Source code for sympy.solvers.inequalities
"""Tools for solving inequalities and systems of inequalities. """
from __future__ import print_function, division
from sympy.core import Symbol, Interval
from sympy.core.relational import Relational, Eq, Ge, Lt
from sympy.core.sets import FiniteSet, Union
from sympy.core.singleton import S
from sympy.assumptions import ask, AppliedPredicate, Q
from sympy.functions import re, im, Abs
from sympy.logic import And
from sympy.polys import Poly, PolynomialError, parallel_poly_from_expr
[docs]def solve_poly_inequality(poly, rel):
"""Solve a polynomial inequality with rational coefficients.
Examples
========
>>> from sympy import Poly
>>> from sympy.abc import x
>>> from sympy.solvers.inequalities import solve_poly_inequality
>>> solve_poly_inequality(Poly(x, x, domain='ZZ'), '==')
[{0}]
>>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '!=')
[(-oo, -1), (-1, 1), (1, oo)]
>>> solve_poly_inequality(Poly(x**2 - 1, x, domain='ZZ'), '==')
[{-1}, {1}]
See Also
========
solve_poly_inequalities
"""
reals, intervals = poly.real_roots(multiple=False), []
if rel == '==':
for root, _ in reals:
interval = Interval(root, root)
intervals.append(interval)
elif rel == '!=':
left = S.NegativeInfinity
for right, _ in reals + [(S.Infinity, 1)]:
interval = Interval(left, right, True, True)
intervals.append(interval)
left = right
else:
if poly.LC() > 0:
sign = +1
else:
sign = -1
eq_sign, equal = None, False
if rel == '>':
eq_sign = +1
elif rel == '<':
eq_sign = -1
elif rel == '>=':
eq_sign, equal = +1, True
elif rel == '<=':
eq_sign, equal = -1, True
else:
raise ValueError("'%s' is not a valid relation" % rel)
right, right_open = S.Infinity, True
reals.sort(key=lambda w: w[0], reverse=True)
for left, multiplicity in reals:
if multiplicity % 2:
if sign == eq_sign:
intervals.insert(
0, Interval(left, right, not equal, right_open))
sign, right, right_open = -sign, left, not equal
else:
if sign == eq_sign and not equal:
intervals.insert(
0, Interval(left, right, True, right_open))
right, right_open = left, True
elif sign != eq_sign and equal:
intervals.insert(0, Interval(left, left))
if sign == eq_sign:
intervals.insert(
0, Interval(S.NegativeInfinity, right, True, right_open))
return intervals
def solve_poly_inequalities(polys):
"""Solve polynomial inequalities with rational coefficients.
Examples
========
>>> from sympy.solvers.inequalities import solve_poly_inequalities
>>> from sympy.polys import Poly
>>> from sympy.abc import x
>>> solve_poly_inequalities(((
... Poly(x**2 - 3), ">"), (
... Poly(-x**2 + 1), ">")))
(-oo, -sqrt(3)) U (-1, 1) U (sqrt(3), oo)
"""
from sympy import Union
return Union(*[solve_poly_inequality(*p) for p in polys])
[docs]def solve_rational_inequalities(eqs):
"""Solve a system of rational inequalities with rational coefficients.
Examples
========
>>> from sympy.abc import x
>>> from sympy import Poly
>>> from sympy.solvers.inequalities import solve_rational_inequalities
>>> solve_rational_inequalities([[
... ((Poly(-x + 1), Poly(1, x)), '>='),
... ((Poly(-x + 1), Poly(1, x)), '<=')]])
{1}
>>> solve_rational_inequalities([[
... ((Poly(x), Poly(1, x)), '!='),
... ((Poly(-x + 1), Poly(1, x)), '>=')]])
(-oo, 0) U (0, 1]
See Also
========
solve_poly_inequality
"""
result = S.EmptySet
for _eqs in eqs:
global_intervals = None
for (numer, denom), rel in _eqs:
numer_intervals = solve_poly_inequality(numer*denom, rel)
denom_intervals = solve_poly_inequality(denom, '==')
if global_intervals is None:
global_intervals = numer_intervals
else:
intervals = []
for numer_interval in numer_intervals:
for global_interval in global_intervals:
interval = numer_interval.intersect(global_interval)
if interval is not S.EmptySet:
intervals.append(interval)
global_intervals = intervals
intervals = []
for global_interval in global_intervals:
for denom_interval in denom_intervals:
global_interval -= denom_interval
if global_interval is not S.EmptySet:
intervals.append(global_interval)
global_intervals = intervals
if not global_intervals:
break
for interval in global_intervals:
result = result.union(interval)
return result
[docs]def reduce_rational_inequalities(exprs, gen, assume=True, relational=True):
"""Reduce a system of rational inequalities with rational coefficients.
