1 #include <u.h>
2 #include <libc.h>
3 #include "map.h"
4 
5 /* elliptic integral routine, R.Bulirsch,
6  *	Numerische Mathematik 7(1965) 78-90
7  *	calculate integral from 0 to x+iy of
8  *	(a+b*t^2)/((1+t^2)*sqrt((1+t^2)*(1+kc^2*t^2)))
9  *	yields about D valid figures, where CC=10e-D
10  *	for a*b>=0, except at branchpoints x=0,y=+-i,+-i/kc;
11  *	there the accuracy may be reduced.
12  *	fails for kc=0 or x<0
13  *	return(1) for success, return(0) for fail
14  *
15  *	special case a=b=1 is equivalent to
16  *	standard elliptic integral of first kind
17  *	from 0 to atan(x+iy) of
18  *	1/sqrt(1-k^2*(sin(t))^2) where k^2=1-kc^2
19 */
20 
21 #define ROOTINF 10.e18
22 #define CC 1.e-6
23 
24 int
25 elco2(double x, double y, double kc, double a, double b, double *u, double *v)
26 {
27 	double c,d,dn1,dn2,e,e1,e2,f,f1,f2,h,k,m,m1,m2,sy;
28 	double d1[13],d2[13];
29 	int i,l;
30 	if(kc==0||x<0)
31 		return(0);
32 	sy = y>0? 1: y==0? 0: -1;
33 	y = fabs(y);
34 	csq(x,y,&c,&e2);
35 	d = kc*kc;
36 	k = 1-d;
37 	e1 = 1+c;
38 	cdiv2(1+d*c,d*e2,e1,e2,&f1,&f2);
39 	f2 = -k*x*y*2/f2;
40 	csqr(f1,f2,&dn1,&dn2);
41 	if(f1<0) {
42 		f1 = dn1;
43 		dn1 = -dn2;
44 		dn2 = -f1;
45 	}
46 	if(k<0) {
47 		dn1 = fabs(dn1);
48 		dn2 = fabs(dn2);
49 	}
50 	c = 1+dn1;
51 	cmul(e1,e2,c,dn2,&f1,&f2);
52 	cdiv(x,y,f1,f2,&d1[0],&d2[0]);
53 	h = a-b;
54 	d = f = m = 1;
55 	kc = fabs(kc);
56 	e = a;
57 	a += b;
58 	l = 4;
59 	for(i=1;;i++) {
60 		m1 = (kc+m)/2;
61 		m2 = m1*m1;
62 		k *= f/(m2*4);
63 		b += e*kc;
64 		e = a;
65 		cdiv2(kc+m*dn1,m*dn2,c,dn2,&f1,&f2);
66 		csqr(f1/m1,k*dn2*2/f2,&dn1,&dn2);
67 		cmul(dn1,dn2,x,y,&f1,&f2);
68 		x = fabs(f1);
69 		y = fabs(f2);
70 		a += b/m1;
71 		l *= 2;
72 		c = 1 +dn1;
73 		d *= k/2;
74 		cmul(x,y,x,y,&e1,&e2);
75 		k *= k;
76 
77 		cmul(c,dn2,1+e1*m2,e2*m2,&f1,&f2);
78 		cdiv(d*x,d*y,f1,f2,&d1[i],&d2[i]);
79 		if(k<=CC) 
80 			break;
81 		kc = sqrt(m*kc);
82 		f = m2;
83 		m = m1;
84 	}
85 	f1 = f2 = 0;
86 	for(;i>=0;i--) {
87 		f1 += d1[i];
88 		f2 += d2[i];
89 	}
90 	x *= m1;
91 	y *= m1;
92 	cdiv2(1-y,x,1+y,-x,&e1,&e2);
93 	e2 = x*2/e2;
94 	d = a/(m1*l);
95 	*u = atan2(e2,e1);
96 	if(*u<0)
97 		*u += PI;
98 	a = d*sy/2;
99 	*u = d*(*u) + f1*h;
100 	*v = (-1-log(e1*e1+e2*e2))*a + f2*h*sy + a;
101 	return(1);
102 }
103 
104 void
105 cdiv2(double c1, double c2, double d1, double d2, double *e1, double *e2)
106 {
107 	double t;
108 	if(fabs(d2)>fabs(d1)) {
109 		t = d1, d1 = d2, d2 = t;
110 		t = c1, c1 = c2, c2 = t;
111 	}
112 	if(fabs(d1)>ROOTINF)
113 		*e2 = ROOTINF*ROOTINF;
114 	else
115 		*e2 = d1*d1 + d2*d2;
116 	t = d2/d1;
117 	*e1 = (c1+t*c2)/(d1+t*d2); /* (c1*d1+c2*d2)/(d1*d1+d2*d2) */
118 }
119 
120 /* complex square root of |x|+iy */
121 void
122 csqr(double c1, double c2, double *e1, double *e2)
123 {
124 	double r2;
125 	r2 = c1*c1 + c2*c2;
126 	if(r2<=0) {
127 		*e1 = *e2 = 0;
128 		return;
129 	}
130 	*e1 = sqrt((sqrt(r2) + fabs(c1))/2);
131 	*e2 = c2/(*e1*2);
132 }