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""" 

This module contains functions to: 

 

- solve a single equation for a single variable, in any domain either real or complex. 

 

- solve a system of linear equations with N variables and M equations. 

 

- solve a system of Non Linear Equations with N variables and M equations 

""" 

from __future__ import print_function, division 

 

from sympy.core.sympify import sympify 

from sympy.core import S, Pow, Dummy, pi, Expr, Wild, Mul, Equality 

from sympy.core.containers import Tuple 

from sympy.core.facts import InconsistentAssumptions 

from sympy.core.numbers import I, Number, Rational, oo 

from sympy.core.function import (Lambda, expand_complex, AppliedUndef, Function, 

expand_log) 

from sympy.core.relational import Eq 

from sympy.core.symbol import Symbol 

from sympy.simplify.simplify import simplify, fraction, trigsimp 

from sympy.simplify import powdenest 

from sympy.functions import (log, Abs, tan, cot, sin, cos, sec, csc, exp, 

acos, asin, acsc, asec, arg, 

piecewise_fold, Piecewise) 

from sympy.functions.elementary.trigonometric import (TrigonometricFunction, 

HyperbolicFunction) 

from sympy.functions.elementary.miscellaneous import real_root 

from sympy.logic.boolalg import And 

from sympy.sets import (FiniteSet, EmptySet, imageset, Interval, Intersection, 

Union, ConditionSet, ImageSet, Complement, Contains) 

from sympy.sets.sets import Set 

from sympy.matrices import Matrix 

from sympy.polys import (roots, Poly, degree, together, PolynomialError, 

RootOf, factor) 

from sympy.solvers.solvers import (checksol, denoms, unrad, 

_simple_dens, recast_to_symbols, _ispow) 

from sympy.solvers.polysys import solve_poly_system 

from sympy.solvers.inequalities import solve_univariate_inequality 

from sympy.utilities import filldedent 

from sympy.utilities.iterables import numbered_symbols 

from sympy.calculus.util import periodicity, continuous_domain 

from sympy.core.compatibility import ordered, default_sort_key, is_sequence 

 

from types import GeneratorType 

 

 

def _masked(f, *atoms): 

"""Return ``f``, with all objects given by ``atoms`` replaced with 

Dummy symbols, ``d``, and the list of replacements, ``(d, e)``, 

where ``e`` is an object of type given by ``atoms`` in which 

any other instances of atoms have been recursively replaced with 

Dummy symbols, too. The tuples are ordered so that if they are 

applied in sequence, the origin ``f`` will be restored. 

 

Examples 

======== 

 

>>> from sympy import cos 

>>> from sympy.abc import x 

>>> from sympy.solvers.solveset import _masked 

 

>>> f = cos(cos(x) + 1) 

>>> f, reps = _masked(cos(1 + cos(x)), cos) 

>>> f 

_a1 

>>> reps 

[(_a1, cos(_a0 + 1)), (_a0, cos(x))] 

>>> for d, e in reps: 

... f = f.xreplace({d: e}) 

>>> f 

cos(cos(x) + 1) 

""" 

sym = numbered_symbols('a', cls=Dummy, real=True) 

mask = [] 

for a in ordered(f.atoms(*atoms)): 

for i in mask: 

a = a.replace(*i) 

mask.append((a, next(sym))) 

for i, (o, n) in enumerate(mask): 

f = f.replace(o, n) 

mask[i] = (n, o) 

mask = list(reversed(mask)) 

return f, mask 

 

 

def _invert(f_x, y, x, domain=S.Complexes): 

r""" 

Reduce the complex valued equation ``f(x) = y`` to a set of equations 

``{g(x) = h_1(y), g(x) = h_2(y), ..., g(x) = h_n(y) }`` where ``g(x)`` is 

a simpler function than ``f(x)``. The return value is a tuple ``(g(x), 

set_h)``, where ``g(x)`` is a function of ``x`` and ``set_h`` is 

the set of function ``{h_1(y), h_2(y), ..., h_n(y)}``. 

Here, ``y`` is not necessarily a symbol. 

 

The ``set_h`` contains the functions, along with the information 

about the domain in which they are valid, through set 

operations. For instance, if ``y = Abs(x) - n`` is inverted 

in the real domain, then ``set_h`` is not simply 

`{-n, n}` as the nature of `n` is unknown; rather, it is: 

`Intersection([0, oo) {n}) U Intersection((-oo, 0], {-n})` 

 

By default, the complex domain is used which means that inverting even 

seemingly simple functions like ``exp(x)`` will give very different 

results from those obtained in the real domain. 

(In the case of ``exp(x)``, the inversion via ``log`` is multi-valued 

in the complex domain, having infinitely many branches.) 

 

If you are working with real values only (or you are not sure which 

function to use) you should probably set the domain to 

``S.Reals`` (or use `invert\_real` which does that automatically). 

 

 

Examples 

======== 

 

>>> from sympy.solvers.solveset import invert_complex, invert_real 

>>> from sympy.abc import x, y 

>>> from sympy import exp, log 

 

When does exp(x) == y? 

 

>>> invert_complex(exp(x), y, x) 

(x, ImageSet(Lambda(_n, I*(2*_n*pi + arg(y)) + log(Abs(y))), S.Integers)) 

>>> invert_real(exp(x), y, x) 

(x, Intersection(S.Reals, {log(y)})) 

 

When does exp(x) == 1? 

 

>>> invert_complex(exp(x), 1, x) 

(x, ImageSet(Lambda(_n, 2*_n*I*pi), S.Integers)) 

>>> invert_real(exp(x), 1, x) 

(x, {0}) 

 

See Also 

======== 

invert_real, invert_complex 

""" 

x = sympify(x) 

if not x.is_Symbol: 

raise ValueError("x must be a symbol") 

f_x = sympify(f_x) 

if x not in f_x.free_symbols: 

raise ValueError("Inverse of constant function doesn't exist") 

y = sympify(y) 

if x in y.free_symbols: 

raise ValueError("y should be independent of x ") 

 

if domain.is_subset(S.Reals): 

x1, s = _invert_real(f_x, FiniteSet(y), x) 

else: 

x1, s = _invert_complex(f_x, FiniteSet(y), x) 

 

if not isinstance(s, FiniteSet) or x1 != x: 

return x1, s 

 

return x1, s.intersection(domain) 

 

 

invert_complex = _invert 

 

 

def invert_real(f_x, y, x, domain=S.Reals): 

""" 

Inverts a real-valued function. Same as _invert, but sets 

the domain to ``S.Reals`` before inverting. 

""" 

return _invert(f_x, y, x, domain) 

 

 

def _invert_real(f, g_ys, symbol): 

"""Helper function for _invert.""" 

 

if f == symbol: 

return (f, g_ys) 

 

n = Dummy('n', real=True) 

 

if hasattr(f, 'inverse') and not isinstance(f, ( 

TrigonometricFunction, 

HyperbolicFunction, 

)): 

if len(f.args) > 1: 

raise ValueError("Only functions with one argument are supported.") 

return _invert_real(f.args[0], 

imageset(Lambda(n, f.inverse()(n)), g_ys), 

symbol) 

 

if isinstance(f, Abs): 

return _invert_abs(f.args[0], g_ys, symbol) 

 

if f.is_Add: 

# f = g + h 

g, h = f.as_independent(symbol) 

if g is not S.Zero: 

return _invert_real(h, imageset(Lambda(n, n - g), g_ys), symbol) 

 

if f.is_Mul: 

# f = g*h 

g, h = f.as_independent(symbol) 

 

if g is not S.One: 

return _invert_real(h, imageset(Lambda(n, n/g), g_ys), symbol) 

 

if f.is_Pow: 

base, expo = f.args 

base_has_sym = base.has(symbol) 

expo_has_sym = expo.has(symbol) 

 

if not expo_has_sym: 

res = imageset(Lambda(n, real_root(n, expo)), g_ys) 

if expo.is_rational: 

numer, denom = expo.as_numer_denom() 

if denom % 2 == 0: 

base_positive = solveset(base >= 0, symbol, S.Reals) 

res = imageset(Lambda(n, real_root(n, expo) 

), g_ys.intersect( 

Interval.Ropen(S.Zero, S.Infinity))) 

_inv, _set = _invert_real(base, res, symbol) 

return (_inv, _set.intersect(base_positive)) 

 

elif numer % 2 == 0: 

n = Dummy('n') 

neg_res = imageset(Lambda(n, -n), res) 

return _invert_real(base, res + neg_res, symbol) 

 

else: 

return _invert_real(base, res, symbol) 

else: 

if not base.is_positive: 

raise ValueError("x**w where w is irrational is not " 

"defined for negative x") 

return _invert_real(base, res, symbol) 

 

if not base_has_sym: 

rhs = g_ys.args[0] 

if base > S.Zero: 

return _invert_real(expo, 

imageset(Lambda(n, log(n, base, evaluate=False)), g_ys), symbol) 

elif base < S.Zero: 

from sympy.core.power import integer_log 

s, b = integer_log(rhs, base) 

if b: 

return _invert_real(expo, FiniteSet(s), symbol) 

elif rhs is S.One: 

#special case: 0**x - 1 

return (expo, FiniteSet(0)) 

return (expo, S.EmptySet) 

 

 

if isinstance(f, TrigonometricFunction): 

if isinstance(g_ys, FiniteSet): 

def inv(trig): 

if isinstance(f, (sin, csc)): 

F = asin if isinstance(f, sin) else acsc 

return (lambda a: n*pi + (-1)**n*F(a),) 

if isinstance(f, (cos, sec)): 

F = acos if isinstance(f, cos) else asec 

return ( 

lambda a: 2*n*pi + F(a), 

lambda a: 2*n*pi - F(a),) 

if isinstance(f, (tan, cot)): 

return (lambda a: n*pi + f.inverse()(a),) 

 

n = Dummy('n', integer=True) 

invs = S.EmptySet 

for L in inv(f): 

invs += Union(*[imageset(Lambda(n, L(g)), S.Integers) for g in g_ys]) 

return _invert_real(f.args[0], invs, symbol) 

 

return (f, g_ys) 

 

 

def _invert_complex(f, g_ys, symbol): 

"""Helper function for _invert.""" 

 

if f == symbol: 

return (f, g_ys) 

 

n = Dummy('n') 

 

if f.is_Add: 

# f = g + h 

g, h = f.as_independent(symbol) 

if g is not S.Zero: 

return _invert_complex(h, imageset(Lambda(n, n - g), g_ys), symbol) 

 

if f.is_Mul: 

# f = g*h 

g, h = f.as_independent(symbol) 

 

if g is not S.One: 

if g in set([S.NegativeInfinity, S.ComplexInfinity, S.Infinity]): 

return (h, S.EmptySet) 

return _invert_complex(h, imageset(Lambda(n, n/g), g_ys), symbol) 

 

if hasattr(f, 'inverse') and \ 

not isinstance(f, TrigonometricFunction) and \ 

not isinstance(f, HyperbolicFunction) and \ 

not isinstance(f, exp): 

if len(f.args) > 1: 

raise ValueError("Only functions with one argument are supported.") 

return _invert_complex(f.args[0], 

imageset(Lambda(n, f.inverse()(n)), g_ys), symbol) 

 

if isinstance(f, exp): 

if isinstance(g_ys, FiniteSet): 

exp_invs = Union(*[imageset(Lambda(n, I*(2*n*pi + arg(g_y)) + 

log(Abs(g_y))), S.Integers) 

for g_y in g_ys if g_y != 0]) 

return _invert_complex(f.args[0], exp_invs, symbol) 

 

return (f, g_ys) 

 

 

def _invert_abs(f, g_ys, symbol): 

"""Helper function for inverting absolute value functions. 

 

Returns the complete result of inverting an absolute value 

function along with the conditions which must also be satisfied. 

