 2.0 / fNum;
double fBkm = Besselk0( fNum );
double fBk = Besselk1( fNum );
for( sal_Int32 n = 1 ; n < nOrder ; n++ )
{
fBkp = fBkm + double( n ) * fTox * fBk;
fBkm = fBk;
fBk = fBkp;
}
return fBk;
}
}
}
double Bessely0( double fNum ) throw( IllegalArgumentException )
{
double fRet;
if( fNum < 8.0 )
{
double y = fNum * fNum;
double f1 = -2957821389.0 + y * ( 7062834065.0 + y * ( -512359803.6 +
y * ( 10879881.29 + y * ( -86327.92757 + y * 228.4622733 ) ) ) );
double f2 = 40076544269.0 + y * ( 745249964.8 + y * ( 7189466.438 +
y * ( 47447.26470 + y * ( 226.1030244 + y ) ) ) );
fRet = f1 / f2 + 0.636619772 * BesselJ( fNum, 0 ) * log( fNum );
}
else
{
double z = 8.0 / fNum;
double y = z * z;
double xx = fNum - 0.785398164;
double f1 = 1.0 + y * ( -0.1098628627e-2 + y * ( 0.2734510407e-4 +
y * ( -0.2073370639e-5 + y * 0.2093887211e-6 ) ) );
double f2 = -0.1562499995e-1 + y * ( 0.1430488765e-3 +
y * ( -0.6911147651e-5 + y * ( 0.7621095161e-6 +
y * ( -0.934945152e-7 ) ) ) );
fRet = sqrt( 0.636619772 / fNum ) * ( sin( xx ) * f1 + z * cos( xx ) * f2 );
}
return fRet;
}
double Bessely1( double fNum ) throw( IllegalArgumentException )
{
double fRet;
if( fNum < 8.0 )
{
double y = fNum * fNum;
double f1 = fNum * ( -0.4900604943e13 + y * ( 0.1275274390e13 +
y * ( -0.5153438139e11 + y * ( 0.7349264551e9 +
y * ( -0.4237922726e7 + y * 0.8511937935e4 ) ) ) ) );
double f2 = 0.2499580570e14 + y * ( 0.4244419664e12 +
y * ( 0.3733650367e10 + y * ( 0.2245904002e8 +
y * ( 0.1020426050e6 + y * ( 0.3549632885e3 + y ) ) ) ) );
fRet = f1 / f2 + 0.636619772 * ( BesselJ( fNum, 1 ) * log( fNum ) - 1.0 / fNum );
}
else
{
#if 0
// #i12430# don't know the intention of this piece of code...
double z = 8.0 / fNum;
double y = z * z;
double xx = fNum - 2.356194491;
double f1 = 1.0 + y * ( 0.183105e-2 + y * ( -0.3516396496e-4 +
y * ( 0.2457520174e-5 + y * ( -0.240337019e6 ) ) ) );
double f2 = 0.04687499995 + y * ( -0.2002690873e-3 +
y * ( 0.8449199096e-5 + y * ( -0.88228987e-6 +
y * 0.105787412e-6 ) ) );
fRet = sqrt( 0.636619772 / fNum ) * ( sin( xx ) * f1 + z * cos( xx ) * f2 );
#endif
// #i12430# ...but this seems to work much better.
fRet = sqrt( 0.636619772 / fNum ) * sin( fNum - 2.356194491 );
}
return fRet;
}
double BesselY( double fNum, sal_Int32 nOrder ) throw( IllegalArgumentException )
{
switch( nOrder )
{
case 0: return Bessely0( fNum );
case 1: return Bessely1( fNum );
default:
{
double fByp;
double fTox = 2.0 / fNum;
double fBym = Bessely0( fNum );
double fBy = Bessely1( fNum );
for( sal_Int32 n = 1 ; n < nOrder ; n++ )
{
fByp = double( n ) * fTox * fBy - fBym;
fBym = fBy;
fBy = fByp;
}
return fBy;
}
}
}
// ============================================================================
} // namespace analysis
} // namespace sca
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