def continuous_domain(f, symbol, domain):
"""
Returns the intervals in the given domain for which the function is continuous.
This method is limited by the ability to determine the various
singularities and discontinuities of the given function.
Examples
========
>>> from sympy import Symbol, S, tan, log, pi, sqrt
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import continuous_domain
>>> x = Symbol('x')
>>> continuous_domain(1/x, x, S.Reals)
(-oo, 0) U (0, oo)
>>> continuous_domain(tan(x), x, Interval(0, pi))
[0, pi/2) U (pi/2, pi]
>>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
[2, 5]
>>> continuous_domain(log(2*x - 1), x, S.Reals)
(1/2, oo)
"""
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.solvers.solveset import solveset, _has_rational_power
if domain.is_subset(S.Reals):
constrained_interval = domain
for atom in f.atoms(Pow):
predicate, denom = _has_rational_power(atom, symbol)
constraint = S.EmptySet
if predicate and denom == 2:
constraint = solve_univariate_inequality(
atom.base >= 0, symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
for atom in f.atoms(log):
constraint = solve_univariate_inequality(atom.args[0] > 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
domain = constrained_interval
try:
sings = S.EmptySet
for atom in f.atoms(Pow):
predicate, denom = _has_rational_power(atom, symbol)
if predicate and denom == 2:
sings = solveset(1 / f, symbol, domain)
break
else:
sings = Intersection(solveset(1 / f, symbol), domain)
except:
raise NotImplementedError(
"Methods for determining the continuous domains"
" of this function has not been developed.")
return domain - sings
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 {}
if not pattern_match.get(p, S.Zero).is_zero:
f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r]
q_pos_cond = solve_univariate_inequality(f_q >= 0,
symbol,
relational=False)
q_neg_cond = solve_univariate_inequality(f_q < 0,
symbol,
relational=False)
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 continuous_domain(f, symbol, domain):
"""
Returns the intervals in the given domain for which the function is continuous.
This method is limited by the ability to determine the various
singularities and discontinuities of the given function.
Examples
========
>>> from sympy import Symbol, S, tan, log, pi, sqrt
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import continuous_domain
>>> x = Symbol('x')
>>> continuous_domain(1/x, x, S.Reals)
(-oo, 0) U (0, oo)
>>> continuous_domain(tan(x), x, Interval(0, pi))
[0, pi/2) U (pi/2, pi]
>>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
[2, 5]
>>> continuous_domain(log(2*x - 1), x, S.Reals)
(1/2, oo)
"""
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.solvers.solveset import solveset, _has_rational_power
if domain.is_subset(S.Reals):
constrained_interval = domain
for atom in f.atoms(Pow):
predicate, denom = _has_rational_power(atom, symbol)
constraint = S.EmptySet
if predicate and denom == 2:
constraint = solve_univariate_inequality(atom.base >= 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
for atom in f.atoms(log):
constraint = solve_univariate_inequality(atom.args[0] > 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
domain = constrained_interval
try:
sings = S.EmptySet
for atom in f.atoms(Pow):
predicate, denom = _has_rational_power(atom, symbol)
if predicate and denom == 2:
sings = solveset(1/f, symbol, domain)
break
else:
sings = Intersection(solveset(1/f, symbol), domain)
except:
raise NotImplementedError("Methods for determining the continuous domains"
" of this function has not been developed.")
return domain - sings
def draw_curve_plotly(curve: _ParametricCurve,
num=50,
domain=(-1, 1),
exist_range=[(None, None), (None, None), (None, None)],
color_func=(lambda curve: curve.curvature().simplify()),
fig=None,
line=dict(width=4, showscale=True),
mode="lines",
*arg,
**kwarg):
from numpy import linspace
from sympy import lambdify, Interval
from sympy.sets.sets import Union
from sympy.solvers.inequalities import solve_univariate_inequality
import plotly.graph_objects as go
domain = Interval(*domain).intersect(Interval(*curve.sym_limit(0)))
for i, lim in enumerate(exist_range):
if lim[0] is not None:
domain = solve_univariate_inequality(lim[0] < curve.expr(i),
curve.sym(0),
relational=False,
domain=domain)
if lim[1] is not None:
domain = solve_univariate_inequality(lim[1] > curve.expr(i),
curve.sym(0),
relational=False,
domain=domain)
if isinstance(domain, Union):
domain = domain.args[0]
domain_ = linspace(float(domain.start), float(domain.end), num=num)
values_ = lambdify(curve.sym(0), curve.expr(), 'numpy')(domain_)
if color_func is not None:
colors_ = lambdify(curve.sym(0), color_func(curve), 'numpy')(domain_)
kwarg['x'] = values_[0]
kwarg['y'] = values_[1]
kwarg['z'] = values_[2]
kwarg['line'] = line
if color_func is not None: kwarg['line']['color'] = colors_
kwarg['mode'] = mode
if fig is None:
fig = go.Figure(data=go.Scatter3d(*arg, **kwarg))
else:
fig.add_trace(go.Scatter3d(*arg, **kwarg))
fig.update_layout(
scene=dict(aspectratio=dict(x=1, y=1, z=1), aspectmode='data'))
return fig
def _eval_as_set(self):
# self is univariate and periodicity(self, x) in (0, None)
from sympy.solvers.inequalities import solve_univariate_inequality
syms = self.free_symbols