Examples
========
>>> from sympy import Poly, Symbol
>>> from sympy.solvers.inequalities import reduce_rational_inequalities
>>> x = Symbol('x', real=True)
>>> reduce_rational_inequalities([[x**2 <= 0]], x)
x == 0
>>> reduce_rational_inequalities([[x + 2 > 0]], x)
x > -2
>>> reduce_rational_inequalities([[(x + 2, ">")]], x)
x > -2
>>> reduce_rational_inequalities([[x + 2]], x)
x == -2
"""
exact = True
eqs = []
for _exprs in exprs:
_eqs = []
for expr in _exprs:
if isinstance(expr, tuple):
expr, rel = expr
else:
if expr.is_Relational:
expr, rel = expr.lhs - expr.rhs, expr.rel_op
else:
expr, rel = expr, '=='
try:
(numer, denom), opt = parallel_poly_from_expr(
expr.together().as_numer_denom(), gen)
except PolynomialError:
raise PolynomialError("only polynomials and "
"rational functions are supported in this context")
if not opt.domain.is_Exact:
numer, denom, exact = numer.to_exact(), denom.to_exact(), False
domain = opt.domain.get_exact()
if not (domain.is_ZZ or domain.is_QQ):
raise NotImplementedError(
"inequality solving is not supported over %s" % opt.domain)
_eqs.append(((numer, denom), rel))
eqs.append(_eqs)
solution = solve_rational_inequalities(eqs)
if not exact:
solution = solution.evalf()
if not relational:
return solution
real = ask(Q.real(gen), assumptions=assume)
if not real:
result = And(solution.as_relational(re(gen)), Eq(im(gen), 0))
else:
result = solution.as_relational(gen)
return result
[docs]def reduce_abs_inequality(expr, rel, gen, assume=True):
"""Reduce an inequality with nested absolute values.
Examples
========
>>> from sympy import Q, Abs
>>> from sympy.abc import x
>>> from sympy.solvers.inequalities import reduce_abs_inequality
>>> reduce_abs_inequality(Abs(x - 5) - 3, '<', x, assume=Q.real(x))
And(2 < x, x < 8)
>>> reduce_abs_inequality(Abs(x + 2)*3 - 13, '<', x, assume=Q.real(x))
And(-19/3 < x, x < 7/3)
See Also
========
reduce_abs_inequalities
"""
if not ask(Q.real(gen), assumptions=assume):
raise NotImplementedError("can't solve inequalities with absolute "
"values of a complex variable")
def _bottom_up_scan(expr):
exprs = []
if expr.is_Add or expr.is_Mul:
op = expr.__class__
for arg in expr.args:
_exprs = _bottom_up_scan(arg)
if not exprs:
exprs = _exprs
else:
args = []
for expr, conds in exprs:
for _expr, _conds in _exprs:
args.append((op(expr, _expr), conds + _conds))
exprs = args
elif expr.is_Pow:
n = expr.exp
if not n.is_Integer or n < 0:
raise ValueError(
"only non-negative integer powers are allowed")
_exprs = _bottom_up_scan(expr.base)
for expr, conds in _exprs:
exprs.append((expr**n, conds))
elif isinstance(expr, Abs):
_exprs = _bottom_up_scan(expr.args[0])
for expr, conds in _exprs:
exprs.append(( expr, conds + [Ge(expr, 0)]))
exprs.append((-expr, conds + [Lt(expr, 0)]))
else:
exprs = [(expr, [])]
return exprs
exprs = _bottom_up_scan(expr)
mapping = {'<': '>', '<=': '>='}
inequalities = []
for expr, conds in exprs:
if rel not in mapping.keys():
expr = Relational( expr, 0, rel)
else:
expr = Relational(-expr, 0, mapping[rel])
inequalities.append([expr] + conds)
return reduce_rational_inequalities(inequalities, gen, assume)
[docs]def reduce_abs_inequalities(exprs, gen, assume=True):
"""Reduce a system of inequalities with nested absolute values.
