/* j0.c */ asm (" export Libm"); /* floating point Bessel's function of the first and second kinds of order zero j0(x) returns the value of J0(x) for all real values of x. There are no error returns. Calls sin, cos, sqrt. There is a niggling bug in J0 which causes errors up to 2e-16 for x in the interval [-8,8]. The bug is caused by an inappropriate order of summation of the series. rhm will fix it someday. Coefficients are from Hart & Cheney. #5849 (19.22D) #6549 (19.25D) #6949 (19.41D) y0(x) returns the value of Y0(x) for positive real values of x. For x<=0, if on the VAX, error number EDOM is set and the reserved operand fault is generated; otherwise (an IEEE machine) an invalid operation is performed. Calls sin, cos, sqrt, log, j0. The values of Y0 have not been checked to more than ten places. Coefficients are from Hart & Cheney. #6245 (18.78D) #6549 (19.25D) #6949 (19.41D) */ mesa double Libm←sin(), Libm←cos(), Libm←tan(); mesa double Libm←asin(), Libm←acos(), Libm←atan(), Libm←atan2(); mesa double Libm←sinh(), Libm←cosh(), Libm←tanh(); mesa double Libm←asinh(), Libm←acosh(), Libm←atanh(); mesa double Libm←log(), Libm←log1p(), Libm←lgamma(); mesa double Libm←exp(), Libm←frexp(), Libm←ldexp(), Libm←expm1(), Libm←pow(); mesa double Libm←fmod(), Libm←modf(); mesa double Libm←floor(), Libm←ceil(), Libm←rint(); mesa double Libm←cabs(), Libm←hypot(); mesa double Libm←sqrt(), Libm←cbrt(); mesa double Libm←j1(), Libm←jn(); mesa double Libm←fabs(); mesa int Libm←abs(); #define sin(x) (Libm←sin(x)) #define cos(x) (Libm←cos(x)) #define tan(x) (Libm←tan(x)) #define asin(x) (Libm←asin(x)) #define acos(x) (Libm←acos(x)) #define atan(x) (Libm←atan(x)) #define atan2(x,y) (Libm←atan2(x,y)) #define sinh(x) (Libm←sinh(x)) #define cosh(x) (Libm←cosh(x)) #define tanh(x) (Libm←tanh(x)) #define asinh(x) (Libm←asinh(x)) #define acosh(x) (Libm←acosh(x)) #define atanh(x) (Libm←atanh(x)) #define log(x) (Libm←log(x)) #define log1p(x) (Libm←log1p(x)) #define lgamma(a) (Libm←lgamma(a)) #define exp(x) (Libm←exp(x)) #define frexp(x,i) (Libm←frexp(x,i)) #define ldexp(x,e) (Libm←ldexp(x,e)) #define expm1(x) (Libm←expm1(x)) #define pow(x,y) (Libm←pow(x,y)) #define fmod(x,y) (Libm←fmod(x,y)) #define modf(d,i) (Libm←modf(d,i)) #define floor(d) (Libm←floor(d)) #define ceil(d) (Libm←ceil(d)) #define rint(x) (Libm←rint(x)) #define cabs(z) (Libm←cabs(z)) #define hypot(x,y) (Libm←hypot(x,y)) #define sqrt(x) (Libm←sqrt(x)) #define cbrt(x) (Libm←cbrt(x)) #define j1(a) (Libm←j1(a)) #define jn(n,x) (Libm←jn(n,x)) #define fabs(d) (Libm←fabs(d)) #define abs(i) (Libm←abs(i)) #define HUGE 1.701411733192644270e38 static double zero = 0.e0; static double pzero, qzero; static double tpi = .6366197723675813430755350535e0; static double pio4 = .7853981633974483096156608458e0; static double p1[] = { 0.4933787251794133561816813446e21, -.1179157629107610536038440800e21, 0.6382059341072356562289432465e19, -.1367620353088171386865416609e18, 0.1434354939140344111664316553e16, -.8085222034853793871199468171e13, 