airy.c

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/*                                                     airy.c
 *
 *     Airy function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, ai, aip, bi, bip;
 * int airy();
 *
 * airy( x, _&ai, _&aip, _&bi, _&bip );
 *
 *
 *
 * DESCRIPTION:
 *
 * Solution of the differential equation
 *
 *     y"(x) = xy.
 *
 * The function returns the two independent solutions Ai, Bi
 * and their first derivatives Ai'(x), Bi'(x).
 *
 * Evaluation is by power series summation for small x,
 * by rational minimax approximations for large x.
 *
 *
 *
 * ACCURACY:
 * Error criterion is absolute when function <= 1, relative
 * when function > 1, except * denotes relative error criterion.
 * For large negative x, the absolute error increases as x^1.5.
 * For large positive x, the relative error increases as x^1.5.
 *
 * Arithmetic  domain   function  # trials      peak         rms
 * IEEE        -10, 0     Ai        10000       1.6e-15     2.7e-16
 * IEEE          0, 10    Ai        10000       2.3e-14*    1.8e-15*
 * IEEE        -10, 0     Ai'       10000       4.6e-15     7.6e-16
 * IEEE          0, 10    Ai'       10000       1.8e-14*    1.5e-15*
 * IEEE        -10, 10    Bi        30000       4.2e-15     5.3e-16
 * IEEE        -10, 10    Bi'       30000       4.9e-15     7.3e-16
 *
 */
/*                                                     airy.c */
 
/*
 * Cephes Math Library Release 2.8:  June, 2000
 * Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
 */
 
#include "mconf.h"
 
static double c1 = 0.35502805388781723926;
static double c2 = 0.258819403792806798405;
static double sqrt3 = 1.732050807568877293527;
static double sqpii = 5.64189583547756286948E-1;
 
extern double MACHEP;
 
#ifdef UNK
#define MAXAIRY 25.77
#endif
#ifdef IBMPC
#define MAXAIRY 103.892
#endif
#ifdef MIEEE
#define MAXAIRY 103.892
#endif
 
 
static double AN[8] = {
    3.46538101525629032477E-1,
    1.20075952739645805542E1,
    7.62796053615234516538E1,
    1.68089224934630576269E2,
    1.59756391350164413639E2,
    7.05360906840444183113E1,
    1.40264691163389668864E1,
    9.99999999999999995305E-1,
};
 
static double AD[8] = {
    5.67594532638770212846E-1,
    1.47562562584847203173E1,
    8.45138970141474626562E1,
    1.77318088145400459522E2,
    1.64234692871529701831E2,
    7.14778400825575695274E1,
    1.40959135607834029598E1,
    1.00000000000000000470E0,
};
 
static double APN[8] = {
    6.13759184814035759225E-1,
    1.47454670787755323881E1,
    8.20584123476060982430E1,
    1.71184781360976385540E2,
    1.59317847137141783523E2,
    6.99778599330103016170E1,
    1.39470856980481566958E1,
    1.00000000000000000550E0,
};
 
static double APD[8] = {
    3.34203677749736953049E-1,
    1.11810297306158156705E1,
    7.11727352147859965283E1,
    1.58778084372838313640E2,
    1.53206427475809220834E2,
    6.86752304592780337944E1,
    1.38498634758259442477E1,
    9.99999999999999994502E-1,
};
 
static double BN16[5] = {
    -2.53240795869364152689E-1,
    5.75285167332467384228E-1,
    -3.29907036873225371650E-1,
    6.44404068948199951727E-2,
    -3.82519546641336734394E-3,
};
 
static double BD16[5] = {
    /* 1.00000000000000000000E0, */
    -7.15685095054035237902E0,
    1.06039580715664694291E1,
    -5.23246636471251500874E0,
    9.57395864378383833152E-1,
    -5.50828147163549611107E-2,
};
 
static double BPPN[5] = {
    4.65461162774651610328E-1,
    -1.08992173800493920734E0,
    6.38800117371827987759E-1,
    -1.26844349553102907034E-1,
    7.62487844342109852105E-3,
};
 
static double BPPD[5] = {
    /* 1.00000000000000000000E0, */
    -8.70622787633159124240E0,
    1.38993162704553213172E1,
    -7.14116144616431159572E0,
    1.34008595960680518666E0,
    -7.84273211323341930448E-2,
};
 
static double AFN[9] = {
    -1.31696323418331795333E-1,
    -6.26456544431912369773E-1,
    -6.93158036036933542233E-1,
    -2.79779981545119124951E-1,
    -4.91900132609500318020E-2,
    -4.06265923594885404393E-3,
    -1.59276496239262096340E-4,
    -2.77649108155232920844E-6,
    -1.67787698489114633780E-8,
};
 
static double AFD[9] = {
    /* 1.00000000000000000000E0, */
    1.33560420706553243746E1,
    3.26825032795224613948E1,
    2.67367040941499554804E1,
    9.18707402907259625840E0,
    1.47529146771666414581E0,
    1.15687173795188044134E-1,
    4.40291641615211203805E-3,
    7.54720348287414296618E-5,
    4.51850092970580378464E-7,
};
 
static double AGN[11] = {
    1.97339932091685679179E-2,
    3.91103029615688277255E-1,
    1.06579897599595591108E0,
    9.39169229816650230044E-1,
    3.51465656105547619242E-1,
    6.33888919628925490927E-2,
    5.85804113048388458567E-3,
    2.82851600836737019778E-4,
    6.98793669997260967291E-6,
    8.11789239554389293311E-8,
    3.41551784765923618484E-10,
};
 
