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|
////////////////////////////////////////////////////////////////////////////////
// File: exponential_integral_Ei.c //
// Routine(s): //
// Exponential_Integral_Ei //
// xExponential_Integral_Ei //
////////////////////////////////////////////////////////////////////////////////
/* #include <math.h> // required for fabsl(), expl() and logl() */
/* #include <float.h> // required for LDBL_EPSILON, DBL_MAX */
/* #include <mpfr.h> */
/* #include <stdlib.h> */
/* #include <stdio.h> */
/* #include <string.h> */
/* #define ACC 1000 */
#include "SweeD.h"
// Internally Defined Routines //
double Exponential_Integral_Ei( double x );
long double xExponential_Integral_Ei( long double x );
static long double Continued_Fraction_Ei( long double x );
static long double Power_Series_Ei( long double x );
static long double Argument_Addition_Series_Ei( long double x);
static void mpfr_Continued_Fraction_Ei( mpfr_t ei, mpfr_t x);
static void mpfr_Power_Series_Ei(mpfr_t ei, mpfr_t x);
// Internally Defined Constants //
static const long double epsilon = 10.0 * LDBL_EPSILON;
////////////////////////////////////////////////////////////////////////////////
// double Exponential_Integral_Ei( double x ) //
// //
// Description: //
// The exponential integral Ei(x) is the integral with integrand //
// exp(t) / t //
// where the integral extends from -inf to x. //
// Note that there is a singularity at t = 0. Therefore for x > 0, the //
// integral is defined to be the Cauchy principal value: //
// lim { I[-inf, -eta] exp(-t) dt / t + I[eta, x] exp(-t) dt / t } //
// in which the limit is taken as eta > 0 approaches 0 and I[a,b] //
// denotes the integral from a to b. //
// //
// Arguments: //
// double x The argument of the exponential integral Ei(). //
// //
// Return Value: //
// The value of the exponential integral Ei evaluated at x. //
// If x = 0.0, then Ei is -inf and -DBL_MAX is returned. //
// //
// Example: //
// double y, x; //
// //
// ( code to initialize x ) //
// //
// y = Exponential_Integral_Ei( x ); //
////////////////////////////////////////////////////////////////////////////////
double Exponential_Integral_Ei( double x )
{
return (double) xExponential_Integral_Ei( (long double) x);
}
////////////////////////////////////////////////////////////////////////////////
// long double xExponential_Integral_Ei( long double x ) //
// //
// Description: //
// The exponential integral Ei(x) is the integral with integrand //
// exp(t) / t //
// where the integral extends from -inf to x. //
// Note that there is a singularity at t = 0. Therefore for x > 0, the //
// integral is defined to be the Cauchy principal value: //
// lim { I[-inf, -eta] exp(-t) dt / t + I[eta, x] exp(-t) dt / t } //
// in which the limit is taken as eta > 0 approaches 0 and I[a,b] //
// denotes the integral from a to b. //
// //
// Arguments: //
// long double x The argument of the exponential integral Ei(). //
// //
// Return Value: //
// The value of the exponential integral Ei evaluated at x. //
// If x = 0.0, then Ei is -inf and -DBL_MAX is returned. //
// //
// Example: //
// long double y, x; //
// //
// ( code to initialize x ) //
// //
// y = xExponential_Integral_Ei( x ); //
////////////////////////////////////////////////////////////////////////////////
long double xExponential_Integral_Ei( long double x )
{
if ( x < -5.0L ) return Continued_Fraction_Ei(x);
if ( x == 0.0L ) return -DBL_MAX;
if ( x < 6.8L ) return Power_Series_Ei(x);
if ( x < 50.0L ) return Argument_Addition_Series_Ei(x);
return Continued_Fraction_Ei(x);
}
void mpfr_Exponential_Integral_Ei( mpfr_t ei, mpfr_t x)
{
if( mpfr_cmp_d(x, -5.0) < 0)
mpfr_Continued_Fraction_Ei(ei, x );
else if( mpfr_cmp_d(x, 6.8) < 0)
mpfr_Power_Series_Ei(ei, x);
else if( mpfr_cmp_d(x, 50.0) < 0)
mpfr_eint(ei, x, GMP_RNDU);
else if (mpfr_cmp_d(x, 50.0) >= 0)
mpfr_Continued_Fraction_Ei(ei, x);
else
assert(0);
}
static void mpfr_Continued_Fraction_Ei( mpfr_t t, mpfr_t x )
{
mpfr_t epsilon1;
mpfr_t Am1, A0, Bm1, B0, a, b, Ap1, Bp1, j;
mpfr_t tmp, tmp1, tmp2, tmp3, tmp4;
mpfr_inits2(ACC, epsilon1, Am1, A0, Bm1, B0, a, b, Ap1, Bp1, tmp, tmp1, tmp2, tmp3, tmp4, j, (mpfr_ptr) 0);
mpfr_set_ld(epsilon1, LDBL_EPSILON, GMP_RNDU);
/* mpfr_set_ld(epsilon1, 1.0L, GMP_RNDU); */
/* mpfr_mul_d(epsilon1, epsilon1, 0.5, GMP_RNDU); */
/* mpfr_exp_t e = mpfr_get_emin(); */
/* mpfr_set_exp(epsilon1, e); */
/* mpfr_out_str(NULL, 10, 20, epsilon1, GMP_RNDU); */
/* putchar('\n'); */
mpfr_set_ld(Am1, 1.0L, GMP_RNDU);
mpfr_set_ld(A0, 0.0L, GMP_RNDU);
mpfr_set_ld(Bm1, 0.0L, GMP_RNDU);
mpfr_set_ld(B0, 1.0L, GMP_RNDU);
mpfr_set_ld(a, 0.0L, GMP_RNDU); //!
