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office-gobmx/scaddins/source/analysis/bessel.cxx
Eike Rathke 0c428b052f dr78: #i43040# removed german umlauts and utf8 bom accidentially added by too smart editor 
# HG changeset patch
# User Daniel Rentz [dr] <daniel.rentz@oracle.com>
# Date 1295613905 -3600
# Node ID 65540563b0d1ba68bd3a9d6e3e8111c6c0e1a71d
# Parent  7c526a3a4dfa801f59eaeeea99e7b7a032bc6fe4
2011-11-28 15:25:32 +01:00

487 lines
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/* -*- Mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*************************************************************************
*
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
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* Copyright 2000, 2010 Oracle and/or its affiliates.
*
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* it under the terms of the GNU Lesser General Public License version 3
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* 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.
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************************************************************************/
#include "bessel.hxx"
#include "analysishelper.hxx"
#include <rtl/math.hxx>
using ::com::sun::star::lang::IllegalArgumentException;
using ::com::sun::star::sheet::NoConvergenceException;
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:
The algorithm follows
http://www.reference-global.com/isbn/978-3-11-020354-7
Numerical Mathematics 1 / Numerische Mathematik 1,
An algorithm-based introduction / Eine algorithmisch orientierte Einfuehrung
Deuflhard, Peter; Hohmann, Andreas
Berlin, New York (Walter de Gruyter) 2008
4. ueberarb. u. erw. Aufl. 2008
eBook ISBN: 978-3-11-020355-4
Chapter 6.3.2 , algorithm 6.24
The source is in German.
The BesselJ-function is a special case of the adjoint summation with
a_k = 2*(k-1)/x for k=1,...
b_k = -1, for all k, directly substituted
m_0=1, m_k=2 for k even, and m_k=0 for k odd, calculated on the fly
alpha_k=1 for k=N and alpha_k=0 otherwise
*/
// ----------------------------------------------------------------------------
double BesselJ( double x, sal_Int32 N ) throw (IllegalArgumentException, NoConvergenceException)
{
if( N < 0 )
throw IllegalArgumentException();
if (x==0.0)
return (N==0) ? 1.0 : 0.0;
/* The algorithm works only for x>0, therefore remember sign. BesselJ
with integer order N is an even function for even N (means J(-x)=J(x))
and an odd function for odd N (means J(-x)=-J(x)).*/
double fSign = (N % 2 == 1 && x < 0) ? -1.0 : 1.0;
double fX = fabs(x);
const double fMaxIteration = 9000000.0; //experimental, for to return in < 3 seconds
double fEstimateIteration = fX * 1.5 + N;
bool bAsymptoticPossible = pow(fX,0.4) > N;
if (fEstimateIteration > fMaxIteration)
{
if (bAsymptoticPossible)
return fSign * sqrt(f_2_DIV_PI/fX)* cos(fX-N*f_PI_DIV_2-f_PI_DIV_4);
else
throw NoConvergenceException();
}
double epsilon = 1.0e-15; // relative error
bool bHasfound = false;
double k= 0.0;
// e_{-1} = 0; e_0 = alpha_0 / b_2
double u ; // u_0 = e_0/f_0 = alpha_0/m_0 = alpha_0
// first used with k=1
double m_bar; // m_bar_k = m_k * f_bar_{k-1}
double g_bar; // g_bar_k = m_bar_k - a_{k+1} + g_{k-1}
double g_bar_delta_u; // g_bar_delta_u_k = f_bar_{k-1} * alpha_k
// - g_{k-1} * delta_u_{k-1} - m_bar_k * u_{k-1}
// f_{-1} = 0.0; f_0 = m_0 / b_2 = 1/(-1) = -1
double g = 0.0; // g_0= f_{-1} / f_0 = 0/(-1) = 0
double delta_u = 0.0; // dummy initialize, first used with * 0
double f_bar = -1.0; // f_bar_k = 1/f_k, but only used for k=0
if (N==0)
{
//k=0; alpha_0 = 1.0
u = 1.0; // u_0 = alpha_0