assert len(syms) == 1
x = syms.pop()
return solve_univariate_inequality(self, x, relational=False)
def as_set(self):
"""
Rewrites univariate inequality in terms of real sets
Examples
========
>>> from sympy import Symbol, Eq
>>> x = Symbol('x', real=True)
>>> (x > 0).as_set()
Interval.open(0, oo)
>>> Eq(x, 0).as_set()
{0}
"""
from sympy.solvers.inequalities import solve_univariate_inequality
syms = self.free_symbols
if len(syms) == 1:
sym = syms.pop()
else:
raise NotImplementedError("Sorry, Relational.as_set procedure"
" is not yet implemented for"
" multivariate expressions")
return solve_univariate_inequality(self, sym, relational=False)
def as_set(self):
"""
Rewrites univariate inequality in terms of real sets
Examples
========
>>> from sympy import Symbol, Eq
>>> x = Symbol('x', real=True)
>>> (x > 0).as_set()
Interval.open(0, oo)
>>> Eq(x, 0).as_set()
{0}
"""
from sympy.solvers.inequalities import solve_univariate_inequality
syms = self.free_symbols
if len(syms) == 1:
sym = syms.pop()
else:
raise NotImplementedError("Sorry, Relational.as_set procedure"
" is not yet implemented for"
" multivariate expressions")
return solve_univariate_inequality(self, sym, relational=False)
def _eval_as_set(self):
# self is univariate and periodicity(self, x) in (0, None)
from sympy.solvers.inequalities import solve_univariate_inequality
syms = self.free_symbols
assert len(syms) == 1
x = syms.pop()
return solve_univariate_inequality(self, x, relational=False)
def _solve_abs(f, symbol):
""" Helper function to solve equation involving absolute value function """
p, q, r = Wild('p'), Wild('q'), Wild('r')
pattern_match = f.match(p*Abs(q) + r) or {}
if not pattern_match.get(p, S.Zero).is_zero:
f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r]
q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
relational=False)
q_neg_cond = solve_univariate_inequality(f_q < 0, symbol,
relational=False)
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), S.Complexes)
def _solve_abs(f, symbol):
""" Helper function to solve equation involving absolute value function """
p, q, r = Wild('p'), Wild('q'), Wild('r')
pattern_match = f.match(p*Abs(q) + r) or {}
if not pattern_match.get(p, S.Zero).is_zero:
f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r]
q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
relational=False)
q_neg_cond = solve_univariate_inequality(f_q < 0, symbol,
relational=False)
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), S.Complexes)
def _solve_abs(f, symbol):
""" Helper function to solve equation involving absolute value function """
from sympy.solvers.inequalities import solve_univariate_inequality
assert f.has(Abs)
p, q, r = Wild('p'), Wild('q'), Wild('r')
pattern_match = f.match(p*Abs(q) + r)
if not pattern_match[p].is_zero:
f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r]
q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
relational=False)
q_neg_cond = solve_univariate_inequality(f_q < 0, symbol,
relational=False)
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:
raise NotImplementedError
def _solve_abs(f, symbol):
""" Helper function to solve equation involving absolute value function """
from sympy.solvers.inequalities import solve_univariate_inequality
assert f.has(Abs)
p, q, r = Wild('p'), Wild('q'), Wild('r')
pattern_match = f.match(p*Abs(q) + r)
if not pattern_match[p].is_zero:
f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r]
q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
relational=False)
q_neg_cond = solve_univariate_inequality(f_q < 0, symbol,
relational=False)
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:
raise NotImplementedError
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 {}
if not pattern_match.get(p, S.Zero).is_zero:
f_p, f_q, f_r = pattern_match[p], pattern_match[q], pattern_match[r]
q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
relational=False)
q_neg_cond = solve_univariate_inequality(f_q < 0, symbol,
relational=False)
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 _eval_as_set(self):
# self is univariate and periodicity(self, x) in (0, None)
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.sets.conditionset import ConditionSet
syms = self.free_symbols
assert len(syms) == 1
x = syms.pop()
try:
xset = solve_univariate_inequality(self, x, relational=False)
except NotImplementedError:
# solve_univariate_inequality raises NotImplementedError for
# unsolvable equations/inequalities.
xset = ConditionSet(x, self, S.Reals)
return xset
def is_convex(f, *syms, **kwargs):
"""Determines the convexity of the function passed in the argument.
Parameters
==========
f : Expr
The concerned function.
syms : Tuple of symbols
The variables with respect to which the convexity is to be determined.
domain : Interval, optional
The domain over which the convexity of the function has to be checked.
If unspecified, S.Reals will be the default domain.
Returns
=======
Boolean
The method returns `True` if the function is convex otherwise it
returns `False`.
Raises
======
NotImplementedError
The check for the convexity of multivariate functions is not implemented yet.
Notes
=====
To determine concavity of a function pass `-f` as the concerned function.
To determine logarithmic convexity of a function pass log(f) as
concerned function.
