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julia/laguerre-conway.jl

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+function laguerre_conway(state::Vector{Float64}, μ::Float64, time::Float64)
+
+  n = 5                               # Choose LaGuerre-Conway "n"
+  i = 0
+
+  r0 = state[1:3]                     # Are we in elliptical or hyperbolic orbit?
+  r0_mag = norm(r0)
+  v0 = state[4:6]
+  v0_mag = norm(v0)
+  σ0 = (r0 ⋅ v0)/√(μ)
+  a = 1 / ( 2/r0_mag - v0_mag^2/μ )
+  coeff = 1 - r0_mag/a
+
+  if a > 0    # Elliptical
+    ΔM = ΔE_new = √(μ) / sqrt(a^3) * time
+    ΔE = 1000
+    while abs(ΔE - ΔE_new) > 1e-12
+      ΔE = ΔE_new
+      F = ΔE - ΔM + σ0 / √(a) * (1-cos(ΔE)) - coeff * sin(ΔE)
+      dF = 1 + σ0 / √(a) * sin(ΔE) - coeff * cos(ΔE)
+      d2F = σ0 / √(a) * cos(ΔE) + coeff * sin(ΔE)
+      sign = dF >= 0 ? 1 : -1
+      ΔE_new = ΔE - n*F / ( dF + sign * √(abs((n-1)^2*dF^2 - n*(n-1)*F*d2F )))
+      i += 1
+    end
+    F = 1 - a/r0_mag * (1-cos(ΔE))
+    G = a * σ0/ √(μ) * (1-cos(ΔE)) + r0_mag * √(a) / √(μ) * sin(ΔE)
+    r = a + (r0_mag - a) * cos(ΔE) + σ0 * √(a) * sin(ΔE)
+    Ft = -√(a)*√(μ) / (r*r0_mag) * sin(ΔE)
+    Gt = 1 - a/r * (1-cos(ΔE))
+  else      # Hyperbolic or Parabolic
+    ΔN = √(μ) / sqrt(-a^3) * time
+    ΔH = 0
+    ΔH_new = time < 0 ? -1 : 1
+    while abs(ΔH - ΔH_new) > 1e-12
+      ΔH = ΔH_new
+      F = -ΔN - ΔH + σ0 / √(-a) * (cos(ΔH)-1) + coeff * sin(ΔH)
+      dF = -1 + σ0 / √(-a) * sin(ΔH) + coeff * cos(ΔH)
+      d2F = σ0 / √(-a) * cos(ΔH) + coeff * sin(ΔH)
+      sign = dF >= 0 ? 1 : -1
+      ΔH_new = ΔH - n*F / ( dF + sign * √(abs((n-1)^2*dF^2 - n*(n-1)*F*d2F )))
+      i += 1
+    end
+    F = 1 - a/r0_mag * (1-cos(ΔH))
+    G = a * σ0/ √(μ) * (1-cos(ΔH)) + r0_mag * √(-a) / √(μ) * sin(ΔH)
+    r = a + (r0_mag - a) * cos(ΔH) + σ0 * √(-a) * sin(ΔH)
+    Ft = -√(-a)*√(μ) / (r*r0_mag) * sin(ΔH)
+    Gt = 1 - a/r * (1-cos(ΔH))
+  end
+
+  if i > 100
+    throw(ErrorException("LaGuerre-Conway did not converge!"))
+  end
+
+  return [ F*state[1:3] + G*state[4:6]; Ft*state[1:3] + Gt*state[4:6]]
+
+end