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
elco2(double x,double y,double kc,double a,double b,double * u,double * v)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
cdiv2(double c1,double c2,double d1,double d2,double * e1,double * e2)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
csqr(double c1,double c2,double * e1,double * e2)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 }
133