def asintotas(f):
# Igual usar singularities mejor: asintotas, verticales, horizontales y oblícuas (a,b) en oo y -oo
asint = []
asintex = r'Asíntotas:\\'
asint.append([
(i, limit(f, x, i))
for i in list(S.Reals - continuous_domain(f, x, domain=S.Reals))
])
asintex += ', '.join(
r'A.V. $x=' + str(i) + r'$\\'
for i in list(S.Reals - continuous_domain(f, x, domain=S.Reals)))
if abs(limit(f, x, oo)) != oo:
asintex += r'A.H. $y=' + latex(limit(f, x, oo)) + r'$\\'
if abs(limit(f, x, -oo)) != oo:
asintex += r'A.H. $y=' + latex(limit(f, x, -oo)) + r'$\\'
asint.append([(oo, limit(f, x, oo)), (-oo, limit(f, x, -oo))])
oblicuas = []
if abs(limit(f / x, x, oo)) != oo:
oblicuas.append((oo, limit(f / x, x, oo) * x +
limit(f - limit(f / x, x, oo) * x, x, oo)))
#display(latex(limit(f/x,x,oo)*x+limit(f-limit(f/x,x,oo)*x,x,oo)))
asintex += r'A.O. $y=' + latex(
limit(f / x, x, oo) * x +
limit(f - limit(f / x, x, oo) * x, x, oo)) + r'$ \\'
if abs(limit(f / x, x, -oo)) != oo:
oblicuas.append((-oo, limit(f / x, x, -oo) * x +
limit(f - limit(f / x, x, -oo) * x, x, -oo)))
asintex += r'A.O. $y=' + latex(
limit(f / x, x, -oo) * x +
limit(f - limit(f / x, x, -oo) * x, x, -oo)) + r'$ \\'
asint.append(oblicuas)
return asint, asintex
def test_continuous_domain():
x = Symbol('x')
assert continuous_domain(sin(x), x, Interval(0, 2*pi)) == Interval(0, 2*pi)
assert continuous_domain(tan(x), x, Interval(0, 2*pi)) == \
Union(Interval(0, pi/2, False, True), Interval(pi/2, 3*pi/2, True, True),
Interval(3*pi/2, 2*pi, True, False))
assert continuous_domain((x - 1)/((x - 1)**2), x, S.Reals) == \
Union(Interval(-oo, 1, True, True), Interval(1, oo, True, True))
assert continuous_domain(log(x) + log(4*x - 1), x, S.Reals) == \
Interval(1/4, oo, True, True)
assert continuous_domain(1/sqrt(x - 3), x, S.Reals) == Interval(3, oo, True, True)
def test_continuous_domain():
x = Symbol('x')
assert continuous_domain(sin(x), x,
Interval(0, 2 * pi)) == Interval(0, 2 * pi)
assert continuous_domain(tan(x), x, Interval(0, 2*pi)) == \
Union(Interval(0, pi/2, False, True), Interval(pi/2, 3*pi/2, True, True),
Interval(3*pi/2, 2*pi, True, False))
assert continuous_domain((x - 1)/((x - 1)**2), x, S.Reals) == \
Union(Interval(-oo, 1, True, True), Interval(1, oo, True, True))
assert continuous_domain(log(x) + log(4*x - 1), x, S.Reals) == \
Interval(1/4, oo, True, True)
assert continuous_domain(1 / sqrt(x - 3), x,
S.Reals) == Interval(3, oo, True, True)
def construct(self):
expr2 = self.expr.subs(e, math.e)
f = lambdify(x, expr2, 'numpy')
domains = continuous_domain(self.expr, x,
Interval(self.x_min, self.x_max))
if type(domains) is Union:
domains = domains.args
else:
domains = [domains]
self.setup_axes(animate=True)
func_graph = VGroup()
for domain in domains:
graph = self.get_graph(f, self.function_color,
get_left_bound(domain),
get_right_bound(domain))
func_graph.add(graph)
graph_lab = self.get_graph_label(func_graph[0], label=latex(self.expr))
self.play(ShowCreation(func_graph, run_time=3))
self.play(ShowCreation(graph_lab))
self.wait(5)
def test_continuous_domain():
x = Symbol('x')
assert continuous_domain(sin(x), x, Interval(0, 2*pi)) == Interval(0, 2*pi)
assert continuous_domain(tan(x), x, Interval(0, 2*pi)) == \
Union(Interval(0, pi/2, False, True), Interval(pi/2, pi*Rational(3, 2), True, True),
Interval(pi*Rational(3, 2), 2*pi, True, False))
assert continuous_domain((x - 1)/((x - 1)**2), x, S.Reals) == \
Union(Interval(-oo, 1, True, True), Interval(1, oo, True, True))
assert continuous_domain(log(x) + log(4*x - 1), x, S.Reals) == \
Interval(Rational(1, 4), oo, True, True)
assert continuous_domain(1/sqrt(x - 3), x, S.Reals) == Interval(3, oo, True, True)
assert continuous_domain(1/x - 2, x, S.Reals) == \
Union(Interval.open(-oo, 0), Interval.open(0, oo))
assert continuous_domain(1/(x**2 - 4) + 2, x, S.Reals) == \
Union(Interval.open(-oo, -2), Interval.open(-2, 2), Interval.open(2, oo))
domain = continuous_domain(log(tan(x)**2 + 1), x, S.Reals)
assert not domain.contains(3*pi/2)
assert domain.contains(5)
d = Symbol('d', even=True, zero=False)
assert continuous_domain(x**(1/d), x, S.Reals) == Interval(0, oo)
def find_real_domain(f, symbol):
from sympy import symbols
try:
# We first need to make sure all variables are registered as being real, before
# we pass into continuous_domain, as the Sage conversion doesn't preserved Reals.
for sym in f.free_symbols.difference({symbol}):
f = f.subs(sym, symbols(str(sym), real=True))
new_symbol = symbols(str(symbol), real=True)
f = f.subs(symbol, new_symbol)
symbol = new_symbol
return continuous_domain(f, symbol, S.Reals)
except Exception as e:
warnings.warn(
"Failed to find the domain of: " + str(f) + "\n" + str(e),
RuntimeWarning)
return S.Reals
def estudio(f):
# Estudio en una función a trozos
set = S.Reals
conj_singular = S.EmptySet
for j, t in enumerate(f.args):
#display(singularities(t[0],x))
#conj_singular = Union(conj_singular,singularities(t[0],x))
#display(Union(conj_singular,singularities(t[0],x)))
#conj_singular = Union(conj_singular,Complement(S.Reals,continuous_domain(t[0],x,S.Reals)))
#conj_singular = Union(conj_singular,Intersection(t[1].as_set(),Complement(S.Reals,continuous_domain(t[0],x,S.Reals))))
set2 = Intersection(set, t[1].as_set())
conj_singular = Union(
conj_singular,
Intersection(
set2, Complement(S.Reals, continuous_domain(t[0], x,
S.Reals))))
set = Complement(S.Reals, t[1].as_set())
sol = r"Singularidades de las expresiones analíticas: $" + latex(
conj_singular) + "$"
sol += r".\\ Posibles discontinuidades en los extremos de los trozos:"
xs = []
estudio = []
for j, t in enumerate(trozos(f)):
xs.append(str(t[0]))
if (t[1] == t[2]):
estudio.append(
r"\\En {} es continua ya que hay límite y $\lim = f({})={}$".
format(t[0], t[0], t[3]))
display(r"En $x_0={}$ hay límite y f({})={}".format(
t[0], t[0], t[3]))
else:
estudio.append(
r"\\En {} no es continua porque no existe límite. Límites laterales: ${}$ y ${}$"
.format(t[0], latex(t[1]), latex(t[2])))
display(
r"En {} no existe límite. Límites laterales: {} y {}".format(
t[0], t[1], t[2]))
sol += ', '.join(xs) + r"." + '. '.join(estudio)
return (sol)
def __init__(self, function):
self.function = Explorer.get_sym(function)
self.derivative = self.function.diff(x)
self.derived_second = self.derivative.diff(x)
self.domain = [continuous_domain(self.function, x, S.Reals)]
self.y_point_of_intersection = (0, self.function.subs(x, 0))
self.x_point_of_intersections = []
self.extremums = []
self.inflection_points = []
self.vertical_asymptote = []
self.horizontal_asymptotes = []
self.place_ud_ = []
self.place_pn_ = []
self.x_point_of_intersection()
self.get_extremum()
self.get_inflection_point()
self.find_vertical_asymptote()
self.find_horizontal_asymptote()
# למשתנה שבשורה הבאה נוספו ערכי נקודת פיתול מכיוון שיכול להיות שהממוצע יצא אחד מהם
self.list_of_changes = sorted([-oo, oo] + [xy[0] for m,xy in self.extremums] + self.vertical_asymptote + [xy[0] for un, xy, nu in self.inflection_points] + [xy[0] for xy in self.x_point_of_intersections])
self.place_pn()
self.place_ud()
def test_continuous_domain():
x = Symbol("x")
assert continuous_domain(sin(x), x,
Interval(0, 2 * pi)) == Interval(0, 2 * pi)
assert continuous_domain(tan(x), x, Interval(0, 2 * pi)) == Union(
Interval(0, pi / 2, False, True),
Interval(pi / 2, pi * Rational(3, 2), True, True),
Interval(pi * Rational(3, 2), 2 * pi, True, False),
)
assert continuous_domain((x - 1) / ((x - 1)**2), x,
S.Reals) == Union(Interval(-oo, 1, True, True),
Interval(1, oo, True, True))
assert continuous_domain(log(x) + log(4 * x - 1), x,
S.Reals) == Interval(Rational(1, 4), oo, True,
True)
assert continuous_domain(1 / sqrt(x - 3), x,
S.Reals) == Interval(3, oo, True, True)
assert continuous_domain(1 / x - 2, x,
S.Reals) == Union(Interval.open(-oo, 0),
Interval.open(0, oo))
assert continuous_domain(1 / (x**2 - 4) + 2, x,
S.Reals) == Union(Interval.open(-oo, -2),
Interval.open(-2, 2),
Interval.open(2, oo))
from sympy import Symbol, Interval, Range
from sympy.calculus.util import continuous_domain
