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/*
Implementation of methods given in
B. C. Carlson
Computation of elliptic integrals
Numerical Algorithms 10, 13-26 (1995)
Copyright (c) Charles Karney (2009-2013) <karney@alum.mit.edu> and
licensed under the MIT/X11 License. For more information, see
https://geographiclib.sourceforge.io/
*/
/* fpprec:120$ Should be set outside */
etol:0.1b0^fpprec$ /* For Carlson */
tolRF : (3 * etol)^(1/8b0)$
tolRD : (etol/5)^(1/8b0)$
tolRG0 : 2.7b0 * sqrt(etol)$
tolJAC:sqrt(etol)$ /* For Bulirsch */
pi:bfloat(%pi)$
atan2x(y,x) := -atan2(-y, x)$ /* fix atan2(0b0,-1b0) = -pi */
rf(x,y,z) := block([a0:(x+y+z)/3, q,x0:x,y0:y,z0:z,
an,ln,xx,yy,zz,e2,e3,mul:1b0],
q:max(abs(a0-x),abs(a0-y),abs(a0-z))/tolRF,
an:a0,
while q >= mul * abs(an) do (
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,
mul:mul*4),
xx:(a0-x)/(mul*an),
yy:(a0-y)/(mul*an),
zz:-xx-yy,
e2:xx*yy-zz^2,
e3:xx*yy*zz,
(e3 * (6930 * e3 + e2 * (15015 * e2 - 16380) + 17160) +
e2 * ((10010 - 5775 * e2) * e2 - 24024) + 240240) /
(240240 * 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,e2,e3,e4,e5,s:0b0,mul:1b0],
q:max(abs(a0-x),abs(a0-y),abs(a0-z))/tolRD,
an:a0,
while q >= mul*abs(an) do (
ln:sqrt(x0)*sqrt(y0)+sqrt(y0)*sqrt(z0)+sqrt(z0)*sqrt(x0),
s:s+1/(mul*sqrt(z0)*(z0+ln)),
an:(an+ln)/4,
x0:(x0+ln)/4,
y0:(y0+ln)/4,
z0:(z0+ln)/4,
mul:4*mul),
xx:(a0-x)/(mul*an),
yy:(a0-y)/(mul*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,
((471240 - 540540 * e2) * e5 +
(612612 * e2 - 540540 * e3 - 556920) * e4 +
e3 * (306306 * e3 + e2 * (675675 * e2 - 706860) + 680680) +
e2 * ((417690 - 255255 * e2) * e2 - 875160) + 4084080) /
(4084080 * mul * an * sqrt(an)) + 3 * s)$
/* R_G(x,y,0) */
rg0(x,y) := block([x0:sqrt(max(x,y)),y0:sqrt(min(x,y)),xn,yn,
t,s:0b0,mul:0.25b0],
xn:x0,
yn:y0,
while abs(xn-yn) > tolRG0 * xn do (
t:(xn+yn)/2,
yn:sqrt(xn*yn),
xn:t,
mul : 2*mul,
s:s+mul*(xn-yn)^2),
((x0+y0)^2/4 - s)*pi/(2*(xn+yn)) )$
rg(x, y, z) := (
if z = 0b0 then (z:y, y:0b0),
(z * rf(x, y, z) - (x-z) * (y-z) * rd(x, y, z) / 3
+ sqrt(x * y / z)) / 2)$
rj(x, y, z, p) := block([A0:(x + y + z + 2*p)/5,An,delta:(p-x) * (p-y) * (p-z),
Q,x0:x,y0:y,z0:z,p0:p,mul:1b0,mul3:1b0,s:0b0,X,Y,Z,P,E2,E3,E4,E5],
An : A0,
Q : max(abs(A0-x), abs(A0-y), abs(A0-z), abs(A0-p)) / tolRD,
while Q >= mul * abs(An) do block([lam,d0,e0],
lam : sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0),
d0 : (sqrt(p0)+sqrt(x0)) * (sqrt(p0)+sqrt(y0)) * (sqrt(p0)+sqrt(z0)),
e0 : delta/(mul3 * d0^2),
s : s + rc(1b0, 1b0 + e0)/(mul * d0),
An : (An + lam)/4,
x0 : (x0 + lam)/4,
y0 : (y0 + lam)/4,
z0 : (z0 + lam)/4,
p0 : (p0 + lam)/4,
mul : 4*mul,
mul3 : 64*mul3),
X : (A0 - x) / (mul * An),
Y : (A0 - y) / (mul * An),
Z : (A0 - z) / (mul * An),
P : -(X + Y + Z) / 2,
E2 : X*Y + X*Z + Y*Z - 3*P*P,
E3 : X*Y*Z + 2*P * (E2 + 2*P*P),
E4 : (2*X*Y*Z + P * (E2 + 3*P*P)) * P,
E5 : X*Y*Z*P*P,
((471240 - 540540 * E2) * E5 +
(612612 * E2 - 540540 * E3 - 556920) * E4 +
E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
(4084080 * mul * An * sqrt(An)) + 6 * s)$
rd(x, y, z) := block([A0:(x + y + 3*z)/5,An,Q,x0:x,y0:y,z0:z,
mul:1b0,s:0b0,X,Y,Z,E2,E3,E4,E5],
An : A0,
Q : max(abs(A0-x), abs(A0-y), abs(A0-z)) / tolRD,
while Q >= mul * abs(An) do block([lam],
lam : sqrt(x0)*sqrt(y0) + sqrt(y0)*sqrt(z0) + sqrt(z0)*sqrt(x0),