 

If it is certain that all these conditions are met, a `FiniteSet` 

of all possible solutions is returned. If any condition cannot be 

satisfied, an `EmptySet` is returned. Otherwise, a `ConditionSet` 

of the solutions, with all the required conditions specified, is 

returned. 

 

""" 

if not g_ys.is_FiniteSet: 

# this could be used for FiniteSet, but the 

# results are more compact if they aren't, e.g. 

# ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n}) vs 

# Union(Intersection(Interval(0, oo), {n}), Intersection(Interval(-oo, 0), {-n})) 

# for the solution of abs(x) - n 

pos = Intersection(g_ys, Interval(0, S.Infinity)) 

parg = _invert_real(f, pos, symbol) 

narg = _invert_real(-f, pos, symbol) 

if parg[0] != narg[0]: 

raise NotImplementedError 

return parg[0], Union(narg[1], parg[1]) 

 

# check conditions: all these must be true. If any are unknown 

# then return them as conditions which must be satisfied 

unknown = [] 

for a in g_ys.args: 

ok = a.is_nonnegative if a.is_Number else a.is_positive 

if ok is None: 

unknown.append(a) 

elif not ok: 

return symbol, S.EmptySet 

if unknown: 

conditions = And(*[Contains(i, Interval(0, oo)) 

for i in unknown]) 

else: 

conditions = True 

n = Dummy('n', real=True) 

# this is slightly different than above: instead of solving 

# +/-f on positive values, here we solve for f on +/- g_ys 

g_x, values = _invert_real(f, Union( 

imageset(Lambda(n, n), g_ys), 

imageset(Lambda(n, -n), g_ys)), symbol) 

return g_x, ConditionSet(g_x, conditions, values) 

 

 

def domain_check(f, symbol, p): 

"""Returns False if point p is infinite or any subexpression of f 

is infinite or becomes so after replacing symbol with p. If none of 

these conditions is met then True will be returned. 

 

Examples 

======== 

 

>>> from sympy import Mul, oo 

>>> from sympy.abc import x 

>>> from sympy.solvers.solveset import domain_check 

>>> g = 1/(1 + (1/(x + 1))**2) 

>>> domain_check(g, x, -1) 

False 

>>> domain_check(x**2, x, 0) 

True 

>>> domain_check(1/x, x, oo) 

False 

 

* The function relies on the assumption that the original form 

of the equation has not been changed by automatic simplification. 

 

>>> domain_check(x/x, x, 0) # x/x is automatically simplified to 1 

True 

 

* To deal with automatic evaluations use evaluate=False: 

 

>>> domain_check(Mul(x, 1/x, evaluate=False), x, 0) 

False 

""" 

f, p = sympify(f), sympify(p) 

if p.is_infinite: 

return False 

return _domain_check(f, symbol, p) 

 

 

def _domain_check(f, symbol, p): 

# helper for domain check 

if f.is_Atom and f.is_finite: 

return True 

elif f.subs(symbol, p).is_infinite: 

return False 

else: 

return all([_domain_check(g, symbol, p) 

for g in f.args]) 

 

 

def _is_finite_with_finite_vars(f, domain=S.Complexes): 

""" 

Return True if the given expression is finite. For symbols that 

don't assign a value for `complex` and/or `real`, the domain will 

be used to assign a value; symbols that don't assign a value 

for `finite` will be made finite. All other assumptions are 

left unmodified. 

""" 

def assumptions(s): 

A = s.assumptions0 

A.setdefault('finite', A.get('finite', True)) 

if domain.is_subset(S.Reals): 

# if this gets set it will make complex=True, too 

A.setdefault('real', True) 

else: 

# don't change 'real' because being complex implies 

# nothing about being real 

A.setdefault('complex', True) 

return A 

 

reps = {s: Dummy(**assumptions(s)) for s in f.free_symbols} 

return f.xreplace(reps).is_finite 

 

 

def _is_function_class_equation(func_class, f, symbol): 

""" Tests whether the equation is an equation of the given function class. 

 

The given equation belongs to the given function class if it is 

comprised of functions of the function class which are multiplied by 

or added to expressions independent of the symbol. In addition, the 

arguments of all such functions must be linear in the symbol as well. 

 

Examples 

======== 

 

>>> from sympy.solvers.solveset import _is_function_class_equation 

>>> from sympy import tan, sin, tanh, sinh, exp 

>>> from sympy.abc import x 

>>> from sympy.functions.elementary.trigonometric import (TrigonometricFunction, 

... HyperbolicFunction) 

>>> _is_function_class_equation(TrigonometricFunction, exp(x) + tan(x), x) 

False 

>>> _is_function_class_equation(TrigonometricFunction, tan(x) + sin(x), x) 

True 

>>> _is_function_class_equation(TrigonometricFunction, tan(x**2), x) 

False 

>>> _is_function_class_equation(TrigonometricFunction, tan(x + 2), x) 

True 

>>> _is_function_class_equation(HyperbolicFunction, tanh(x) + sinh(x), x) 

True 

""" 

if f.is_Mul or f.is_Add: 

return all(_is_function_class_equation(func_class, arg, symbol) 

for arg in f.args) 

 

if f.is_Pow: 

if not f.exp.has(symbol): 

return _is_function_class_equation(func_class, f.base, symbol) 

else: 

return False 

 

if not f.has(symbol): 

return True 

 

if isinstance(f, func_class): 

try: 

g = Poly(f.args[0], symbol) 

return g.degree() <= 1 

except PolynomialError: 

return False 

else: 

return False 

 

 

def _solve_as_rational(f, symbol, domain): 

""" solve rational functions""" 

f = together(f, deep=True) 

g, h = fraction(f) 

if not h.has(symbol): 

try: 

return _solve_as_poly(g, symbol, domain) 

except NotImplementedError: 

# The polynomial formed from g could end up having 

# coefficients in a ring over which finding roots 

# isn't implemented yet, e.g. ZZ[a] for some symbol a 

return ConditionSet(symbol, Eq(f, 0), domain) 

else: 

valid_solns = _solveset(g, symbol, domain) 

invalid_solns = _solveset(h, symbol, domain) 

return valid_solns - invalid_solns 

 

 

def _solve_trig(f, symbol, domain): 

"""Function to call other helpers to solve trigonometric equations """ 

sol1 = sol = None 

try: 

sol1 = _solve_trig1(f, symbol, domain) 

except BaseException as error: 

pass 

if sol1 is None or isinstance(sol1, ConditionSet): 

try: 

sol = _solve_trig2(f, symbol, domain) 

except BaseException as error: 

sol = sol1 

if isinstance(sol1, ConditionSet) and isinstance(sol, ConditionSet): 

if sol1.count_ops() < sol.count_ops(): 

sol = sol1 

else: 

sol = sol1 

if sol is None: 

raise NotImplementedError(filldedent(''' 

Solution to this kind of trigonometric equations 

is yet to be implemented''')) 

return sol 

 

 

def _solve_trig1(f, symbol, domain): 

"""Primary Helper to solve trigonometric equations """ 

f = trigsimp(f) 

f_original = f 

f = f.rewrite(exp) 

f = together(f) 

g, h = fraction(f) 

y = Dummy('y') 

g, h = g.expand(), h.expand() 

g, h = g.subs(exp(I*symbol), y), h.subs(exp(I*symbol), y) 

if g.has(symbol) or h.has(symbol): 

return ConditionSet(symbol, Eq(f, 0), S.Reals) 

 

solns = solveset_complex(g, y) - solveset_complex(h, y) 

if isinstance(solns, ConditionSet): 

raise NotImplementedError 

 

if isinstance(solns, FiniteSet): 

if any(isinstance(s, RootOf) for s in solns): 

raise NotImplementedError 

result = Union(*[invert_complex(exp(I*symbol), s, symbol)[1] 

for s in solns]) 

return Intersection(result, domain) 

elif solns is S.EmptySet: 

return S.EmptySet 

else: 

return ConditionSet(symbol, Eq(f_original, 0), S.Reals) 

 

 

def _solve_trig2(f, symbol, domain): 

"""Secondary helper to solve trigonometric equations, 

called when first helper fails """ 

from sympy import ilcm, igcd, expand_trig, degree, simplify 

f = trigsimp(f) 

f_original = f 

trig_functions = f.atoms(sin, cos, tan, sec, cot, csc) 

trig_arguments = [e.args[0] for e in trig_functions] 

denominators = [] 

numerators = [] 

 

for ar in trig_arguments: 

try: 

poly_ar = Poly(ar, symbol) 

 

except ValueError: 

raise ValueError("give up, we can't solve if this is not a polynomial in x") 

if poly_ar.degree() > 1: # degree >1 still bad 

raise ValueError("degree of variable inside polynomial should not exceed one") 

if poly_ar.degree() == 0: # degree 0, don't care 

continue 

c = poly_ar.all_coeffs()[0] # got the coefficient of 'symbol' 

numerators.append(Rational(c).p) 

denominators.append(Rational(c).q) 

 

x = Dummy('x') 

 

# ilcm() and igcd() require more than one argument 

if len(numerators) > 1: 

mu = Rational(2)*ilcm(*denominators)/igcd(*numerators) 

else: 

assert len(numerators) == 1 

mu = Rational(2)*denominators[0]/numerators[0] 

 

f = f.subs(symbol, mu*x) 

f = f.rewrite(tan) 

f = expand_trig(f) 

f = together(f) 

 

g, h = fraction(f) 

y = Dummy('y') 

g, h = g.expand(), h.expand() 

g, h = g.subs(tan(x), y), h.subs(tan(x), y) 

 

if g.has(x) or h.has(x): 

return ConditionSet(symbol, Eq(f_original, 0), domain) 

solns = solveset(g, y, S.Reals) - solveset(h, y, S.Reals) 

 

if isinstance(solns, FiniteSet): 

result = Union(*[invert_real(tan(symbol/mu), s, symbol)[1] 

for s in solns]) 

dsol = invert_real(tan(symbol/mu), oo, symbol)[1] 

if degree(h) > degree(g): # If degree(denom)>degree(num) then there 

result = Union(result, dsol) # would be another sol at Lim(denom-->oo) 

return Intersection(result, domain) 

elif solns is S.EmptySet: 

return S.EmptySet 

else: 

return ConditionSet(symbol, Eq(f_original, 0), S.Reals) 

 

 

def _solve_as_poly(f, symbol, domain=S.Complexes): 

""" 

Solve the equation using polynomial techniques if it already is a 

polynomial equation or, with a change of variables, can be made so. 

""" 

result = None 

if f.is_polynomial(symbol): 

solns = roots(f, symbol, cubics=True, quartics=True, 

quintics=True, domain='EX') 

num_roots = sum(solns.values()) 

if degree(f, symbol) <= num_roots: 

result = FiniteSet(*solns.keys()) 

else: 

poly = Poly(f, symbol) 

solns = poly.all_roots() 

if poly.degree() <= len(solns): 

result = FiniteSet(*solns) 

else: 

result = ConditionSet(symbol, Eq(f, 0), domain) 

else: 

poly = Poly(f) 

if poly is None: 

result = ConditionSet(symbol, Eq(f, 0), domain) 

gens = [g for g in poly.gens if g.has(symbol)] 

 

if len(gens) == 1: 

poly = Poly(poly, gens[0]) 

gen = poly.gen 

deg = poly.degree() 

poly = Poly(poly.as_expr(), poly.gen, composite=True) 

poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True, 

quintics=True).keys()) 

 

if len(poly_solns) < deg: 

result = ConditionSet(symbol, Eq(f, 0), domain) 

 

if gen != symbol: 

y = Dummy('y') 

inverter = invert_real if domain.is_subset(S.Reals) else invert_complex 

lhs, rhs_s = inverter(gen, y, symbol) 

if lhs == symbol: 

result = Union(*[rhs_s.subs(y, s) for s in poly_solns]) 

else: 

result = ConditionSet(symbol, Eq(f, 0), domain) 

else: 

result = ConditionSet(symbol, Eq(f, 0), domain) 

 

if result is not None: 

if isinstance(result, FiniteSet): 

# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2 

# - sqrt(2)*I/2. We are not expanding for solution with symbols 

# or undefined functions because that makes the solution more complicated. 

# For example, expand_complex(a) returns re(a) + I*im(a) 

if all([s.atoms(Symbol, AppliedUndef) == set() and not isinstance(s, RootOf) 

for s in result]): 

s = Dummy('s') 

result = imageset(Lambda(s, expand_complex(s)), result) 

if isinstance(result, FiniteSet): 

result = result.intersection(domain) 

return result 

else: 

return ConditionSet(symbol, Eq(f, 0), domain) 

 

 

def _has_rational_power(expr, symbol): 

""" 

Returns (bool, den) where bool is True if the term has a 

non-integer rational power and den is the denominator of the 

expression's exponent. 