mpfr_set_ld(b, 0.0L, GMP_RNDU);
mpfr_set_ld(Ap1, 0.0L, GMP_RNDU);
mpfr_set_ld(Bp1, 0.0L, GMP_RNDU);
/* printf("*x: "); */
/* mpfr_out_str(NULL, 10, 10, x, GMP_RNDU); */
mpfr_exp(a, x, GMP_RNDU);
mpfr_set(b, x, GMP_RNDU);
mpfr_mul_si(b, b, -1, GMP_RNDU);
mpfr_add_d(b, b, 1.0, GMP_RNDU);
mpfr_set(Ap1, b, GMP_RNDU);
mpfr_mul(Ap1, Ap1, A0, GMP_RNDU);
mpfr_set(tmp, a, GMP_RNDU);
mpfr_mul(tmp, tmp, Am1, GMP_RNDU);
mpfr_add(Ap1, Ap1, tmp, GMP_RNDU);
mpfr_set(Bp1, b, GMP_RNDU);
mpfr_mul(Bp1, Bp1, B0, GMP_RNDU);
mpfr_set(tmp, a, GMP_RNDU);
mpfr_mul(tmp, tmp, Bm1, GMP_RNDU);
mpfr_add(Bp1, Bp1, tmp, GMP_RNDU);
mpfr_set_ld(j, 1.0L, GMP_RNDU);
mpfr_set_ld(a, 1.0L, GMP_RNDU);
/* a = 1.0L; */
mpfr_set(tmp, Ap1, GMP_RNDU);
mpfr_mul(tmp, tmp, B0, GMP_RNDU);
mpfr_set(tmp1, A0, GMP_RNDU);
mpfr_mul(tmp1, tmp1, Bp1, GMP_RNDU);
mpfr_sub(tmp, tmp, tmp1, GMP_RNDU);
mpfr_abs(tmp, tmp, GMP_RNDU);
mpfr_abs(tmp1, tmp1, GMP_RNDU);
mpfr_mul(tmp1, tmp1, epsilon1, GMP_RNDU);
while ( mpfr_cmp(tmp, tmp1) > 0)
{
/* printf("** "); */
/* mpfr_out_str(NULL, 10, 10, tmp, GMP_RNDU); */
/* printf("\t"); */
/* mpfr_out_str(NULL, 10, 10, tmp1, GMP_RNDU); */
/* printf("\n"); */
mpfr_set(tmp2, Bp1, GMP_RNDU);
mpfr_abs(tmp2, tmp2, GMP_RNDU);
if( mpfr_cmp_ld(tmp2, 1.0L) > 0){
mpfr_set(tmp3, A0, GMP_RNDU);
mpfr_div(tmp3, tmp3, Bp1, GMP_RNDU);
mpfr_set(Am1, tmp3, GMP_RNDU);
mpfr_set(tmp3, Ap1, GMP_RNDU);
mpfr_div(tmp3, tmp3, Bp1, GMP_RNDU);
mpfr_set(A0, tmp3, GMP_RNDU);
mpfr_set(tmp3, B0, GMP_RNDU);
mpfr_div(tmp3, tmp3, Bp1, GMP_RNDU);
mpfr_set(Bm1, tmp3, GMP_RNDU);
mpfr_set_ld(B0, 1.0L, GMP_RNDU);
}
else {
mpfr_set(Am1, A0, GMP_RNDU);
mpfr_set(A0, Ap1, GMP_RNDU);
mpfr_set(Bm1, B0, GMP_RNDU);
mpfr_set(B0, Bp1, GMP_RNDU);
}
mpfr_set(tmp3, j, GMP_RNDU);
mpfr_mul(tmp3, tmp3, tmp3, GMP_RNDU);
mpfr_mul_si(tmp3, tmp3, -1, GMP_RNDU);
mpfr_set(a, tmp3, GMP_RNDU);
mpfr_add_d(b, b, 2.0, GMP_RNDU);
mpfr_set(tmp3, b, GMP_RNDU);
mpfr_mul(tmp3, tmp3, A0, GMP_RNDU);
mpfr_set(Ap1, tmp3, GMP_RNDU);
mpfr_set(tmp3, a, GMP_RNDU);
mpfr_mul(tmp3, tmp3, Am1, GMP_RNDU);
mpfr_add(Ap1, Ap1, tmp3, GMP_RNDU);
mpfr_set(tmp3, b, GMP_RNDU);
mpfr_mul(tmp3, tmp3, B0, GMP_RNDU);
mpfr_set(Bp1, tmp3, GMP_RNDU);
mpfr_set(tmp3, a, GMP_RNDU);
mpfr_mul(tmp3, tmp3, Bm1, GMP_RNDU);
mpfr_add(Bp1, Bp1, tmp3, GMP_RNDU);
mpfr_add_d(j, j, 1.0, GMP_RNDU);
mpfr_set(tmp, Ap1, GMP_RNDU);
mpfr_mul(tmp, tmp, B0, GMP_RNDU);
mpfr_set(tmp1, A0, GMP_RNDU);
mpfr_mul(tmp1, tmp1, Bp1, GMP_RNDU);
mpfr_sub(tmp, tmp, tmp1, GMP_RNDU);
mpfr_abs(tmp, tmp, GMP_RNDU);
mpfr_abs(tmp1, tmp1, GMP_RNDU);
mpfr_mul(tmp1, tmp1, epsilon1, GMP_RNDU);
}
mpfr_set(t, Ap1, GMP_RNDU);
mpfr_mul_si(t, t, -1, GMP_RNDU);