To determine logartihmic concavity of a function pass -log(f) as
concerned function.
Currently, convexity check of multivariate functions is not handled.
Examples
========
>>> from sympy import symbols, exp, oo, Interval
>>> from sympy.calculus.util import is_convex
>>> x = symbols('x')
>>> is_convex(exp(x), x)
True
>>> is_convex(x**3, x, domain = Interval(-1, oo))
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Convex_function
.. [2] http://www.ifp.illinois.edu/~angelia/L3_convfunc.pdf
.. [3] https://en.wikipedia.org/wiki/Logarithmically_convex_function
.. [4] https://en.wikipedia.org/wiki/Logarithmically_concave_function
.. [5] https://en.wikipedia.org/wiki/Concave_function
"""
if len(syms) > 1:
raise NotImplementedError(
"The check for the convexity of multivariate functions is not implemented yet."
)
f = _sympify(f)
domain = kwargs.get('domain', S.Reals)
var = syms[0]
condition = f.diff(var, 2) < 0
if solve_univariate_inequality(condition, var, False, domain):
return False
return True
def ewma():
import plottool as pt
import ubelt as ub
import numpy as np
pt.qtensure()
# Investigate the span parameter
span = 20
alpha = 2 / (span + 1)
# how long does it take for the estimation to hit 0?
# (ie, it no longer cares about the initial 1?)
# about 93 iterations to get to 1e-4
# about 47 iterations to get to 1e-2
# about 24 iterations to get to 1e-1
# 20 iterations goes to .135
data = ([1] + [0] * 20 + [1] * 40 + [0] * 20 + [1] * 50 + [0] * 20 +
[1] * 60 + [0] * 20 + [1] * 165 + [0] * 20 + [0])
mave = []
iter_ = iter(data)
current = next(iter_)
mave += [current]
for x in iter_:
current = (alpha * x) + (1 - alpha) * current
mave += [current]
if False:
pt.figure(fnum=1, doclf=True)
pt.plot(data)
pt.plot(mave)
np.where(np.array(mave) < 1e-1)
import sympy as sym
# span, alpha, n = sym.symbols('span, alpha, n')
n = sym.symbols('n', integer=True, nonnegative=True, finite=True)
span = sym.symbols('span', integer=True, nonnegative=True, finite=True)
thresh = sym.symbols('thresh', real=True, nonnegative=True, finite=True)
# alpha = 2 / (span + 1)
a, b, c = sym.symbols('a, b, c', real=True, nonnegative=True, finite=True)
sym.solve(sym.Eq(b**a, c), a)
current = 1
x = 0
steps = []
for _ in range(10):
current = (alpha * x) + (1 - alpha) * current
steps.append(current)
alpha = sym.symbols('alpha', real=True, nonnegative=True, finite=True)
base = sym.symbols('base', real=True, finite=True)
alpha = 2 / (span + 1)
thresh_expr = (1 - alpha)**n
thresthresh_exprh_expr = base**n
n_expr = sym.ceiling(sym.log(thresh) / sym.log(1 - 2 / (span + 1)))
sym.pprint(sym.simplify(thresh_expr))
sym.pprint(sym.simplify(n_expr))
print(sym.latex(sym.simplify(n_expr)))
# def calc_n2(span, thresh):
# return np.log(thresh) / np.log(1 - 2 / (span + 1))
def calc_n(span, thresh):
return np.log(thresh) / np.log((span - 1) / (span + 1))
def calc_thresh_val(n, span):
alpha = 2 / (span + 1)
return (1 - alpha)**n
span = np.arange(2, 200)
n_frac = calc_n(span, thresh=.5)
n = np.ceil(n_frac)
calc_thresh_val(n, span)
pt.figure(fnum=1, doclf=True)
ydatas = ut.odict([('thresh=%f' % thresh,
np.ceil(calc_n(span, thresh=thresh)))
for thresh in [1e-3, .01, .1, .2, .3, .4, .5]])
pt.multi_plot(
span,
ydatas,
xlabel='span',
ylabel='n iters to acheive thresh',
marker='',
# num_xticks=len(span),
fnum=1)
pt.gca().set_aspect('equal')
def both_sides(eqn, func):
return sym.Eq(func(eqn.lhs), func(eqn.rhs))
eqn = sym.Eq(thresh_expr, thresh)
n_expr = sym.solve(eqn,
n)[0].subs(base,
(1 - alpha)).subs(alpha, (2 / (span + 1)))
eqn = both_sides(eqn, lambda x: sym.log(x, (1 - alpha)))
lhs = eqn.lhs
from sympy.solvers.inequalities import solve_univariate_inequality
def eval_expr(span_value, n_value):
return np.array(
[thresh_expr.subs(span, span_value).subs(n, n_) for n_ in n_value],
dtype=np.float)
eval_expr(20, np.arange(20))
def linear(x, a, b):
return a * x + b
def sigmoidal_4pl(x, a, b, c, d):
return d + (a - d) / (1 + (x / c)**b)
def exponential(x, a, b, c):
return a + b * np.exp(-c * x)
import scipy.optimize
# Determine how to choose span, such that you get to .01 from 1
# in n timesteps
thresh_to_span_to_n = []
thresh_to_n_to_span = []
for thresh_value in ub.ProgIter([.0001, .001, .01, .1, .2, .3, .4, .5]):
print('')
test_vals = sorted([2, 3, 4, 5, 6])
n_to_span = []
for n_value in ub.ProgIter(test_vals):
