# SasView/sasmodels

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 /* j1.c * * Bessel function of order one * * * * SYNOPSIS: * * double x, y, j1(); * * y = j1( x ); * * * * DESCRIPTION: * * Returns Bessel function of order one of the argument. * * The domain is divided into the intervals [0, 8] and * (8, infinity). In the first interval a 24 term Chebyshev * expansion is used. In the second, the asymptotic * trigonometric representation is employed using two * rational functions of degree 5/5. * * * * ACCURACY: * * Absolute error: * arithmetic domain # trials peak rms * DEC 0, 30 10000 4.0e-17 1.1e-17 * IEEE 0, 30 30000 2.6e-16 1.1e-16 * * */ /* Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier */ #if FLOAT_SIZE>4 //Cephes double pression function double cephes_j1(double x); constant double RPJ1[8] = { -8.99971225705559398224E8, 4.52228297998194034323E11, -7.27494245221818276015E13, 3.68295732863852883286E15, 0.0, 0.0, 0.0, 0.0 }; constant double RQJ1[8] = { 6.20836478118054335476E2, 2.56987256757748830383E5, 8.35146791431949253037E7, 2.21511595479792499675E10, 4.74914122079991414898E12, 7.84369607876235854894E14, 8.95222336184627338078E16, 5.32278620332680085395E18 }; constant double PPJ1[8] = { 7.62125616208173112003E-4, 7.31397056940917570436E-2, 1.12719608129684925192E0, 5.11207951146807644818E0, 8.42404590141772420927E0, 5.21451598682361504063E0, 1.00000000000000000254E0, 0.0} ; constant double PQJ1[8] = { 5.71323128072548699714E-4, 6.88455908754495404082E-2, 1.10514232634061696926E0, 5.07386386128601488557E0, 8.39985554327604159757E0, 5.20982848682361821619E0, 9.99999999999999997461E-1, 0.0 }; constant double QPJ1[8] = { 5.10862594750176621635E-2, 4.98213872951233449420E0, 7.58238284132545283818E1, 3.66779609360150777800E2, 7.10856304998926107277E2, 5.97489612400613639965E2, 2.11688757100572135698E2, 2.52070205858023719784E1 }; constant double QQJ1[8] = { 7.42373277035675149943E1, 1.05644886038262816351E3, 4.98641058337653607651E3, 9.56231892404756170795E3, 7.99704160447350683650E3, 2.82619278517639096600E3, 3.36093607810698293419E2, 0.0 }; double cephes_j1(double x) { double w, z, p, q, abs_x, sign_x; const double Z1 = 1.46819706421238932572E1; const double Z2 = 4.92184563216946036703E1; // 2017-05-18 PAK - mathematica and mpmath use J1(-x) = -J1(x) if (x < 0) { abs_x = -x; sign_x = -1.0; } else { abs_x = x; sign_x = 1.0; } if( abs_x <= 5.0 ) { z = abs_x * abs_x; w = polevl( z, RPJ1, 3 ) / p1evl( z, RQJ1, 8 ); w = w * abs_x * (z - Z1) * (z - Z2); return( sign_x * w ); } w = 5.0/abs_x; z = w * w; p = polevl( z, PPJ1, 6)/polevl( z, PQJ1, 6 ); q = polevl( z, QPJ1, 7)/p1evl( z, QQJ1, 7 ); // 2017-05-19 PAK improve accuracy using trig identies // original: // const double THPIO4 = 2.35619449019234492885; // const double SQ2OPI = 0.79788456080286535588; // double sin_xn, cos_xn; // SINCOS(abs_x - THPIO4, sin_xn, cos_xn); // p = p * cos_xn - w * q * sin_xn; // return( sign_x * p * SQ2OPI / sqrt(abs_x) ); // expanding p*cos(a - 3 pi/4) - wq sin(a - 3 pi/4) // [ p(sin(a) - cos(a)) + wq(sin(a) + cos(a)) / sqrt(2) // note that sqrt(1/2) * sqrt(2/pi) = sqrt(1/pi) const double SQRT1_PI = 0.56418958354775628; double sin_x, cos_x; SINCOS(abs_x, sin_x, cos_x); p = p*(sin_x - cos_x) + w*q*(sin_x + cos_x); return( sign_x * p * SQRT1_PI / sqrt(abs_x) ); } #else //Single precission version of cephes float cephes_j1f(float x); constant float JPJ1[8] = { -4.878788132172128E-009, 6.009061827883699E-007, -4.541343896997497E-005, 1.937383947804541E-003, -3.405537384615824E-002, 0.0, 0.0, 0.0 }; constant float MO1J1[8] = { 6.913942741265801E-002, -2.284801500053359E-001, 3.138238455499697E-001, -2.102302420403875E-001, 5.435364690523026E-003, 1.493389585089498E-001, 4.976029650847191E-006, 7.978845453073848E-001 }; constant float PH1J1[8] = { -4.497014141919556E+001, 5.073465654089319E+001, -2.485774108720340E+001, 7.222973196770240E+000, -1.544842782180211E+000, 3.503787691653334E-001, -1.637986776941202E-001, 3.749989509080821E-001 }; float cephes_j1f(float xx) { float x, w, z, p, q, xn; const float Z1 = 1.46819706421238932572E1; // 2017-05-18 PAK - mathematica and mpmath use J1(-x) = -J1(x) x = xx; if( x < 0 ) x = -xx; if( x <= 2.0 ) { z = x * x; p = (z-Z1) * x * polevl( z, JPJ1, 4 ); return( xx < 0. ? -p : p ); } q = 1.0/x; w = sqrt(q); p = w * polevl( q, MO1J1, 7); w = q*q; // 2017-05-19 PAK improve accuracy using trig identies // original: // const float THPIO4F = 2.35619449019234492885; /* 3*pi/4 */ // xn = q * polevl( w, PH1J1, 7) - THPIO4F; // p = p * cos(xn + x); // return( xx < 0. ? -p : p ); // expanding cos(a + b - 3 pi/4) is // [sin(a)sin(b) + sin(a)cos(b) + cos(a)sin(b)-cos(a)cos(b)] / sqrt(2) xn = q * polevl( w, PH1J1, 7); float cos_xn, sin_xn; float cos_x, sin_x; SINCOS(xn, sin_xn, cos_xn); // about xn and 1 SINCOS(x, sin_x, cos_x); p *= M_SQRT1_2*(sin_xn*(sin_x+cos_x) + cos_xn*(sin_x-cos_x)); return( xx < 0. ? -p : p ); } #endif #if FLOAT_SIZE>4 #define sas_J1 cephes_j1 #else #define sas_J1 cephes_j1f #endif //Finally J1c function that equals 2*J1(x)/x double sas_2J1x_x(double x); double sas_2J1x_x(double x) { return (x != 0.0 ) ? 2.0*sas_J1(x)/x : 1.0; }