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diff --git a/scaddins/source/analysis/bessel.cxx b/scaddins/source/analysis/bessel.cxx new file mode 100644 index 000000000..44b79e798 --- /dev/null +++ b/scaddins/source/analysis/bessel.cxx @@ -0,0 +1,452 @@ +/* -*- Mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */ +/* + * This file is part of the LibreOffice project. + * + * This Source Code Form is subject to the terms of the Mozilla Public + * License, v. 2.0. If a copy of the MPL was not distributed with this + * file, You can obtain one at http://mozilla.org/MPL/2.0/. + * + * This file incorporates work covered by the following license notice: + * + * Licensed to the Apache Software Foundation (ASF) under one or more + * contributor license agreements. See the NOTICE file distributed + * with this work for additional information regarding copyright + * ownership. The ASF licenses this file to you under the Apache + * License, Version 2.0 (the "License"); you may not use this file + * except in compliance with the License. You may obtain a copy of + * the License at http://www.apache.org/licenses/LICENSE-2.0 . + */ + +#include "bessel.hxx" +#include <cmath> +#include <rtl/math.hxx> + +#include <com/sun/star/lang/IllegalArgumentException.hpp> +#include <com/sun/star/sheet/NoConvergenceException.hpp> + +using ::com::sun::star::lang::IllegalArgumentException; +using ::com::sun::star::sheet::NoConvergenceException; + +namespace sca::analysis { + +// 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 ) + +{ + 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) + throw NoConvergenceException(); + return fSign * sqrt(M_2_PI/fX)* cos(fX-N*M_PI_2-M_PI_4); + } + + double const 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) + throw NoConvergenceException(); // unlikely to happen + + return u * fSign; +} + + +// 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 ) +{ + 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; + const double fEpsilon = 1.0E-15; + 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; +} + +/// @throws IllegalArgumentException +/// @throws NoConvergenceException +static double Besselk0( double fNum ) +{ + 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; +} + +/// @throws IllegalArgumentException +/// @throws NoConvergenceException +static double Besselk1( double fNum ) +{ + 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 ) +{ + switch( nOrder ) + { + case 0: return Besselk0( fNum ); + case 1: return Besselk1( fNum ); + default: + { + double fTox = 2.0 / fNum; + double fBkm = Besselk0( fNum ); + double fBk = Besselk1( fNum ); + + for( sal_Int32 n = 1 ; n < nOrder ; n++ ) + { + const double 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 + https://bz.apache.org/ooo/attachment.cgi?id=63609 +*/ + +/// @throws IllegalArgumentException +/// @throws NoConvergenceException +static double Bessely0( double fX ) +{ + // If fX > 2^64 then sin and cos fail + if (fX <= 0 || !rtl::math::isValidArcArg(fX)) + 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/M_PI/fX) + *(std::sin(fX)-std::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 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; + bool bHasFound = false; + k = k + 1; + do + { + double km1mod2 = fmod(k-1.0, 2.0); + double 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) + throw NoConvergenceException(); // not likely to happen + return u*M_2_PI; +} + +// See #i31656# for a commented version of this implementation, attachment #desc6 +// https://bz.apache.org/ooo/attachment.cgi?id=63609 +/// @throws IllegalArgumentException +/// @throws NoConvergenceException +static double Bessely1( double fX ) +{ + // If fX > 2^64 then sin and cos fail + if (fX <= 0 || !rtl::math::isValidArcArg(fX)) + 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/M_PI/fX) + *(std::sin(fX)+std::cos(fX)); + const double epsilon = 1.0e-15; + const double EulerGamma = 0.57721566490153286060; + double alpha = 1.0/fX; + double f_bar = -1.0; + double u = alpha; + double k = 1.0; + 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; + double g = -1.0/g_bar; + f_bar = f_bar * g; + double sign_alpha = -1.0; + bool bHasFound = false; + k = k + 1.0; + do + { + double km1mod2 = fmod(k-1.0,2.0); + double m_bar = (2.0*km1mod2) * f_bar; + double 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) + throw NoConvergenceException(); + return -u*2.0/M_PI; +} + +double BesselY( double fNum, sal_Int32 nOrder ) +{ + switch( nOrder ) + { + case 0: return Bessely0( fNum ); + case 1: return Bessely1( fNum ); + default: + { + double fTox = 2.0 / fNum; + double fBym = Bessely0( fNum ); + double fBy = Bessely1( fNum ); + + for( sal_Int32 n = 1 ; n < nOrder ; n++ ) + { + const double fByp = double( n ) * fTox * fBy - fBym; + fBym = fBy; + fBy = fByp; + } + + return fBy; + } + } +} + +} // namespace sca::analysis + +/* vim:set shiftwidth=4 softtabstop=4 expandtab: */ |