 

Examples 

======== 

 

>>> from sympy.solvers.solveset import _has_rational_power 

>>> from sympy import sqrt 

>>> from sympy.abc import x 

>>> _has_rational_power(sqrt(x), x) 

(True, 2) 

>>> _has_rational_power(x**2, x) 

(False, 1) 

""" 

a, p, q = Wild('a'), Wild('p'), Wild('q') 

pattern_match = expr.match(a*p**q) or {} 

if pattern_match.get(a, S.Zero) is S.Zero: 

return (False, S.One) 

elif p not in pattern_match.keys(): 

return (False, S.One) 

elif isinstance(pattern_match[q], Rational) \ 

and pattern_match[p].has(symbol): 

if not pattern_match[q].q == S.One: 

return (True, pattern_match[q].q) 

 

if not isinstance(pattern_match[a], Pow) \ 

or isinstance(pattern_match[a], Mul): 

return (False, S.One) 

else: 

return _has_rational_power(pattern_match[a], symbol) 

 

 

def _solve_radical(f, symbol, solveset_solver): 

""" Helper function to solve equations with radicals """ 

eq, cov = unrad(f) 

if not cov: 

result = solveset_solver(eq, symbol) - \ 

Union(*[solveset_solver(g, symbol) for g in denoms(f, symbol)]) 

else: 

y, yeq = cov 

if not solveset_solver(y - I, y): 

yreal = Dummy('yreal', real=True) 

yeq = yeq.xreplace({y: yreal}) 

eq = eq.xreplace({y: yreal}) 

y = yreal 

g_y_s = solveset_solver(yeq, symbol) 

f_y_sols = solveset_solver(eq, y) 

result = Union(*[imageset(Lambda(y, g_y), f_y_sols) 

for g_y in g_y_s]) 

 

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)) 

 

return solution_set 

 

 

def _solve_abs(f, symbol, domain): 

""" Helper function to solve equation involving absolute value function """ 

if not domain.is_subset(S.Reals): 

raise ValueError(filldedent(''' 

Absolute values cannot be inverted in the 

complex domain.''')) 

p, q, r = Wild('p'), Wild('q'), Wild('r') 

pattern_match = f.match(p*Abs(q) + r) or {} 

f_p, f_q, f_r = [pattern_match.get(i, S.Zero) for i in (p, q, r)] 

 

if not (f_p.is_zero or f_q.is_zero): 

domain = continuous_domain(f_q, symbol, domain) 

q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol, 

relational=False, domain=domain, continuous=True) 

q_neg_cond = q_pos_cond.complement(domain) 

 

sols_q_pos = solveset_real(f_p*f_q + f_r, 

symbol).intersect(q_pos_cond) 

sols_q_neg = solveset_real(f_p*(-f_q) + f_r, 

symbol).intersect(q_neg_cond) 

return Union(sols_q_pos, sols_q_neg) 

else: 

return ConditionSet(symbol, Eq(f, 0), domain) 

 

 

def solve_decomposition(f, symbol, domain): 

""" 

Function to solve equations via the principle of "Decomposition 

and Rewriting". 

 

Examples 

======== 

>>> from sympy import exp, sin, Symbol, pprint, S 

>>> from sympy.solvers.solveset import solve_decomposition as sd 

>>> x = Symbol('x') 

>>> f1 = exp(2*x) - 3*exp(x) + 2 

>>> sd(f1, x, S.Reals) 

{0, log(2)} 

>>> f2 = sin(x)**2 + 2*sin(x) + 1 

>>> pprint(sd(f2, x, S.Reals), use_unicode=False) 

3*pi 

{2*n*pi + ---- | n in S.Integers} 

2 

>>> f3 = sin(x + 2) 

>>> pprint(sd(f3, x, S.Reals), use_unicode=False) 

{2*n*pi - 2 | n in S.Integers} U {2*n*pi - 2 + pi | n in S.Integers} 

 

""" 

from sympy.solvers.decompogen import decompogen 

from sympy.calculus.util import function_range 

# decompose the given function 

g_s = decompogen(f, symbol) 

# `y_s` represents the set of values for which the function `g` is to be 

# solved. 

# `solutions` represent the solutions of the equations `g = y_s` or 

# `g = 0` depending on the type of `y_s`. 

# As we are interested in solving the equation: f = 0 

y_s = FiniteSet(0) 

for g in g_s: 

frange = function_range(g, symbol, domain) 

y_s = Intersection(frange, y_s) 

result = S.EmptySet 

if isinstance(y_s, FiniteSet): 

for y in y_s: 

solutions = solveset(Eq(g, y), symbol, domain) 

if not isinstance(solutions, ConditionSet): 

result += solutions 

 

else: 

if isinstance(y_s, ImageSet): 

iter_iset = (y_s,) 

 

elif isinstance(y_s, Union): 

iter_iset = y_s.args 

 

elif isinstance(y_s, EmptySet): 

# y_s is not in the range of g in g_s, so no solution exists 

#in the given domain 

return y_s 

 

for iset in iter_iset: 

new_solutions = solveset(Eq(iset.lamda.expr, g), symbol, domain) 

dummy_var = tuple(iset.lamda.expr.free_symbols)[0] 

base_set = iset.base_set 

if isinstance(new_solutions, FiniteSet): 

new_exprs = new_solutions 

 

elif isinstance(new_solutions, Intersection): 

if isinstance(new_solutions.args[1], FiniteSet): 

new_exprs = new_solutions.args[1] 

 

for new_expr in new_exprs: 

result += ImageSet(Lambda(dummy_var, new_expr), base_set) 

 

if result is S.EmptySet: 

return ConditionSet(symbol, Eq(f, 0), domain) 

 

y_s = result 

 

return y_s 

 

 

def _solveset(f, symbol, domain, _check=False): 

"""Helper for solveset to return a result from an expression 

that has already been sympify'ed and is known to contain the 

given symbol.""" 

# _check controls whether the answer is checked or not 

from sympy.simplify.simplify import signsimp 

 

orig_f = f 

if f.is_Mul: 

coeff, f = f.as_independent(symbol, as_Add=False) 

if coeff in set([S.ComplexInfinity, S.NegativeInfinity, S.Infinity]): 

f = together(orig_f) 

elif f.is_Add: 

a, h = f.as_independent(symbol) 

m, h = h.as_independent(symbol, as_Add=False) 

if m not in set([S.ComplexInfinity, S.Zero, S.Infinity, 

S.NegativeInfinity]): 

f = a/m + h # XXX condition `m != 0` should be added to soln 

 

# assign the solvers to use 

solver = lambda f, x, domain=domain: _solveset(f, x, domain) 

inverter = lambda f, rhs, symbol: _invert(f, rhs, symbol, domain) 

 

result = EmptySet() 

 

if f.expand().is_zero: 

return domain 

elif not f.has(symbol): 

return EmptySet() 

elif f.is_Mul and all(_is_finite_with_finite_vars(m, domain) 

for m in f.args): 

# if f(x) and g(x) are both finite we can say that the solution of 

# f(x)*g(x) == 0 is same as Union(f(x) == 0, g(x) == 0) is not true in 

# general. g(x) can grow to infinitely large for the values where 

# f(x) == 0. To be sure that we are not silently allowing any 

# wrong solutions we are using this technique only if both f and g are 

# finite for a finite input. 

result = Union(*[solver(m, symbol) for m in f.args]) 

elif _is_function_class_equation(TrigonometricFunction, f, symbol) or \ 

_is_function_class_equation(HyperbolicFunction, f, symbol): 

result = _solve_trig(f, symbol, domain) 

elif isinstance(f, arg): 

a = f.args[0] 

result = solveset_real(a > 0, symbol) 

elif f.is_Piecewise: 

result = EmptySet() 

expr_set_pairs = f.as_expr_set_pairs(domain) 

for (expr, in_set) in expr_set_pairs: 

if in_set.is_Relational: 

in_set = in_set.as_set() 

solns = solver(expr, symbol, in_set) 

result += solns 

elif isinstance(f, Eq): 

from sympy.core import Add 

result = solver(Add(f.lhs, - f.rhs, evaluate=False), symbol, domain) 

 

elif f.is_Relational: 

if not domain.is_subset(S.Reals): 

raise NotImplementedError(filldedent(''' 

Inequalities in the complex domain are 

not supported. Try the real domain by 

setting domain=S.Reals''')) 

try: 

result = solve_univariate_inequality( 

f, symbol, domain=domain, relational=False) 

except NotImplementedError: 

result = ConditionSet(symbol, f, domain) 

return result 

else: 

lhs, rhs_s = inverter(f, 0, symbol) 

if lhs == symbol: 

# do some very minimal simplification since 

# repeated inversion may have left the result 

# in a state that other solvers (e.g. poly) 

# would have simplified; this is done here 

# rather than in the inverter since here it 

# is only done once whereas there it would 

# be repeated for each step of the inversion 

if isinstance(rhs_s, FiniteSet): 

rhs_s = FiniteSet(*[Mul(* 

signsimp(i).as_content_primitive()) 

for i in rhs_s]) 

result = rhs_s 

 

elif isinstance(rhs_s, FiniteSet): 

for equation in [lhs - rhs for rhs in rhs_s]: 

if equation == f: 

if any(_has_rational_power(g, symbol)[0] 

for g in equation.args) or _has_rational_power( 

equation, symbol)[0]: 

result += _solve_radical(equation, 

symbol, 

solver) 

elif equation.has(Abs): 

result += _solve_abs(f, symbol, domain) 

else: 

new_result = _solve_as_rational(equation, symbol, domain) 

if isinstance(new_result, ConditionSet): 

# may be a transcendental type equation 

result += _transolve(equation, symbol, domain) 

else: 

result += new_result 

else: 

result += solver(equation, symbol) 

 

elif rhs_s is not S.EmptySet: 

result = ConditionSet(symbol, Eq(f, 0), domain) 

 

if isinstance(result, ConditionSet): 

num, den = f.as_numer_denom() 

if den.has(symbol): 

_result = _solveset(num, symbol, domain) 

if not isinstance(_result, ConditionSet): 

singularities = _solveset(den, symbol, domain) 

result = _result - singularities 

 

if _check: 

if isinstance(result, ConditionSet): 

# it wasn't solved or has enumerated all conditions 

# -- leave it alone 

return result 

 

# whittle away all but the symbol-containing core 

# to use this for testing 

fx = orig_f.as_independent(symbol, as_Add=True)[1] 

fx = fx.as_independent(symbol, as_Add=False)[1] 

 

if isinstance(result, FiniteSet): 

# check the result for invalid solutions 

result = FiniteSet(*[s for s in result 

if isinstance(s, RootOf) 

or domain_check(fx, symbol, s)]) 

 

return result 

 

 

def _expo_solver(f): 

""" 

Helper function for solving exponential equations. 

 

Exponential equations are the type of equations which includes a 

variable located in the exponent. 

For example `5**(2*x + 3) - 5**(3*x - 1)`. 