# In n iterations I want to choose a span that the expression go
# less than a threshold
constraint = thresh_expr.subs(n, n_value) < thresh_value
solution = solve_univariate_inequality(constraint, span)
try:
lowbound = np.ceil(float(solution.args[0].lhs))
highbound = np.floor(float(solution.args[1].rhs))
assert lowbound <= highbound
span_value = lowbound
except AttributeError:
span_value = np.floor(float(solution.rhs))
n_to_span.append((n_value, span_value))
# Given a threshold, find a minimum number of steps
# that brings you up to that threshold given a span
test_vals = sorted(set(list(range(2, 1000, 50)) + [2, 3, 4, 5, 6]))
span_to_n = []
for span_value in ub.ProgIter(test_vals):
constraint = thresh_expr.subs(span, span_value) < thresh_value
solution = solve_univariate_inequality(constraint, n)
n_value = solution.lhs
span_to_n.append((span_value, n_value))
thresh_to_n_to_span.append((thresh_value, n_to_span))
thresh_to_span_to_n.append((thresh_value, span_to_n))
thresh_to_params = []
for thresh_value, span_to_n in thresh_to_span_to_n:
xdata, ydata = [np.array(_, dtype=np.float) for _ in zip(*span_to_n)]
p0 = (1 / np.diff((ydata - ydata[0])[1:]).mean(), ydata[0])
func = linear
popt, pcov = scipy.optimize.curve_fit(func, xdata, ydata, p0)
# popt, pcov = scipy.optimize.curve_fit(exponential, xdata, ydata)
if False:
yhat = func(xdata, *popt)
pt.figure(fnum=1, doclf=True)
pt.plot(xdata, ydata, label='measured')
pt.plot(xdata, yhat, label='predicteed')
pt.legend()
# slope = np.diff(ydata).mean()
# pt.plot(d)
thresh_to_params.append((thresh_value, popt))
# pt.plt.plot(*zip(*thresh_to_slope), 'x-')
# for thresh_value=.01, we get a rough line with slop ~2.302,
# for thresh_value=.5, we get a line with slop ~34.66
# if we want to get to 0 in n timesteps, with a thresh_value of
# choose span=f(thresh_value) * (n + 2))
# f is some inverse exponential
# 0.0001, 460.551314197147
# 0.001, 345.413485647860,
# 0.01, 230.275657098573,
# 0.1, 115.137828549287,
# 0.2, 80.4778885203347,
# 0.3, 60.2031233261536,
# 0.4, 45.8179484913827,
# 0.5, 34.6599400289520
# Seems to be 4PL symetrical sigmoid
# f(x) = -66500.85 + (66515.88 - -66500.85) / (1 + (x/0.8604672)^0.001503716)
# f(x) = -66500.85 + (66515.88 - -66500.85)/(1 + (x/0.8604672)^0.001503716)
def f(x):
return -66500.85 + (66515.88 -
-66500.85) / (1 + (x / 0.8604672)**0.001503716)
# return (10000 * (-6.65 + (13.3015) / (1 + (x/0.86) ** 0.00150)))
# f(.5) * (n - 1)
# f(
solve_rational_inequalities(thresh_expr < .01, n)
def solveset(f, symbol=None, domain=S.Complexes):
"""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 evaluatee 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 Integers()}
>>> x = Symbol('x', real=True)
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in 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 Integers()} \ {0}) U ({2*n*pi + pi | n in Integers()} \ {0})
>>> p = Symbol('p', positive=True)
>>> pprint(solveset(sin(p)/p, p), use_unicode=False)
{2*n*pi | n in Integers()} U {2*n*pi + pi | n in 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)
(0, oo)
"""
f = sympify(f)
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))
free_symbols = f.free_symbols
if 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()
else:
raise ValueError(filldedent('''
The independent variable must be specified for a
multivariate equation.'''))
elif not getattr(symbol, 'is_Symbol', False):
raise ValueError('A Symbol must be given, not type %s: %s' %
(type(symbol), symbol))
if isinstance(f, Eq):
from sympy.core import Add
f = Add(f.lhs, - f.rhs, evaluate=False)
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, relational=False) - _invalid_solutions(
f, symbol, domain)
except NotImplementedError:
result = ConditionSet(symbol, f, domain)
return result
return _solveset(f, symbol, domain, _check=True)
def continuous_domain(f, symbol, domain):
"""
Returns the intervals in the given domain for which the function
is continuous.
This method is limited by the ability to determine the various
singularities and discontinuities of the given function.
Parameters
==========
f : Expr
The concerned function.
symbol : Symbol
The variable for which the intervals are to be determined.
domain : Interval
The domain over which the continuity of the symbol has to be checked.
