# FindRoot::jsing: Encountered a singular Jacobian at the point when solving NONLINEAR EQUATIONS

I have tried many methods to solve the following nonlinear equations.

equa={AmI CmI - (BmI CmI)/R3^2 == k1 u0,
2 R3 AmI CmI - (2 BmI CmI)/R3^3 == 0,
AmI DmI - (BmI DmI)/R3^2 == 0,
2 R3 AmI DmI - (2 BmI DmI)/R3^3 == 0,
AmI CmI - (BmI CmI)/R2^2 ==
R2^(-1 - (
2 \[Pi])/\[Beta]) \[Beta] (-((
2 Aki (Sin[thetai] - Sin[thetai + \[Beta]]))/(
4 \[Pi]^2 - \[Beta]^2)) + (
2 R1^(-((4 \[Pi])/\[Beta])) R2^((4 \[Pi])/\[Beta])
Aki (Sin[thetai] - Sin[thetai + \[Beta]]))/(
4 \[Pi]^2 - \[Beta]^2) - (
R2^(\[Pi]/\[Beta])
Aki (Sin[thetai] +
Sin[thetai + \[Beta]]))/((\[Pi] - \[Beta]) (\[Pi] + \[Beta])) \
+ (R1^(-((2 \[Pi])/\[Beta])) R2^((3 \[Pi])/\[Beta])
Aki (Sin[thetai] +
Sin[thetai + \[Beta]]))/((\[Pi] - \[Beta]) (\[Pi] + \
\[Beta]))),
2 R2 AmI CmI - (2 BmI CmI)/R2^3 == -((
2 R1^(-((4 \[Pi])/\[Beta]))
R2^(-1 - (
2 \[Pi])/\[Beta]) (-R1^(((2 \[Pi])/\[Beta])) + R2^((
2 \[Pi])/\[Beta])) \[Beta] ((R1^((2 \[Pi])/\[Beta]) + R2^((
2 \[Pi])/\[Beta])) (\[Pi]^2 - 4 \[Beta]^2) Aki Cos[
2 thetai + \[Beta]] Sin[\[Beta]] -
2 R1^((2 \[Pi])/\[Beta])
R2^(\[Pi]/\[Beta]) (\[Pi]^2 - \[Beta]^2) Aki[
1] Cos[\[Beta]] Sin[2 thetai + \[Beta]]))/(\[Pi]^4 -
5 \[Pi]^2 \[Beta]^2 + 4 \[Beta]^4)),
AmI DmI - (BmI DmI)/R2^2 ==
R2^(-1 - (
2 \[Pi])/\[Beta]) \[Beta] ((
2 Aki (Cos[thetai] - Cos[thetai + \[Beta]]))/(
4 \[Pi]^2 - \[Beta]^2) + (
4 R1^(-((4 \[Pi])/\[Beta])) R2^((4 \[Pi])/\[Beta])
Aki (-Cos[thetai] + Cos[thetai + \[Beta]]))/(
8 \[Pi]^2 - 2 \[Beta]^2) + (
R2^(\[Pi]/\[Beta])
Aki (Cos[thetai] +
Cos[thetai + \[Beta]]))/((\[Pi] - \[Beta]) (\[Pi] + \[Beta])) \
- (R1^(-((2 \[Pi])/\[Beta])) R2^((3 \[Pi])/\[Beta])
Aki (Cos[thetai] +
Cos[thetai + \[Beta]]))/((\[Pi] - \[Beta]) (\[Pi] + \
\[Beta]))),
2 R2 AmI DmI - (2 BmI DmI)/R2^3 == (
2 R1^(-((4 \[Pi])/\[Beta]))
R2^(-1 - (
2 \[Pi])/\[Beta]) (R1^((2 \[Pi])/\[Beta]) - R2^((
2 \[Pi])/\[Beta])) \[Beta] (2 R1^((2 \[Pi])/\[Beta])
R2^(\[Pi]/\[Beta]) (\[Pi]^2 - \[Beta]^2) Aki[
1] Cos[\[Beta]] Cos[
2 thetai + \[Beta]] + (R1^((2 \[Pi])/\[Beta]) + R2^((
2 \[Pi])/\[Beta])) (\[Pi]^2 - 4 \[Beta]^2) Aki[
2] Sin[\[Beta]] Sin[2 thetai + \[Beta]]))/(\[Pi]^4 -
5 \[Pi]^2 \[Beta]^2 + 4 \[Beta]^4),
Aki == (1/(R2^2 (\[Pi]^4 - 5 \[Pi]^2 \[Beta]^2 + 4 \[Beta]^4)))