// k = 1.0; at least one step is necessary
// m_bar_k = m_k * f_bar_{k-1} ==> m_bar_1 = 0.0
g_bar_delta_u = 0.0; // alpha_k = 0.0, m_bar = 0.0; g= 0.0
g_bar = - 2.0/fX; // k = 1.0, g = 0.0
delta_u = g_bar_delta_u / g_bar;
u = u + delta_u ; // u_k = u_{k-1} + delta_u_k
g = -1.0 / g_bar; // g_k=b_{k+2}/g_bar_k
f_bar = f_bar * g; // f_bar_k = f_bar_{k-1}* g_k
k = 2.0;
// From now on all alpha_k = 0.0 and k > N+1
}
else
{ // N >= 1 and alpha_k = 0.0 for k<N
u=0.0; // u_0 = alpha_0
for (k =1.0; k<= N-1; k = k + 1.0)
{
m_bar=2.0 * fmod(k-1.0, 2.0) * f_bar;
g_bar_delta_u = - g * delta_u - m_bar * u; // alpha_k = 0.0
g_bar = m_bar - 2.0*k/fX + g;
delta_u = g_bar_delta_u / g_bar;
u = u + delta_u;
g = -1.0/g_bar;
f_bar=f_bar * g;
}
// Step alpha_N = 1.0
m_bar=2.0 * fmod(k-1.0, 2.0) * f_bar;
g_bar_delta_u = f_bar - g * delta_u - m_bar * u; // alpha_k = 1.0
g_bar = m_bar - 2.0*k/fX + g;
delta_u = g_bar_delta_u / g_bar;
u = u + delta_u;
g = -1.0/g_bar;
f_bar = f_bar * g;
k = k + 1.0;
}
// Loop until desired accuracy, always alpha_k = 0.0
do
{
m_bar = 2.0 * fmod(k-1.0, 2.0) * f_bar;
g_bar_delta_u = - g * delta_u - m_bar * u;
g_bar = m_bar - 2.0*k/fX + g;
delta_u = g_bar_delta_u / g_bar;
u = u + delta_u;
g = -1.0/g_bar;
f_bar = f_bar * g;
bHasfound = (fabs(delta_u)<=fabs(u)*epsilon);
k = k + 1.0;
}
while (!bHasfound && k <= fMaxIteration);
if (bHasfound)
return u * fSign;
else
throw NoConvergenceException(); // unlikely to happen
}
// ============================================================================
// 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)!
No asymptotic approximation used, see issue 43040.
*/
// ----------------------------------------------------------------------------
double BesselI( double x, sal_Int32 n ) throw( IllegalArgumentException, NoConvergenceException )
{
const double fEpsilon = 1.0E-15;
const sal_Int32 nMaxIteration = 2000;
const double fXHalf = x / 2.0;
if( n < 0 )
throw IllegalArgumentException();
double fResult = 0.0;
/* Start the iteration without TERM(n,0), which is set here.
TERM(n,0) = (x/2)^n / n!
*/
sal_Int32 nK = 0;
double fTerm = 1.0;
// avoid overflow in Fak(n)
for( nK = 1; nK <= n; ++nK )
{
fTerm = fTerm / static_cast< double >( nK ) * fXHalf;
}
fResult = fTerm; // Start result with TERM(n,0).
if( fTerm != 0.0 )
{
nK = 1;
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 = fTerm * fXHalf / static_cast<double>(nK) * fXHalf / static_cast<double>(nK+n);
fResult += fTerm;
nK++;
}
while( (fabs( fTerm ) > fabs(fResult) * fEpsilon) && (nK < nMaxIteration) );
}
return fResult;
}
// ============================================================================
double Besselk0( double fNum ) throw( IllegalArgumentException, NoConvergenceException )
{
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, NoConvergenceException )
{
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, NoConvergenceException )
{
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;
}
}
}
// ============================================================================
// BESSEL Y
// ============================================================================
/* The BESSEL function, second kind, unmodified:
The algorithm for order 0 and for order 1 follows
http://www.reference-global.com/isbn/978-3-11-020354-7
Numerical Mathematics 1 / Numerische Mathematik 1,
An algorithm-based introduction / Eine algorithmisch orientierte Einfuehrung
Deuflhard, Peter; Hohmann, Andreas
Berlin, New York (Walter de Gruyter) 2008
4. ueberarb. u. erw. Aufl. 2008
eBook ISBN: 978-3-11-020355-4
Chapter 6.3.2 , algorithm 6.24
The source is in German.