Examples
========
>>> from sympy import Q, Abs
>>> from sympy.abc import x
>>> from sympy.solvers.inequalities import reduce_abs_inequalities
>>> reduce_abs_inequalities([(Abs(3*x - 5) - 7, '<'),
... (Abs(x + 25) - 13, '>')], x, assume=Q.real(x))
And(-2/3 < x, Or(x < -38, x > -12), x < 4)
>>> reduce_abs_inequalities([(Abs(x - 4) + Abs(3*x - 5) - 7, '<')], x,
... assume=Q.real(x))
And(1/2 < x, x < 4)
See Also
========
reduce_abs_inequality
"""
return And(*[ reduce_abs_inequality(expr, rel, gen, assume)
for expr, rel in exprs ])
[docs]def solve_univariate_inequality(expr, gen, assume=True, relational=True):
"""Solves a real univariate inequality.
Examples
========
>>> from sympy.solvers.inequalities import solve_univariate_inequality
>>> from sympy.core.symbol import Symbol
>>> x = Symbol('x', real=True)
>>> solve_univariate_inequality(x**2 >= 4, x)
Or(x <= -2, x >= 2)
>>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
(-oo, -2] U [2, oo)
"""
# Implementation for continous functions
from sympy.solvers.solvers import solve
solns = solve(expr.lhs - expr.rhs, gen, assume=assume)
oo = S.Infinity
start = -oo
sol_sets = [S.EmptySet]
for x in sorted(s for s in solns if s.is_real):
end = x
if expr.subs(gen, (start + end)/2 if start != -oo else end - 1):
sol_sets.append(Interval(start, end, True, True))
if expr.subs(gen, x):
sol_sets.append(FiniteSet(x))
start = end
end = oo
if expr.subs(gen, start + 1):
sol_sets.append(Interval(start, end, True, True))
rv = Union(*sol_sets)
return rv if not relational else rv.as_relational(gen)
def _solve_inequality(ie, s, assume=True):
""" A hacky replacement for solve, since the latter only works for
univariate inequalities. """
if not ie.rel_op in ('>', '>=', '<', '<='):
raise NotImplementedError
expr = ie.lhs - ie.rhs
try:
p = Poly(expr, s)
if p.degree() != 1:
raise NotImplementedError
except (PolynomialError, NotImplementedError):
try:
n, d = expr.as_numer_denom()
return reduce_rational_inequalities([[ie]], s, assume=assume)
except PolynomialError:
return solve_univariate_inequality(ie, s, assume=assume)
a, b = p.all_coeffs()
if a.is_positive:
return ie.func(s, -b/a)
elif a.is_negative:
return ie.func(-b/a, s)
else:
raise NotImplementedError
[docs]def reduce_inequalities(inequalities, assume=True, symbols=[]):
"""Reduce a system of inequalities with rational coefficients.
Examples
========
>>> from sympy import Q, sympify as S
>>> from sympy.abc import x, y
>>> from sympy.solvers.inequalities import reduce_inequalities
>>> reduce_inequalities(S(0) <= x + 3, Q.real(x), [])
x >= -3
>>> reduce_inequalities(S(0) <= x + y*2 - 1, True, [x])
-2*y + 1 <= x
"""
if not hasattr(inequalities, '__iter__'):
inequalities = [inequalities]
if len(inequalities) == 1 and len(symbols) == 1 \
and inequalities[0].is_Relational:
try:
return _solve_inequality(inequalities[0], symbols[0],
assume=assume)
except NotImplementedError:
pass
poly_part, abs_part, extra_assume = {}, {}, []
for inequality in inequalities:
if isinstance(inequality, bool):
if inequality is False:
return False
else:
continue
if isinstance(inequality, AppliedPredicate):
extra_assume.append(inequality)
continue
if inequality.is_Relational:
expr, rel = inequality.lhs - inequality.rhs, inequality.rel_op
else:
expr, rel = inequality, '=='
gens = expr.atoms(Symbol)
if not gens:
return False
elif len(gens) == 1:
gen = gens.pop()
else:
raise NotImplementedError(
"only univariate inequalities are supported")
components = expr.find(lambda u: u.is_Function)
if not components:
if gen in poly_part:
poly_part[gen].append((expr, rel))
else:
poly_part[gen] = [(expr, rel)]
else:
if all(isinstance(comp, Abs) for comp in components):
if gen in abs_part:
abs_part[gen].append((expr, rel))
else:
abs_part[gen] = [(expr, rel)]
else:
raise NotImplementedError("can't reduce %s" % inequalities)
extra_assume = And(*extra_assume)
if assume is not None:
assume = And(assume, extra_assume)
else:
assume = extra_assume
poly_reduced = []
abs_reduced = []
for gen, exprs in poly_part.items():
poly_reduced.append(reduce_rational_inequalities([exprs], gen, assume))
for gen, exprs in abs_part.items():
abs_reduced.append(reduce_abs_inequalities(exprs, gen, assume))
return And(*(poly_reduced + abs_reduced))
