template <typename Scalar> Scalar i0e(Scalar x) {
/* Chebyshev coefficients for exp(-x) I0(x)
* in the interval [0,8].
*
* lim(x->0){ exp(-x) I0(x) } = 1.
*/
static const Scalar A[] = {
-4.41534164647933937950E-18, 3.33079451882223809783E-17,
-2.43127984654795469359E-16, 1.71539128555513303061E-15,
-1.16853328779934516808E-14, 7.67618549860493561688E-14,
-4.85644678311192946090E-13, 2.95505266312963983461E-12,
-1.72682629144155570723E-11, 9.67580903537323691224E-11,
-5.18979560163526290666E-10, 2.65982372468238665035E-9,
-1.30002500998624804212E-8, 6.04699502254191894932E-8,
-2.67079385394061173391E-7, 1.11738753912010371815E-6,
-4.41673835845875056359E-6, 1.64484480707288970893E-5,
-5.75419501008210370398E-5, 1.88502885095841655729E-4,
-5.76375574538582365885E-4, 1.63947561694133579842E-3,
-4.32430999505057594430E-3, 1.05464603945949983183E-2,
-2.37374148058994688156E-2, 4.93052842396707084878E-2,
-9.49010970480476444210E-2, 1.71620901522208775349E-1,
-3.04682672343198398683E-1, 6.76795274409476084995E-1
};
/* Chebyshev coefficients for exp(-x) sqrt(x) I0(x)
* in the inverted interval [8,infinity].
*
* lim(x->inf){ exp(-x) sqrt(x) I0(x) } = 1/sqrt(2pi).
*/
static const Scalar B[] = {
-7.23318048787475395456E-18, -4.83050448594418207126E-18,
4.46562142029675999901E-17, 3.46122286769746109310E-17,
-2.82762398051658348494E-16, -3.42548561967721913462E-16,
1.77256013305652638360E-15, 3.81168066935262242075E-15,
-9.55484669882830764870E-15, -4.15056934728722208663E-14,
1.54008621752140982691E-14, 3.85277838274214270114E-13,
7.18012445138366623367E-13, -1.79417853150680611778E-12,
-1.32158118404477131188E-11, -3.14991652796324136454E-11,
1.18891471078464383424E-11, 4.94060238822496958910E-10,
3.39623202570838634515E-9, 2.26666899049817806459E-8,
2.04891858946906374183E-7, 2.89137052083475648297E-6,
6.88975834691682398426E-5, 3.36911647825569408990E-3,
8.04490411014108831608E-1
};
if (x < 0)
x = -x;
if (x <= 8) {
Scalar y = (x * (Scalar) 0.5) - (Scalar) 2.0;
return chbevl(y, A, 30);
} else {
return chbevl((Scalar) 32.0/x - (Scalar) 2.0, B, 25) / std::sqrt(x);
}
}
double i0_approx(double x)
{
double y;
if (x < 0)
x = -x;
if (x <= 8.0) {
y = (x / 2.0) - 2.0;
return(exp(x) * chbevl(y, A_approx, 15));
}
return(exp(x) * chbevl(32.0 / x - 2.0, B_approx, 4) / sqrt(x));
}
double i0e(double x)
{
double y;
if (x < 0)
x = -x;
if (x <= 8.0) {
y = (x / 2.0) - 2.0;
return(chbevl(y, A, 30));
}
return(chbevl(32.0 / x - 2.0, B, 25) / sqrt(x));
}
double i0(double x)
{
double y;
if( x < 0 )
x = -x;
if( x <= 8.0 )
{
y = (x/2.0) - 2.0;
return( exp(x) * chbevl( y, A_i0, 30 ) );
}
return( exp(x) * chbevl( 32.0/x - 2.0, B_i0, 25 ) / sqrt(x) );
}
double k1e (double x)
{
double y;
if (x <= 0.0) {
mtherr("k1e", CEPHES_DOMAIN);
return MAXNUM;
}
if (x <= 2.0) {
y = x * x - 2.0;
y = log(0.5 * x) * cephes_bessel_I1(x) + chbevl(y, A, 11) / x;
return y * exp(x);
}
return chbevl(8.0/x - 2.0, B, 25) / sqrt(x);
}
double i1e( double x, int prec1, int prec2 )
