1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
|
#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);
}
|