mpfr_div(t, t, Bp1, GMP_RNDU);
mpfr_clears(epsilon1, Am1, A0, Bm1, B0, a, b, Ap1, Bp1, tmp, tmp1, tmp2, tmp3, tmp4, j, (mpfr_ptr) 0);
}
////////////////////////////////////////////////////////////////////////////////
// static long double Continued_Fraction_Ei( long double x ) //
// //
// Description: //
// For x < -5 or x > 50, the continued fraction representation of Ei //
// converges fairly rapidly. //
// //
// The continued fraction expansion of Ei(x) is: //
// Ei(x) = -exp(x) { 1/(-x+1-) 1/(-x+3-) 4/(-x+5-) 9/(-x+7-) ... }. //
// //
// //
// Arguments: //
// long double x //
// The argument of the exponential integral Ei(). //
// //
// Return Value: //
// The value of the exponential integral Ei evaluated at x. //
////////////////////////////////////////////////////////////////////////////////
static long double Continued_Fraction_Ei( long double x )
{
long double Am1 = 1.0L;
long double A0 = 0.0L;
long double Bm1 = 0.0L;
long double B0 = 1.0L;
long double a = expl(x);
long double b = -x + 1.0L;
long double Ap1 = b * A0 + a * Am1;
long double Bp1 = b * B0 + a * Bm1;
int j = 1;
a = 1.0L;
while ( fabsl(Ap1 * B0 - A0 * Bp1) > epsilon * fabsl(A0 * Bp1) ) {
printf("ORIG %Le\n", fabsl(Ap1 * B0 - A0 * Bp1));
if ( fabsl(Bp1) > 1.0L) {
Am1 = A0 / Bp1;
A0 = Ap1 / Bp1;
Bm1 = B0 / Bp1;
B0 = 1.0L;
} else {
Am1 = A0;
A0 = Ap1;
Bm1 = B0;
B0 = Bp1;
}
a = -j * j;
b += 2.0L;
Ap1 = b * A0 + a * Am1;
Bp1 = b * B0 + a * Bm1;
j += 1;
}
return (-Ap1 / Bp1);
}
////////////////////////////////////////////////////////////////////////////////
// static long double Power_Series_Ei( long double x ) //
// //
// Description: //
// For -5 < x < 6.8, the power series representation for //
// (Ei(x) - gamma - ln|x|)/exp(x) is used, where gamma is Euler's gamma //
// constant. //
// Note that for x = 0.0, Ei is -inf. In which case -DBL_MAX is //
// returned. //
// //
// The power series expansion of (Ei(x) - gamma - ln|x|) / exp(x) is //
// - Sum(1 + 1/2 + ... + 1/j) (-x)^j / j!, where the Sum extends //
// from j = 1 to inf. //
// //
// Arguments: //
// long double x //
// The argument of the exponential integral Ei(). //
// //
// Return Value: //
// The value of the exponential integral Ei evaluated at x. //
////////////////////////////////////////////////////////////////////////////////
static long double Power_Series_Ei( long double x )
{
long double xn = -x;
long double Sn = -x;
long double Sm1 = 0.0L;
long double hsum = 1.0L;
long double g = 0.5772156649015328606065121L;
long double y = 1.0L;
long double factorial = 1.0L;
if ( x == 0.0L ) return (long double) -DBL_MAX;
while ( fabsl(Sn - Sm1) > epsilon * fabsl(Sm1) ) {
