bessik.cpp

这是C++数值算法(第二版)的源代码,其中包含了目前一些比较常用的数值计算的算法.

#include <cmath>
#include <limits>
#include "nr.h"
using namespace std;

void NR::bessik(const DP x, const DP xnu, DP &ri, DP &rk, DP &rip, DP &rkp)
{
	const int MAXIT=10000;
	const DP EPS=numeric_limits<DP>::epsilon();
	const DP FPMIN=numeric_limits<DP>::min()/EPS;
	const DP XMIN=2.0, PI=3.141592653589793;
	DP a,a1,b,c,d,del,del1,delh,dels,e,f,fact,fact2,ff,gam1,gam2,
		gammi,gampl,h,p,pimu,q,q1,q2,qnew,ril,ril1,rimu,rip1,ripl,
		ritemp,rk1,rkmu,rkmup,rktemp,s,sum,sum1,x2,xi,xi2,xmu,xmu2;
	int i,l,nl;

	if (x <= 0.0 || xnu < 0.0) nrerror("bad arguments in bessik");
	nl=int(xnu+0.5);
	xmu=xnu-nl;
	xmu2=xmu*xmu;
	xi=1.0/x;
	xi2=2.0*xi;
	h=xnu*xi;
	if (h < FPMIN) h=FPMIN;
	b=xi2*xnu;
	d=0.0;
	c=h;
	for (i=0;i<MAXIT;i++) {
		b += xi2;
		d=1.0/(b+d);
		c=b+1.0/c;
		del=c*d;
		h=del*h;
		if (fabs(del-1.0) <= EPS) break;
	}
	if (i >= MAXIT)
		nrerror("x too large in bessik; try asymptotic expansion");
	ril=FPMIN;
	ripl=h*ril;
	ril1=ril;
	rip1=ripl;
	fact=xnu*xi;
	for (l=nl-1;l >= 0;l--) {
		ritemp=fact*ril+ripl;
		fact -= xi;
		ripl=fact*ritemp+ril;
		ril=ritemp;
	}
	f=ripl/ril;
	if (x < XMIN) {
		x2=0.5*x;
		pimu=PI*xmu;
		fact = (fabs(pimu) < EPS ? 1.0 : pimu/sin(pimu));
		d = -log(x2);
		e=xmu*d;
		fact2 = (fabs(e) < EPS ? 1.0 : sinh(e)/e);
		beschb(xmu,gam1,gam2,gampl,gammi);
		ff=fact*(gam1*cosh(e)+gam2*fact2*d);
		sum=ff;
		e=exp(e);
		p=0.5*e/gampl;
		q=0.5/(e*gammi);
		c=1.0;
		d=x2*x2;
		sum1=p;
		for (i=1;i<=MAXIT;i++) {
			ff=(i*ff+p+q)/(i*i-xmu2);
			c *= (d/i);
			p /= (i-xmu);
			q /= (i+xmu);
			del=c*ff;
			sum += del;
			del1=c*(p-i*ff);
			sum1 += del1;
			if (fabs(del) < fabs(sum)*EPS) break;
		}
		if (i > MAXIT) nrerror("bessk series failed to converge");
		rkmu=sum;
		rk1=sum1*xi2;
	} else {
		b=2.0*(1.0+x);
		d=1.0/b;
		h=delh=d;
		q1=0.0;
		q2=1.0;
		a1=0.25-xmu2;
		q=c=a1;
		a = -a1;
		s=1.0+q*delh;
		for (i=1;i<MAXIT;i++) {
			a -= 2*i;
			c = -a*c/(i+1.0);
			qnew=(q1-b*q2)/a;
			q1=q2;
			q2=qnew;
			q += c*qnew;
			b += 2.0;
			d=1.0/(b+a*d);
			delh=(b*d-1.0)*delh;
			h += delh;
			dels=q*delh;
			s += dels;
			if (fabs(dels/s) <= EPS) break;
		}
		if (i >= MAXIT) nrerror("bessik: failure to converge in cf2");
		h=a1*h;
		rkmu=sqrt(PI/(2.0*x))*exp(-x)/s;
		rk1=rkmu*(xmu+x+0.5-h)*xi;
	}
	rkmup=xmu*xi*rkmu-rk1;
	rimu=xi/(f*rkmu-rkmup);
	ri=(rimu*ril1)/ril;
	rip=(rimu*rip1)/ril;
	for (i=1;i <= nl;i++) {
		rktemp=(xmu+i)*xi2*rk1+rkmu;
		rkmu=rk1;
		rk1=rktemp;
	}
	rk=rkmu;
	rkp=xnu*xi*rkmu-rk1;
}