add numerical recipes library

lib/nr/cpp/recipes/frenel.cpp

@@ -0,0 +1,73 @@
+#include <cmath>
+#include <complex>
+#include <limits>
+#include "nr.h"
+using namespace std;
+
+void NR::frenel(const DP x, complex<DP> &cs)
+{
+	const int MAXIT=100;
+	const DP EPS=numeric_limits<DP>::epsilon();
+	const DP FPMIN=numeric_limits<DP>::min();
+	const DP BIG=numeric_limits<DP>::max()*EPS;
+	const DP PI=3.141592653589793238, PIBY2=(PI/2.0), XMIN=1.5;
+	bool odd;
+	int k,n;
+	DP a,ax,fact,pix2,sign,sum,sumc,sums,term,test;
+	complex<DP> b,cc,d,h,del;
+
+	ax=fabs(x);
+	if (ax < sqrt(FPMIN)) {
+		cs=ax;
+	} else if (ax <= XMIN) {
+		sum=sums=0.0;
+		sumc=ax;
+		sign=1.0;
+		fact=PIBY2*ax*ax;
+		odd=true;
+		term=ax;
+		n=3;
+		for (k=1;k<=MAXIT;k++) {
+			term *= fact/k;
+			sum += sign*term/n;
+			test=fabs(sum)*EPS;
+			if (odd) {
+				sign = -sign;
+				sums=sum;
+				sum=sumc;
+			} else {
+				sumc=sum;
+				sum=sums;
+			}
+			if (term < test) break;
+			odd=!odd;
+			n += 2;
+		}
+		if (k > MAXIT) nrerror("series failed in frenel");
+		cs=complex<DP>(sumc,sums);
+	} else {
+		pix2=PI*ax*ax;
+		b=complex<DP>(1.0,-pix2);
+		cc=BIG;
+		d=h=1.0/b;
+		n = -1;
+		for (k=2;k<=MAXIT;k++) {
+			n += 2;
+			a = -n*(n+1);
+			b += 4.0;
+			d=1.0/(a*d+b);
+			cc=b+a/cc;
+			del=cc*d;
+			h *= del;
+			if (fabs(real(del)-1.0)+fabs(imag(del)) <= EPS) break;
+		}
+		if (k > MAXIT) nrerror("cf failed in frenel");
+		h *= complex<DP>(ax,-ax);
+		cs=complex<DP>(0.5,0.5)
+			*(1.0-complex<DP>(cos(0.5*pix2),sin(0.5*pix2))*h);
+	}
+	if (x < 0.0) {
+		cs = -cs;
+	}
+	return;
+}