2 \[Beta] (-R2 (\[Pi]^2 - 4 \[Beta]^2) (R2^2 AmI + BmI) Cos[
thetai] DmI -
R2 (\[Pi]^2 - 4 \[Beta]^2) (R2^2 AmI + BmI) Cos[
thetai + \[Beta]] DmI -
2 \[Pi]^2 R2^4 AmI Cos[2 thetai] DmI +
2 R2^4 \[Beta]^2 AmI Cos[2 thetai] DmI -
2 \[Pi]^2 BmI Cos[2 thetai] DmI +
2 \[Beta]^2 BmI Cos[2 thetai] DmI -
2 \[Pi]^2 R2^4 AmI Cos[2 (thetai + \[Beta])] DmI +
2 R2^4 \[Beta]^2 AmI Cos[2 (thetai + \[Beta])] DmI -
2 \[Pi]^2 BmI Cos[2 (thetai + \[Beta])] DmI +
2 \[Beta]^2 BmI Cos[2 (thetai + \[Beta])] DmI[
2] + \[Pi]^2 R2^3 AmI CmI Sin[thetai] -
4 R2^3 \[Beta]^2 AmI CmI Sin[thetai] + \[Pi]^2 R2 BmI[
1] CmI Sin[thetai] -
4 R2 \[Beta]^2 BmI CmI Sin[thetai] +
2 \[Pi]^2 R2^4 AmI CmI Sin[2 thetai] -
2 R2^4 \[Beta]^2 AmI CmI Sin[2 thetai] +
2 \[Pi]^2 BmI CmI Sin[2 thetai] -
2 \[Beta]^2 BmI CmI Sin[2 thetai] + \[Pi]^2 R2^3 AmI[
1] CmI Sin[thetai + \[Beta]] -
4 R2^3 \[Beta]^2 AmI CmI Sin[
thetai + \[Beta]] + \[Pi]^2 R2 BmI CmI Sin[
thetai + \[Beta]] -
4 R2 \[Beta]^2 BmI CmI Sin[thetai + \[Beta]] +
2 \[Pi]^2 R2^4 AmI CmI Sin[2 (thetai + \[Beta])] -
2 R2^4 \[Beta]^2 AmI CmI Sin[2 (thetai + \[Beta])] +
2 \[Pi]^2 BmI CmI Sin[2 (thetai + \[Beta])] -
2 \[Beta]^2 BmI CmI Sin[2 (thetai + \[Beta])]),
Aki == (1/(R2^2 (4 \[Pi]^4 - 5 \[Pi]^2 \[Beta]^2 + \[Beta]^4)))
2 \[Beta] Sin[\[Beta]/
2] (-2 R2 (\[Pi]^2 - \[Beta]^2) (R2^2 AmI + BmI) CmI Cos[
thetai + \[Beta]/2] - (4 \[Pi]^2 - \[Beta]^2) (R2^4 AmI +
BmI) CmI Cos[1/2 (4 thetai + \[Beta])] -
4 \[Pi]^2 R2^4 AmI CmI Cos[2 thetai + (3 \[Beta])/2] +
R2^4 \[Beta]^2 AmI CmI Cos[2 thetai + (3 \[Beta])/2] -
4 \[Pi]^2 BmI CmI Cos[
2 thetai + (3 \[Beta])/2] + \[Beta]^2 BmI CmI Cos[
2 thetai + (3 \[Beta])/2] -
2 \[Pi]^2 R2^3 AmI DmI Sin[thetai + \[Beta]/2] +
2 R2^3 \[Beta]^2 AmI DmI Sin[thetai + \[Beta]/2] -
2 \[Pi]^2 R2 BmI DmI Sin[thetai + \[Beta]/2] +
2 R2 \[Beta]^2 BmI DmI Sin[thetai + \[Beta]/2] -
4 \[Pi]^2 R2^4 AmI DmI Sin[1/2 (4 thetai + \[Beta])] +
R2^4 \[Beta]^2 AmI DmI Sin[1/2 (4 thetai + \[Beta])] -
4 \[Pi]^2 BmI DmI Sin[
1/2 (4 thetai + \[Beta])] + \[Beta]^2 BmI DmI Sin[
1/2 (4 thetai + \[Beta])] -
4 \[Pi]^2 R2^4 AmI DmI Sin[2 thetai + (3 \[Beta])/2] +
R2^4 \[Beta]^2 AmI DmI Sin[2 thetai + (3 \[Beta])/2] -
4 \[Pi]^2 BmI DmI Sin[
2 thetai + (3 \[Beta])/2] + \[Beta]^2 BmI DmI Sin[
2 thetai + (3 \[Beta])/2])}