The function evaluates exponetial equations having two arguments, 

i.e, exponential equations having only two exponent terms 

(like the one above). 

 

The helper takes the equation as the input, tries to reduce the 

equation to log form (if possible) and returns the modified 

equation. 

 

* Proof of correctness of the method 

 

The logarithm is the inverse function to exponentiation. 

The defining relation between exponentiation and logarithm is: 

 

log(x, b) = y if b**y = x 

 

Therefore if we are given an equation with exponent terms, we can 

convert every term to its corresponding log form. This is achieved by 

taking logarithms and expanding them using log identities so that the 

equations can be easily handled by `solveset`. 

For example: 3**(2*x) = 2**(x + 3). Taking log both sides will reduce 

the equation to `2*x*log(3) - (x + 3)*log(2)`. This is easily handled by 

solveset. 

""" 

 

try: 

arg1, arg2 = ordered(f.args) 

except TypeError: 

raise ValueError("Equations with more than two arguments are not\ 

supported") 

 

lhs = arg1 

rhs = -arg2 

 

lhs = expand_log(log(lhs)) 

rhs = expand_log(log(rhs)) 

 

return lhs - rhs 

 

 

def _check_expo(f, symbol): 

""" 

Helper to check whether an equation is exponential or not. 

 

Returns True if it is of exponential type otherwise False. 

This helper is specifically designed to determine two argument 

type exponential equations. Two argument type exponential 

equations are the ones which consists of only two exponent terms. 

Therefore it returns False if `f` does not contains exactly two 

arguments. 

 

It checks whether any of the symbol dependent term contains 

power. If so there is a chance that the equation is of exponential 

type, hence returns True otherwise False. 

 

Examples 

======== 

 

>>> from sympy import symbols 

>>> from sympy.solvers.solveset import _check_expo as check 

>>> x = symbols('x') 

>>> check(3**x - 2, x) 

True 

>>> check(2*x, x) 

False 

""" 

try: 

arg1, arg2 = ordered(f.args) 

except ValueError: 

return False 

 

arg1_dep = arg1.as_independent(symbol)[1] 

arg2_dep = arg2.as_independent(symbol)[1] 

 

return any(_ispow(i) for i in (arg1_dep, arg2_dep)) 

 

 

def _transolve(f, symbol, domain): 

""" 

Function to solve transcendental equations. It is a helper to 

solveset and should be used internally as of now. `transolve` 

currently supports the following class of equations: 

 

- Exponential equations 

 

Parameters 

========== 

 

``f`` can be any (supported) transcendental equation that needs 

to be solved. This needs to be an expression, which is assumed 

to be equal to 0. 

 

``symbol`` is variable for which the equation is solved. 

This needs to be of class `Symbol`. 

 

``domain`` is a set over which the equation is solved. 

This needs to be of class `Set`. 

 

 

How to use transolve 

==================== 

 

`transolve` should be used as an internal function. There should be 

an internal call to `transolve`, passing all the three above 

arguments as parameters. These three parameters should be sympy 

instances. 

 

 

How transolve works 

=================== 

 

The main idea behind `transolve` is to make it modular, extensible, 

robust and easy to understand. For this to be achieved `transolve` 

uses two types of helper functions to solve equation of a particular 

class: 

 

Identifying helpers: These helpers are used to determine whether a given 

equation belongs to a certain class of equation or not. 

Heuristics are implemented to determine this. These functions return 

either True or False. 

 

Solving helpers: Once identified that the equation belongs to a 

particular class, another helper is invoked which will be responsible 

to either solve the equation fully or reduce to a better form for 

`solveset` to handle it. A generalized algorithm or heuristic is 

implemented to get the output. 

 

* Philosophy behind the module 

 

`transolve` comes into action when solveset is unable to solve the 

equation as a last resort to get the solutions. So the idea is that, 

first it tries to invert the equation to get the `lhs` and the `rhs`. 

Depending on the general form of the class of the equation the 

equations are sent to different cases, like for example, logarithmic 

and exponential equations takes the form as `a*f(x) + b*g(x)`, where 

f(x) and g(x) are power (or log) terms, therefore they are included in 

`Add` case. Similarly different classes of equation can be included 

in cases depending on their general form. Once figuring out, different 

identifying helper conditions are included to check which class the 

equation belongs. If the equation belongs to a particular class it is 

solved by a call to its respective solving helper. This helper either 

solves the equation completely or transforms to a better form for 

`solveset` to handle. If the equation is solved the result is returned 

otherwise a `ConditionSet` is returned. 

 

 

How transolve is better than _tsolve 

==================================== 

 

1) Better output 

 

`transolve` provides a better output to some equations than 

`_tsolve`. Though result from both the functions are correct 

it's just that the one from transolve is easy to understand and 

appropriate. This is because transolve smartly evaluates the 

result minimising it wherever necessary. 

 

Consider a simple exponential equation 3**(2*x) - 2**(x + 3) 

transolve gives the output as {(-3*log(2)/(-2*log(3) + log(2))}, 

whereas _tsolve would return [-log(2**(3/log(2/9)))].`transolve's` 

output seems more simple, readable and appropriate as compared to 

the latter one. 

 

 

2) Extensible 

 

The API of `transolve` is designed such that it is easily extensible, 

i.e, it is easy to add a new class of equation solver without having 

to mess with the API or solving code of other helpers. The idea is 

that the function itself contains call to different helpers that 

identifies and solves the equation. So to make new class of equation 

solve, you must add a helper that identifies that particular class 

of the equation and another helper that solves that type of 

equations. You should read the below section to get more insight 

on how to add new class of equations in transolve. 

 

 

3) Modular 

 

`transolve` is designed to be modular i.e, for every class of 

equation a separate helper for identification and solving is 

implemented. This makes it easy to change or modify any of the 

method implemented directly in the helpers without interfering 

with the actual structure of the API. 

 

 

4) Less complex API 

 

Since modular approach is used in transolve, the API automatically 

becomes less complex to read and understand unlike in `_tsolve` 

where the API structure is a mess and lots of recursive and function 

calls makes it more hard to understand. 

 

 

5) Faster Computation 

 

Equation solving via transolve is much faster as compared to 

`_tsolve`. The way equations are treated in solve is complicated 

which make it a bit time consuming. `solve` tries every possibility 

to solve the equation and in process it gets into each and every 

condition and starts solving the equation to get an answer, 

if the result is not achieved the equation is solved by another 

means. Therefore this series of solving makes it a bit slow. 

Whereas in `transolve` computation begins only when the equation 

is identified of being a particular type. 

 

 

How to add new class of equations 

================================= 

 

`transolve` is designed in such a way that it becomes an easy task to 

add a new class of equation solver. 

 

The first task that needs to be done for adding your own solver 

is to decide from where its `identificaion helper` will be 

invoked. To do so determine the general form of the class of the 

equation to place the invocation of the identification helper to 

an appropriate place within transolve, for example the general 

form of the exponential equations is `a*f(x) + b*g(x)` so we 

need to place the invocation condition inside `Add` case. Once 

the place for identification helper is determined, you can add a 

call to solving helper. For your class of equation you need to 

define your own identification and solving helpers. The 

identification helper should be implemented for generalised cases 

and should return either `True` if the given equation belongs to the 

class otherwise `False`. Solving helpers needs to be implemented with 

such heuristics or algorithms that solves most of the equations 

belonging to that class. The value returned from this helper should 

be handled properly in the main (transolve) function itself. It could 

be either the exact solution or a reduced form of the equation which 

`solveset` can handle. 

 

Apart from this, few other things needs to be taken care while adding 

an equation solver: 

 

- Naming conventions 

Name of the identification helper should be like `_check_class`, 

where `class` will be the name or abbreviation of the class of 

equation. The solving helper will be named as `_class_solver`. 

For eg: for exponential equation it becomes `_check_expo` and 

`_expo_solver_`. 

- Be sure to consider consider corner cases. 

- Adding tests for each method (if necessary). 

- Adding a docstring to your helper that describes the method 

implemented with its proof of correctness by explaining with few 

examples. Since most of the methods devised to identify and solve 

particular class of equations are mostly heuristics therefore it 

becomes necessary to provide proof of correctness of these methods. 

Following things needs to be included while writing the 

documentation for the helpers: 

- What is the purpose of the helper. 

- How it solves the equation and how does it identifies the equation. 

- Examples should be included to prove its correctness. 

- What are the input parameters and what does the helper returns. 

 

 

Examples 

======== 

 

>>> from sympy.solvers.solveset import _transolve 

>>> from sympy import symbols, S 

 

>>> x = symbols('x', real=True) 

>>> _transolve(5**(x-3) - 3**(2*x + 1), x, S.Reals) 

{-log(375)/(-log(5) + 2*log(3))} 

""" 

 

# invert_complex handles the call to the desired inverter based 

# on the domain specified. 

lhs, rhs_s = invert_complex(f, 0, symbol, domain) 

 

result = ConditionSet(symbol, Eq(f, 0), domain) 

 

if rhs_s is S.EmptySet: 

result = S.EmptySet 

else: 

rhs = rhs_s.args[0] 

# do we need a loop for rhs_s? 

# Can't determine a case as of now. 

if lhs == symbol: 

result = rhs_s 

 

elif lhs.is_Add: 

# trying to convert to P.O.S form 

equation = factor(powdenest(lhs - rhs)) 

 

if equation.is_Mul: 

result = _solveset(equation, symbol, domain) 

 

# check if it is exponential type equation 

elif _check_expo(equation, symbol): 

new_f = _expo_solver(equation) 

result = _solveset(new_f, symbol, domain) 

 

elif lhs.is_Pow: 

new_f = _expo_solver(lhs - rhs) 

result = _solveset(new_f, symbol, domain) 

 

if isinstance(result, ConditionSet): 

# there is chance that ConditionSet is returned from 

# any of the helper which will have modified equation 

# therefore to get the original equation in the ConditionSet 

# this condition is necessary. 

result = ConditionSet(symbol, Eq(f, 0), domain) 

 

return result 

 

 

def solveset(f, symbol=None, domain=S.Complexes): 

r"""Solves a given inequality or equation with set as output 

 

Parameters 

========== 

 

f : Expr or a relational. 

The target equation or inequality 

symbol : Symbol 

The variable for which the equation is solved 

domain : Set 

The domain over which the equation is solved 

 

Returns 

======= 

 

Set 

A set of values for `symbol` for which `f` is True or is equal to 

zero. An `EmptySet` is returned if `f` is False or nonzero. 

A `ConditionSet` is returned as unsolved object if algorithms 

to evaluate complete solution are not yet implemented. 

 

`solveset` claims to be complete in the solution set that it returns. 

 

Raises 

====== 

 

NotImplementedError 

The algorithms to solve inequalities in complex domain are 

not yet implemented. 

ValueError 

The input is not valid. 

RuntimeError 

It is a bug, please report to the github issue tracker. 

 

 

Notes 

===== 

 

Python interprets 0 and 1 as False and True, respectively, but 

in this function they refer to solutions of an expression. So 0 and 1 

return the Domain and EmptySet, respectively, while True and False 

return the opposite (as they are assumed to be solutions of relational 

expressions). 

 

 

See Also 

======== 

 

solveset_real: solver for real domain 

solveset_complex: solver for complex domain 

 

Examples 

======== 

 

>>> from sympy import exp, sin, Symbol, pprint, S 

>>> from sympy.solvers.solveset import solveset, solveset_real 

 

* The default domain is complex. Not specifying a domain will lead 

to the solving of the equation in the complex domain (and this 

is not affected by the assumptions on the symbol): 

 

>>> x = Symbol('x') 

>>> pprint(solveset(exp(x) - 1, x), use_unicode=False) 

{2*n*I*pi | n in S.Integers} 

 

>>> x = Symbol('x', real=True) 

>>> pprint(solveset(exp(x) - 1, x), use_unicode=False) 

{2*n*I*pi | n in S.Integers} 

 

* If you want to use `solveset` to solve the equation in the 

real domain, provide a real domain. (Using `solveset\_real` 

does this automatically.) 