s : s + 1/(mul * sqrt(z0) * (z0 + lam)),
An : (An + lam)/4,
x0 : (x0 + lam)/4,
y0 : (y0 + lam)/4,
z0 : (z0 + lam)/4,
mul : 4*mul),
X : (A0 - x) / (mul * An),
Y : (A0 - y) / (mul * An),
Z : -(X + Y) / 3,
E2 : X*Y - 6*Z*Z,
E3 : (3*X*Y - 8*Z*Z)*Z,
E4 : 3 * (X*Y - Z*Z) * Z*Z,
E5 : X*Y*Z*Z*Z,
((471240 - 540540 * E2) * E5 +
(612612 * E2 - 540540 * E3 - 556920) * E4 +
E3 * (306306 * E3 + E2 * (675675 * E2 - 706860) + 680680) +
E2 * ((417690 - 255255 * E2) * E2 - 875160) + 4084080) /
(4084080 * mul * An * sqrt(An)) + 3 * s)$
rc(x, y):=if x < y then atan(sqrt((y - x) / x)) / sqrt(y - x)
else ( if x = y and y > 0b0 then 1 / sqrt(y) else
atanh( if y > 0b0 then sqrt((x - y) / x)
else sqrt(x / (x - y)) ) / sqrt(x - y) )$
/* k^2 = m */
ec(k2):=2*rg0(1b0-k2,1b0)$
kc(k2):=rf(0b0,1b0-k2,1b0)$
dc(k2):=rd(0b0,1b0-k2,1b0)/3$
pic(k2,alpha2):=if alpha2 # 0b0 then
kc(k2)+alpha2*rj(0b0,1b0-k2,1b0,1b0-alpha2)/3
else kc(k2)$
gc(k2,alpha2):=if alpha2 # 0b0 then (if k2 # 1b0 then
kc(k2) + (alpha2-k2)*rj(0b0,1b0-k2,1b0,1b0-alpha2)/3
else rc(1b0,1b0-alpha2))
else ec(k2)$
hc(k2,alpha2):=if alpha2 # 0b0 then (if k2 # 1b0 then
kc(k2) - (1b0-alpha2)*rj(0b0,1b0-k2,1b0,1b0-alpha2)/3
else rc(1b0,1b0-alpha2))
else kc(k2) - dc(k2)$
/* 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)<=tolJAC*a then return(false),
mc:a*mc,
a:c
),
x:c*x,
sn:sin(x),
cn:cos(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]
)$
Delta(sn, k2):=sqrt(1 - k2 * sn*sn)$
/* Versions of incomplete functions in terms of Jacobi elliptic function
with u = am(phi) real and in [0,K(k2)] */
eirx(sn,cn,dn,k2,ec):=block([t,cn2:cn*cn,dn2:dn*dn,sn2:sn*sn,ei,kp2:1b0-k2],
ei : (if k2 <= 0b0 then
rf(cn2, dn2, 1b0) - k2 * sn2 * rd(cn2, dn2, 1b0) / 3 else
(if kp2 >= 0b0 then
kp2 * rf(cn2, dn2, 1b0) +
k2 * kp2 * sn2 * rd(cn2, 1b0, dn2) / 3 +
k2 * abs(cn) / dn else
- kp2 * sn2 * rd(dn2, 1b0, cn2) / 3 + dn / abs(cn) ) ),
ei : ei * abs(sn),
if cn < 0b0 then ei : 2 * ec - ei,
if sn < 0 then ei : -ei,
ei)$
dirx(sn, cn, dn, k2, dc):=block(
[di:abs(sn) * sn*sn * rd(cn*cn, dn*dn, 1b0) / 3],
if cn < 0b0 then di : 2 * dc - di,
if sn < 0b0 then di : -di,
di)$
hirx(sn, cn, dn, k2, alpha2, hc):=block(
[cn2 : cn*cn, dn2 : dn*dn, sn2 : sn*sn, hi, alphap2:1b0-alpha2],
hi : abs(sn) * (rf(cn2, dn2, 1b0) -
alphap2 * sn2 *
rj(cn2, dn2, 1b0, cn2 + alphap2 * sn2) / 3),
if cn < 0 then hi : 2 * hc - hi,
if sn < 0 then hi : -hi,
hi)$
deltae(sn, cn, dn, k2, ec):=(
if cn < 0 then (cn : -cn, sn : -sn),
eirx(sn, cn, dn, k2, ec) * (pi/2) / ec - atan2x(sn, cn))$
deltad(sn, cn, dn, k2, dc):=(
if cn < 0 then (cn : -cn, sn : -sn),
dirx(sn, cn, dn, k2, ec) * (pi/2) / dc - atan2x(sn, cn))$
deltah(sn, cn, dn, k2, alpha2, hc):=(
if cn < 0 then (cn : -cn, sn : -sn),
hirx(sn, cn, dn, k2, alpha2, hc) * (pi/2) / hc - atan2x(sn, cn))$
einv(x, k2, ec):=block(
[n : floor(x / (2 * ec) + 0.5b0), phi, eps : k2/(sqrt(1b0-k2) + 1b0)^2,
err:1b0],
x : x - 2 * ec * n,
phi : pi * x / (2 * ec),
phi : phi - eps * sin(2 * phi) / 2,
while abs(err) > tolJAC do block([sn:sin(phi), cn:cos(phi), dn],
dn : Delta(sn, k2),
err : (eirx(sn, cn, dn, k2, ec) - x)/dn,
phi : phi - err),
n * pi + phi)$
deltaeinv(stau, ctau, k2, ec) := block([tau],
if ctau < 0 then (ctau : -ctau, stau : -stau),
tau : atan2x(stau, ctau),
einv(tau * ec / (pi/2), k2, ec) - tau)$
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