Why I can't solve these equations using some specific parameters? (Sympy)

Question

Here's the code with equations I'm trying to solve using sympy:

# Import sympy
from math import remainder, tau
from sympy import *
from sympy.solvers import solve

# Define the parameters
nx, ny, nz = [0,0,1]
a,b,c = [0,0,0]
d = 1
t = 9

# Solve the equations
theta = remainder(2*t*sqrt(b**2+c**2+d**2), tau)
delta, beta, gamma = symbols('delta beta gamma')
beta = 7 # Randomly assigned. 
eq1 = Eq(tan((delta+beta)/2),nz*tan(theta/2))
eq2 = Eq(ny*tan((delta-beta)/2),nx)
eq3 = Eq(cos((delta+beta)/2)*cos(gamma/2),cos(theta/2))
result = solve([eq1, eq2, eq3], [delta, beta, gamma])

My question is based on the last parameter t. For the current value, the function should be solvable but it doesn't return any results. If I change the value of t to some other values, then I can get the result. For instance, if t=99, then the results look like

[(-12.0619298297468, 9.00000000000000, 7.46580816699663e-8*I), 
(-12.0619298297468, 9.00000000000000, 12.5663706143592 - 7.46580045560101e-8*I)]

Why I'm unable to get the result from the first t value? How can I fix the issue? Thanks!!

It's better if you post a single complete piece of code to demonstrate the problem — Oscar Benjamin, Mar 12 '21 at 13:56
@Oscar Benjamin Sorry I had a typo in my original code, just fixed — ZR-, Mar 12 '21 at 15:20
You create `beta` as a `symbol` but then replace with a number. — hpaulj, Mar 12 '21 at 15:58
@hpaulj Thanks for the comment! In my current setting `beta` has to be specified to make `delta` meaningful. — ZR-, Mar 12 '21 at 16:44
If `nx` and `ny` are both zero then the second equation is `0=0` so you've really only got two equations for 3 unknowns — Oscar Benjamin, Mar 12 '21 at 17:50
`beta` and `eq2` don't do anything with these values. How about making `theta` a symbol, get the `result`, and then evaluate with a specific numeric values (via `.subs`)? — hpaulj, Mar 13 '21 at 00:15
Wait are you asking to solve for `9` as one of the unknowns? — Oscar Benjamin, Mar 13 '21 at 01:26
@Oscar Benjamin I'm asking if I could obtain the solutions of `[delta, beta, gamma]` if I set `t=9`:) In my jupyter notebook that doesn't work. — ZR-, Mar 13 '21 at 02:22
@hpaulj Thanks, what do you mean by 'evaluate with a specific numeric values'? — ZR-, Mar 13 '21 at 02:24

hpaulj · Accepted Answer · 2021-03-13T03:28:05.663

 In [59]: 
    ...: delta, gamma = symbols('delta gamma')
    ...: theta = symbols('theta')
    ...: beta = 9
    ...: eq1 = Eq(tan((delta+beta)/2),nz*tan(theta/2))
    ...: #eq2 = Eq(ny*tan((delta-beta)/2),nx)
    ...: eq3 = Eq(cos((delta+beta)/2)*cos(gamma/2),cos(theta/2))
In [60]: eq1
Out[60]: 
   ⎛δ   9⎞      ⎛θ⎞
tan⎜─ + ─⎟ = tan⎜─⎟
   ⎝2   2⎠      ⎝2⎠
In [61]: eq3
Out[61]: 
   ⎛γ⎞    ⎛δ   9⎞      ⎛θ⎞
cos⎜─⎟⋅cos⎜─ + ─⎟ = cos⎜─⎟
   ⎝2⎠    ⎝2   2⎠      ⎝2⎠
In [62]: result = solve([eq1, eq3], [delta, gamma])
In [63]: result
Out[63]: 
⎡⎛                            ⎛       ____________       ⎞      ⎞  ⎛                          ⎛       
⎢⎜      ⎛   ⎛θ⎞⎞              ⎜      ╱     1          ⎛θ⎞⎟      ⎟  ⎜      ⎛   ⎛θ⎞⎞            ⎜      ╱
⎢⎜2⋅atan⎜tan⎜─⎟⎟ - 9, - 2⋅acos⎜√2⋅  ╱  ────────── ⋅cos⎜─⎟⎟ + 4⋅π⎟, ⎜2⋅atan⎜tan⎜─⎟⎟ - 9, 2⋅acos⎜√2⋅  ╱ 
⎣⎝      ⎝   ⎝2⎠⎠              ⎝   ╲╱   cos(θ) + 1     ⎝2⎠⎠      ⎠  ⎝      ⎝   ⎝2⎠⎠            ⎝   ╲╱  

____________       ⎞⎞⎤
     1          ⎛θ⎞⎟⎟⎥
 ────────── ⋅cos⎜─⎟⎟⎟⎥
 cos(θ) + 1     ⎝2⎠⎠⎠⎦

Now evaluate some of the result terms with specific theta values.

In [64]: result[0][0].subs({theta:-0.849})
Out[64]: -9.84900000000000
In [65]: result[0][1].subs({theta:-0.849})
Out[65]: -2⋅acos(0.707106781186547⋅√2) + 4⋅π

I don't see any clues as to why result would be empty (no solution) for certain values of t, and hence theta. But it should be easier to explore those issues with this algebraic solution.

In [66]: t=9
In [67]: remainder(2*t*sqrt(b**2+c**2+d**2), tau)
Out[67]: -0.8495559215387587

That acos term might be the problem:

acos(0.707106781186547⋅√2)

That's approximately acos(1), but nan if the acos arg is at all larger than 1.

With numpy:

In [299]: def foo(theta):
     ...:     return np.sqrt(2)*np.sqrt(1/(np.cos(theta)+1))*np.cos(theta/2)
     ...: 
In [300]: foo(0)
Out[300]: 1.0000000000000002
In [301]: foo(-1)
Out[301]: 1.0000000000000002
In [302]: foo(-.89)
Out[302]: 1.0000000000000002
In [303]: foo(-3.0)
Out[303]: 0.999999999999998

So the acos argument is algebraically 1, but numerically may be a bit larger, resulting in a nan.

Thanks a lot for the detailed answer! That's super helpful:) — ZR-, Mar 13 '21 at 06:47

Why I can't solve these equations using some specific parameters? (Sympy)

1 Answers1