 

>>> R = S.Reals 

>>> x = Symbol('x') 

>>> solveset(exp(x) - 1, x, R) 

{0} 

>>> solveset_real(exp(x) - 1, x) 

{0} 

 

The solution is mostly unaffected by assumptions on the symbol, 

but there may be some slight difference: 

 

>>> pprint(solveset(sin(x)/x,x), use_unicode=False) 

({2*n*pi | n in S.Integers} \ {0}) U ({2*n*pi + pi | n in S.Integers} \ {0}) 

 

>>> p = Symbol('p', positive=True) 

>>> pprint(solveset(sin(p)/p, p), use_unicode=False) 

{2*n*pi | n in S.Integers} U {2*n*pi + pi | n in S.Integers} 

 

* Inequalities can be solved over the real domain only. Use of a complex 

domain leads to a NotImplementedError. 

 

>>> solveset(exp(x) > 1, x, R) 

Interval.open(0, oo) 

 

""" 

f = sympify(f) 

symbol = sympify(symbol) 

 

if f is S.true: 

return domain 

 

if f is S.false: 

return S.EmptySet 

 

if not isinstance(f, (Expr, Number)): 

raise ValueError("%s is not a valid SymPy expression" % f) 

 

if not isinstance(symbol, Expr) and symbol is not None: 

raise ValueError("%s is not a valid SymPy symbol" % symbol) 

 

if not isinstance(domain, Set): 

raise ValueError("%s is not a valid domain" %(domain)) 

 

free_symbols = f.free_symbols 

 

if symbol is None and not free_symbols: 

b = Eq(f, 0) 

if b is S.true: 

return domain 

elif b is S.false: 

return S.EmptySet 

else: 

raise NotImplementedError(filldedent(''' 

relationship between value and 0 is unknown: %s''' % b)) 

 

if symbol is None: 

if len(free_symbols) == 1: 

symbol = free_symbols.pop() 

elif free_symbols: 

raise ValueError(filldedent(''' 

The independent variable must be specified for a 

multivariate equation.''')) 

elif not isinstance(symbol, Symbol): 

f, s, swap = recast_to_symbols([f], [symbol]) 

# the xreplace will be needed if a ConditionSet is returned 

return solveset(f[0], s[0], domain).xreplace(swap) 

 

if domain.is_subset(S.Reals): 

if not symbol.is_real: 

assumptions = symbol.assumptions0 

assumptions['real'] = True 

try: 

r = Dummy('r', **assumptions) 

return solveset(f.xreplace({symbol: r}), r, domain 

).xreplace({r: symbol}) 

except InconsistentAssumptions: 

pass 

# Abs has its own handling method which avoids the 

# rewriting property that the first piece of abs(x) 

# is for x >= 0 and the 2nd piece for x < 0 -- solutions 

# can look better if the 2nd condition is x <= 0. Since 

# the solution is a set, duplication of results is not 

# an issue, e.g. {y, -y} when y is 0 will be {0} 

f, mask = _masked(f, Abs) 

f = f.rewrite(Piecewise) # everything that's not an Abs 

for d, e in mask: 

# everything *in* an Abs 

e = e.func(e.args[0].rewrite(Piecewise)) 

f = f.xreplace({d: e}) 

f = piecewise_fold(f) 

 

return _solveset(f, symbol, domain, _check=True) 

 

 

def solveset_real(f, symbol): 

return solveset(f, symbol, S.Reals) 

 

 

def solveset_complex(f, symbol): 

return solveset(f, symbol, S.Complexes) 

 

 

def solvify(f, symbol, domain): 

"""Solves an equation using solveset and returns the solution in accordance 

with the `solve` output API. 

 

Returns 

======= 

 

We classify the output based on the type of solution returned by `solveset`. 

 

Solution | Output 

---------------------------------------- 

FiniteSet | list 

 

ImageSet, | list (if `f` is periodic) 

Union | 

 

EmptySet | empty list 

 

Others | None 

 

 

Raises 

====== 

 

NotImplementedError 

A ConditionSet is the input. 

 

Examples 

======== 

 

>>> from sympy.solvers.solveset import solvify, solveset 

>>> from sympy.abc import x 

>>> from sympy import S, tan, sin, exp 

>>> solvify(x**2 - 9, x, S.Reals) 

[-3, 3] 

>>> solvify(sin(x) - 1, x, S.Reals) 

[pi/2] 

>>> solvify(tan(x), x, S.Reals) 

[0] 

>>> solvify(exp(x) - 1, x, S.Complexes) 

 

>>> solvify(exp(x) - 1, x, S.Reals) 

[0] 

 

""" 

solution_set = solveset(f, symbol, domain) 

result = None 

if solution_set is S.EmptySet: 

result = [] 

 

elif isinstance(solution_set, ConditionSet): 

raise NotImplementedError('solveset is unable to solve this equation.') 

 

elif isinstance(solution_set, FiniteSet): 

result = list(solution_set) 

 

else: 

period = periodicity(f, symbol) 

if period is not None: 

solutions = S.EmptySet 

iter_solutions = () 

if isinstance(solution_set, ImageSet): 

iter_solutions = (solution_set,) 

elif isinstance(solution_set, Union): 

if all(isinstance(i, ImageSet) for i in solution_set.args): 

iter_solutions = solution_set.args 

 

for solution in iter_solutions: 

solutions += solution.intersect(Interval(0, period, False, True)) 

 

if isinstance(solutions, FiniteSet): 

result = list(solutions) 

 

else: 

solution = solution_set.intersect(domain) 

if isinstance(solution, FiniteSet): 

result += solution 

 

return result 

 

 

############################################################################### 

################################ LINSOLVE ##################################### 

############################################################################### 

 

 

def linear_eq_to_matrix(equations, *symbols): 

r""" 

Converts a given System of Equations into Matrix form. 

Here `equations` must be a linear system of equations in 

`symbols`. The order of symbols in input `symbols` will 

determine the order of coefficients in the returned 

Matrix. 

 

The Matrix form corresponds to the augmented matrix form. 

For example: 

 

.. math:: 4x + 2y + 3z = 1 

.. math:: 3x + y + z = -6 

.. math:: 2x + 4y + 9z = 2 

 

This system would return `A` & `b` as given below: 

 

:: 

 

[ 4 2 3 ] [ 1 ] 

A = [ 3 1 1 ] b = [-6 ] 

[ 2 4 9 ] [ 2 ] 

 

Examples 

======== 

 

>>> from sympy import linear_eq_to_matrix, symbols 

>>> x, y, z = symbols('x, y, z') 

>>> eqns = [x + 2*y + 3*z - 1, 3*x + y + z + 6, 2*x + 4*y + 9*z - 2] 

>>> A, b = linear_eq_to_matrix(eqns, [x, y, z]) 

>>> A 

Matrix([ 

[1, 2, 3], 

[3, 1, 1], 

[2, 4, 9]]) 

>>> b 

Matrix([ 

[ 1], 

[-6], 

[ 2]]) 

>>> eqns = [x + z - 1, y + z, x - y] 

>>> A, b = linear_eq_to_matrix(eqns, [x, y, z]) 

>>> A 

Matrix([ 

[1, 0, 1], 

[0, 1, 1], 

[1, -1, 0]]) 

>>> b 

Matrix([ 

[1], 

[0], 

[0]]) 

 

* Symbolic coefficients are also supported 

 

>>> a, b, c, d, e, f = symbols('a, b, c, d, e, f') 

>>> eqns = [a*x + b*y - c, d*x + e*y - f] 

>>> A, B = linear_eq_to_matrix(eqns, x, y) 

>>> A 

Matrix([ 

[a, b], 

[d, e]]) 

>>> B 

Matrix([ 

[c], 

[f]]) 

 

""" 

 

if not symbols: 

raise ValueError('Symbols must be given, for which coefficients \ 

are to be found.') 

 

if hasattr(symbols[0], '__iter__'): 

symbols = symbols[0] 

 

M = Matrix([symbols]) 

# initialize Matrix with symbols + 1 columns 

M = M.col_insert(len(symbols), Matrix([1])) 

row_no = 1 

 

for equation in equations: 

f = sympify(equation) 

if isinstance(f, Equality): 

f = f.lhs - f.rhs 

 

# Extract coeff of symbols 

coeff_list = [] 

for symbol in symbols: 

coeff_list.append(f.coeff(symbol)) 

 

# append constant term (term free from symbols) 

coeff_list.append(-f.as_coeff_add(*symbols)[0]) 

 

# insert equations coeff's into rows 

M = M.row_insert(row_no, Matrix([coeff_list])) 

row_no += 1 

 

# delete the initialized (Ist) trivial row 

M.row_del(0) 

A, b = M[:, :-1], M[:, -1:] 

return A, b 

 

 

def linsolve(system, *symbols): 

r""" 

Solve system of N linear equations with M variables, which 

means both under - and overdetermined systems are supported. 

The possible number of solutions is zero, one or infinite. 

Zero solutions throws a ValueError, where as infinite 

solutions are represented parametrically in terms of given 

symbols. For unique solution a FiniteSet of ordered tuple 

is returned. 

 

All Standard input formats are supported: 

For the given set of Equations, the respective input types 

are given below: 

 

.. math:: 3x + 2y - z = 1 

.. math:: 2x - 2y + 4z = -2 

.. math:: 2x - y + 2z = 0 

 

* Augmented Matrix Form, `system` given below: 

 

:: 

 

[3 2 -1 1] 

system = [2 -2 4 -2] 

[2 -1 2 0] 

 

* List Of Equations Form 

 

`system = [3x + 2y - z - 1, 2x - 2y + 4z + 2, 2x - y + 2z]` 

 

* Input A & b Matrix Form (from Ax = b) are given as below: 

 

:: 

 

[3 2 -1 ] [ 1 ] 

A = [2 -2 4 ] b = [ -2 ] 

[2 -1 2 ] [ 0 ] 

 

`system = (A, b)` 

 

Symbols can always be passed but are actually only needed 

when 1) a system of equations is being passed and 2) the 

system is passed as an underdetermined matrix and one wants 

to control the name of the free variables in the result. 

An error is raised if no symbols are used for case 1, but if 

no symbols are provided for case 2, internally generated symbols 

will be provided. When providing symbols for case 2, there should 

be at least as many symbols are there are columns in matrix A. 

 

The algorithm used here is Gauss-Jordan elimination, which 

results, after elimination, in a row echelon form matrix. 

 

Returns 

======= 

 

A FiniteSet containing an ordered tuple of values for the 

unknowns for which the `system` has a solution. (Wrapping 

the tuple in FiniteSet is used to maintain a consistent 

output format throughout solveset.) 

 

Returns EmptySet(), if the linear system is inconsistent. 

 

Raises 

====== 

 

ValueError 

The input is not valid. 

The symbols are not given. 

 

Examples 

======== 

 

>>> from sympy import Matrix, S, linsolve, symbols 

>>> x, y, z = symbols("x, y, z") 

>>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]]) 

>>> b = Matrix([3, 6, 9]) 

>>> A 

Matrix([ 

[1, 2, 3], 

[4, 5, 6], 

[7, 8, 10]]) 

>>> b 

Matrix([ 

[3], 

[6], 

[9]]) 

>>> linsolve((A, b), [x, y, z]) 

{(-1, 2, 0)} 

 

* Parametric Solution: In case the system is underdetermined, the 

function will return a parametric solution in terms of the given 

symbols. Those that are free will be returned unchanged. e.g. in 

the system below, `z` is returned as the solution for variable z; 

it can take on any value. 

 

>>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) 

>>> b = Matrix([3, 6, 9]) 

>>> linsolve((A, b), x, y, z) 

{(z - 1, -2*z + 2, z)} 

 

If no symbols are given, internally generated symbols will be used. 

