/usr/share/doc/geographiclib/html/ellint.mac is in geographiclib-tools 1.8-2.
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Written by Charles Karney <charles@karney.com>
http://geographiclib.sourceforge.net/
$Id: ellint.mac 6720 2009-10-17 23:13:57Z ckarney $
*/
/* Implementation of methods given in
B. C. Carlson
Computation of elliptic integrals
Numerical Algorithms 10, 13-26 (1995)
*/
/* fpprec:120$ Should be set outside */
etol:0.1b0^fpprec$ /* For Carlson */
ca:sqrt(etol)$ /* For Bulirsch */
eps:0.1b0^fpprec$ /* For eirx */
pi:bfloat(%pi)$
ratprint:false$
rf(x,y,z) := block(
[a0:(x+y+z)/3, q,x0:x,y0:y,z0:z,an,ln,xx,yy,zz,n,e2,e3],
q:(3*etol)^(-1/6)*max(abs(a0-x),abs(a0-y),abs(a0-z)),
an:a0,
n:0,
while q >= abs(an) do (
n:n+1,
ln:sqrt(x0)*sqrt(y0)+sqrt(y0)*sqrt(z0)+sqrt(z0)*sqrt(x0),
an:(an+ln)/4,
x0:(x0+ln)/4,
y0:(y0+ln)/4,
z0:(z0+ln)/4,
q:q/4),
xx:(a0-x)/(4^n*an),
yy:(a0-y)/(4^n*an),
zz:-xx-yy,
e2:xx*yy-zz^2,
e3:xx*yy*zz,
(1-e2/10+e3/14+e2^2/24-3*e2*e3/44) / sqrt(an))$
rd(x,y,z) := block(
[a0:(x+y+3*z)/5, q,x0:x,y0:y,z0:z,an,ln,xx,yy,zz,n,e2,e3,e4,e5,s],
q:(etol/4)^(-1/6)*max(abs(a0-x),abs(a0-y),abs(a0-z)),
an:a0,
n:0,
s:0,
while q >= abs(an) do (
ln:sqrt(x0)*sqrt(y0)+sqrt(y0)*sqrt(z0)+sqrt(z0)*sqrt(x0),
s:s+1/(4^n*sqrt(z0)*(z0+ln)),
n:n+1,
an:(an+ln)/4,
x0:(x0+ln)/4,
y0:(y0+ln)/4,
z0:(z0+ln)/4,
q:q/4),
xx:(a0-x)/(4^n*an),
yy:(a0-y)/(4^n*an),
zz:-(xx+yy)/3,
e2:xx*yy-6*zz^2,
e3:(3*xx*yy-8*zz^2)*zz,
e4:3*(xx*yy-zz^2)*zz^2,
e5:xx*yy*zz^3,
(1-3*e2/14+e3/6+9*e2^2/88-3*e4/22-9*e2*e3/52+3*e5/26)/(4^n*an*sqrt(an))
+3*s)$
/* R_G(x,y,0) */
rg0(x,y) := block(
[x0:sqrt(x),y0:sqrt(y),xn,yn,t,s,n],
xn:x0,
yn:y0,
n:0,
s:0,
while abs(xn-yn) >= 2.7b0 * sqrt(etol) * abs(xn) do (
t:(xn+yn)/2,
yn:sqrt(xn*yn),
xn:t,
n:n+1,
s:s+(xn-yn)^2*2^(n-2)),
((x0+y0)^2/4 - s)*pi/(2*(xn+yn)) )$
/* k^2 = m */
ec(m):=2*rg0(1b0-m,1b0)$
kc(m):=rf(0b0,1b0-m,1b0)$
/* Implementation of methods given in
Roland Bulirsch
Numerical Calculation of Elliptic Integrals and Elliptic Functions
Numericshe Mathematik 7, 78-90 (1965)
*/
sncndn(x,mc):=block([bo, a, b, c, d, l, sn, cn, dn, m, n],
local(m, n),
if mc # 0 then (
bo:is(mc < 0b0),
if bo then (
d:1-mc,
mc:-mc/d,
d:sqrt(d),
x:d*x),
dn:a:1,
for i:0 thru 12 do (
l:i,
m[i]:a,
n[i]:mc:sqrt(mc),
c:(a+mc)/2,
if abs(a-mc)<=ca*a then return(false),
mc:a*mc,
a:c
),
x:c*x,
sn:sin(x),
cn:sin(pi/2-x),
if sn#0b0 then (
a:cn/sn,
c:a*c,
for i:l step -1 thru 0 do (
b:m[i],
a:c*a,
c:dn*c,
dn:(n[i]+a)/(b+a),
a:c/b
),
a:1/sqrt(c*c+1b0),
sn:if sn<0b0 then -a else a,
cn:c*sn
),
if bo then (
a:dn,
dn:cn,
cn:a,
sn:sn/d
)
) else /* mc = 0 */ (
sn:tanh(x),
dn:cn:sech(x)
/* d:exp(x), a:1/d, b:a+d, cn:dn:2/b,
if x < 0.3b0 then (
d:x*x*x*x,
d:(d*(d*(d*(d+93024b0)+3047466240b0)+24135932620800b0)+
20274183401472000b0)/60822550204416000b0,
sn:cn*(x*x*x*d+sin(x))
) else
sn:(d-a)/b */
),
[sn,cn,dn]
)$
/* Versions of incomplete functions in terms of Jacobi elliptic function
with u = am(phi) real and in [0,K(m)] */
eirx(sn,cn,dn,m,ec):=block([t],
t:if abs(sn) < eps then abs(sn) else
(rf((cn/sn)^2,(dn/sn)^2,1/sn^2)-m/3b0*rd((cn/sn)^2,(dn/sn)^2,1/sn^2)),
if cn < 0 then t:2*ec - t,
if sn < 0 then t:-t,
t)$
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