ref: 9ced712a2ad70557a8ceb89ff3b969b92c85f68c
dir: /sys/src/cmd/map/libmap/elco2.c/
#include <u.h>
#include <libc.h>
#include "map.h"
/* elliptic integral routine, R.Bulirsch,
* Numerische Mathematik 7(1965) 78-90
* calculate integral from 0 to x+iy of
* (a+b*t^2)/((1+t^2)*sqrt((1+t^2)*(1+kc^2*t^2)))
* yields about D valid figures, where CC=10e-D
* for a*b>=0, except at branchpoints x=0,y=+-i,+-i/kc;
* there the accuracy may be reduced.
* fails for kc=0 or x<0
* return(1) for success, return(0) for fail
*
* special case a=b=1 is equivalent to
* standard elliptic integral of first kind
* from 0 to atan(x+iy) of
* 1/sqrt(1-k^2*(sin(t))^2) where k^2=1-kc^2
*/
#define ROOTINF 10.e18
#define CC 1.e-6
int
elco2(double x, double y, double kc, double a, double b, double *u, double *v)
{
double c,d,dn1,dn2,e,e1,e2,f,f1,f2,h,k,m,m1,m2,sy;
double d1[13],d2[13];
int i,l;
if(kc==0||x<0)
return(0);
sy = y>0? 1: y==0? 0: -1;
y = fabs(y);
csq(x,y,&c,&e2);
d = kc*kc;
k = 1-d;
e1 = 1+c;
cdiv2(1+d*c,d*e2,e1,e2,&f1,&f2);
f2 = -k*x*y*2/f2;
csqr(f1,f2,&dn1,&dn2);
if(f1<0) {
f1 = dn1;
dn1 = -dn2;
dn2 = -f1;
}
if(k<0) {
dn1 = fabs(dn1);
dn2 = fabs(dn2);
}
c = 1+dn1;
cmul(e1,e2,c,dn2,&f1,&f2);
cdiv(x,y,f1,f2,&d1[0],&d2[0]);
h = a-b;
d = f = m = 1;
kc = fabs(kc);
e = a;
a += b;
l = 4;
for(i=1;;i++) {
m1 = (kc+m)/2;
m2 = m1*m1;
k *= f/(m2*4);
b += e*kc;
e = a;
cdiv2(kc+m*dn1,m*dn2,c,dn2,&f1,&f2);
csqr(f1/m1,k*dn2*2/f2,&dn1,&dn2);
cmul(dn1,dn2,x,y,&f1,&f2);
x = fabs(f1);
y = fabs(f2);
a += b/m1;
l *= 2;
c = 1 +dn1;
d *= k/2;
cmul(x,y,x,y,&e1,&e2);
k *= k;
cmul(c,dn2,1+e1*m2,e2*m2,&f1,&f2);
cdiv(d*x,d*y,f1,f2,&d1[i],&d2[i]);
if(k<=CC)
break;
kc = sqrt(m*kc);
f = m2;
m = m1;
}
f1 = f2 = 0;
for(;i>=0;i--) {
f1 += d1[i];
f2 += d2[i];
}
x *= m1;
y *= m1;
cdiv2(1-y,x,1+y,-x,&e1,&e2);
e2 = x*2/e2;
d = a/(m1*l);
*u = atan2(e2,e1);
if(*u<0)
*u += PI;
a = d*sy/2;
*u = d*(*u) + f1*h;
*v = (-1-log(e1*e1+e2*e2))*a + f2*h*sy + a;
return(1);
}
void
cdiv2(double c1, double c2, double d1, double d2, double *e1, double *e2)
{
double t;
if(fabs(d2)>fabs(d1)) {
t = d1, d1 = d2, d2 = t;
t = c1, c1 = c2, c2 = t;
}
if(fabs(d1)>ROOTINF)
*e2 = ROOTINF*ROOTINF;
else
*e2 = d1*d1 + d2*d2;
t = d2/d1;
*e1 = (c1+t*c2)/(d1+t*d2); /* (c1*d1+c2*d2)/(d1*d1+d2*d2) */
}
/* complex square root of |x|+iy */
void
csqr(double c1, double c2, double *e1, double *e2)
{
double r2;
r2 = c1*c1 + c2*c2;
if(r2<=0) {
*e1 = *e2 = 0;
return;
}
*e1 = sqrt((sqrt(r2) + fabs(c1))/2);
*e2 = c2/(*e1*2);
}