The `tau0` in the 3rd position indicates (as before) that the 3rd 

variable -- whatever it's named -- can take on any value: 

 

>>> linsolve((A, b)) 

{(tau0 - 1, -2*tau0 + 2, tau0)} 

 

* List of Equations as input 

 

>>> Eqns = [3*x + 2*y - z - 1, 2*x - 2*y + 4*z + 2, - x + y/2 - z] 

>>> linsolve(Eqns, x, y, z) 

{(1, -2, -2)} 

 

* Augmented Matrix as input 

 

>>> aug = Matrix([[2, 1, 3, 1], [2, 6, 8, 3], [6, 8, 18, 5]]) 

>>> aug 

Matrix([ 

[2, 1, 3, 1], 

[2, 6, 8, 3], 

[6, 8, 18, 5]]) 

>>> linsolve(aug, x, y, z) 

{(3/10, 2/5, 0)} 

 

* Solve for symbolic coefficients 

 

>>> a, b, c, d, e, f = symbols('a, b, c, d, e, f') 

>>> eqns = [a*x + b*y - c, d*x + e*y - f] 

>>> linsolve(eqns, x, y) 

{((-b*f + c*e)/(a*e - b*d), (a*f - c*d)/(a*e - b*d))} 

 

* A degenerate system returns solution as set of given 

symbols. 

 

>>> system = Matrix(([0, 0, 0], [0, 0, 0], [0, 0, 0])) 

>>> linsolve(system, x, y) 

{(x, y)} 

 

* For an empty system linsolve returns empty set 

 

>>> linsolve([], x) 

EmptySet() 

 

""" 

if not system: 

return S.EmptySet 

 

# If second argument is an iterable 

if symbols and hasattr(symbols[0], '__iter__'): 

symbols = symbols[0] 

sym_gen = isinstance(symbols, GeneratorType) 

 

swap = {} 

b = None # if we don't get b the input was bad 

syms_needed_msg = None 

 

# unpack system 

 

if hasattr(system, '__iter__'): 

 

# 1). (A, b) 

if len(system) == 2 and isinstance(system[0], Matrix): 

A, b = system 

 

# 2). (eq1, eq2, ...) 

if not isinstance(system[0], Matrix): 

if sym_gen or not symbols: 

raise ValueError(filldedent(''' 

When passing a system of equations, the explicit 

symbols for which a solution is being sought must 

be given as a sequence, too. 

''')) 

system = list(system) 

for i, eq in enumerate(system): 

try: 

# since we are checking it, we might as well take the 

# expanded expr that it will give 

system[i] = Poly(eq, symbols).as_expr() 

if any (degree(eq, sym) > 1 for sym in symbols): 

raise PolynomialError 

except PolynomialError: 

raise ValueError(filldedent(''' 

%s contains non-linear terms in the 

variables to be evaluated 

''') % eq) 

system, symbols, swap = recast_to_symbols(system, symbols) 

A, b = linear_eq_to_matrix(system, symbols) 

syms_needed_msg = 'free symbols in the equations provided' 

 

elif isinstance(system, Matrix) and not ( 

symbols and not isinstance(symbols, GeneratorType) and 

isinstance(symbols[0], Matrix)): 

# 3). A augmented with b 

A, b = system[:, :-1], system[:, -1:] 

 

if b is None: 

raise ValueError("Invalid arguments") 

 

syms_needed_msg = syms_needed_msg or 'columns of A' 

 

if sym_gen: 

symbols = [next(symbols) for i in range(A.cols)] 

if any(set(symbols) & (A.free_symbols | b.free_symbols)): 

raise ValueError(filldedent(''' 

At least one of the symbols provided 

already appears in the system to be solved. 

One way to avoid this is to use Dummy symbols in 

the generator, e.g. numbered_symbols('%s', cls=Dummy) 

''' % symbols[0].name.rstrip('1234567890'))) 

 

try: 

solution, params, free_syms = A.gauss_jordan_solve(b, freevar=True) 

except ValueError: 

# No solution 

return S.EmptySet 

 

# Replace free parameters with free symbols 

if params: 

if not symbols: 

symbols = [_ for _ in params] 

# re-use the parameters but put them in order 

# params [x, y, z] 

# free_symbols [2, 0, 4] 

# idx [1, 0, 2] 

idx = list(zip(*sorted(zip(free_syms, range(len(free_syms))))))[1] 

# simultaneous replacements {y: x, x: y, z: z} 

replace_dict = dict(zip(symbols, [symbols[i] for i in idx])) 

elif len(symbols) >= A.cols: 

replace_dict = {v: symbols[free_syms[k]] for k, v in enumerate(params)} 

else: 

raise IndexError(filldedent(''' 

the number of symbols passed should have a length 

equal to the number of %s. 

''' % syms_needed_msg)) 

solution = [sol.xreplace(replace_dict) for sol in solution] 

 

solution = [simplify(sol).xreplace(swap) for sol in solution] 

return FiniteSet(tuple(solution)) 

 

 

 

############################################################################## 

# ------------------------------nonlinsolve ---------------------------------# 

############################################################################## 

 

def _return_conditionset(eqs, symbols): 

# return conditionset 

condition_set = ConditionSet( 

Tuple(*symbols), 

FiniteSet(*eqs), 

S.Complexes) 

return condition_set 

 

 

def substitution(system, symbols, result=[{}], known_symbols=[], 

exclude=[], all_symbols=None): 

r""" 

Solves the `system` using substitution method. It is used in 

`nonlinsolve`. This will be called from `nonlinsolve` when any 

equation(s) is non polynomial equation. 

 

Parameters 

========== 

 

system : list of equations 

The target system of equations 

symbols : list of symbols to be solved. 

The variable(s) for which the system is solved 

known_symbols : list of solved symbols 

Values are known for these variable(s) 

result : An empty list or list of dict 

If No symbol values is known then empty list otherwise 

symbol as keys and corresponding value in dict. 

exclude : Set of expression. 

Mostly denominator expression(s) of the equations of the system. 

Final solution should not satisfy these expressions. 

all_symbols : known_symbols + symbols(unsolved). 

 

Returns 

======= 

 

A FiniteSet of ordered tuple of values of `all_symbols` for which the 

`system` has solution. Order of values in the tuple is same as symbols 

present in the parameter `all_symbols`. If parameter `all_symbols` is None 

then same as symbols present in the parameter `symbols`. 

 

Please note that general FiniteSet is unordered, the solution returned 

here is not simply a FiniteSet of solutions, rather it is a FiniteSet of 

ordered tuple, i.e. the first & only argument to FiniteSet is a tuple of 

solutions, which is ordered, & hence the returned solution is ordered. 

 

Also note that solution could also have been returned as an ordered tuple, 

FiniteSet is just a wrapper `{}` around the tuple. It has no other 

significance except for the fact it is just used to maintain a consistent 

output format throughout the solveset. 

 

Raises 

====== 

 

ValueError 

The input is not valid. 

The symbols are not given. 

AttributeError 

The input symbols are not `Symbol` type. 

 

Examples 

======== 

 

>>> from sympy.core.symbol import symbols 

>>> x, y = symbols('x, y', real=True) 

>>> from sympy.solvers.solveset import substitution 

>>> substitution([x + y], [x], [{y: 1}], [y], set([]), [x, y]) 

{(-1, 1)} 

 

* when you want soln should not satisfy eq `x + 1 = 0` 

 

>>> substitution([x + y], [x], [{y: 1}], [y], set([x + 1]), [y, x]) 

EmptySet() 

>>> substitution([x + y], [x], [{y: 1}], [y], set([x - 1]), [y, x]) 

{(1, -1)} 

>>> substitution([x + y - 1, y - x**2 + 5], [x, y]) 

{(-3, 4), (2, -1)} 

 

* Returns both real and complex solution 

 

>>> x, y, z = symbols('x, y, z') 

>>> from sympy import exp, sin 

>>> substitution([exp(x) - sin(y), y**2 - 4], [x, y]) 

{(log(sin(2)), 2), (ImageSet(Lambda(_n, I*(2*_n*pi + pi) + 

log(sin(2))), S.Integers), -2), (ImageSet(Lambda(_n, 2*_n*I*pi + 

Mod(log(sin(2)), 2*I*pi)), S.Integers), 2)} 

 

>>> eqs = [z**2 + exp(2*x) - sin(y), -3 + exp(-y)] 

>>> substitution(eqs, [y, z]) 

{(-log(3), -sqrt(-exp(2*x) - sin(log(3)))), 

(-log(3), sqrt(-exp(2*x) - sin(log(3)))), 

(ImageSet(Lambda(_n, 2*_n*I*pi + Mod(-log(3), 2*I*pi)), S.Integers), 

ImageSet(Lambda(_n, -sqrt(-exp(2*x) + sin(2*_n*I*pi + 

Mod(-log(3), 2*I*pi)))), S.Integers)), 

(ImageSet(Lambda(_n, 2*_n*I*pi + Mod(-log(3), 2*I*pi)), S.Integers), 

ImageSet(Lambda(_n, sqrt(-exp(2*x) + sin(2*_n*I*pi + 

Mod(-log(3), 2*I*pi)))), S.Integers))} 

 

""" 

 

from sympy import Complement 

from sympy.core.compatibility import is_sequence 

 

if not system: 

return S.EmptySet 

 

if not symbols: 

msg = ('Symbols must be given, for which solution of the ' 

'system is to be found.') 

raise ValueError(filldedent(msg)) 

 

if not is_sequence(symbols): 

msg = ('symbols should be given as a sequence, e.g. a list.' 

'Not type %s: %s') 

raise TypeError(filldedent(msg % (type(symbols), symbols))) 

 

try: 

sym = symbols[0].is_Symbol 

except AttributeError: 

sym = False 

 

if not sym: 

msg = ('Iterable of symbols must be given as ' 

'second argument, not type %s: %s') 

raise ValueError(filldedent(msg % (type(symbols[0]), symbols[0]))) 

 

# By default `all_symbols` will be same as `symbols` 

if all_symbols is None: 

all_symbols = symbols 

 

old_result = result 

# storing complements and intersection for particular symbol 

complements = {} 

intersections = {} 

 

# when total_solveset_call is equals to total_conditionset 

# means solvest fail to solve all the eq. 

total_conditionset = -1 

total_solveset_call = -1 

 

def _unsolved_syms(eq, sort=False): 

"""Returns the unsolved symbol present 

in the equation `eq`. 

""" 

free = eq.free_symbols 

unsolved = (free - set(known_symbols)) & set(all_symbols) 

if sort: 

unsolved = list(unsolved) 

unsolved.sort(key=default_sort_key) 

return unsolved 

# end of _unsolved_syms() 

 

# sort such that equation with the fewest potential symbols is first. 

# means eq with less number of variable first in the list. 

eqs_in_better_order = list( 

ordered(system, lambda _: len(_unsolved_syms(_)))) 

 

def add_intersection_complement(result, sym_set, **flags): 

# If solveset have returned some intersection/complement 

# for any symbol. It will be added in final solution. 

final_result = [] 

for res in result: 

res_copy = res 

for key_res, value_res in res.items(): 

# Intersection/complement is in Interval or Set. 

intersection_true = flags.get('Intersection', True) 

complements_true = flags.get('Complement', True) 

for key_sym, value_sym in sym_set.items(): 

if key_sym == key_res: 

if intersection_true: 

# testcase is not added for this line(intersection) 

new_value = \ 

Intersection(FiniteSet(value_res), value_sym) 

if new_value is not S.EmptySet: 

res_copy[key_res] = new_value 

if complements_true: 

new_value = \ 

Complement(FiniteSet(value_res), value_sym) 

if new_value is not S.EmptySet: 

res_copy[key_res] = new_value 

final_result.append(res_copy) 

return final_result 

# end of def add_intersection_complement() 

 

def _extract_main_soln(sol, soln_imageset): 

"""separate the Complements, Intersections, ImageSet lambda expr 

and it's base_set. 