import numpy as np
import math
import matplotlib.pyplot as plt
x = Symbol("x")
y = input("Wprowadź wzór funkcji: f(x) = ")
y = eval(y)
x1 = float(input("Podaj początek przedziału: "))
x2 = float(input("Podaj koniec przedziału: "))
domain = continuous_domain(y, x, Interval(x1, x2))
view_range = np.arange(x1, x2 + 0.1, 0.01)
index = 0
for i in view_range:
view_range[index] = round(view_range[index], 2)
if i not in domain:
np.delete(view_range, index)
index -= 1
index += 1
#end for
x = view_range
y = eval(str(y))
plt.plot(x, y)
plt.show()
def solve_univariate_inequality(expr, gen, relational=True, domain=S.Reals, continuous=False):
"""Solves a real univariate inequality.
Parameters
==========
expr : Relational
The target inequality
gen : Symbol
The variable for which the inequality is solved
relational : bool
A Relational type output is expected or not
domain : Set
The domain over which the equation is solved
continuous: bool
True if expr is known to be continuous over the given domain
(and so continuous_domain() doesn't need to be called on it)
Raises
======
NotImplementedError
The solution of the inequality cannot be determined due to limitation
in `solvify`.
Notes
=====
Currently, we cannot solve all the inequalities due to limitations in
`solvify`. Also, the solution returned for trigonometric inequalities
are restricted in its periodic interval.
See Also
========
solvify: solver returning solveset solutions with solve's output API
Examples
========
>>> from sympy.solvers.inequalities import solve_univariate_inequality
>>> from sympy import Symbol, sin, Interval, S
>>> x = Symbol('x')
>>> solve_univariate_inequality(x**2 >= 4, x)
((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))
>>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
Union(Interval(-oo, -2), Interval(2, oo))
>>> domain = Interval(0, S.Infinity)
>>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
Interval(2, oo)
>>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
Interval.open(0, pi)
"""
from sympy import im
from sympy.calculus.util import (continuous_domain, periodicity,
function_range)
from sympy.solvers.solvers import denoms
from sympy.solvers.solveset import solveset_real, solvify, solveset
from sympy.solvers.solvers import solve
# This keeps the function independent of the assumptions about `gen`.
# `solveset` makes sure this function is called only when the domain is
# real.
_gen = gen
_domain = domain
if gen.is_real is False:
rv = S.EmptySet
return rv if not relational else rv.as_relational(_gen)
elif gen.is_real is None:
gen = Dummy('gen', real=True)
try:
expr = expr.xreplace({_gen: gen})
except TypeError:
raise TypeError(filldedent('''
When gen is real, the relational has a complex part
which leads to an invalid comparison like I < 0.
'''))
rv = None
if expr is S.true:
rv = domain
elif expr is S.false:
rv = S.EmptySet
else:
e = expr.lhs - expr.rhs
period = periodicity(e, gen)
if period is not None:
frange = function_range(e, gen, domain)
rel = expr.rel_op
if rel == '<' or rel == '<=':
if expr.func(frange.sup, 0):
rv = domain
elif not expr.func(frange.inf, 0):
rv = S.EmptySet
elif rel == '>' or rel == '>=':
if expr.func(frange.inf, 0):
rv = domain
elif not expr.func(frange.sup, 0):
rv = S.EmptySet
inf, sup = domain.inf, domain.sup
if sup - inf is S.Infinity:
domain = Interval(0, period, False, True)
if rv is None:
n, d = e.as_numer_denom()
try:
if gen not in n.free_symbols and len(e.free_symbols) > 1:
raise ValueError
# this might raise ValueError on its own
# or it might give None...
solns = solvify(e, gen, domain)
if solns is None:
# in which case we raise ValueError
raise ValueError
except (ValueError, NotImplementedError):
raise NotImplementedError(filldedent('''
The inequality cannot be solved using solve_univariate_inequality.
