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/* expn.c
*
* Exponential integral En
*
*
*
* SYNOPSIS:
*
* int n;
* double x, y, expn();
*
* y = expn( n, x );
*
*
*
* DESCRIPTION:
*
* Evaluates the exponential integral
*
* inf.
* -
* | | -xt
* | e
* E (x) = | ---- dt.
* n | n
* | | t
* -
* 1
*
*
* Both n and x must be nonnegative.
*
* The routine employs either a power series, a continued
* fraction, or an asymptotic formula depending on the
* relative values of n and x.
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0, 30 5000 2.0e-16 4.6e-17
* IEEE 0, 30 10000 1.7e-15 3.6e-16
*
*/
/* expn.c */
/* Cephes Math Library Release 2.8: June, 2000
Copyright 1985, 2000 by Stephen L. Moshier */
#include <math.h>
#ifdef ANSIPROT
extern double pow ( double, double );
extern double gamma ( double );
extern double log ( double );
extern double exp ( double );
extern double fabs ( double );
#else
double pow(), gamma(), log(), exp(), fabs();
#endif
#define EUL 0.57721566490153286060
#define BIG 1.44115188075855872E+17
extern double MAXNUM, MACHEP, MAXLOG;
double expn( n, x )
int n;
double x;
{
double ans, r, t, yk, xk;
double pk, pkm1, pkm2, qk, qkm1, qkm2;
double psi, z;
int i, k;
static double big = BIG;
if( n < 0 )
goto domerr;
if( x < 0 )
{
domerr: mtherr( "expn", DOMAIN );
return( MAXNUM );
}
if( x > MAXLOG )
return( 0.0 );
if( x == 0.0 )
{
if( n < 2 )
{
mtherr( "expn", SING );
return( MAXNUM );
}
else
return( 1.0/(n-1.0) );
}
if( n == 0 )
return( exp(-x)/x );
/* expn.c */
/* Expansion for large n */
if( n > 5000 )
{
xk = x + n;
yk = 1.0 / (xk * xk);
t = n;
ans = yk * t * (6.0 * x * x - 8.0 * t * x + t * t);
ans = yk * (ans + t * (t - 2.0 * x));
ans = yk * (ans + t);
ans = (ans + 1.0) * exp( -x ) / xk;
goto done;
}
if( x > 1.0 )
goto cfrac;
/* expn.c */
/* Power series expansion */
psi = -EUL - log(x);
for( i=1; i<n; i++ )
psi = psi + 1.0/i;
z = -x;
xk = 0.0;
yk = 1.0;
pk = 1.0 - n;
if( n == 1 )
ans = 0.0;
else
ans = 1.0/pk;
do
{
xk += 1.0;
yk *= z/xk;
pk += 1.0;
if( pk != 0.0 )
{
ans += yk/pk;
}
if( ans != 0.0 )
t = fabs(yk/ans);
else
t = 1.0;
}
while( t > MACHEP );
k = xk;
t = n;
r = n - 1;
ans = (pow(z, r) * psi / gamma(t)) - ans;
goto done;
/* expn.c */
/* continued fraction */
cfrac:
k = 1;
pkm2 = 1.0;
qkm2 = x;
pkm1 = 1.0;
qkm1 = x + n;
ans = pkm1/qkm1;
do
{
k += 1;
if( k & 1 )
{
yk = 1.0;
xk = n + (k-1)/2;
}
else
{
yk = x;
xk = k/2;
}
pk = pkm1 * yk + pkm2 * xk;
qk = qkm1 * yk + qkm2 * xk;
if( qk != 0 )
{
r = pk/qk;
t = fabs( (ans - r)/r );
ans = r;
}
else
t = 1.0;
pkm2 = pkm1;
pkm1 = pk;
qkm2 = qkm1;
qkm1 = qk;
if( fabs(pk) > big )
{
pkm2 /= big;
pkm1 /= big;
qkm2 /= big;
qkm1 /= big;
}
}
while( t > MACHEP );
ans *= exp( -x );
done:
return( ans );
}
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