""" 

# if there is union, then need to check 

# Complement, Intersection, Imageset. 

# Order should not be changed. 

if isinstance(sol, Complement): 

# extract solution and complement 

complements[sym] = sol.args[1] 

sol = sol.args[0] 

# complement will be added at the end 

# using `add_intersection_complement` method 

if isinstance(sol, Intersection): 

# Interval/Set will be at 0th index always 

if sol.args[0] != Interval(-oo, oo): 

# sometimes solveset returns soln 

# with intersection `S.Reals`, to confirm that 

# soln is in `domain=S.Reals` or not. We don't consider 

# that intersection. 

intersections[sym] = sol.args[0] 

sol = sol.args[1] 

# after intersection and complement Imageset should 

# be checked. 

if isinstance(sol, ImageSet): 

soln_imagest = sol 

expr2 = sol.lamda.expr 

sol = FiniteSet(expr2) 

soln_imageset[expr2] = soln_imagest 

 

# if there is union of Imageset or other in soln. 

# no testcase is written for this if block 

if isinstance(sol, Union): 

sol_args = sol.args 

sol = S.EmptySet 

# We need in sequence so append finteset elements 

# and then imageset or other. 

for sol_arg2 in sol_args: 

if isinstance(sol_arg2, FiniteSet): 

sol += sol_arg2 

else: 

# ImageSet, Intersection, complement then 

# append them directly 

sol += FiniteSet(sol_arg2) 

 

if not isinstance(sol, FiniteSet): 

sol = FiniteSet(sol) 

return sol, soln_imageset 

# end of def _extract_main_soln() 

 

# helper function for _append_new_soln 

def _check_exclude(rnew, imgset_yes): 

rnew_ = rnew 

if imgset_yes: 

# replace all dummy variables (Imageset lambda variables) 

# with zero before `checksol`. Considering fundamental soln 

# for `checksol`. 

rnew_copy = rnew.copy() 

dummy_n = imgset_yes[0] 

for key_res, value_res in rnew_copy.items(): 

rnew_copy[key_res] = value_res.subs(dummy_n, 0) 

rnew_ = rnew_copy 

# satisfy_exclude == true if it satisfies the expr of `exclude` list. 

try: 

# something like : `Mod(-log(3), 2*I*pi)` can't be 

# simplified right now, so `checksol` returns `TypeError`. 

# when this issue is fixed this try block should be 

# removed. Mod(-log(3), 2*I*pi) == -log(3) 

satisfy_exclude = any( 

checksol(d, rnew_) for d in exclude) 

except TypeError: 

satisfy_exclude = None 

return satisfy_exclude 

# end of def _check_exclude() 

 

# helper function for _append_new_soln 

def _restore_imgset(rnew, original_imageset, newresult): 

restore_sym = set(rnew.keys()) & \ 

set(original_imageset.keys()) 

for key_sym in restore_sym: 

img = original_imageset[key_sym] 

rnew[key_sym] = img 

if rnew not in newresult: 

newresult.append(rnew) 

# end of def _restore_imgset() 

 

def _append_eq(eq, result, res, delete_soln, n=None): 

u = Dummy('u') 

if n: 

eq = eq.subs(n, 0) 

satisfy = checksol(u, u, eq, minimal=True) 

if satisfy is False: 

delete_soln = True 

res = {} 

else: 

result.append(res) 

return result, res, delete_soln 

 

def _append_new_soln(rnew, sym, sol, imgset_yes, soln_imageset, 

original_imageset, newresult, eq=None): 

"""If `rnew` (A dict <symbol: soln>) contains valid soln 

append it to `newresult` list. 

`imgset_yes` is (base, dummy_var) if there was imageset in previously 

calculated result(otherwise empty tuple). `original_imageset` is dict 

of imageset expr and imageset from this result. 

`soln_imageset` dict of imageset expr and imageset of new soln. 

""" 

satisfy_exclude = _check_exclude(rnew, imgset_yes) 

delete_soln = False 

# soln should not satisfy expr present in `exclude` list. 

if not satisfy_exclude: 

local_n = None 

# if it is imageset 

if imgset_yes: 

local_n = imgset_yes[0] 

base = imgset_yes[1] 

if sym and sol: 

# when `sym` and `sol` is `None` means no new 

# soln. In that case we will append rnew directly after 

# substituting original imagesets in rnew values if present 

# (second last line of this function using _restore_imgset) 

dummy_list = list(sol.atoms(Dummy)) 

# use one dummy `n` which is in 

# previous imageset 

local_n_list = [ 

local_n for i in range( 

0, len(dummy_list))] 

 

dummy_zip = zip(dummy_list, local_n_list) 

lam = Lambda(local_n, sol.subs(dummy_zip)) 

rnew[sym] = ImageSet(lam, base) 

if eq is not None: 

newresult, rnew, delete_soln = _append_eq( 

eq, newresult, rnew, delete_soln, local_n) 

elif eq is not None: 

newresult, rnew, delete_soln = _append_eq( 

eq, newresult, rnew, delete_soln) 

elif soln_imageset: 

rnew[sym] = soln_imageset[sol] 

# restore original imageset 

_restore_imgset(rnew, original_imageset, newresult) 

else: 

newresult.append(rnew) 

elif satisfy_exclude: 

delete_soln = True 

rnew = {} 

_restore_imgset(rnew, original_imageset, newresult) 

return newresult, delete_soln 

# end of def _append_new_soln() 

 

def _new_order_result(result, eq): 

# separate first, second priority. `res` that makes `eq` value equals 

# to zero, should be used first then other result(second priority). 

# If it is not done then we may miss some soln. 

first_priority = [] 

second_priority = [] 

for res in result: 

if not any(isinstance(val, ImageSet) for val in res.values()): 

if eq.subs(res) == 0: 

first_priority.append(res) 

else: 

second_priority.append(res) 

if first_priority or second_priority: 

return first_priority + second_priority 

return result 

 

def _solve_using_known_values(result, solver): 

"""Solves the system using already known solution 

(result contains the dict <symbol: value>). 

solver is `solveset_complex` or `solveset_real`. 

""" 

# stores imageset <expr: imageset(Lambda(n, expr), base)>. 

soln_imageset = {} 

total_solvest_call = 0 

total_conditionst = 0 

 

# sort such that equation with the fewest potential symbols is first. 

# means eq with less variable first 

for index, eq in enumerate(eqs_in_better_order): 

newresult = [] 

original_imageset = {} 

# if imageset expr is used to solve other symbol 

imgset_yes = False 

result = _new_order_result(result, eq) 

for res in result: 

got_symbol = set() # symbols solved in one iteration 

if soln_imageset: 

# find the imageset and use its expr. 

for key_res, value_res in res.items(): 

if isinstance(value_res, ImageSet): 

res[key_res] = value_res.lamda.expr 

original_imageset[key_res] = value_res 

dummy_n = value_res.lamda.expr.atoms(Dummy).pop() 

base = value_res.base_set 

imgset_yes = (dummy_n, base) 

# update eq with everything that is known so far 

eq2 = eq.subs(res) 

unsolved_syms = _unsolved_syms(eq2, sort=True) 

if not unsolved_syms: 

if res: 

newresult, delete_res = _append_new_soln( 

res, None, None, imgset_yes, soln_imageset, 

original_imageset, newresult, eq2) 

if delete_res: 

# `delete_res` is true, means substituting `res` in 

# eq2 doesn't return `zero` or deleting the `res` 

# (a soln) since it staisfies expr of `exclude` 

# list. 

result.remove(res) 

continue # skip as it's independent of desired symbols 

depen = eq2.as_independent(unsolved_syms)[0] 

if depen.has(Abs) and solver == solveset_complex: 

# Absolute values cannot be inverted in the 

# complex domain 

continue 

soln_imageset = {} 

for sym in unsolved_syms: 

not_solvable = False 

try: 

soln = solver(eq2, sym) 

total_solvest_call += 1 

soln_new = S.EmptySet 

if isinstance(soln, Complement): 

# separate solution and complement 

complements[sym] = soln.args[1] 

soln = soln.args[0] 

# complement will be added at the end 

if isinstance(soln, Intersection): 

# Interval will be at 0th index always 

if soln.args[0] != Interval(-oo, oo): 

# sometimes solveset returns soln 

# with intersection S.Reals, to confirm that 

# soln is in domain=S.Reals 

intersections[sym] = soln.args[0] 

soln_new += soln.args[1] 

soln = soln_new if soln_new else soln 

if index > 0 and solver == solveset_real: 

# one symbol's real soln , another symbol may have 

# corresponding complex soln. 

if not isinstance(soln, (ImageSet, ConditionSet)): 

soln += solveset_complex(eq2, sym) 

except NotImplementedError: 

# If sovleset is not able to solve equation `eq2`. Next 

# time we may get soln using next equation `eq2` 

continue 

if isinstance(soln, ConditionSet): 

soln = S.EmptySet 

# don't do `continue` we may get soln 

# in terms of other symbol(s) 

not_solvable = True 

total_conditionst += 1 

 

if soln is not S.EmptySet: 

soln, soln_imageset = _extract_main_soln( 

soln, soln_imageset) 

 

for sol in soln: 

# sol is not a `Union` since we checked it 

# before this loop 

sol, soln_imageset = _extract_main_soln( 

sol, soln_imageset) 

sol = set(sol).pop() 

free = sol.free_symbols 

if got_symbol and any([ 

ss in free for ss in got_symbol 

]): 

# sol depends on previously solved symbols 

# then continue 

continue 

rnew = res.copy() 

# put each solution in res and append the new result 

# in the new result list (solution for symbol `s`) 

# along with old results. 

for k, v in res.items(): 

if isinstance(v, Expr): 

# if any unsolved symbol is present 

# Then subs known value 

rnew[k] = v.subs(sym, sol) 

# and add this new solution 

if soln_imageset: 

# replace all lambda variables with 0. 

imgst = soln_imageset[sol] 

rnew[sym] = imgst.lamda( 

*[0 for i in range(0, len( 

imgst.lamda.variables))]) 

else: 

rnew[sym] = sol 

newresult, delete_res = _append_new_soln( 

rnew, sym, sol, imgset_yes, soln_imageset, 

original_imageset, newresult) 

if delete_res: 

# deleting the `res` (a soln) since it staisfies 

# eq of `exclude` list 

result.remove(res) 

# solution got for sym 

if not not_solvable: 

got_symbol.add(sym) 

# next time use this new soln 

if newresult: 

result = newresult 

return result, total_solvest_call, total_conditionst 

# end def _solve_using_know_values() 

 

new_result_real, solve_call1, cnd_call1 = _solve_using_known_values( 

old_result, solveset_real) 

new_result_complex, solve_call2, cnd_call2 = _solve_using_known_values( 

old_result, solveset_complex) 

 

# when `total_solveset_call` is equals to `total_conditionset` 

# means solvest fails to solve all the eq. 

# return conditionset in this case 

total_conditionset += (cnd_call1 + cnd_call2) 

total_solveset_call += (solve_call1 + solve_call2) 

 

if total_conditionset == total_solveset_call and total_solveset_call != -1: 

return _return_conditionset(eqs_in_better_order, all_symbols) 

 

# overall result 

result = new_result_real + new_result_complex 

 

result_all_variables = [] 

result_infinite = [] 

for res in result: 

if not res: 

# means {None : None} 

continue 

# If length < len(all_symbols) means infinite soln. 

# Some or all the soln is dependent on 1 symbol. 

# eg. {x: y+2} then final soln {x: y+2, y: y} 

if len(res) < len(all_symbols): 

solved_symbols = res.keys() 

unsolved = list(filter( 

lambda x: x not in solved_symbols, all_symbols)) 

for unsolved_sym in unsolved: 

res[unsolved_sym] = unsolved_sym 

result_infinite.append(res) 

if res not in result_all_variables: 

result_all_variables.append(res) 