Examples
========
>>> from sympy import Symbol, S, tan, log, pi, sqrt
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import continuous_domain
>>> x = Symbol('x')
>>> continuous_domain(1/x, x, S.Reals)
Union(Interval.open(-oo, 0), Interval.open(0, oo))
>>> continuous_domain(tan(x), x, Interval(0, pi))
Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi))
>>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
Interval(2, 5)
>>> continuous_domain(log(2*x - 1), x, S.Reals)
Interval.open(1/2, oo)
Returns
=======
Interval
Union of all intervals where the function is continuous.
Raises
======
NotImplementedError
If the method to determine continuity of such a function
has not yet been developed.
"""
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.solvers.solveset import solveset, _has_rational_power
if domain.is_subset(S.Reals):
constrained_interval = domain
for atom in f.atoms(Pow):
predicate, denomin = _has_rational_power(atom, symbol)
constraint = S.EmptySet
if predicate and denomin == 2:
constraint = solve_univariate_inequality(
atom.base >= 0, symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
for atom in f.atoms(log):
constraint = solve_univariate_inequality(atom.args[0] > 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
domain = constrained_interval
try:
sings = S.EmptySet
if f.has(Abs):
sings = solveset(1/f, symbol, domain) + \
solveset(denom(together(f)), symbol, domain)
else:
for atom in f.atoms(Pow):
predicate, denomin = _has_rational_power(atom, symbol)
if predicate and denomin == 2:
sings = solveset(1/f, symbol, domain) +\
solveset(denom(together(f)), symbol, domain)
break
else:
sings = Intersection(solveset(1/f, symbol), domain) + \
solveset(denom(together(f)), symbol, domain)
except NotImplementedError:
import sys
raise (NotImplementedError(
"Methods for determining the continuous domains"
" of this function have not been developed."), None,
sys.exc_info()[2])
return domain - sings
def solveset(f, symbol=None, domain=S.Complexes):
"""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 no solution is found.
A `ConditionSet` is returned as unsolved object if algorithms
to evaluatee 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.
`solveset` uses two underlying functions `solveset_real` and
`solveset_complex` to solve equations. They are the solvers for real and
complex domain respectively. `solveset` ignores the assumptions on the
variable being solved for and instead, uses the `domain` parameter to
decide which solver to use.
See Also
========
solveset_real: solver for real domain
solveset_complex: solver for complex domain
Examples
========
>>> from sympy import exp, Symbol, Eq, pprint, S, solveset
>>> from sympy.abc import x
* The default domain is complex. Not specifying a domain will lead to the
solving of the equation in the complex domain.
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in Integers()}
* If you want to solve equation in real domain by the `solveset`
interface, then specify that the domain is real. Alternatively use
`solveset\_real`.
>>> x = Symbol('x')
>>> solveset(exp(x) - 1, x, S.Reals)
{0}
>>> solveset(Eq(exp(x), 1), x, S.Reals)
{0}
* Inequalities can be solved over the real domain only. Use of a complex
domain leads to a NotImplementedError.
>>> solveset(exp(x) > 1, x, S.Reals)
(0, oo)
"""
from sympy.solvers.inequalities import solve_univariate_inequality
if symbol is None:
free_symbols = f.free_symbols
if len(free_symbols) == 1:
symbol = free_symbols.pop()
else:
raise ValueError(
filldedent('''
The independent variable must be specified for a
multivariate equation.'''))
elif not symbol.is_Symbol:
raise ValueError('A Symbol must be given, not type %s: %s' %
(type(symbol), symbol))
f = sympify(f)
if f is S.false:
return EmptySet()
if f is S.true:
return domain
if isinstance(f, Eq):
from sympy.core import Add
f = Add(f.lhs, -f.rhs, evaluate=False)
if f.is_Relational:
if not domain.is_subset(S.Reals):
raise NotImplementedError("Inequalities in the complex domain are "
"not supported. Try the real domain by"
"setting domain=S.Reals")
try:
result = solve_univariate_inequality(
f, symbol, relational=False).intersection(domain)
except NotImplementedError:
result = ConditionSet(symbol, f, domain)
return result
if isinstance(f, (Expr, Number)):
if domain is S.Reals:
return solveset_real(f, symbol)
elif domain is S.Complexes:
return solveset_complex(f, symbol)
elif domain.is_subset(S.Reals):
return Intersection(solveset_real(f, symbol), domain)
else:
return Intersection(solveset_complex(f, symbol), domain)
def continuous_domain(f, symbol, domain):
"""
Returns the intervals in the given domain for which the function
is continuous.
This method is limited by the ability to determine the various
singularities and discontinuities of the given function.
Parameters
==========
f : :py:class:`~.Expr`
The concerned function.
symbol : :py:class:`~.Symbol`
The variable for which the intervals are to be determined.
domain : :py:class:`~.Interval`
The domain over which the continuity of the symbol has to be checked.
Examples
========
>>> from sympy import Interval, Symbol, S, tan, log, pi, sqrt
>>> from sympy.calculus.util import continuous_domain
>>> x = Symbol('x')
>>> continuous_domain(1/x, x, S.Reals)
Union(Interval.open(-oo, 0), Interval.open(0, oo))
>>> continuous_domain(tan(x), x, Interval(0, pi))
Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi))
>>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
Interval(2, 5)
>>> continuous_domain(log(2*x - 1), x, S.Reals)
Interval.open(1/2, oo)
Returns
=======
:py:class:`~.Interval`
Union of all intervals where the function is continuous.
