Source code for sympy.sets.fancysets
from __future__ import print_function, division
from sympy.logic.boolalg import And
from sympy.core.basic import Basic
from sympy.core.compatibility import as_int, with_metaclass, range, PY3
from sympy.sets.sets import (Set, Interval, Intersection, EmptySet, Union,
FiniteSet)
from sympy.core.singleton import Singleton, S, sympify
from sympy.core.sympify import _sympify, converter
from sympy.core.function import Lambda
from sympy.utilities.misc import filldedent, func_name
[docs]class Naturals(with_metaclass(Singleton, Set)):
"""
Represents the natural numbers (or counting numbers) which are all
positive integers starting from 1. This set is also available as
the Singleton, S.Naturals.
Examples
========
>>> from sympy import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Naturals)
>>> next(iterable)
1
>>> next(iterable)
2
>>> next(iterable)
3
>>> pprint(S.Naturals.intersect(Interval(0, 10)))
{1, 2, ..., 10}
See Also
========
Naturals0 : non-negative integers (i.e. includes 0, too)
Integers : also includes negative integers
"""
is_iterable = True
_inf = S.One
_sup = S.Infinity
def _intersect(self, other):
if other.is_Interval:
return Intersection(
S.Integers, other, Interval(self._inf, S.Infinity))
return None
def _contains(self, other):
if other.is_positive and other.is_integer:
return S.true
elif other.is_integer is False or other.is_positive is False:
return S.false
def __iter__(self):
i = self._inf
while True:
yield i
i = i + 1
@property
def _boundary(self):
return self
[docs]class Naturals0(Naturals):
"""Represents the whole numbers which are all the non-negative integers,
inclusive of zero.
See Also
========
Naturals : positive integers; does not include 0
Integers : also includes the negative integers
"""
_inf = S.Zero
def _contains(self, other):
if other.is_integer and other.is_nonnegative:
return S.true
elif other.is_integer is False or other.is_nonnegative is False:
return S.false
[docs]class Integers(with_metaclass(Singleton, Set)):
"""
Represents all integers: positive, negative and zero. This set is also
available as the Singleton, S.Integers.
Examples
========
>>> from sympy import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Integers)
>>> next(iterable)
0
>>> next(iterable)
1
>>> next(iterable)
-1
>>> next(iterable)
2
>>> pprint(S.Integers.intersect(Interval(-4, 4)))
{-4, -3, ..., 4}
See Also
========
Naturals0 : non-negative integers
Integers : positive and negative integers and zero
"""
is_iterable = True
def _intersect(self, other):
from sympy.functions.elementary.integers import floor, ceiling
if other is Interval(S.NegativeInfinity, S.Infinity) or other is S.Reals:
return self
elif other.is_Interval:
s = Range(ceiling(other.left), floor(other.right) + 1)
return s.intersect(other) # take out endpoints if open interval
return None
def _contains(self, other):
if other.is_integer:
return S.true
elif other.is_integer is False:
return S.false
def __iter__(self):
yield S.Zero
i = S.One
while True:
yield i
yield -i
i = i + 1
@property
def _inf(self):
return -S.Infinity
@property
def _sup(self):
return S.Infinity
@property
def _boundary(self):
return self
def _eval_imageset(self, f):
from sympy import Wild
expr = f.expr
if len(f.variables) > 1:
return
n = f.variables[0]