 

if result_infinite: 

# we have general soln 

# eg : [{x: -1, y : 1}, {x : -y , y: y}] then 

# return [{x : -y, y : y}] 

result_all_variables = result_infinite 

if intersections and complements: 

# no testcase is added for this block 

result_all_variables = add_intersection_complement( 

result_all_variables, intersections, 

Intersection=True, Complement=True) 

elif intersections: 

result_all_variables = add_intersection_complement( 

result_all_variables, intersections, Intersection=True) 

elif complements: 

result_all_variables = add_intersection_complement( 

result_all_variables, complements, Complement=True) 

 

# convert to ordered tuple 

result = S.EmptySet 

for r in result_all_variables: 

temp = [r[symb] for symb in all_symbols] 

result += FiniteSet(tuple(temp)) 

return result 

# end of def substitution() 

 

 

def _solveset_work(system, symbols): 

soln = solveset(system[0], symbols[0]) 

if isinstance(soln, FiniteSet): 

_soln = FiniteSet(*[tuple((s,)) for s in soln]) 

return _soln 

else: 

return FiniteSet(tuple(FiniteSet(soln))) 

 

 

def _handle_positive_dimensional(polys, symbols, denominators): 

from sympy.polys.polytools import groebner 

# substitution method where new system is groebner basis of the system 

_symbols = list(symbols) 

_symbols.sort(key=default_sort_key) 

basis = groebner(polys, _symbols, polys=True) 

new_system = [] 

for poly_eq in basis: 

new_system.append(poly_eq.as_expr()) 

result = [{}] 

result = substitution( 

new_system, symbols, result, [], 

denominators) 

return result 

# end of def _handle_positive_dimensional() 

 

 

def _handle_zero_dimensional(polys, symbols, system): 

# solve 0 dimensional poly system using `solve_poly_system` 

result = solve_poly_system(polys, *symbols) 

# May be some extra soln is added because 

# we used `unrad` in `_separate_poly_nonpoly`, so 

# need to check and remove if it is not a soln. 

result_update = S.EmptySet 

for res in result: 

dict_sym_value = dict(list(zip(symbols, res))) 

if all(checksol(eq, dict_sym_value) for eq in system): 

result_update += FiniteSet(res) 

return result_update 

# end of def _handle_zero_dimensional() 

 

 

def _separate_poly_nonpoly(system, symbols): 

polys = [] 

polys_expr = [] 

nonpolys = [] 

denominators = set() 

poly = None 

for eq in system: 

# Store denom expression if it contains symbol 

denominators.update(_simple_dens(eq, symbols)) 

# try to remove sqrt and rational power 

without_radicals = unrad(simplify(eq)) 

if without_radicals: 

eq_unrad, cov = without_radicals 

if not cov: 

eq = eq_unrad 

if isinstance(eq, Expr): 

eq = eq.as_numer_denom()[0] 

poly = eq.as_poly(*symbols, extension=True) 

elif simplify(eq).is_number: 

continue 

if poly is not None: 

polys.append(poly) 

polys_expr.append(poly.as_expr()) 

else: 

nonpolys.append(eq) 

return polys, polys_expr, nonpolys, denominators 

# end of def _separate_poly_nonpoly() 

 

 

def nonlinsolve(system, *symbols): 

r""" 

Solve system of N non linear equations with M variables, which means both 

under and overdetermined systems are supported. Positive dimensional 

system is also supported (A system with infinitely many solutions is said 

to be positive-dimensional). In Positive dimensional system solution will 

be dependent on at least one symbol. Returns both real solution 

and complex solution(If system have). The possible number of solutions 

is zero, one or infinite. 

 

Parameters 

========== 

 

system : list of equations 

The target system of equations 

symbols : list of Symbols 

symbols should be given as a sequence eg. list 

 

Returns 

======= 

 

A FiniteSet of ordered tuple of values of `symbols` for which the `system` 

has solution. Order of values in the tuple is same as symbols present in 

the parameter `symbols`. 

 

Please note that general FiniteSet is unordered, the solution returned 

here is not simply a FiniteSet of solutions, rather it is a FiniteSet of 

ordered tuple, i.e. the first & only argument to FiniteSet is a tuple of 

solutions, which is ordered, & hence the returned solution is ordered. 

 

Also note that solution could also have been returned as an ordered tuple, 

FiniteSet is just a wrapper `{}` around the tuple. It has no other 

significance except for the fact it is just used to maintain a consistent 

output format throughout the solveset. 

 

For the given set of Equations, the respective input types 

are given below: 

 

.. math:: x*y - 1 = 0 

.. math:: 4*x**2 + y**2 - 5 = 0 

 

`system = [x*y - 1, 4*x**2 + y**2 - 5]` 

`symbols = [x, y]` 

 

Raises 

====== 

 

ValueError 

The input is not valid. 

The symbols are not given. 

AttributeError 

The input symbols are not `Symbol` type. 

 

Examples 

======== 

 

>>> from sympy.core.symbol import symbols 

>>> from sympy.solvers.solveset import nonlinsolve 

>>> x, y, z = symbols('x, y, z', real=True) 

>>> nonlinsolve([x*y - 1, 4*x**2 + y**2 - 5], [x, y]) 

{(-1, -1), (-1/2, -2), (1/2, 2), (1, 1)} 

 

1. Positive dimensional system and complements: 

 

>>> from sympy import pprint 

>>> from sympy.polys.polytools import is_zero_dimensional 

>>> a, b, c, d = symbols('a, b, c, d', real=True) 

>>> eq1 = a + b + c + d 

>>> eq2 = a*b + b*c + c*d + d*a 

>>> eq3 = a*b*c + b*c*d + c*d*a + d*a*b 

>>> eq4 = a*b*c*d - 1 

>>> system = [eq1, eq2, eq3, eq4] 

>>> is_zero_dimensional(system) 

False 

>>> pprint(nonlinsolve(system, [a, b, c, d]), use_unicode=False) 

-1 1 1 -1 

{(---, -d, -, {d} \ {0}), (-, -d, ---, {d} \ {0})} 

d d d d 

>>> nonlinsolve([(x+y)**2 - 4, x + y - 2], [x, y]) 

{(-y + 2, y)} 

 

2. If some of the equations are non polynomial equation then `nonlinsolve` 

will call `substitution` function and returns real and complex solutions, 

if present. 

 

>>> from sympy import exp, sin 

>>> nonlinsolve([exp(x) - sin(y), y**2 - 4], [x, y]) 

{(log(sin(2)), 2), (ImageSet(Lambda(_n, I*(2*_n*pi + pi) + 

log(sin(2))), S.Integers), -2), (ImageSet(Lambda(_n, 2*_n*I*pi + 

Mod(log(sin(2)), 2*I*pi)), S.Integers), 2)} 

 

3. If system is Non linear polynomial zero dimensional then it returns 

both solution (real and complex solutions, if present using 

`solve_poly_system`): 

 

>>> from sympy import sqrt 

>>> nonlinsolve([x**2 - 2*y**2 -2, x*y - 2], [x, y]) 

{(-2, -1), (2, 1), (-sqrt(2)*I, sqrt(2)*I), (sqrt(2)*I, -sqrt(2)*I)} 

 

4. `nonlinsolve` can solve some linear(zero or positive dimensional) 

system (because it is using `groebner` function to get the 

groebner basis and then `substitution` function basis as the new `system`). 

But it is not recommended to solve linear system using `nonlinsolve`, 

because `linsolve` is better for all kind of linear system. 

 

>>> nonlinsolve([x + 2*y -z - 3, x - y - 4*z + 9 , y + z - 4], [x, y, z]) 

{(3*z - 5, -z + 4, z)} 

 

5. System having polynomial equations and only real solution is present 

(will be solved using `solve_poly_system`): 

 

>>> e1 = sqrt(x**2 + y**2) - 10 

>>> e2 = sqrt(y**2 + (-x + 10)**2) - 3 

>>> nonlinsolve((e1, e2), (x, y)) 

{(191/20, -3*sqrt(391)/20), (191/20, 3*sqrt(391)/20)} 

>>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [x, y]) 

{(1, 2), (1 + sqrt(5), -sqrt(5) + 2), (-sqrt(5) + 1, 2 + sqrt(5))} 

>>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [y, x]) 

{(2, 1), (2 + sqrt(5), -sqrt(5) + 1), (-sqrt(5) + 2, 1 + sqrt(5))} 

 

6. It is better to use symbols instead of Trigonometric Function or 

Function (e.g. replace `sin(x)` with symbol, replace `f(x)` with symbol 

and so on. Get soln from `nonlinsolve` and then using `solveset` get 

the value of `x`) 

 

How nonlinsolve is better than old solver `_solve_system` : 

=========================================================== 

 

1. A positive dimensional system solver : nonlinsolve can return 

solution for positive dimensional system. It finds the 

Groebner Basis of the positive dimensional system(calling it as 

basis) then we can start solving equation(having least number of 

variable first in the basis) using solveset and substituting that 

solved solutions into other equation(of basis) to get solution in 

terms of minimum variables. Here the important thing is how we 

are substituting the known values and in which equations. 

 

2. Real and Complex both solutions : nonlinsolve returns both real 

and complex solution. If all the equations in the system are polynomial 

then using `solve_poly_system` both real and complex solution is returned. 

If all the equations in the system are not polynomial equation then goes to 

`substitution` method with this polynomial and non polynomial equation(s), 

to solve for unsolved variables. Here to solve for particular variable 

solveset_real and solveset_complex is used. For both real and complex 

solution function `_solve_using_know_values` is used inside `substitution` 

function.(`substitution` function will be called when there is any non 

polynomial equation(s) is present). When solution is valid then add its 

general solution in the final result. 

 

3. Complement and Intersection will be added if any : nonlinsolve maintains 

dict for complements and Intersections. If solveset find complements or/and 

Intersection with any Interval or set during the execution of 

`substitution` function ,then complement or/and Intersection for that 

variable is added before returning final solution. 

 

""" 

from sympy.polys.polytools import is_zero_dimensional 

 

if not system: 

return S.EmptySet 

 

if not symbols: 

msg = ('Symbols must be given, for which solution of the ' 

'system is to be found.') 

raise ValueError(filldedent(msg)) 

 

if hasattr(symbols[0], '__iter__'): 

symbols = symbols[0] 

 

if not is_sequence(symbols) or not symbols: 

msg = ('Symbols must be given, for which solution of the ' 

'system is to be found.') 

raise IndexError(filldedent(msg)) 

 

system, symbols, swap = recast_to_symbols(system, symbols) 

if swap: 

soln = nonlinsolve(system, symbols) 

return FiniteSet(*[tuple(i.xreplace(swap) for i in s) for s in soln]) 

 

if len(system) == 1 and len(symbols) == 1: 

return _solveset_work(system, symbols) 

 

# main code of def nonlinsolve() starts from here 

polys, polys_expr, nonpolys, denominators = _separate_poly_nonpoly( 

system, symbols) 

 

if len(symbols) == len(polys): 

# If all the equations in the system are poly 

if is_zero_dimensional(polys, symbols): 

# finite number of soln (Zero dimensional system) 

try: 

return _handle_zero_dimensional(polys, symbols, system) 

except NotImplementedError: 

# Right now it doesn't fail for any polynomial system of 

# equation. If `solve_poly_system` fails then `substitution` 

# method will handle it. 

result = substitution( 

polys_expr, symbols, exclude=denominators) 

return result 

 

# positive dimensional system 

return _handle_positive_dimensional(polys, symbols, denominators) 

 

else: 

# If all the equations are not polynomial. 

# Use `substitution` method for the system 

result = substitution( 

polys_expr + nonpolys, symbols, exclude=denominators) 

return result