Solve system of equations with NSolve?

Questions
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1

I need help with NSolve.. below I explain the maths of what I am doing so the question becomes clear..

I have two points, P1 and P2, defined in Euler angles relative to the origin of the spherical coordinates frame:

P1[alpha_,beta_,gamma_]:={-0.5207928328826381` Cos[alpha] Sin[beta] - 
   0.7566899220728581` (Cos[beta] Cos[alpha] Cos[gamma] - Sin[alpha] Sin[gamma]), -0.5207928328826381` Sin[beta] Sin[alpha] - 0.7566899220728581` (Cos[beta] Cos[gamma] Sin[alpha] +Cos[alpha] Sin[gamma]), -0.5207928328826381`Cos[beta] + 0.7566899220728581` Cos[gamma] Sin[beta]};

P2[alpha_,beta_,gamma_]:={-0.5207928328826381` Cos[alpha] Sin[beta] + 
       0.7566899220728581` (Cos[beta] Cos[alpha] Cos[gamma] - Sin[alpha] Sin[gamma]), -0.5207928328826381` Sin[beta] Sin[alpha] + 0.7566899220728581` (Cos[beta] Cos[gamma] Sin[alpha] +Cos[alpha] Sin[gamma]), -0.5207928328826381`Cos[beta] - 0.7566899220728581` Cos[gamma] Sin[beta]};

I want to shift these two points to a general position, so I define the spherical coordinates of a centre as

R[r_,theta_,phi_]:=r {Sin[theta] Cos[phi], Sin[theta] Sin[phi], Cos[theta]};

However I change the Euler angles, I want to keep the radial distance of the R fixed at 0.0949009, in other words, I always have

  R[0.0949009,theta,phi]

and at the same time I want the radial distances of the two points, P1 and P2 equal 0.753964 at all values of Euler angles, so I solve for theta and phi values that satisfy this condition

functoSolve[alpha_, beta_, gamma_] := Module[{a1, a2}, a1 = Chop[NSolve[{(P1[alpha, beta, gamma]+ R[0.0949009,theta,phi]).(P1[alpha, beta, gamma]+ R[0.0949009,theta,phi]) ==0.753964, (P2[alpha, beta, gamma]+ R[0.0949009,theta,phi]).(P2[alpha, beta, gamma]+ R[0.0949009,theta,phi]) ==0.753964}, {theta,phi}], 10^-7]; a2 = Union[a1, SameTest -> (Abs[ (theta /. #1) - (theta /. #2)] < 10^-6 && Abs[(phi. #1) - (phi /. #2)] < 10^-6 &)]; Chop[P[0.0949009,theta,phi] /. a2[[1]], 10^-7]]

where a2 picks the first solution since NSolve returns all possible solution. So the functoSolve is the function I need to use many times to find theta and gamma for any {alpha,beta,gamma} combination. It works for when I put {alpha, beta, gamma}={0,0,0}. However, for any other combination, such as {10 Degree, 4 Degree, 0},NSolve fails to find the solution and returns the error messages

NSolve::ifun: Inverse functions are being used by NSolve, so some solutions may not be found; use Reduce for complete solution information. >>

NSolve::svars: Equations may not give solutions for all "solve" variables. >>

I cannot understand why Mathematica complains (!!) the maths is OK and the problem is simple enough. I do not need a high precision solution but I need a solution.

I really need help fix this problem.

Thanks,

eft

2

It’s better to use CodeSample (the first icon in the row) as copying as text can cause problems. For example, there seems to be a missing ] at near the end of your code. In any case, you’ve got one equation and two variables, which probably explains why NSolve doesn’t do what you expect.

3

Yes, indeed some parenthesis are missing. If corrected/added (hopefully as intended in the OP it appears to work.

NSolve[(-0.38148299350912984 + 
      0.09490089588967025 Cos[x])^2 + (-0.8335932784285498 + 
      0.09490089588967025 Cos[y] Sin[x])^2 + (-0.05829052041804752 + 
      0.09490089588967025 Sin[x] Sin[y])^2 == 0.753964, {x, y}]

The solution is somewhat long and does not seem to contribute to this. Actually, I suppose that the thread will be closed because the question is only caused by a simple typo.

Anyway, here’s a representation of the solutions:

ContourPlot[(-0.38148299350912984 + 
      0.09490089588967025 Cos[x])^2 + (-0.8335932784285498 + 
      0.09490089588967025 Cos[y] Sin[x])^2 + (-0.05829052041804752 + 
      0.09490089588967025 Sin[x] Sin[y])^2 == 0.753964, {x, -10, 10}, {y, -15, 15}, PlotPoints -> 50]

ScreenShot2016-08-20at22.57.36.png
ScreenShot2016-08-20at22.57.36.png375×366 19.4 KB

Cheers,

Marco

PS: Note that Mathematica suggests to use Reduce instead of NSolve. You should consider to follow that advice.

4

well, I have to apologise for the ill expressed question. I chopped half the equation while editing.. I have been trying to add the full equations using CodeSample with no successful so far..Would you kindly explain how I should use it?

eft

5

Hi,

this here is a good starting point. The moderation team explain how to post and format everything correctly:

Cheers,

Marco

6

I have edited the question and added details of the problem. I really hope to get help come over it.

eft

7

The documentation on DSolve says that “NSolve deals primarily with linear and polynomial equations.” You may try FindRoot if your problem is mainly numerical.