Raises
======
NotImplementedError
If the method to determine continuity of such a function
has not yet been developed.
"""
from sympy.solvers.inequalities import solve_univariate_inequality
if domain.is_subset(S.Reals):
constrained_interval = domain
for atom in f.atoms(Pow):
den = atom.exp.as_numer_denom()[1]
if den.is_even and den.is_nonzero:
constraint = solve_univariate_inequality(
atom.base >= 0, symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
for atom in f.atoms(log):
constraint = solve_univariate_inequality(atom.args[0] > 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
return constrained_interval - singularities(f, symbol, domain)
def solveset(f, symbol=None):
"""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
Returns
=======
Set
A set of values for `symbol` for which `f` is True or is equal to
zero. An `EmptySet` is returned if no solution is found.
`solveset` claims to be complete in the solution set that it returns.
Raises
======
NotImplementedError
The algorithms for to find the solution of the given equation are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
`solveset` uses two underlying functions `solveset_real` and
`solveset_complex` to solve equations. They are
the solvers for real and complex domain respectively. The domain of
the solver is decided by the assumption on the variable for which the
equation is being solved.
See Also
========
solveset_real: solver for real domain
solveset_complex: solver for complex domain
Examples
========
>>> from sympy import exp, Symbol, Eq, pprint
>>> from sympy.solvers.solveset import solveset
>>> from sympy.abc import x
Symbols in Sympy are complex by default. A complex variable
will lead to the solving of the equation in complex domain
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in Integers()}
If you want to solve equation in real domain by the `solveset`
interface, then specify the variable to real. Alternatively use
`solveset_real`.
>>> x = Symbol('x', real=True)
>>> solveset(exp(x) - 1, x)
{0}
>>> solveset(Eq(exp(x), 1), x)
{0}
Inequalities are always solved in the real domain irrespective of
the assumption on the variable for which the inequality is solved.
>>> solveset(exp(x) > 1, x)
(0, oo)
"""
from sympy.solvers.inequalities import solve_univariate_inequality
if symbol is None:
free_symbols = f.free_symbols
if len(free_symbols) == 1:
symbol = free_symbols.pop()
else:
raise ValueError(filldedent('''
The independent variable must be specified for a
multivariate equation.'''))
elif not symbol.is_Symbol:
raise ValueError('A Symbol must be given, not type %s: %s' % (type(symbol), symbol))
real = (symbol.is_real is True)
f = sympify(f)
if isinstance(f, Eq):
f = f.lhs - f.rhs
if f.is_Relational:
if real is False:
warnings.warn(filldedent('''
The variable you are solving for is complex
but will assumed to be real since solving complex
inequalities is not supported.
'''))
return solve_univariate_inequality(f, symbol, relational=False)
if isinstance(f, (Expr, Number)):
if real is True:
return solveset_real(f, symbol)
else:
return solveset_complex(f, symbol)
a[i, j, :] = f(np.linspace(-3, 30, 20))
assert (len(a[a < 0]) == 0)
# good sign...
e = (post(n * x) * u(W) - u(c / x)).subs(W, 500).subs(n, 1)
f = lambdify(x, e, 'numpy')
g = lambdify(rho, f(np.linspace(1.01, 20, 10)), 'numpy')
m = g(np.linspace(1.2, 20, 20))
domain = Interval.Lopen(1, 4)
e = (post(n * x) * u(W) > u(c / x)).subs(x, 5).subs(n, 6).subs(W, 50)
solve_univariate_inequality(e, rho, domain=domain)
c = exp(post(n) * log(W))
from sympy import Interval
rho = Symbol('rho')
x, n, W = symbols('x n W')
e = (n / ((2 + n * x)**2)) * (1 / (rho - 1)) <= 1 / (x**rho)
from sympy import Union
domain = Union(Interval.Ropen(-50, 1), Interval.Lopen(1, 4))
solve_univariate_inequality(e.subs(n, 6).subs(x, 50), rho, domain=domain)
solveset(e, rho, domain=domain)
def is_convex(f, *syms, domain=S.Reals):
r"""Determines the convexity of the function passed in the argument.
Parameters
==========
f : :py:class:`~.Expr`
The concerned function.
syms : Tuple of :py:class:`~.Symbol`
The variables with respect to which the convexity is to be determined.
domain : :py:class:`~.Interval`, optional
The domain over which the convexity of the function has to be checked.
If unspecified, S.Reals will be the default domain.
Returns
=======
bool
The method returns ``True`` if the function is convex otherwise it
returns ``False``.
Raises
======
NotImplementedError
The check for the convexity of multivariate functions is not implemented yet.
Notes
=====
To determine concavity of a function pass `-f` as the concerned function.
To determine logarithmic convexity of a function pass `\log(f)` as
concerned function.
To determine logartihmic concavity of a function pass `-\log(f)` as
concerned function.
Currently, convexity check of multivariate functions is not handled.