'''))
expanded_e = expand_mul(e)
def valid(x):
# this is used to see if gen=x satisfies the
# relational by substituting it into the
# expanded form and testing against 0, e.g.
# if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2
# and expanded_e = x**2 + x - 2; the test is
# whether a given value of x satisfies
# x**2 + x - 2 < 0
#
# expanded_e, expr and gen used from enclosing scope
v = expanded_e.subs(gen, x)
try:
r = expr.func(v, 0)
except TypeError:
r = S.false
if r in (S.true, S.false):
return r
if v.is_real is False:
return S.false
else:
v = v.n(2)
if v.is_comparable:
return expr.func(v, 0)
# not comparable or couldn't be evaluated
raise NotImplementedError(
'relationship did not evaluate: %s' % r)
singularities = []
for d in denoms(expr, gen):
singularities.extend(solvify(d, gen, domain))
if not continuous:
domain = continuous_domain(e, gen, domain)
include_x = '=' in expr.rel_op and expr.rel_op != '!='
try:
discontinuities = set(domain.boundary -
FiniteSet(domain.inf, domain.sup))
# remove points that are not between inf and sup of domain
critical_points = FiniteSet(*(solns + singularities + list(
discontinuities))).intersection(
Interval(domain.inf, domain.sup,
domain.inf not in domain, domain.sup not in domain))
if all(r.is_number for r in critical_points):
reals = _nsort(critical_points, separated=True)[0]
else:
from sympy.utilities.iterables import sift
sifted = sift(critical_points, lambda x: x.is_real)
if sifted[None]:
# there were some roots that weren't known
# to be real
raise NotImplementedError
try:
reals = sifted[True]
if len(reals) > 1:
reals = list(sorted(reals))
except TypeError:
raise NotImplementedError
except NotImplementedError:
raise NotImplementedError('sorting of these roots is not supported')
#If expr contains imaginary coefficients
#Only real values of x for which the imaginary part is 0 are taken
make_real = S.Reals
if im(expanded_e) != S.Zero:
check = True
im_sol = FiniteSet()
try:
a = solveset(im(expanded_e), gen, domain)
if not isinstance(a, Interval):
for z in a:
if z not in singularities and valid(z) and z.is_real:
im_sol += FiniteSet(z)
else:
start, end = a.inf, a.sup
for z in _nsort(critical_points + FiniteSet(end)):
valid_start = valid(start)
if start != end:
valid_z = valid(z)
pt = _pt(start, z)
if pt not in singularities and pt.is_real and valid(pt):
if valid_start and valid_z:
im_sol += Interval(start, z)
elif valid_start:
im_sol += Interval.Ropen(start, z)
elif valid_z:
im_sol += Interval.Lopen(start, z)
else:
im_sol += Interval.open(start, z)
start = z
for s in singularities:
im_sol -= FiniteSet(s)
except (TypeError):
im_sol = S.Reals
check = False
if isinstance(im_sol, EmptySet):
raise ValueError(filldedent('''
%s contains imaginary parts which cannot be made 0 for any value of %s
satisfying the inequality, leading to relations like I < 0.
''' % (expr.subs(gen, _gen), _gen)))
make_real = make_real.intersect(im_sol)
empty = sol_sets = [S.EmptySet]
start = domain.inf
if valid(start) and start.is_finite:
sol_sets.append(FiniteSet(start))
for x in reals:
end = x
if valid(_pt(start, end)):
sol_sets.append(Interval(start, end, True, True))
if x in singularities:
singularities.remove(x)
else:
if x in discontinuities:
discontinuities.remove(x)
_valid = valid(x)
else: # it's a solution
_valid = include_x
if _valid:
sol_sets.append(FiniteSet(x))
start = end
end = domain.sup
if valid(end) and end.is_finite:
sol_sets.append(FiniteSet(end))
if valid(_pt(start, end)):
sol_sets.append(Interval.open(start, end))
if im(expanded_e) != S.Zero and check:
rv = (make_real).intersect(_domain)
else:
rv = Intersection(
(Union(*sol_sets)), make_real, _domain).subs(gen, _gen)
return rv if not relational else rv.as_relational(_gen)
from sympy import *
from fractions import *
from sympy import Symbol, S
from sympy.calculus.util import continuous_domain
import math
x = Symbol("x")
f = 1 / x - 2
z = continuous_domain(f, x, S.Reals)
def dominio_rango(R):
x = []
y = []
for a, b in R:
x.append(a)
y.append(b)
return "El dominio es: ", sorted(
set(x), key=x.index), " El rango es: ", sorted(set(y), key=y.index)
print(
"En caso de ingresar una relación el dominio y rango estarán definidos por sus ejes X y Y"
)
print("Relación ingresada")
print("[(1,1), (1,2), (2,3), (4,2)]")
print(dominio_rango([(1, 1), (1, 2), (2, 3), (4, 2)]))
def solve_univariate_inequality(expr,
gen,
relational=True,
domain=S.Reals,
continuous=False):
"""Solves a real univariate inequality.
