'solve' function lost an answer?

[loopxia]

In [1]: eq=(2-x)**(1/S(3))+sqrt(x-1)-1

In [2]: eq
Out[2]:
3 _______     _______
╲╱ 2 - x  + ╲╱ x - 1  - 1

In [3]: solve(eq)
Out[3]: [1, 2]

why another answer x=10 is lost?
is this a bug or I lost something?

'solveset' is similar lost answer 10.

[oscarbenjamin]

SymPy uses the principle root so the cube root of -8 is not -2:

In [8]: cbrt(-8)
Out[8]: 
  3 ____
2⋅╲╱ -1 

In [9]: cbrt(-8).n()
Out[9]: 1.0 + 1.73205080756888⋅ⅈ

You should be able to solve this using real_root but that crashes out with recursion error:

In [16]: x = Symbol('x', real=True)

In [17]: eq = real_root(2-x, 3) + sqrt(x-1)-1

In [18]: eq
Out[18]: 
  _______   3 _________                
╲╱ x - 1  + ╲╱ │x - 2│ ⋅sign(2 - x) - 1

In [19]: solve(eq, x)
...
RecursionError: maximum recursion depth exceeded

An explicit Piecewise rewrite works:

In [20]: eq.rewrite(Piecewise)
Out[20]: 
            ⎛⎧3 _______               ⎞ ⎛⎧1   for x - 2 < 0⎞    
  _______   ⎜⎪╲╱ x - 2   for x - 2 ≥ 0⎟ ⎜⎪                 ⎟    
╲╱ x - 1  + ⎜⎨                        ⎟⋅⎜⎨-1  for x - 2 > 0⎟ - 1
            ⎜⎪3 _______               ⎟ ⎜⎪                 ⎟    
            ⎝⎩╲╱ 2 - x     otherwise  ⎠ ⎝⎩0     otherwise  ⎠    

In [21]: piecewise_fold(eq.rewrite(Piecewise))
Out[21]: 
⎧  3 _______     _______                   
⎪- ╲╱ x - 2  + ╲╱ x - 1  - 1    for x > 2  
⎪                                          
⎪         _______                          
⎨       ╲╱ x - 1  - 1         for x - 2 ≥ 0
⎪                                          
⎪ 3 _______     _______                    
⎪ ╲╱ 2 - x  + ╲╱ x - 1  - 1     otherwise  
⎩                                          

In [22]: solve(piecewise_fold(eq.rewrite(Piecewise)), x)
Out[22]: [1, 2, 10]

The recursion error should be fixed.

[loopxia]

Hi @oscarbenjamin , thanks for your reply.
I got a different print of 'eq' and error (Sympy:1.7.1 ):

In [219]: x = Symbol('x', real=True)

In [220]: eq = real_root(2-x, 3) + sqrt(x-1)-1

In [221]: eq
Out[221]:
            ⎛⎧3 _________                           ⎞
  _______   ⎜⎪╲╱ │x - 2│ ⋅sign(2 - x)  for im(x) = 0⎟
╲╱ x - 1  + ⎜⎨                                      ⎟ - 1
            ⎜⎪       3 _______                      ⎟
            ⎝⎩       ╲╱ 2 - x            otherwise  ⎠

In [222]: solve(eq)
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-222-4c33efa621d3> in <module>
----> 1 solve(eq)

/usr/local/anaconda3/lib/python3.8/site-packages/sympy/solvers/solvers.py in solve(f, *symbols, **flags)
    954         for e in fi.find(Abs):
    955             if e.has(*symbols):
--> 956                 raise NotImplementedError('solving %s when the argument '
    957                     'is not real or imaginary.' % e)
    958

NotImplementedError: solving Abs(x - 2) when the argument is not real or imaginary.

