How well do these methods for finding the eccentric anomaly work?
e : eccentricity
M : mean anomaly
u : eccentric anomaly
Method #1.
u = M + e sin(M)
i = 1
REPEAT
i = i+1
ú = u
u = M + e sin(ú)
UNTIL |u−ú| < 1ᴇ-11
Method #2.
i = 1
u = M + (M + e sin(M) − M) / (1 − e cos(M))
REPEAT
i = i+1
ú = u
u = ú + (M + e sin(ú) − ú) / (1 − e cos(ú))
UNTIL |u−ú| < 1ᴇ-11
Method #3.
i = 1
u = M + e sin(M)
REPEAT
i = i+1
ú = u
u = M + e sin(u)
UNTIL i=3
REPEAT
i = i+1
ú = u
u = ú + (M + e sin(ú) − ú) / (1 − e cos(ú))
UNTIL |u−ú| < 1ᴇ-11
Method #4.
u = M + (e − e³/8 + e⁵/192) sin(M) + (e²/2 − e⁴/6) sin(2M) + (3e³/8 − 27e⁵/128) sin(3M) + (e⁴/3) sin(4M)
REPEAT
ú = u
A = u − e sin ú − M
B = 1 − e cos ú
C = e sin ú
D = e cos ú
E = −A / B
F = −A / (B + EC/2)
G = −A / (B + EC/2 + DF²/6)
u = ú+G
UNTIL |u−ú|/u < 1ᴇ-11