Parameters
==========
expr : Relational
The target inequality
gen : Symbol
The variable for which the inequality is solved
relational : bool
A Relational type output is expected or not
domain : Set
The domain over which the equation is solved
continuous: bool
True if expr is known to be continuous over the given domain
(and so continuous_domain() doesn't need to be called on it)
Raises
======
NotImplementedError
The solution of the inequality cannot be determined due to limitation
in `solvify`.
Notes
=====
Currently, we cannot solve all the inequalities due to limitations in
`solvify`. Also, the solution returned for trigonometric inequalities
are restricted in its periodic interval.
See Also
========
solvify: solver returning solveset solutions with solve's output API
Examples
========
>>> from sympy.solvers.inequalities import solve_univariate_inequality
>>> from sympy import Symbol, sin, Interval, S
>>> x = Symbol('x')
>>> solve_univariate_inequality(x**2 >= 4, x)
((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))
>>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
(-oo, -2] U [2, oo)
>>> domain = Interval(0, S.Infinity)
>>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
[2, oo)
>>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
(0, pi)
"""
from sympy.calculus.util import (continuous_domain, periodicity,
function_range)
from sympy.solvers.solvers import denoms
from sympy.solvers.solveset import solveset_real, solvify
# This keeps the function independent of the assumptions about `gen`.
# `solveset` makes sure this function is called only when the domain is
# real.
d = Dummy(real=True)
expr = expr.subs(gen, d)
_gen = gen
gen = d
rv = None
if expr is S.true:
rv = domain
elif expr is S.false:
rv = S.EmptySet
else:
e = expr.lhs - expr.rhs
period = periodicity(e, gen)
if period is not None:
frange = function_range(e, gen, domain)
rel = expr.rel_op
if rel == '<' or rel == '<=':
if expr.func(frange.sup, 0):
rv = domain
elif not expr.func(frange.inf, 0):
rv = S.EmptySet
elif rel == '>' or rel == '>=':
if expr.func(frange.inf, 0):
rv = domain
elif not expr.func(frange.sup, 0):
rv = S.EmptySet
inf, sup = domain.inf, domain.sup
if sup - inf is S.Infinity:
domain = Interval(0, period, False, True)
if rv is None:
singularities = []
for d in denoms(e):
singularities.extend(solvify(d, gen, domain))
if not continuous:
domain = continuous_domain(e, gen, domain)
solns = solvify(e, gen, domain)
if solns is None:
raise NotImplementedError(
filldedent('''The inequality cannot be
solved using solve_univariate_inequality.'''))
include_x = expr.func(0, 0)
def valid(x):
v = e.subs(gen, x)
try:
r = expr.func(v, 0)
except TypeError:
r = S.false
if r in (S.true, S.false):
return r
if v.is_real is False:
return S.false
else:
v = v.n(2)
if v.is_comparable:
return expr.func(v, 0)
return S.false
start = domain.inf
sol_sets = [S.EmptySet]
try:
discontinuities = domain.boundary - FiniteSet(
domain.inf, domain.sup)
critical_points = set(solns + singularities +
list(discontinuities))
reals = _nsort(critical_points, separated=True)[0]
except NotImplementedError:
raise NotImplementedError(
'sorting of these roots is not supported')
if valid(start) and start.is_finite:
sol_sets.append(FiniteSet(start))
for x in reals:
end = x
if end in [S.NegativeInfinity, S.Infinity]:
if valid(S(0)):
sol_sets.append(Interval(start, S.Infinity, True,
True))
break
pt = ((start + end) / 2 if start is not S.NegativeInfinity else
(end / 2 if end.is_positive else
(2 * end if end.is_negative else end - 1)))
if valid(pt):
sol_sets.append(Interval(start, end, True, True))
if x in singularities:
singularities.remove(x)
elif include_x:
sol_sets.append(FiniteSet(x))
start = end
end = domain.sup
# in case start == -oo then there were no solutions so we just
# check a point between -oo and oo (e.g. 0) else pick a point
# past the last solution (which is start after the end of the
# for-loop above
pt = (0 if start is S.NegativeInfinity else
(start / 2 if start.is_negative else
(2 * start if start.is_positive else start + 1)))
if pt >= end:
pt = (start + end) / 2
if valid(pt):
sol_sets.append(Interval(start, end, True, True))
rv = Union(*sol_sets).subs(gen, _gen)
return rv if not relational else rv.as_relational(_gen)
def solve_univariate_inequality(expr, gen, relational=True, domain=S.Reals, continuous=False):
"""Solves a real univariate inequality.
Parameters
==========
expr : Relational
The target inequality
gen : Symbol
The variable for which the inequality is solved
relational : bool
A Relational type output is expected or not
domain : Set
The domain over which the equation is solved
continuous: bool
True if expr is known to be continuous over the given domain
(and so continuous_domain() doesn't need to be called on it)
Raises
======
NotImplementedError
The solution of the inequality cannot be determined due to limitation
in `solvify`.