a = Wild('a')
b = Wild('b')
match = expr.match(a*n + b)
if match[a].is_negative:
expr = -expr
match = expr.match(a*n + b)
if match[a] is S.One and match[b].is_integer:
expr = expr - match[b]
return ImageSet(Lambda(n, expr), S.Integers)
class Reals(with_metaclass(Singleton, Interval)):
def __new__(cls):
return Interval.__new__(cls, -S.Infinity, S.Infinity)
def __eq__(self, other):
return other == Interval(-S.Infinity, S.Infinity)
def __hash__(self):
return hash(Interval(-S.Infinity, S.Infinity))
[docs]class ImageSet(Set):
"""
Image of a set under a mathematical function
Examples
========
>>> from sympy import Symbol, S, ImageSet, FiniteSet, Lambda
>>> x = Symbol('x')
>>> N = S.Naturals
>>> squares = ImageSet(Lambda(x, x**2), N) # {x**2 for x in N}
>>> 4 in squares
True
>>> 5 in squares
False
>>> FiniteSet(0, 1, 2, 3, 4, 5, 6, 7, 9, 10).intersect(squares)
{1, 4, 9}
>>> square_iterable = iter(squares)
>>> for i in range(4):
... next(square_iterable)
1
4
9
16
"""
def __new__(cls, lamda, base_set):
return Basic.__new__(cls, lamda, base_set)
lamda = property(lambda self: self.args[0])
base_set = property(lambda self: self.args[1])
def __iter__(self):
already_seen = set()
for i in self.base_set:
val = self.lamda(i)
if val in already_seen:
continue
else:
already_seen.add(val)
yield val
def _is_multivariate(self):
return len(self.lamda.variables) > 1
def _contains(self, other):
from sympy.solvers.solveset import solveset, linsolve
L = self.lamda
if self._is_multivariate():
solns = list(linsolve([expr - val for val, expr in
zip(other, L.expr)], L.variables).args[0])
else:
solnsSet = solveset(L.expr-other, L.variables[0])
if solnsSet.is_FiniteSet:
solns = list(solveset(L.expr - other, L.variables[0]))
else:
raise NotImplementedError(filldedent('''
Determining whether an ImageSet contains %s has not
been implemented.''' % func_name(other)))
for soln in solns:
try:
if soln in self.base_set:
return S.true
except TypeError:
return self.base_set.contains(soln.evalf())
return S.false
@property
def is_iterable(self):
return self.base_set.is_iterable
def _intersect(self, other):
from sympy import Dummy
from sympy.solvers.diophantine import diophantine
from sympy.sets.sets import imageset
if self.base_set is S.Integers:
if isinstance(other, ImageSet) and other.base_set is S.Integers:
f, g = self.lamda.expr, other.lamda.expr
n, m = self.lamda.variables[0], other.lamda.variables[0]
# Diophantine sorts the solutions according to the alphabetic
# order of the variable names, since the result should not depend
# on the variable name, they are replaced by the dummy variables
# below
a, b = Dummy('a'), Dummy('b')
f, g = f.subs(n, a), g.subs(m, b)
solns_set = diophantine(f - g)
if solns_set == set():
return EmptySet()
solns = list(diophantine(f - g))
if len(solns) == 1:
t = list(solns[0][0].free_symbols)[0]
else:
return None
# since 'a' < 'b'
return imageset(Lambda(t, f.subs(a, solns[0][0])), S.Integers)
if other == S.Reals:
from sympy.solvers.solveset import solveset_real
from sympy.core.function import expand_complex
if len(self.lamda.variables) > 1:
return None
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
return imageset(Lambda(n_, re),
self.base_set.intersect(
solveset_real(im, n_)))
[docs]class Range(Set):
"""
Represents a range of integers.
