Continue nonlinsolve :
PR #11111
 Removed the dependency of old solvers
_invert
. It was basically helpful for removingsqrt
or Rational power. Now usingunrad
and little changes insubstitution
function can handle the system havningsqrt
orcube root
.
Continue  Diophantine in Solveset :
PR 11234

Some points we discussed :

Solveset
should return all the solution, so we need to check for which kind of eq.diophantine
returns all the solution. We can get the equation type usingclassify_diop
defined indiophantine.py
. 
If there is the case where
diophantine
doesn’t return all the solutions then just returnConditionSet
or try to make all solution (e.g. if just need to permute sign in values then usepermute_signs
).


After reading the diophantine Documentation and
diophantine
doctests and examples, I found that : Currently, following five types of Diophantine equations can be solved using
~sympy.solvers.diophantine.diophantine
and other helper functions of the Diophantine module.
 Linear Diophantine equations:
a1*x1 + a2*x2 + ... + an*xn = b
. General binary quadratic equation:
ax^2 + bxy + cy^2 + dx + ey + f = 0
 Homogeneous ternary quadratic equation:
ax^2 + by^2 + cz^2 + dxy + eyz + fzx = 0
 Extended Pythagorean equation:
a_{1} * x_{1}^2 + a_{2} * x_{2}^2 + .... + a_{n} * x_{n}^2 = a_{n+1} * x_{n+1}^2
 General sum of squares:
x_{1}^2 + x_{2}^2 + ... + x_{n}^2 = k

If I am correct then
Diophantine
returns all the soln if eq is Linear Diophantine equations, General binary quadratic equation and Homogeneous ternary quadratic equation. I tried some Linear Diophantine equations and cross checked with online Diophantine solvers(one of the solver is here). 
In last 2 (Extended Pythagorean equation and General sum of squares) we need to do
permute_signs
to get all the soln.
E.g.
``` >>> from sympy.utilities.iterables import permute_signs >>> list(permute_signs((1, 12))) [(1, 12), (1, 12), (1, 12), (1, 12)] ```
In general if all variables have even powers then we should do
permute_signs
.Diophantine
solves the factors of the eq so if we have eq like thisx**2  y**2
, diophantine solves factorsx + y
andx  y
.``` >>> from sympy.solvers.diophantine import diophantine >>> from sympy.abc import x, y, z >>> diophantine(x**2  y**2) set([(t_0, t_0), (t_0, t_0)]) ```
so we should check even powers and permute sign.
other examples :
```
>>> from sympy.solvers.diophantine import diop_general_sum_of_squares >>> from sympy.abc import a, b, c, d, e, f >>> diop_general_sum_of_squares(a**2 + b**2 + c**2 + d**2 + e**2  2345) set([(15, 22, 22, 24, 24)]) >>> from sympy.solvers.diophantine import diop_general_pythagorean >>> diop_general_pythagorean(a**2 + b**2 + c**2  d**2) (m1**2 + m2**2  m3**2, 2*m1*m3, 2*m2*m3, m1**2 + m2**2 + m3**2) >>> diop_general_pythagorean(9*a**2  4*b**2 + 16*c**2 + 25*d**2 + e**2) (10*m1**2 + 10*m2**2 + 10*m3**2  10*m4**2, 15*m1**2 + 15*m2**2 + 15*m3**2 + 15*m4**2, 15*m1*m4, 12*m2*m4, 60*m3*m4) >>> from sympy.solvers.diophantine import diop_general_sum_of_even_powers >>> diop_general_sum_of_even_powers(a**4 + b**4  (2**4 + 3**4)) set([(2, 3)])
```
In above these types of cases we need
permute_signs
. If we check these solution you can see thatpermute_signs
is needed when solutions is not parameterized.e.g.
```
>>> from sympy.solvers.diophantine import diophantine >>> from sympy.abc import x, y, z >>> diophantine(x**2  y**2) set([(t_0, t_0), (t_0, t_0)])
```
solution
set([(t_0, t_0), (t_0, t_0)])
is same asset([(t_0, t_0), (t_0, t_0), (t_0, t_0), (t_0, t_0)])
.(becauset_0
can take any integer value.)I discussed these things with @thilinarmtb (He have worked on
diophantine
. Blog link : https://thilinaatsympy.wordpress.com/). Main points are :Only the linear solver is incomplete, the algorithm in Diophantine module should be fixed (`permute_signs` or something like that won't help). We can use `permute_signs` when we have even powers in all variables. We should update Diophantine module to use permute sign. But we should not returns `ConditionSet` for linear diophantine eq. because for linear diophantine eq `diophantine()`returns parameterized solution which is complete most of the time.

classify_diop
can returns thesediop_type
: linear
 univariate
 binary_quadratic
 inhomogeneous_ternary_quadratic
 homogeneous_ternary_quadratic_normal
 homogeneous_ternary_quadratic
 inhomogeneous_general_quadratic
 inhomogeneous_general_quadratic
 homogeneous_general_quadratic
 general_sum_of_squares
 general_pythagorean
 cubic_thue
 general_sum_of_even_powers
If the equation type is none of these then
solveset_integers
should returnsConditionSet
.Because currentlydiophantine
can handle these kinds of eq only.  Currently, following five types of Diophantine equations can be solved using

PR Update diophantine to get some missing solution: 11334
Continue simplified Trigonometric eq solution :
PR #11188

Shifted the
_union_simplify
function code to_union
infancyset/ImageSet
with some changes. 
Still facing problem (It sometimes pass all cases sometimes fail).
Meanwhile :
 I found a problem in
checksol
defined insympy/solvers/solvers.py
and opened a PR that fixes the small issue, #11339
continue…
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