Sm1 = Sn;
y += 1.0L;
xn *= (-x);
factorial *= y;
hsum += (1.0 / y);
Sn += hsum * xn / factorial;
}
return (g + logl(fabsl(x)) - expl(x) * Sn);
}
static void mpfr_Power_Series_Ei(mpfr_t t, mpfr_t x)
{
mpfr_t epsilon1;
mpfr_t xn, Sn, Sm1, hsum, g, y, factorial;
mpfr_t tmp, tmp1, tmp2, tmp3, tmp4;
mpfr_inits2(ACC, epsilon1, xn, Sn, Sm1, hsum, g, y, factorial, tmp, tmp1, tmp2, tmp3, tmp4, (mpfr_ptr) 0);
mpfr_set_ld(epsilon1, LDBL_EPSILON, GMP_RNDU);
/* long double xn = -x; */
mpfr_set(xn, x, GMP_RNDU);
mpfr_mul_si(xn, xn, -1, GMP_RNDU);
/* long double Sn = -x; */
mpfr_set(Sn, xn, GMP_RNDU);
/* long double Sm1 = 0.0L; */
mpfr_set_ld(Sm1, 0.0L, GMP_RNDU);
/* long double hsum = 1.0L; */
mpfr_set_ld(hsum, 1.0L, GMP_RNDU);
/*long double g = 0.5772156649015328606065121L; */
//mpfr_set_ld(g, 0.5772156649015328606065121L, GMP_RNDU);
mpfr_const_euler(g, GMP_RNDU);
/*long double y = 1.0L;*/
mpfr_set_ld(y, 1.0L, GMP_RNDU);
/* long double factorial = 1.0L; */
mpfr_set_ui(factorial, 1, GMP_RNDU);
/* if ( x == 0.0L ) return (long double) -DBL_MAX; */
assert( mpfr_cmp_ld(x, 0.0L) != 0);
mpfr_set(tmp, Sn, GMP_RNDU);
mpfr_sub(tmp, tmp, Sm1, GMP_RNDU);
mpfr_abs(tmp, tmp, GMP_RNDU);
mpfr_set(tmp1, Sm1, GMP_RNDU);
mpfr_abs(tmp1, tmp1, GMP_RNDU);
mpfr_mul(tmp1, tmp1, epsilon1, GMP_RNDU);
while ( (mpfr_cmp(tmp, tmp1) > 0) ) {
/*Sm1 = Sn;*/
mpfr_set(Sm1, Sn, GMP_RNDU);
/* y += 1.0L;*/
mpfr_add_ui(y, y, 1, GMP_RNDU);
/* xn *= (-x); */
mpfr_set(tmp, x, GMP_RNDU);
mpfr_mul_si(tmp, tmp, -1, GMP_RNDU);
mpfr_mul(xn, xn, tmp, GMP_RNDU);
/* factorial *= y; */
mpfr_mul(factorial, factorial, y, GMP_RNDU);
/* hsum += (1.0 / y); */
mpfr_set_ld(tmp, 1.0L, GMP_RNDU);
mpfr_div(tmp, tmp, y, GMP_RNDU);
mpfr_add(hsum, hsum, tmp, GMP_RNDU);
/*Sn += hsum * xn / factorial;*/
mpfr_set(tmp, hsum, GMP_RNDU);
mpfr_mul(tmp, tmp, xn, GMP_RNDU);
mpfr_div(tmp, tmp, factorial, GMP_RNDU);
mpfr_add(Sn, Sn, tmp, GMP_RNDU);
/* update the values */
mpfr_set(tmp, Sn, GMP_RNDU);
mpfr_sub(tmp, tmp, Sm1, GMP_RNDU);
mpfr_abs(tmp, tmp, GMP_RNDU);
mpfr_set(tmp1, Sm1, GMP_RNDU);
mpfr_abs(tmp1, tmp1, GMP_RNDU);
mpfr_mul(tmp1, tmp1, epsilon1, GMP_RNDU);
}
/*return (g + logl(fabsl(x)) - expl(x) * Sn);*/
mpfr_set(tmp, x, GMP_RNDU);
mpfr_abs(tmp, tmp, GMP_RNDU);
mpfr_log(tmp, tmp, GMP_RNDU);
mpfr_add(tmp, tmp, g, GMP_RNDU);
mpfr_set(t, tmp, GMP_RNDU);
mpfr_exp(tmp, x, GMP_RNDU);
mpfr_mul(tmp, tmp, Sn, GMP_RNDU);
mpfr_sub(t, t, tmp, GMP_RNDU);
mpfr_clears(epsilon1, xn, Sn, Sm1, hsum, g, y, factorial, tmp, tmp1, tmp2, tmp3, tmp4, (mpfr_ptr) 0);
}
////////////////////////////////////////////////////////////////////////////////