Notes
=====
Currently, we cannot solve all the inequalities due to limitations in
`solvify`. Also, the solution returned for trigonometric inequalities
are restricted in its periodic interval.
See Also
========
solvify: solver returning solveset solutions with solve's output API
Examples
========
>>> from sympy.solvers.inequalities import solve_univariate_inequality
>>> from sympy import Symbol, sin, Interval, S
>>> x = Symbol('x')
>>> solve_univariate_inequality(x**2 >= 4, x)
((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))
>>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
(-oo, -2] U [2, oo)
>>> domain = Interval(0, S.Infinity)
>>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
[2, oo)
>>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
(0, pi)
"""
from sympy.calculus.util import (continuous_domain, periodicity,
function_range)
from sympy.solvers.solvers import denoms
from sympy.solvers.solveset import solveset_real, solvify
# This keeps the function independent of the assumptions about `gen`.
# `solveset` makes sure this function is called only when the domain is
# real.
d = Dummy(real=True)
expr = expr.subs(gen, d)
_gen = gen
gen = d
rv = None
if expr is S.true:
rv = domain
elif expr is S.false:
rv = S.EmptySet
else:
e = expr.lhs - expr.rhs
period = periodicity(e, gen)
if period is not None:
frange = function_range(e, gen, domain)
rel = expr.rel_op
if rel == '<' or rel == '<=':
if expr.func(frange.sup, 0):
rv = domain
elif not expr.func(frange.inf, 0):
rv = S.EmptySet
elif rel == '>' or rel == '>=':
if expr.func(frange.inf, 0):
rv = domain
elif not expr.func(frange.sup, 0):
rv = S.EmptySet
inf, sup = domain.inf, domain.sup
if sup - inf is S.Infinity:
domain = Interval(0, period, False, True)
if rv is None:
singularities = []
for d in denoms(e):
singularities.extend(solvify(d, gen, domain))
if not continuous:
domain = continuous_domain(e, gen, domain)
solns = solvify(e, gen, domain)
if solns is None:
raise NotImplementedError(filldedent('''The inequality cannot be
solved using solve_univariate_inequality.'''))
include_x = expr.func(0, 0)
def valid(x):
v = e.subs(gen, x)
try:
r = expr.func(v, 0)
except TypeError:
r = S.false
if r in (S.true, S.false):
return r
if v.is_real is False:
return S.false
else:
v = v.n(2)
if v.is_comparable:
return expr.func(v, 0)
return S.false
start = domain.inf
sol_sets = [S.EmptySet]
try:
discontinuities = domain.boundary - FiniteSet(domain.inf, domain.sup)
critical_points = set(solns + singularities + list(discontinuities))
reals = _nsort(critical_points, separated=True)[0]
except NotImplementedError:
raise NotImplementedError('sorting of these roots is not supported')
if valid(start) and start.is_finite:
sol_sets.append(FiniteSet(start))
for x in reals:
end = x
if end in [S.NegativeInfinity, S.Infinity]:
if valid(S(0)):
sol_sets.append(Interval(start, S.Infinity, True, True))
break
pt = ((start + end)/2 if start is not S.NegativeInfinity else
(end/2 if end.is_positive else
(2*end if end.is_negative else
end - 1)))
if valid(pt):
sol_sets.append(Interval(start, end, True, True))
if x in singularities:
singularities.remove(x)
elif include_x:
sol_sets.append(FiniteSet(x))
start = end
end = domain.sup
# in case start == -oo then there were no solutions so we just
# check a point between -oo and oo (e.g. 0) else pick a point
# past the last solution (which is start after the end of the
# for-loop above
pt = (0 if start is S.NegativeInfinity else
(start/2 if start.is_negative else
(2*start if start.is_positive else
start + 1)))
if pt >= end:
pt = (start + end)/2
if valid(pt):
sol_sets.append(Interval(start, end, True, True))
rv = Union(*sol_sets).subs(gen, _gen)
return rv if not relational else rv.as_relational(_gen)
def compute_admissible_domain(self, sym_function):
"""Deprecated, as continuos_domain is superslow"""
x = self.symbol
res = continuous_domain(sym_function, self.symbol, S.Reals)
return res
def solve_univariate_inequality(expr,
gen,
relational=True,
domain=S.Reals,
continuous=False):
"""Solves a real univariate inequality.
Parameters
==========
expr : Relational
The target inequality
gen : Symbol
The variable for which the inequality is solved
relational : bool
A Relational type output is expected or not
domain : Set
The domain over which the equation is solved
continuous: bool
True if expr is known to be continuous over the given domain
(and so continuous_domain() doesn't need to be called on it)
Raises
======
NotImplementedError
The solution of the inequality cannot be determined due to limitation
in `solvify`.