In [224]: eq.rewrite(Piecewise)
Out[224]:
            ⎛⎧3 _________                           ⎞
  _______   ⎜⎪╲╱ │x - 2│ ⋅sign(2 - x)  for im(x) = 0⎟
╲╱ x - 1  + ⎜⎨                                      ⎟ - 1
            ⎜⎪       3 _______                      ⎟
            ⎝⎩       ╲╱ 2 - x            otherwise  ⎠

In [225]: piecewise_fold(_)
Out[225]:
⎧  _______   3 _________
⎪╲╱ x - 1  + ╲╱ │x - 2│ ⋅sign(2 - x) - 1  for im(x) = 0
⎨
⎪       3 _______     _______
⎩       ╲╱ 2 - x  + ╲╱ x - 1  - 1           otherwise

In [226]: solve(_)
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
<ipython-input-226-2ba847fb376d> in <module>
----> 1 solve(_)

/usr/local/anaconda3/lib/python3.8/site-packages/sympy/solvers/solvers.py in solve(f, *symbols, **flags)
    954         for e in fi.find(Abs):
    955             if e.has(*symbols):
--> 956                 raise NotImplementedError('solving %s when the argument '
    957                     'is not real or imaginary.' % e)
    958

NotImplementedError: solving Abs(x - 2) when the argument is not real or imaginary.

[oscarbenjamin]

Oh, sorry I missed this part (edited now):

x = Symbol('x', real=True)

[loopxia]

oops, that's my fault. I missed it.
Thank you @oscarbenjamin

[loopxia]

@oscarbenjamin don't forget function solveset cannot solve it too. Thank you.

[ghost]

I think it should iterate over all the roots, and then select the ones which satisfy the relation, instead of only taking the principal roots of the number. This seems necessary since solveset/solve should in principle give all the possible solutions.

[oscarbenjamin]

The definition in sympy of (-1)**(1/3) is that it means the principal root. It is not for solve to apply a different interpretation of the meaning of rational powers.

[ghost]

Which leads me to this:
Case 1:

>>> x = symbols('x')
>>> f = x**4 - 1
>>> solveset(f,x)
FiniteSet(-1, 1, I, -I)

Case 2:

>>> f = x -root(1,4)
>>> solveset(f,x)
FiniteSet(1)

I would expect solveset to give all the possible answers since, I haven't explicitly demanded real or positive roots

[oscarbenjamin]

I would expect solveset to give all the possible answers since, I haven't explicitly demanded real or positive roots

Look at what you are passing in to solveset:

In [1]: root(1, 4)                                                              
Out[1]: 1

In [2]: solveset(x - 1, x)                                                      
Out[2]: {1}

https://docs.sympy.org/latest/modules/functions/elementary.html#sympy.functions.elementary.miscellaneous.root

[ghost]

@oscarbenjamin Thanks a lot for clarifying! , I should definitely refer to the docs more.

[ghost]

sympy/sympy/solvers/solveset.py

Lines 873 to 883 in b564d9b

	if isinstance(result, Complement) or isinstance(result,ConditionSet):
	solution_set = result
	else:
	f_set = [] # solutions for FiniteSet
	c_set = [] # solutions for ConditionSet
	for s in result:
	if checksol(f, symbol, s):
	f_set.append(s)
	else:
	c_set.append(s)
	solution_set = FiniteSet(*f_set) + ConditionSet(symbol, Eq(f, 0), FiniteSet(*c_set))

I think i found, where exactly the problem lies. Actually sympy is able to find all the possible solutions, but checksol rejects the value 10 because (-8)**Rational(1,3) by default evaluates to the value 1 + 1.7320i . I believe that something has to be done on evalf so that it allows evaluation to other possible roots as well, maybe a flag that forces evalf to not just use the principal value?
@oscarbenjamin what are your thoughts?

[oscarbenjamin]

I've already explained above:

SymPy uses the principal root so the cube root of -8 is not -2:

That means that 10 is not a solution of the equation.

[ghost]

I've already explained above:

SymPy uses the principal root so the cube root of -8 is not -2:

That means that 10 is not a solution of the equation.

the example given below is contrary to what we would expect , if we consider sympy to evaluate only principal roots

>>> x = symbols('x')
>>> eq = (x-2)**Rational(1,3) + 2
>>> solveset(eq,x,S.Reals)
FiniteSet(-6)

[oscarbenjamin]

the example given below is contrary to what we would expect , if we consider sympy to evaluate only principal roots

Okat that result is incorrect.

[smichr]

Okat that result is incorrect.

real_root can be used to get the real, rather than principle, root. So when the domain is Real then that is used there, I believe:

>>> solveset(eq,x)
EmptySet
>>> solveset(eq,x,Reals)
FiniteSet(-6)
>>> ((-8)**Rational(1,3))
2*(-1)**(1/3)
>>> real_root(_)
-2