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buchmann |
1.1 |
#include <math.h>
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#include <TMath.h>
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//************** EXPONENTIAL INTEGRALS En *****
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// define: E_n(x) = \int_1^infty{exp(-xt)/t^n}dt, x>0, n=0,1,...
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double expint(int n, double x) {
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// based on Numerical Recipes in C
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const double euler = 0.57721566; // Euler's constant, gamma
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const int maxit = 100; // max. no. of iterations allowed
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const double fpmin = 1.0e-30; // close to smallest floating-point number
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const double eps = 6.0e-8; // relative error, or absolute error near
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// the zero of Ei at x=0.3725
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int i, ii, nm1;
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double a,b,c,d,del,fact,h,psi,ans;
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nm1=n-1;
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if(n<0 || x<0 || (x==0 && (n==0 || n==1))) {
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cout << "Bad argument for expint(n,x)" << endl; return -1;
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}
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else {
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if(n==0) ans=exp(-x)/x;
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else {
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if(x==0) ans=1.0/nm1;
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else {
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if(x>1) {
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b=x+n;
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c=1.0/fpmin;
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d=1.0/b;
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h=d;
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for(i=1; i<maxit; i++) {
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a = -i*(nm1+i);
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b += 2.0;
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d=1.0/(a*d+b);
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c=b+a/c;
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del=c*d;
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h *= del;
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if(fabs(del-1.0)<eps) {
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ans=h*exp(-x);
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return ans;
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}
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}
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cout << "***continued fraction failed in expint(n,x)!!!" << endl;
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return -1;
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} else {
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ans = (nm1!=0 ? 1.0/nm1 : -log(x)-euler);
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fact=1;
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for(i=1; i<=maxit; i++) {
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fact *= -x/i;
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if(i!=nm1) del = -fact/(i-nm1);
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else {
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psi = -euler;
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for(ii=1; ii<=nm1; ii++) psi += 1.0/ii;
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del = fact*(-log(x)+psi);
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}
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ans += del;
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if(fabs(del)<fabs(ans)*eps) return ans;
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}
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cout << "***series failed in expint!!!" << endl;
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return -1;
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}
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}
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}
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}
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return ans;
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}
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//*********************************************
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//************** EXPONENTIAL INTEGRAL Ei ******
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// define: ei(x) = -\int_{-x}^{\infty}{exp(-t)/t}dt, for x>0
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// power series: ei(x) = eulerconst + ln(x) + x/(1*1!) + x^2/(2*2!) + ...
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double ei(double x)
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{ // taken from Numerical Recipes in C
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const double euler = 0.57721566; // Euler's constant, gamma
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const int maxit = 100; // max. no. of iterations allowed
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const double fpmin = 1.0e-40; // close to smallest floating-point number
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const double eps = 1.0e-30; // relative error, or absolute error near
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// the zero of Ei at x=0.3725
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// I actually changed fpmin and eps into smaller values than in NR
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int k;
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double fact, prev, sum, term;
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// special case
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if(x < 0) return -expint(1,-x);
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if(x == 0.0) { cout << "Bad argument for ei(x)" << endl; return -1; }
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if(x < fpmin) return log(x)+euler;
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if(x <= -log(eps)) {
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sum = 0;
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fact = 1;
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for(k=1; k<=maxit; k++) {
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fact *= x/k;
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term = fact/k;
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sum += term;
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if(term < eps*sum) break;
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}
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if(k>maxit) { cout << "Series failed in ei(x)" << endl; return -1; }
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return sum+log(x)+euler;
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} else {
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sum = 0;
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term = 1;
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for(k=1; k<=maxit; k++) {
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prev = term;
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term *= k/x;
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if(term<eps) break;
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if(term<prev) sum+=term;
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else {
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sum -= prev;
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break;
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}
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}
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return exp(x)*(1.0+sum)/x;
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}
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}
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//*********************************************
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