static double AGD[10] = {
    /*  1.00000000000000000000E0, */
    9.30892908077441974853E0,
    1.98352928718312140417E1,
    1.55646628932864612953E1,
    5.47686069422975497931E0,
    9.54293611618961883998E-1,
    8.64580826352392193095E-2,
    4.12656523824222607191E-3,
    1.01259085116509135510E-4,
    1.17166733214413521882E-6,
    4.91834570062930015649E-9,
};
 
static double APFN[9] = {
    1.85365624022535566142E-1,
    8.86712188052584095637E-1,
    9.87391981747398547272E-1,
    4.01241082318003734092E-1,
    7.10304926289631174579E-2,
    5.90618657995661810071E-3,
    2.33051409401776799569E-4,
    4.08718778289035454598E-6,
    2.48379932900442457853E-8,
};
 
static double APFD[9] = {
    /*  1.00000000000000000000E0, */
    1.47345854687502542552E1,
    3.75423933435489594466E1,
    3.14657751203046424330E1,
    1.09969125207298778536E1,
    1.78885054766999417817E0,
    1.41733275753662636873E-1,
    5.44066067017226003627E-3,
    9.39421290654511171663E-5,
    5.65978713036027009243E-7,
};
 
static double APGN[11] = {
    -3.55615429033082288335E-2,
    -6.37311518129435504426E-1,
    -1.70856738884312371053E0,
    -1.50221872117316635393E0,
    -5.63606665822102676611E-1,
    -1.02101031120216891789E-1,
    -9.48396695961445269093E-3,
    -4.60325307486780994357E-4,
    -1.14300836484517375919E-5,
    -1.33415518685547420648E-7,
    -5.63803833958893494476E-10,
};
 
static double APGD[11] = {
    /*  1.00000000000000000000E0, */
    9.85865801696130355144E0,
    2.16401867356585941885E1,
    1.73130776389749389525E1,
    6.17872175280828766327E0,
    1.08848694396321495475E0,
    9.95005543440888479402E-2,
    4.78468199683886610842E-3,
    1.18159633322838625562E-4,
    1.37480673554219441465E-6,
    5.79912514929147598821E-9,
};
 
int airy(double x, double *ai, double *aip, double *bi, double *bip)
{
    double z, zz, t, f, g, uf, ug, k, zeta, theta;
    int domflg;
 
    domflg = 0;
    if (x > MAXAIRY) {
        *ai = 0;
        *aip = 0;
        *bi = INFINITY;
        *bip = INFINITY;
        return (-1);
    }
 
    if (x < -2.09) {
        domflg = 15;
        t = sqrt(-x);
        zeta = -2.0 * x * t / 3.0;
        t = sqrt(t);
        k = sqpii / t;
        z = 1.0 / zeta;
        zz = z * z;
        uf = 1.0 + zz * polevl(zz, AFN, 8) / p1evl(zz, AFD, 9);
        ug = z * polevl(zz, AGN, 10) / p1evl(zz, AGD, 10);
        theta = zeta + 0.25 * M_PI;
        f = sin(theta);
        g = cos(theta);
        *ai = k * (f * uf - g * ug);
        *bi = k * (g * uf + f * ug);
        uf = 1.0 + zz * polevl(zz, APFN, 8) / p1evl(zz, APFD, 9);
        ug = z * polevl(zz, APGN, 10) / p1evl(zz, APGD, 10);
        k = sqpii * t;
        *aip = -k * (g * uf + f * ug);
        *bip = k * (f * uf - g * ug);
        return (0);
    }
 
    if (x >= 2.09) {            /* cbrt(9) */
        domflg = 5;
        t = sqrt(x);
        zeta = 2.0 * x * t / 3.0;
        g = exp(zeta);
        t = sqrt(t);
        k = 2.0 * t * g;
        z = 1.0 / zeta;
        f = polevl(z, AN, 7) / polevl(z, AD, 7);
        *ai = sqpii * f / k;
        k = -0.5 * sqpii * t / g;
        f = polevl(z, APN, 7) / polevl(z, APD, 7);
        *aip = f * k;
 
        if (x > 8.3203353) {    /* zeta > 16 */
            f = z * polevl(z, BN16, 4) / p1evl(z, BD16, 5);
            k = sqpii * g;
            *bi = k * (1.0 + f) / t;
            f = z * polevl(z, BPPN, 4) / p1evl(z, BPPD, 5);
            *bip = k * t * (1.0 + f);
            return (0);
        }
    }
 
    f = 1.0;
    g = x;
    t = 1.0;
    uf = 1.0;
    ug = x;
    k = 1.0;
    z = x * x * x;
    while (t > MACHEP) {
        uf *= z;
        k += 1.0;
        uf /= k;
        ug *= z;
        k += 1.0;
        ug /= k;
        uf /= k;
        f += uf;
        k += 1.0;
        ug /= k;
        g += ug;
        t = fabs(uf / f);
    }
    uf = c1 * f;
    ug = c2 * g;
    if ((domflg & 1) == 0)
        *ai = uf - ug;
    if ((domflg & 2) == 0)
        *bi = sqrt3 * (uf + ug);
 
    /* the deriviative of ai */
    k = 4.0;
    uf = x * x / 2.0;
    ug = z / 3.0;
    f = uf;
    g = 1.0 + ug;
    uf /= 3.0;
    t = 1.0;
 
    while (t > MACHEP) {
        uf *= z;
        ug /= k;
        k += 1.0;
        ug *= z;
        uf /= k;
        f += uf;
        k += 1.0;
        ug /= k;
        uf /= k;
        g += ug;
        k += 1.0;
        t = fabs(ug / g);
    }
 
    uf = c1 * f;
    ug = c2 * g;
    if ((domflg & 4) == 0)
        *aip = uf - ug;
    if ((domflg & 8) == 0)
        *bip = sqrt3 * (uf + ug);
    return (0);
}