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See the License for the * specific language governing permissions and limitations * under the License. * *************************************************************/ #include "bessel.hxx" #include "analysishelper.hxx" #include 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( 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(nK) * fXHalf / static_cast(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 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