Examples
========
>>> from sympy import is_convex, symbols, exp, oo, Interval
>>> x = symbols('x')
>>> is_convex(exp(x), x)
True
>>> is_convex(x**3, x, domain = Interval(-1, oo))
False
>>> is_convex(1/x**2, x, domain=Interval.open(0, oo))
True
References
==========
.. [1] https://en.wikipedia.org/wiki/Convex_function
.. [2] http://www.ifp.illinois.edu/~angelia/L3_convfunc.pdf
.. [3] https://en.wikipedia.org/wiki/Logarithmically_convex_function
.. [4] https://en.wikipedia.org/wiki/Logarithmically_concave_function
.. [5] https://en.wikipedia.org/wiki/Concave_function
"""
if len(syms) > 1:
raise NotImplementedError(
"The check for the convexity of multivariate functions is not implemented yet."
)
from sympy.solvers.inequalities import solve_univariate_inequality
f = _sympify(f)
var = syms[0]
if any(s in domain for s in singularities(f, var)):
return False
condition = f.diff(var, 2) < 0
if solve_univariate_inequality(condition, var, False, domain):
return False
return True
def solveset(f, symbol=None):
"""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
Returns
=======
Set
A set of values for `symbol` for which `f` is True or is equal to
zero. An `EmptySet` is returned if no solution is found.
`solveset` claims to be complete in the solution set that it returns.
Raises
======
NotImplementedError
The algorithms for to find the solution of the given equation are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
`solveset` uses two underlying functions `solveset_real` and
`solveset_complex` to solve equations. They are
the solvers for real and complex domain respectively. The domain of
the solver is decided by the assumption on the variable for which the
equation is being solved.
See Also
========
solveset_real: solver for real domain
solveset_complex: solver for complex domain
Examples
========
>>> from sympy import exp, Symbol, Eq, pprint
>>> from sympy.solvers.solveset import solveset
>>> from sympy.abc import x
* Symbols in Sympy are complex by default. A complex variable
will lead to the solving of the equation in complex domain.
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in Integers()}
* If you want to solve equation in real domain by the `solveset`
interface, then specify the variable to real. Alternatively use
`solveset\_real`.
>>> x = Symbol('x', real=True)
>>> solveset(exp(x) - 1, x)
{0}
>>> solveset(Eq(exp(x), 1), x)
{0}
* Inequalities are always solved in the real domain irrespective of
the assumption on the variable for which the inequality is solved.
>>> solveset(exp(x) > 1, x)
(0, oo)
"""
from sympy.solvers.inequalities import solve_univariate_inequality
if symbol is None:
free_symbols = f.free_symbols
if len(free_symbols) == 1:
symbol = free_symbols.pop()
else:
raise ValueError(
filldedent('''
The independent variable must be specified for a
multivariate equation.'''))
elif not symbol.is_Symbol:
raise ValueError('A Symbol must be given, not type %s: %s' %
(type(symbol), symbol))
real = (symbol.is_real is True)
f = sympify(f)
if f is S.false:
return EmptySet()
if f is S.true:
if real:
return S.Reals
else:
return S.Complex
if isinstance(f, Eq):
from sympy.core import Add
f = Add(f.lhs, -f.rhs, evaluate=False)
if f.is_Relational:
if real is False:
warnings.warn(
filldedent('''
The variable you are solving for is complex
but will assumed to be real since solving complex
inequalities is not supported.
'''))
return solve_univariate_inequality(f, symbol, relational=False)
if isinstance(f, (Expr, Number)):
if real is True:
return solveset_real(f, symbol)
else:
return solveset_complex(f, symbol)
def solveset(f, symbol=None, domain=S.Complexes):
"""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 no solution is found.
A `ConditionSet` is returned as unsolved object if algorithms
to evaluatee 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.
`solveset` uses two underlying functions `solveset_real` and
`solveset_complex` to solve equations. They are the solvers for real and
complex domain respectively. `solveset` ignores the assumptions on the
variable being solved for and instead, uses the `domain` parameter to
decide which solver to use.
See Also
========
solveset_real: solver for real domain
solveset_complex: solver for complex domain
Examples
========
>>> from sympy import exp, Symbol, Eq, pprint, S, solveset
>>> from sympy.abc import x
* The default domain is complex. Not specifying a domain will lead to the
solving of the equation in the complex domain.
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in Integers()}
* If you want to solve equation in real domain by the `solveset`
interface, then specify that the domain is real. Alternatively use
`solveset\_real`.
>>> x = Symbol('x')
>>> solveset(exp(x) - 1, x, S.Reals)
{0}
>>> solveset(Eq(exp(x), 1), x, S.Reals)
{0}
* Inequalities can be solved over the real domain only. Use of a complex
domain leads to a NotImplementedError.
