# Solving a system of Non-linear equations by Fixed Point method or Newton Raphson method

I have a system of five equations with ten variables, i.e., the radial distance $$r$$, polar angle $$\theta$$; four spin vector components ($$St, Sr, S\theta, S\phi$$) and four momentum components ($$Pt, Pr, P\theta, P\phi$$). We can set $$r=15$$ and $$\theta = \pi/2$$. $$EE$$ and $$LL$$ are energy and angular momentum. We can also give numerical value to these objects, i.e., $$EE=0.95$$, and $$LL=1.5$$

Now, we have five equations with eight variables. So, we can give numerical values to any three, i.e., $$Pr, Sr$$, and $$S\theta$$.

I want to find the initial data using these five equations using Fixed Point method or Newton Raphson method. I am finding difficulties. Can anyone help me please to find the data which satisfy these equations.

    EQ1 = EE + Pt +
1/2 (-((2 Pϕ r^2 Sθ Sin[θ]^2)/Sqrt[
r^4 Sin[θ]^2]) + (
2 Pθ r^2 Sϕ Sin[θ]^2)/Sqrt[
r^4 Sin[θ]^2]);
EQ2 = -LL + Pϕ +
1/2 ((2 Pt (-1 + 2/r) r^2 Sr Cos[θ] Sin[θ])/((1 - 2/
r) Sqrt[r^4 Sin[θ]^2]) - (
2 Pr (-1 + 2/r) r^2 St Cos[θ] Sin[θ])/((1 - 2/
r) Sqrt[r^4 Sin[θ]^2]) + (
2 Pθ (-1 + 2/r) r^3 St Sin[θ]^2)/Sqrt[
r^4 Sin[θ]^2] - (
2 Pt (-1 + 2/r) r^3 Sθ Sin[θ]^2)/Sqrt[
r^4 Sin[θ]^2]);
EQ3 = 1 + (Pr^2 (-2 + r))/((1 - 2/r)^2 r) + (Pt^2 (-1 + 2/r)^2 r)/(
2 - r) + Pθ^2 r^2 + Pϕ^2 r^2 Sin[θ]^2;
EQ4 = (Pr (-2 + r) Sr)/((1 - 2/r)^2 r) + (Pt (-1 + 2/r)^2 r St)/(
2 - r) + Pθ r^2 Sθ +
Pϕ r^2 Sϕ Sin[θ]^2;
EQ5 = -(((-2 + r) Sr^2)/((1 - 2/r)^2 r)) +
SS^2 - ((-1 + 2/r)^2 r St^2)/(2 - r) - r^2 Sθ^2 -
r^2 Sϕ^2 Sin[θ]^2;


Using $$EE = 0.95; LL = 1.5; r = 9; \theta = Pi/2; Pr = 0; Sr = 0.01; S\theta = 0.001; SS = 1;$$ I simpilifed the equations and get

EQ1[Pt_, Pθ_, Pϕ_, Sϕ_] :=
0.95 + Pt + 1/2 (-0.002 Pϕ + 2 Pθ Sϕ);
EQ2[Pt_, Pθ_, Pϕ_, St_] := -1.5 + Pϕ +
1/2 (0. + 0.013999999999999999 Pt - 14 Pθ St);
EQ3[Pt_, Pθ_, Pϕ_] :=
1 - (7 Pt^2)/9 + 81 Pθ^2 + 81 Pϕ^2;
EQ4[Pt_, Pθ_, Pϕ_ , St_, Sϕ_] :=
0. + 0.081 Pθ - (7 Pt St)/9 + 81 Pϕ Sϕ;
EQ5[St_, Sϕ_] := 0.9997904285714285 + (7 St^2)/9 - 81 Sϕ^2;


I was taking help from the help section, where FixedPoint[f, expr] command was used for one equation. That's why, first I was trying to to convert the system in such a way that one equation contains one variable using

Solve[{EQ1[Pt, Pθ, Pϕ, Sϕ] == 0,
EQ2[Pt, Pθ, Pϕ, St] == 0,
EQ3[Pt, Pθ, Pϕ] == 0,
EQ4[Pt, Pθ, Pϕ , St, Sϕ] == 0,
EQ5[St, Sϕ] == 0}, {Pt, Pθ, Pϕ, St, Sϕ}]


But, there was some error. Actually I want to find the initial data which satisfy EQ1, EQ2,..., EQ5.

I shall be highly thankful if someone ca help me.

• I see undefined objects EE~ and LL there. Feb 3, 2020 at 15:44
• How have you tried to use the fixed point method? Can you provide the code you have tried? Feb 3, 2020 at 17:26
• What are FF1[] etc.? Feb 4, 2020 at 14:16
• These are the equations EQ1, EQ2, ... I just rename. EQ1 to FF1[ ].
– MMS
Feb 4, 2020 at 15:13

I take your preset values and turn them into rules

presets = {EE -> 0.95, LL -> 1.5, r -> 9, θ -> Pi/2, Pr -> 0, Sr -> 0.01, Sθ -> 0.001, SS -> 1}


Then

eqs = {EQ1, EQ2, EQ3, EQ4, EQ5} /. presets

Solve[eqs == {0, 0, 0, 0, 0}, {Pt, Pθ, Pϕ, Sϕ, St}]
`

Just one answer shown...

{{Pt -> -0.963824 - 0.0047205 I,

Pθ -> -0.0337024 - 0.132886 I,

Pϕ -> 0.120239 - 0.036884 I,

Sϕ -> -0.0581202 + 0.0901944 I,

St -> 0.392471 - 1.391 I},

...

}

• I was also trying in this way, but I found these imaginary values and some error in mathematica. So, I wanted to apply Newton Raphson or fixed point method.
– MMS
Feb 4, 2020 at 19:18