Notes
=====
Currently, we cannot solve all the inequalities due to limitations in
`solvify`. Also, the solution returned for trigonometric inequalities
are restricted in its periodic interval.
See Also
========
solvify: solver returning solveset solutions with solve's output API
Examples
========
>>> from sympy.solvers.inequalities import solve_univariate_inequality
>>> from sympy import Symbol, sin, Interval, S
>>> x = Symbol('x')
>>> solve_univariate_inequality(x**2 >= 4, x)
((2 <= x) & (x < oo)) | ((x <= -2) & (-oo < x))
>>> solve_univariate_inequality(x**2 >= 4, x, relational=False)
Union(Interval(-oo, -2), Interval(2, oo))
>>> domain = Interval(0, S.Infinity)
>>> solve_univariate_inequality(x**2 >= 4, x, False, domain)
Interval(2, oo)
>>> solve_univariate_inequality(sin(x) > 0, x, relational=False)
Interval.open(0, pi)
"""
from sympy import im
from sympy.calculus.util import (continuous_domain, periodicity,
function_range)
from sympy.solvers.solvers import denoms
from sympy.solvers.solveset import solveset_real, solvify, solveset
from sympy.solvers.solvers import solve
# This keeps the function independent of the assumptions about `gen`.
# `solveset` makes sure this function is called only when the domain is
# real.
_gen = gen
_domain = domain
if gen.is_real is False:
rv = S.EmptySet
return rv if not relational else rv.as_relational(_gen)
elif gen.is_real is None:
gen = Dummy('gen', real=True)
try:
expr = expr.xreplace({_gen: gen})
except TypeError:
raise TypeError(
filldedent('''
When gen is real, the relational has a complex part
which leads to an invalid comparison like I < 0.
'''))
rv = None
if expr is S.true:
rv = domain
elif expr is S.false:
rv = S.EmptySet
else:
e = expr.lhs - expr.rhs
period = periodicity(e, gen)
if period is S.Zero:
e = expand_mul(e)
const = expr.func(e, 0)
if const is S.true:
rv = domain
elif const is S.false:
rv = S.EmptySet
elif period is not None:
frange = function_range(e, gen, domain)
rel = expr.rel_op
if rel == '<' or rel == '<=':
if expr.func(frange.sup, 0):
rv = domain
elif not expr.func(frange.inf, 0):
rv = S.EmptySet
elif rel == '>' or rel == '>=':
if expr.func(frange.inf, 0):
rv = domain
elif not expr.func(frange.sup, 0):
rv = S.EmptySet
inf, sup = domain.inf, domain.sup
if sup - inf is S.Infinity:
domain = Interval(0, period, False, True)
if rv is None:
n, d = e.as_numer_denom()
try:
if gen not in n.free_symbols and len(e.free_symbols) > 1:
raise ValueError
# this might raise ValueError on its own
# or it might give None...
solns = solvify(e, gen, domain)
if solns is None:
# in which case we raise ValueError
raise ValueError
except (ValueError, NotImplementedError):
# replace gen with generic x since it's
# univariate anyway
raise NotImplementedError(
filldedent('''
The inequality, %s, cannot be solved using
solve_univariate_inequality.
''' % expr.subs(gen, Symbol('x'))))
expanded_e = expand_mul(e)
def valid(x):