0.2507158285536881945555156435e11, -.4050412371833132706360663322e8, 0.2685786856980014981415848441e5, }; static double q1[] = { 0.4933787251794133562113278438e21, 0.5428918384092285160200195092e19, 0.3024635616709462698627330784e17, 0.1127756739679798507056031594e15, 0.3123043114941213172572469442e12, 0.6699987672982239671814028660e9, 0.1114636098462985378182402543e7, 0.1363063652328970604442810507e4, 1.0 }; static double p2[] = { 0.5393485083869438325262122897e7, 0.1233238476817638145232406055e8, 0.8413041456550439208464315611e7, 0.2016135283049983642487182349e7, 0.1539826532623911470917825993e6, 0.2485271928957404011288128951e4, 0.0, }; static double q2[] = { 0.5393485083869438325560444960e7, 0.1233831022786324960844856182e8, 0.8426449050629797331554404810e7, 0.2025066801570134013891035236e7, 0.1560017276940030940592769933e6, 0.2615700736920839685159081813e4, 1.0, }; static double p3[] = { -.3984617357595222463506790588e4, -.1038141698748464093880530341e5, -.8239066313485606568803548860e4, -.2365956170779108192723612816e4, -.2262630641933704113967255053e3, -.4887199395841261531199129300e1, 0.0, }; static double q3[] = { 0.2550155108860942382983170882e6, 0.6667454239319826986004038103e6, 0.5332913634216897168722255057e6, 0.1560213206679291652539287109e6, 0.1570489191515395519392882766e5, 0.4087714673983499223402830260e3, 1.0, }; static double p4[] = { -.2750286678629109583701933175e20, 0.6587473275719554925999402049e20, -.5247065581112764941297350814e19, 0.1375624316399344078571335453e18, -.1648605817185729473122082537e16, 0.1025520859686394284509167421e14, -.3436371222979040378171030138e11, 0.5915213465686889654273830069e8, -.4137035497933148554125235152e5, }; static double q4[] = { 0.3726458838986165881989980e21, 0.4192417043410839973904769661e19, 0.2392883043499781857439356652e17, 0.9162038034075185262489147968e14, 0.2613065755041081249568482092e12, 0.5795122640700729537480087915e9, 0.1001702641288906265666651753e7, 0.1282452772478993804176329391e4, 1.0, }; asm (" exportproc ←j0, Libm"); double j0(arg) double arg;{ double argsq, n, d; double sin(), cos(), sqrt(); int i; if(arg < 0.) arg = -arg; if(arg > 8.){ asympt(arg); n = arg - pio4; return(sqrt(tpi/arg)*(pzero*cos(n) - qzero*sin(n))); } argsq = arg*arg; for(n=0,d=0,i=8;i>=0;i--){ n = n*argsq + p1[i]; d = d*argsq + q1[i]; } return(n/d); } asm (" exportproc ←y0, Libm"); double y0(arg) double arg;{ double argsq, n, d; double sin(), cos(), sqrt(), log(), j0(); int i; if(arg <= 0.){ return(zero/zero); /* IEEE machines: invalid operation */ } if(arg > 8.){ asympt(arg); n = arg - pio4; return(sqrt(tpi/arg)*(pzero*sin(n) + qzero*cos(n))); } argsq = arg*arg; for(n=0,d=0,i=8;i>=0;i--){ n = n*argsq + p4[i]; d = d*argsq + q4[i]; } return(n/d + tpi*j0(arg)*log(arg)); } static asympt(arg) double arg;{ double zsq, n, d; int i; zsq = 64./(arg*arg); for(n=0,d=0,i=6;i>=0;i--){ n = n*zsq + p2[i]; d = d*zsq + q2[i]; } pzero = n/d; for(n=0,d=0,i=6;i>=0;i--){ n = n*zsq + p3[i]; d = d*zsq + q3[i]; } qzero = (8./arg)*(n/d); }