>>> solveset(exp(x) > 1, x, S.Reals)
(0, oo)
"""
from sympy.solvers.inequalities import solve_univariate_inequality
if symbol is None:
free_symbols = f.free_symbols
if len(free_symbols) == 1:
symbol = free_symbols.pop()
else:
raise ValueError(filldedent('''
The independent variable must be specified for a
multivariate equation.'''))
elif not symbol.is_Symbol:
raise ValueError('A Symbol must be given, not type %s: %s' % (type(symbol), symbol))
f = sympify(f)
if f is S.false:
return EmptySet()
if f is S.true:
return domain
if isinstance(f, Eq):
from sympy.core import Add
f = Add(f.lhs, - f.rhs, evaluate=False)
if f.is_Relational:
if not domain.is_subset(S.Reals):
raise NotImplementedError("Inequalities in the complex domain are "
"not supported. Try the real domain by"
"setting domain=S.Reals")
try:
result = solve_univariate_inequality(
f, symbol, relational=False).intersection(domain)
except NotImplementedError:
result = ConditionSet(symbol, f, domain)
return result
if isinstance(f, (Expr, Number)):
if domain is S.Reals:
return solveset_real(f, symbol)
elif domain is S.Complexes:
return solveset_complex(f, symbol)
elif domain.is_subset(S.Reals):
return Intersection(solveset_real(f, symbol), domain)
else:
return Intersection(solveset_complex(f, symbol), domain)
def continuous_domain(f, symbol, domain):
"""
Returns the intervals in the given domain for which the function
is continuous.
This method is limited by the ability to determine the various
singularities and discontinuities of the given function.
Parameters
==========
f : Expr
The concerned function.
symbol : Symbol
The variable for which the intervals are to be determined.
domain : Interval
The domain over which the continuity of the symbol has to be checked.
Examples
========
>>> from sympy import Symbol, S, tan, log, pi, sqrt
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import continuous_domain
>>> x = Symbol('x')
>>> continuous_domain(1/x, x, S.Reals)
Union(Interval.open(-oo, 0), Interval.open(0, oo))
>>> continuous_domain(tan(x), x, Interval(0, pi))
Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi))
>>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
Interval(2, 5)
>>> continuous_domain(log(2*x - 1), x, S.Reals)
Interval.open(1/2, oo)
Returns
=======
Interval
Union of all intervals where the function is continuous.
Raises
======
NotImplementedError
If the method to determine continuity of such a function
has not yet been developed.
"""
from sympy.solvers.inequalities import solve_univariate_inequality
from sympy.solvers.solveset import solveset, _has_rational_power
if domain.is_subset(S.Reals):
constrained_interval = domain
for atom in f.atoms(Pow):
predicate, denomin = _has_rational_power(atom, symbol)
constraint = S.EmptySet
if predicate and denomin == 2:
constraint = solve_univariate_inequality(atom.base >= 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
for atom in f.atoms(log):
constraint = solve_univariate_inequality(atom.args[0] > 0,
symbol).as_set()
constrained_interval = Intersection(constraint,
constrained_interval)
domain = constrained_interval
try:
sings = S.EmptySet
if f.has(Abs):
sings = solveset(1/f, symbol, domain) + \
solveset(denom(together(f)), symbol, domain)
else:
for atom in f.atoms(Pow):
predicate, denomin = _has_rational_power(atom, symbol)
if predicate and denomin == 2:
sings = solveset(1/f, symbol, domain) +\
solveset(denom(together(f)), symbol, domain)
break
else:
sings = Intersection(solveset(1/f, symbol), domain) + \
solveset(denom(together(f)), symbol, domain)
except NotImplementedError:
import sys
raise (NotImplementedError("Methods for determining the continuous domains"
" of this function have not been developed."),
None,
sys.exc_info()[2])
return domain - sings
def is_convex(f, *syms, **kwargs):
"""Determines the convexity of the function passed in the argument.
Parameters
==========
f : Expr
The concerned function.
syms : Tuple of symbols
The variables with respect to which the convexity is to be determined.
domain : Interval, optional
The domain over which the convexity of the function has to be checked.
If unspecified, S.Reals will be the default domain.
Returns
=======
Boolean
The method returns `True` if the function is convex otherwise it
returns `False`.
Raises
======
NotImplementedError
The check for the convexity of multivariate functions is not implemented yet.
Notes
=====
To determine concavity of a function pass `-f` as the concerned function.
To determine logarithmic convexity of a function pass log(f) as
concerned function.
To determine logartihmic concavity of a function pass -log(f) as
concerned function.
Currently, convexity check of multivariate functions is not handled.
Examples
========
>>> from sympy import symbols, exp, oo, Interval
>>> from sympy.calculus.util import is_convex
>>> x = symbols('x')
>>> is_convex(exp(x), x)
True
>>> is_convex(x**3, x, domain = Interval(-1, oo))
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Convex_function
.. [2] http://www.ifp.illinois.edu/~angelia/L3_convfunc.pdf
.. [3] https://en.wikipedia.org/wiki/Logarithmically_convex_function
.. [4] https://en.wikipedia.org/wiki/Logarithmically_concave_function
.. [5] https://en.wikipedia.org/wiki/Concave_function
"""
if len(syms) > 1:
raise NotImplementedError(
"The check for the convexity of multivariate functions is not implemented yet.")
f = _sympify(f)
domain = kwargs.get('domain', S.Reals)
var = syms[0]
condition = f.diff(var, 2) < 0
if solve_univariate_inequality(condition, var, False, domain):
return False
return True