# this is used to see if gen=x satisfies the
# relational by substituting it into the
# expanded form and testing against 0, e.g.
# if expr = x*(x + 1) < 2 then e = x*(x + 1) - 2
# and expanded_e = x**2 + x - 2; the test is
# whether a given value of x satisfies
# x**2 + x - 2 < 0
#
# expanded_e, expr and gen used from enclosing scope
v = expanded_e.subs(gen, expand_mul(x))
try:
r = expr.func(v, 0)
except TypeError:
r = S.false
if r in (S.true, S.false):
return r
if v.is_real is False:
return S.false
else:
v = v.n(2)
if v.is_comparable:
return expr.func(v, 0)
# not comparable or couldn't be evaluated
raise NotImplementedError(
'relationship did not evaluate: %s' % r)
singularities = []
for d in denoms(expr, gen):
singularities.extend(solvify(d, gen, domain))
if not continuous:
domain = continuous_domain(expanded_e, gen, domain)
include_x = '=' in expr.rel_op and expr.rel_op != '!='
try:
discontinuities = set(domain.boundary -
FiniteSet(domain.inf, domain.sup))
# remove points that are not between inf and sup of domain
critical_points = FiniteSet(
*(solns + singularities +
list(discontinuities))).intersection(
Interval(domain.inf, domain.sup, domain.inf
not in domain, domain.sup not in domain))
if all(r.is_number for r in critical_points):
reals = _nsort(critical_points, separated=True)[0]
else:
from sympy.utilities.iterables import sift
sifted = sift(critical_points, lambda x: x.is_real)
if sifted[None]:
# there were some roots that weren't known
# to be real
raise NotImplementedError
try:
reals = sifted[True]
if len(reals) > 1:
reals = list(sorted(reals))
except TypeError:
raise NotImplementedError
except NotImplementedError:
raise NotImplementedError(
'sorting of these roots is not supported')
# If expr contains imaginary coefficients, only take real
# values of x for which the imaginary part is 0
make_real = S.Reals
if im(expanded_e) != S.Zero:
check = True
im_sol = FiniteSet()
try:
a = solveset(im(expanded_e), gen, domain)
if not isinstance(a, Interval):
for z in a:
if z not in singularities and valid(
z) and z.is_real:
im_sol += FiniteSet(z)
else:
start, end = a.inf, a.sup
for z in _nsort(critical_points + FiniteSet(end)):
valid_start = valid(start)
if start != end:
valid_z = valid(z)
pt = _pt(start, z)
if pt not in singularities and pt.is_real and valid(
pt):
if valid_start and valid_z:
im_sol += Interval(start, z)
elif valid_start:
im_sol += Interval.Ropen(start, z)
elif valid_z:
im_sol += Interval.Lopen(start, z)
else:
im_sol += Interval.open(start, z)
start = z
for s in singularities:
im_sol -= FiniteSet(s)
except (TypeError):
im_sol = S.Reals
check = False
if isinstance(im_sol, EmptySet):
raise ValueError(
filldedent('''
%s contains imaginary parts which cannot be
made 0 for any value of %s satisfying the
inequality, leading to relations like I < 0.
''' % (expr.subs(gen, _gen), _gen)))
make_real = make_real.intersect(im_sol)
empty = sol_sets = [S.EmptySet]
start = domain.inf
if valid(start) and start.is_finite:
sol_sets.append(FiniteSet(start))
for x in reals:
end = x
if valid(_pt(start, end)):
sol_sets.append(Interval(start, end, True, True))
if x in singularities:
singularities.remove(x)
else:
if x in discontinuities:
discontinuities.remove(x)
_valid = valid(x)
else: # it's a solution
_valid = include_x
if _valid:
sol_sets.append(FiniteSet(x))
start = end
end = domain.sup
if valid(end) and end.is_finite:
sol_sets.append(FiniteSet(end))
if valid(_pt(start, end)):
sol_sets.append(Interval.open(start, end))
if im(expanded_e) != S.Zero and check:
rv = (make_real).intersect(_domain)
else:
rv = Intersection((Union(*sol_sets)), make_real,
_domain).subs(gen, _gen)
return rv if not relational else rv.as_relational(_gen)
def nminmax(func, x, warningctx=None):
# Find minimum and maximum at [0,1].
try:
return [
minimum(func, x, Interval(0, 1)),
maximum(func, x, Interval(0, 1))
]
except:
pw = piecewise_fold(func.rewrite(Piecewise))
if isinstance(pw, Piecewise):
try:
return pwminmax(pw, x, Interval(0, 1))
except:
pass
print("WARNING: Resorting to numerical optimization: %s" % (func),
file=sys.stderr)
if warningctx:
warningctx["warning"] = True
cv = [0, 1]
df = diff(func)
mmh = isinstance(func, Min) or isinstance(func, Max) or isinstance(
func, Heaviside)
n = 40 if mmh else 20
for i in range(n):
try:
ns = nsolve(df, x, (S(i) / n, S(i + 1) / n), solver="bisect")
cv.append(lowerbound(ns))
cv.append(upperbound(ns))
except:
ss = S(i) / n
se = S(i + 1) / n
for k in range(1 + 1):
cv.append(ss + (se - ss) * S(k) / 10)
# Evaluate at critical points, and
# remove incomparable values
cv2 = []
# print(func.__class__)
iscont = continuous_domain(func, x, Interval(0, 1)) == Interval(0, 1)
# print([func.__class__,iscont])
# print(func)
for c in cv:
c2s = [func.subs(x, c)]
# Finding limits for Min, Max, and Heaviside can be
# extremely slow. Also check whether function is continuous
# to avoid unnecessary limit computation
if (not iscont) and (not mmh):
try:
c2s.append(funclimit(func, x, c, "+"))
except:
pass
# XXX: Disabling because it can be extremely slow in some
# cases, especially when Heaviside is involved
# try:
# c2s.append(funclimit(func, x, c, "-"))
# except:
# pass
for c2 in c2s:
# Change complex infinity to infinity
if c2 == zoo:
c2 = oo
try:
Min(c2)
cv2.append(c2)
except:
pass
cv = cv2
return [lowerbound(Min(*cv)), upperbound(Max(*cv))]
