Non solutions returned by NSolve. And why does it return both phi and Cos[phi]?

Question

Similarly to this thread NSolve gives additional solutions that don't satisfy the equations!

NSolve returns "spurious" solutions, even increasing the working precision

fsys={1 - Cos[(11 π)/45] - (3 Cos[ψ1])/4 - 2/3 Sin[φ + π/6] == 0, h + 2/3 Cos[φ + π/6] + Sin[(11 π)/45] - (3 Sin[ψ1])/4 == 0, -0.0436952 - (3 Cos[ψ2])/4 - 2/3 Sin[φ - π/6] == 0,2/5 + h + 2/3 Cos[φ - π/6] + Sin[(17 π)/180] - (3 Sin[ψ2])/4 == 0}

NSolve[N[fsys], {φ, h, ψ1, ψ2}, WorkingPrecision -> 100]

The second solution returned is

    {h -> -1.99515, ψ2 -> -1.22627, ψ1 -> 1.68803, φ -> 0.0618559,
Sin[ψ1] -> -0.993136, Cos[ψ1] -> -0.116967, 
 Sin[ψ2] -> -0.941237, Cos[ψ2] -> 0.337748, Sin[φ] -> 0.0618165,
  Cos[φ] -> 0.998088}, {h -> -1.99515, ψ2 -> 1.22627, 
 ψ1 -> -1.68803, φ -> 0.0618559, Sin[ψ1] -> -0.993136, 
 Cos[ψ1] -> -0.116967, Sin[ψ2] -> -0.941237, 
 Cos[ψ2] -> 0.337748, Sin[φ] -> 0.0618165, Cos[φ] -> 0.998088}

,

I do not understand why both ψ1 and Cos[ψ1] are returned, with Sin[1.68803] actually not equal to 0.993136 (the sign is different)

Good to know (I am using version 7). And do you get the wrong solutions (which do not satisfy the original equations) or no? — Fabio Dalla Libera, Dec 17 '12 at 06:32
Yes I do and I think it's fair as NSolve is not the right tool as stated in the documentation and the warnings. Why not using FindRoot ? — b.gates.you.know.what, Dec 17 '12 at 06:34
Sorry, on v.7 I do not get the warning either. The reason is that I want all the real roots, and not just one — Fabio Dalla Libera, Dec 17 '12 at 06:38

score 2 · Accepted Answer · answered Dec 17 '12 at 06:57

I would use FindRoot rather than NSolve for your problem (this is on version 7):

sol = NSolve[N[fsys], {phi, h, psi1, psi2}, WorkingPrecision -> 100];
fsys /. sol
(* {{False, True, False, False}, {False, True, True, False}, 
    {False, False, False, False}, {False, False, False, False}, 
    {True, True, False, False}, {True, False, False, False}} *)

but

sol2 = FindRoot[fsys, {{phi, 0.5}, {h, 0.21}, {psi1, 0.3}, {psi2, 0.4}},
  WorkingPrecision -> 100];

fsys /. sol2
(* {True, True, True, True} *)

Non solutions returned by NSolve. And why does it return both phi and Cos[phi]?

1 Answers1