{
double y, z;
z = fabs(x);
if( z <= 8.0 )
{
y = (z/2.0) - 2.0;
z = chbevl( y, A, 29, prec1 ) * z;
}
else
{
z = chbevl( 32.0/z - 2.0, B, 25, prec2 ) / sqrt(z);
}
if( x < 0.0 )
z = -z;
return( z );
}
double cephes_bessel_K1 (double x)
{
double y, z;
z = 0.5 * x;
if (z <= 0.0) {
mtherr("k1", CEPHES_DOMAIN);
return MAXNUM;
}
if (x <= 2.0) {
y = x * x - 2.0;
y = log(z) * cephes_bessel_I1(x) + chbevl(y, A, 11) / x;
return y;
}
return exp(-x) * chbevl(8.0/x - 2.0, B, 25) / sqrt(x);
}
double i1(double x)
{
double y, z;
z = fabs(x);
if( z <= 8.0 )
{
y = (z/2.0) - 2.0;
z = chbevl( y, A_i1, 29 ) * z * exp(z);
}
else
{
z = exp(z) * chbevl( 32.0/z - 2.0, B_i1, 25 ) / sqrt(z);
}
if( x < 0.0 )
z = -z;
return( z );
}
double k1(double x)
{
double y, z;
z = 0.5 * x;
if( z <= 0.0 )
/*
{
mtherr( "k1", DOMAIN );
return( MAXNUM );
}
*/
assert(0);
if( x <= 2.0 )
{
y = x * x - 2.0;
y = log(z) * i1(x) + chbevl( y, A_k1, 11 ) / x;
return( y );
}
return( exp(-x) * chbevl( 8.0/x - 2.0, B_k1, 25 ) / sqrt(x) );
}
double k0(double x)
{
double y, z;
if( x <= 0.0 )
/*
{
mtherr( "k0", DOMAIN );
return( MAXNUM );
}
*/
assert(0);
if( x <= 2.0 )
{
y = x * x - 2.0;
y = chbevl( y, A_k0, 10 ) - log( 0.5 * x ) * i0(x);
return( y );
}
z = 8.0/x - 2.0;
y = exp(-x) * chbevl( z, B_k0, 25 ) / sqrt(x);
return(y);
}
double i0e(double x, int prec1, int prec2)
{
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*/
static double A[] =
{
-4.41534164647933937950E-18,
3.33079451882223809783E-17,
-2.43127984654795469359E-16,
1.71539128555513303061E-15,
-1.16853328779934516808E-14,
7.67618549860493561688E-14,
-4.85644678311192946090E-13,
2.95505266312963983461E-12,
-1.72682629144155570723E-11,
9.67580903537323691224E-11,
-5.18979560163526290666E-10,
2.65982372468238665035E-9,
-1.30002500998624804212E-8,
6.04699502254191894932E-8,
-2.67079385394061173391E-7,
1.11738753912010371815E-6,
-4.41673835845875056359E-6,
1.64484480707288970893E-5,
-5.75419501008210370398E-5,
1.88502885095841655729E-4,
-5.76375574538582365885E-4,
1.63947561694133579842E-3,
-4.32430999505057594430E-3,
1.05464603945949983183E-2,
-2.37374148058994688156E-2,
4.93052842396707084878E-2,
-9.49010970480476444210E-2,
1.71620901522208775349E-1,
-3.04682672343198398683E-1,
6.76795274409476084995E-1
};
static double B[] =
{
-7.23318048787475395456E-18,
-4.83050448594418207126E-18,
4.46562142029675999901E-17,
3.46122286769746109310E-17,
-2.82762398051658348494E-16,
-3.42548561967721913462E-16,
1.77256013305652638360E-15,
3.81168066935262242075E-15,
-9.55484669882830764870E-15,
-4.15056934728722208663E-14,
1.54008621752140982691E-14,
3.85277838274214270114E-13,
7.18012445138366623367E-13,
-1.79417853150680611778E-12,
-1.32158118404477131188E-11,
-3.14991652796324136454E-11,
1.18891471078464383424E-11,
4.94060238822496958910E-10,
3.39623202570838634515E-9,
2.26666899049817806459E-8,
2.04891858946906374183E-7,
2.89137052083475648297E-6,
6.88975834691682398426E-5,