See #i31656# for a commented version of the implementation, attachment #desc6
http://www.openoffice.org/nonav/issues/showattachment.cgi/63609/Comments%20to%20the%20implementation%20of%20the%20Bessel%20functions.odt
*/
double Bessely0( double fX ) throw( IllegalArgumentException, NoConvergenceException )
{
if (fX <= 0)
throw IllegalArgumentException();
const double fMaxIteration = 9000000.0; // should not be reached
if (fX > 5.0e+6) // iteration is not considerable better then approximation
return sqrt(1/f_PI/fX)
*(rtl::math::sin(fX)-rtl::math::cos(fX));
const double epsilon = 1.0e-15;
const double EulerGamma = 0.57721566490153286060;
double alpha = log(fX/2.0)+EulerGamma;
double u = alpha;
double k = 1.0;
double m_bar = 0.0;
double g_bar_delta_u = 0.0;
double g_bar = -2.0 / fX;
double delta_u = g_bar_delta_u / g_bar;
double g = -1.0/g_bar;
double f_bar = -1 * g;
double sign_alpha = 1.0;
double km1mod2;
bool bHasFound = false;
k = k + 1;
do
{
km1mod2 = fmod(k-1.0,2.0);
m_bar=(2.0*km1mod2) * f_bar;
if (km1mod2 == 0.0)
alpha = 0.0;
else
{
alpha = sign_alpha * (4.0/k);
sign_alpha = -sign_alpha;
}
g_bar_delta_u = f_bar * alpha - g * delta_u - m_bar * u;
g_bar = m_bar - (2.0*k)/fX + g;
delta_u = g_bar_delta_u / g_bar;
u = u+delta_u;
g = -1.0 / g_bar;
f_bar = f_bar*g;
bHasFound = (fabs(delta_u)<=fabs(u)*epsilon);
k=k+1;
}
while (!bHasFound && k<fMaxIteration);
if (bHasFound)
return u*f_2_DIV_PI;
else
throw NoConvergenceException(); // not likely to happen
}
// See #i31656# for a commented version of this implementation, attachment #desc6
// http://www.openoffice.org/nonav/issues/showattachment.cgi/63609/Comments%20to%20the%20implementation%20of%20the%20Bessel%20functions.odt
double Bessely1( double fX ) throw( IllegalArgumentException, NoConvergenceException )
{
if (fX <= 0)
throw IllegalArgumentException();
const double fMaxIteration = 9000000.0; // should not be reached
if (fX > 5.0e+6) // iteration is not considerable better then approximation
return - sqrt(1/f_PI/fX)
*(rtl::math::sin(fX)+rtl::math::cos(fX));
const double epsilon = 1.0e-15;
const double EulerGamma = 0.57721566490153286060;
double alpha = 1.0/fX;
double f_bar = -1.0;
double g = 0.0;
double u = alpha;
double k = 1.0;
double m_bar = 0.0;
alpha = 1.0 - EulerGamma - log(fX/2.0);
double g_bar_delta_u = -alpha;
double g_bar = -2.0 / fX;
double delta_u = g_bar_delta_u / g_bar;
u = u + delta_u;
g = -1.0/g_bar;
f_bar = f_bar * g;
double sign_alpha = -1.0;
double km1mod2; //will be (k-1) mod 2
double q; // will be (k-1) div 2
bool bHasFound = false;
k = k + 1.0;
do
{
km1mod2 = fmod(k-1.0,2.0);
m_bar=(2.0*km1mod2) * f_bar;
q = (k-1.0)/2.0;
if (km1mod2 == 0.0) // k is odd
{
alpha = sign_alpha * (1.0/q + 1.0/(q+1.0));
sign_alpha = -sign_alpha;
}
else
alpha = 0.0;
g_bar_delta_u = f_bar * alpha - g * delta_u - m_bar * u;
g_bar = m_bar - (2.0*k)/fX + g;
delta_u = g_bar_delta_u / g_bar;
u = u+delta_u;
g = -1.0 / g_bar;
f_bar = f_bar*g;
bHasFound = (fabs(delta_u)<=fabs(u)*epsilon);
k=k+1;
}
while (!bHasFound && k<fMaxIteration);
if (bHasFound)
return -u*2.0/f_PI;
else
throw NoConvergenceException();
}
double BesselY( double fNum, sal_Int32 nOrder ) throw( IllegalArgumentException, NoConvergenceException )
{
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
/* vim:set shiftwidth=4 softtabstop=4 expandtab: */
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