# Solving a system of 8 nonlinear equations

I'm new on Mathematica and have been trying to solve the following system of 8 equations with 7 variable:

However, Mathematica takes too much time trying to solve the system. I left Mathematica working on that for 24 hours without success.

The idea is to find analytical solutions for b3, c,c2, c4, θ, ø (phi) and ω. I have tried with Solve and Reduce, but it didn't work.

I attached the code below:

Clear[\[Alpha]re, \[Alpha]i, \[Beta]re, \[Beta]i, \[Gamma]re , \
\[Gamma]i, \[Delta]re , \[Delta]i]
var = {c, c2, b3, c4, \[Omega], \[Phi], \[Theta]};
Eq1 = \[Alpha]re ==
Cos[b3/2] Cos[
1/4 (2 c + 2 c2 + 2 c4 + \[Pi] -
4 \[Theta])] Cos[\[CurlyPhi] - \[Omega]]
Eq2 = \[Alpha]i == -Cos[b3/2] Cos[\[CurlyPhi] - \[Omega]] Sin[
1/4 (2 c + 2 c2 + 2 c4 + \[Pi] - 4 \[Theta])]
Eq3 = \[Beta]re ==
Sin[b3/2] Sin[
1/4 (-2 c - 2 c2 + 2 c4 + \[Pi] +
4 \[Theta])] Sin[\[CurlyPhi] - \[Omega]]
Eq4 = \[Beta]i == -Sin[b3/2] Sin[
1/4 (2 c + 2 c2 - 2 c4 + \[Pi] -
4 \[Theta])] Sin[\[CurlyPhi] - \[Omega]]
Eq5 = \[Gamma]re ==
Cos[1/4 (2 c + 2 c2 + 2 c4 + \[Pi] -
4 \[Theta])] Cos[\[CurlyPhi] - \[Omega]] Sin[b3/2]
Eq6 = \[Gamma]i == -Cos[\[CurlyPhi] - \[Omega]] Sin[b3/2] Sin[
1/4 (2 c + 2 c2 + 2 c4 + \[Pi] - 4 \[Theta])]
Eq7 = \[Delta]re == -Cos[b3/2] Sin[
1/4 (-2 c - 2 c2 + 2 c4 + \[Pi] +
4 \[Theta])] Sin[\[CurlyPhi] - \[Omega]]
Eq8 = \[Delta]i ==
Cos[b3/2] Sin[
1/4 (2 c + 2 c2 - 2 c4 + \[Pi] -
4 \[Theta])] Sin[\[CurlyPhi] - \[Omega]]
Eqs = {Eq1, Eq2, Eq3, Eq4, Eq5, Eq6, Eq7, Eq8};
Reduce[Eqs, var]

• In your code in var you have Phi and in your equations you have CurlyPhi and I'm guessing that is a typo to fix. Then you want to solve for both Phi and Omega but those two only appear as a difference and no where else. I suggest you replace Phi-Omega with po, solve for po and figure out what to do with that afterwards. Then replace all your b3/2 with b and figure out b3 afterwards. Then get rid of almost all the constants in each of your 1/4 (2 c+2c2+2c4+Pi-4 Theta) by scaling that. Then find a way to have n equations and n unknowns, make it simple. Will it solve?
– Bill
Commented Jan 7 at 6:51

First, according to the @Bill's comment, we make substitutions

Eqs1 = Eqs /. {1/4 (2 c + 2 c2 + 2 c4 + \[Pi] - 4 \[Theta]) -> r,
1/4 (-2 c - 2 c2 + 2 c4 + \[Pi] + 4 \[Theta]) ->  s,
\[CurlyPhi] - \[Omega] -> t,1/4 (2 c + 2 c2 - 2 c4 + \[Pi] - 4 \[Theta]) -> p}


{\[Alpha]re == Cos[b3/2] Cos[r] Cos[t], \[Alpha]i == -Cos[b3/2] Cos[t] Sin[ r], \[Beta]re == Sin[b3/2] Sin[s] Sin[t], \[Beta]i == -Sin[b3/2] Sin[p] Sin[ t], \[Gamma]re == Cos[r] Cos[t] Sin[b3/2], \[Gamma]i == -Cos[t] Sin[b3/2] Sin[ r], \[Delta]re == -Cos[b3/2] Sin[s] Sin[t], \[Delta]i == Cos[b3/2] Sin[p] Sin[t]}

Second, we switch from trigonometry to polynomial algebra by

Eqs2 = Union[{cb^2 + sb^2 == 1, ct^2 + st^2 == 1, cp^2 + sp^2 == 1,
cs^2 + ss^2 == 1, cr^2 + sr^2 == 1},
Eqs1 /. {Cos[b3/2] -> cb, Sin[b3/2] -> sb, Cos[t] -> ct,
Sin[t] -> st, Sin[s] -> ss, Cos[s] -> cs, Sin[p] -> sp,
Cos[p] -> cp, Sin[r] -> sr, Cos[r] -> cr}];


Third,

Reduce[Eqs2, {cp, sp, cr, sr, cs, ss, ct, st, cb, sb}]


produces the output with LeafCount which results in 10534. I leave the rest on your own.

Addition. Another way consists in evaluating all the parameters, for example,

\[Alpha]re = 1/2; \[Alpha]i = -1/3; \[Beta]re = Sqrt[2]/2;
\[Beta]i = 1/3; \[Gamma]re = 0; \[Gamma]i = 5/6; \[Delta]re =  1/3; \[Delta]i = -1/4;


and then

var = {c, c2, b3, c4, \[Omega], \[CurlyPhi], \[Theta]};
Eqs = {Eq1, Eq2, Eq3, Eq4, Eq5, Eq6, Eq7, Eq8};
Reduce[Eqs, var]


False`

Such a result is typical.