// static long double Argument_Addition_Series_Ei(long double x) //
// //
// Description: //
// For 6.8 < x < 50.0, the argument addition series is used to calculate //
// Ei. //
// //
// The argument addition series for Ei(x) is: //
// Ei(x+dx) = Ei(x) + exp(x) Sum j! [exp(j) expj(-dx) - 1] / x^(j+1), //
// where the Sum extends from j = 0 to inf, |x| > |dx| and expj(y) is //
// the exponential polynomial expj(y) = Sum y^k / k!, the Sum extending //
// from k = 0 to k = j. //
// //
// Arguments: //
// long double x //
// The argument of the exponential integral Ei(). //
// //
// Return Value: //
// The value of the exponential integral Ei evaluated at x. //
////////////////////////////////////////////////////////////////////////////////
static long double Argument_Addition_Series_Ei(long double x)
{
static long double ei[] = {
1.915047433355013959531e2L, 4.403798995348382689974e2L,
1.037878290717089587658e3L, 2.492228976241877759138e3L,
6.071406374098611507965e3L, 1.495953266639752885229e4L,
3.719768849068903560439e4L, 9.319251363396537129882e4L,
2.349558524907683035782e5L, 5.955609986708370018502e5L,
1.516637894042516884433e6L, 3.877904330597443502996e6L,
9.950907251046844760026e6L, 2.561565266405658882048e7L,
6.612718635548492136250e7L, 1.711446713003636684975e8L,
4.439663698302712208698e8L, 1.154115391849182948287e9L,
3.005950906525548689841e9L, 7.842940991898186370453e9L,
2.049649711988081236484e10L, 5.364511859231469415605e10L,
1.405991957584069047340e11L, 3.689732094072741970640e11L,
9.694555759683939661662e11L, 2.550043566357786926147e12L,
6.714640184076497558707e12L, 1.769803724411626854310e13L,
4.669055014466159544500e13L, 1.232852079912097685431e14L,
3.257988998672263996790e14L, 8.616388199965786544948e14L,
2.280446200301902595341e15L, 6.039718263611241578359e15L,
1.600664914324504111070e16L, 4.244796092136850759368e16L,
1.126348290166966760275e17L, 2.990444718632336675058e17L,
7.943916035704453771510e17L, 2.111342388647824195000e18L,
5.614329680810343111535e18L, 1.493630213112993142255e19L,
3.975442747903744836007e19L, 1.058563689713169096306e20L
};
int k = (int) (x + 0.5);
int j = 0;
long double xx = (long double) k;
long double dx = x - xx;
long double xxj = xx;
long double edx = expl(dx);
long double Sm = 1.0L;
long double Sn = (edx - 1.0L) / xxj;
long double term = DBL_MAX;
long double factorial = 1.0L;
long double dxj = 1.0L;
while (fabsl(term) > epsilon * fabsl(Sn) ) {
j++;
factorial *= (long double) j;
xxj *= xx;
dxj *= (-dx);
Sm += (dxj / factorial);
term = ( factorial * (edx * Sm - 1.0L) ) / xxj;
Sn += term;
}
return ei[k-7] + Sn * expl(xx);
}
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