First I use Solve and the solution is empty.

In fact when using FindRoot I do not know the initial values, so I do not know if the initial values I have set is proper

Then I use FindRoot and the tips is FindRoot::jsing: Encountered a singular Jacobian at the point.......... Try perturbing the initial point(s). and

FindRoot::lstol: The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the merit function. You may need more than MachinePrecision digits of working precision to meet these tolerances.

Next I have referred the post enter link description here and it seems that the answer in that post is effective,

and changed my code as the listed forms, however the answer is not ideal, so how to solve it?

system = equa;
vars = {AmI, AmI, BmI, BmI, CmI, CmI, DmI,
DmI, Aki, Aki};
parameters = {u0 -> 4*\[Pi]*10^(-7), R1 -> 0.04, R2 -> 0.07,
R3 -> 0.08, \[Beta] -> \[Pi]/6, k1 -> 10^5, L -> 0.1, N1 -> 50,
K -> 50, thetai -> \[Pi]/6};

givenPoint = {{AmI, 0.1}, {AmI, 0.1}, {BmI, 0.1}, {BmI,
0.1}, {CmI, 0.1}, {CmI, 0.1}, {DmI, 0.1}, {DmI,
0.1}, {Aki, 0.1 + I}, {Aki, 0.1 + I}};

s5 = FindRoot[SetPrecision[system /. parameters, 24],
SetPrecision[givenPoint, 24], PrecisionGoal -> 8,
WorkingPrecision -> 24]


FindRoot::precw: The precision of the argument function ....is less than WorkingPrecision. and

FindRoot::jsing: Encountered a singular Jacobian at the point .... Try perturbing the initial point(s).

It looks like your system equa has no solution?

Try NMinimize:

NMinimize[{1, equa /. parameters}, vars]
(*Obtained solution does not satisfy the following constraints...*)


and

NMinimize[# . # &[equa /. Equal -> Subtract /. parameters], vars]
(*{2009.13, {AmI -> -0.0644122, AmI -> 0.0940232,
BmI -> 0.41307, BmI -> 0.0445729, CmI -> 0.048511,
CmI -> 0.0527291, DmI -> -0.155755, DmI -> -0.237591,
Aki -> -7.64135*10^-10, Aki -> -1.55336*10^-18}}*)


To support @Ulrich's answer, Solve indicates there is no solution:

Solve[system /. parameters, vars]
(*  {}  *)
`