3.36911647825569408990E-3,
8.04490411014108831608E-1
};
double y;
if (fabs(x) < DSMALL)
return ONE;
if( x < 0 )
x = -x;
if( x <= 8.0 )
{
y = (x/2.0) - 2.0;
return( chbevl( y, A, 30, prec1 ) );
}
return( chbevl( 32.0/x - 2.0, B, 25, prec2 ) / sqrt(x) );
}
double i1e( double x, int prec1, int prec2 )
{
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*/
static double A[] =
{
2.77791411276104639959E-18,
-2.11142121435816608115E-17,
1.55363195773620046921E-16,
-1.10559694773538630805E-15,
7.60068429473540693410E-15,
-5.04218550472791168711E-14,
3.22379336594557470981E-13,
-1.98397439776494371520E-12,
1.17361862988909016308E-11,
-6.66348972350202774223E-11,
3.62559028155211703701E-10,
-1.88724975172282928790E-9,
9.38153738649577178388E-9,
-4.44505912879632808065E-8,
2.00329475355213526229E-7,
-8.56872026469545474066E-7,
3.47025130813767847674E-6,
-1.32731636560394358279E-5,
4.78156510755005422638E-5,
-1.61760815825896745588E-4,
5.12285956168575772895E-4,
-1.51357245063125314899E-3,
4.15642294431288815669E-3,
-1.05640848946261981558E-2,
2.47264490306265168283E-2,
-5.29459812080949914269E-2,
1.02643658689847095384E-1,
-1.76416518357834055153E-1,
2.52587186443633654823E-1
};
static double B[] =
{
7.51729631084210481353E-18,
4.41434832307170791151E-18,
-4.65030536848935832153E-17,
-3.20952592199342395980E-17,
2.96262899764595013876E-16,
3.30820231092092828324E-16,
-1.88035477551078244854E-15,
-3.81440307243700780478E-15,
1.04202769841288027642E-14,
4.27244001671195135429E-14,
-2.10154184277266431302E-14,
-4.08355111109219731823E-13,
-7.19855177624590851209E-13,
2.03562854414708950722E-12,
1.41258074366137813316E-11,
3.25260358301548823856E-11,
-1.89749581235054123450E-11,
-5.58974346219658380687E-10,
-3.83538038596423702205E-9,
-2.63146884688951950684E-8,
-2.51223623787020892529E-7,
-3.88256480887769039346E-6,
-1.10588938762623716291E-4,
-9.76109749136146840777E-3,
7.78576235018280120474E-1
};
double y, z;
z = fabs(x);
if( z <= 8.0 )
{
y = (z/2.0) - 2.0;
z = chbevl( y, A, 29, prec1 ) * z;
}
else
{
z = chbevl( 32.0/z - 2.0, B, 25, prec2 ) / sqrt(z);
}
if( x < 0.0 )
z = -z;
return( z );
}
int shichi(double x, double *si, double *ci ) {
double k, z, c, s, a;
short sign;
if( x < 0.0 )
{
sign = -1;
x = -x;
}
else
sign = 0;
if( x == 0.0 )
{
*si = 0.0;
*ci = -MAXNUM;
return( 0 );
}
if( x >= 8.0 )
goto chb;
z = x * x;
/* Direct power series expansion */
a = 1.0;
s = 1.0;
c = 0.0;
k = 2.0;
do
{
a *= z/k;
c += a/k;
k += 1.0;
a /= k;
s += a/k;
k += 1.0;
}
while( fabs(a/s) > MACHEP );
s *= x;
goto done;
chb:
if( x < 18.0 )
{
a = (576.0/x - 52.0)/10.0;
k = exp(x) / x;
s = k * chbevl( a, S1, 22 );
c = k * chbevl( a, C1, 23 );
goto done;
}
if( x <= 88.0 )
{
a = (6336.0/x - 212.0)/70.0;
k = exp(x) / x;
s = k * chbevl( a, S2, 23 );
c = k * chbevl( a, C2, 24 );
goto done;
}
else
{
if( sign )
*si = -MAXNUM;
else
*si = MAXNUM;
*ci = MAXNUM;
return(0);
}
done:
if( sign )
s = -s;
*si = s;
*ci = eulers_constant + log(x) + c;
return(0);
}