Examples
========
>>> from sympy import Range
>>> list(Range(5)) # 0 to 5
[0, 1, 2, 3, 4]
>>> list(Range(10, 15)) # 10 to 15
[10, 11, 12, 13, 14]
>>> list(Range(10, 20, 2)) # 10 to 20 in steps of 2
[10, 12, 14, 16, 18]
>>> list(Range(20, 10, -2)) # 20 to 10 backward in steps of 2
[12, 14, 16, 18, 20]
"""
is_iterable = True
def __new__(cls, *args):
from sympy.functions.elementary.integers import ceiling
if len(args) == 1:
if isinstance(args[0], range if PY3 else xrange):
args = args[0].__reduce__()[1] # use pickle method
# expand range
slc = slice(*args)
start, stop, step = slc.start or 0, slc.stop, slc.step or 1
try:
start, stop, step = [w if w in [S.NegativeInfinity, S.Infinity] else sympify(as_int(w))
for w in (start, stop, step)]
except ValueError:
raise ValueError("Inputs to Range must be Integer Valued\n" +
"Use ImageSets of Ranges for other cases")
if not step.is_finite:
raise ValueError("Infinite step is not allowed")
if start == stop:
return S.EmptySet
n = ceiling((stop - start)/step)
if n <= 0:
return S.EmptySet
# normalize args: regardless of how they are entered they will show
# canonically as Range(inf, sup, step) with step > 0
if n.is_finite:
start, stop = sorted((start, start + (n - 1)*step))
else:
start, stop = sorted((start, stop - step))
step = abs(step)
if (start, stop) == (S.NegativeInfinity, S.Infinity):
raise ValueError("Both the start and end value of "
"Range cannot be unbounded")
else:
return Basic.__new__(cls, start, stop + step, step)
start = property(lambda self: self.args[0])
stop = property(lambda self: self.args[1])
step = property(lambda self: self.args[2])
def _intersect(self, other):
from sympy.functions.elementary.integers import floor, ceiling
from sympy.functions.elementary.miscellaneous import Min, Max
if other.is_Interval:
osup = other.sup
oinf = other.inf
# if other is [0, 10) we can only go up to 9
if osup.is_integer and other.right_open:
osup -= 1
if oinf.is_integer and other.left_open:
oinf += 1
# Take the most restrictive of the bounds set by the two sets
# round inwards
inf = ceiling(Max(self.inf, oinf))
sup = floor(Min(self.sup, osup))
# if we are off the sequence, get back on
if inf.is_finite and self.inf.is_finite:
off = (inf - self.inf) % self.step
else:
off = S.Zero
if off:
inf += self.step - off
return Range(inf, sup + 1, self.step)
if other == S.Naturals:
return self._intersect(Interval(1, S.Infinity))
if other == S.Integers:
return self
return None
def _contains(self, other):
if (((self.start - other)/self.step).is_integer or
((self.stop - other)/self.step).is_integer):
return _sympify(other >= self.inf and other <= self.sup)
elif (((self.start - other)/self.step).is_integer is False and
((self.stop - other)/self.step).is_integer is False):
return S.false
def __iter__(self):
if self.start is S.NegativeInfinity:
i = self.stop - self.step
step = -self.step
else:
i = self.start
step = self.step
while(i < self.stop and i >= self.start):
yield i
i += step
def __len__(self):
return (self.stop - self.start)//self.step
def __nonzero__(self):
return True
__bool__ = __nonzero__
def _ith_element(self, i):
return self.start + i*self.step
@property
def _last_element(self):
if self.stop is S.Infinity:
return S.Infinity
elif self.start is S.NegativeInfinity:
return self.stop - self.step
else:
return self._ith_element(len(self) - 1)
@property
def _inf(self):
return self.start
@property
def _sup(self):
return self.stop - self.step
@property
def _boundary(self):
return self
if PY3:
converter[range] = Range
else:
converter[xrange] = Range
[docs]def normalize_theta_set(theta):
"""
Normalize a Real Set `theta` in the Interval [0, 2*pi). It returns
a normalized value of theta in the Set. For Interval, a maximum of
one cycle [0, 2*pi], is returned i.e. for theta equal to [0, 10*pi],
returned normalized value would be [0, 2*pi). As of now intervals
with end points as non-multiples of `pi` is not supported.
Raises
======
NotImplementedError
The algorithms for Normalizing theta Set are not yet
implemented.
ValueError
The input is not valid, i.e. the input is not a real set.
RuntimeError
It is a bug, please report to the github issue tracker.
Examples
========
>>> from sympy.sets.fancysets import normalize_theta_set
>>> from sympy import Interval, FiniteSet, pi
>>> normalize_theta_set(Interval(9*pi/2, 5*pi))
[pi/2, pi]
>>> normalize_theta_set(Interval(-3*pi/2, pi/2))
[0, 2*pi)
>>> normalize_theta_set(Interval(-pi/2, pi/2))
[0, pi/2] U [3*pi/2, 2*pi)
>>> normalize_theta_set(Interval(-4*pi, 3*pi))
[0, 2*pi)
>>> normalize_theta_set(Interval(-3*pi/2, -pi/2))
[pi/2, 3*pi/2]
>>> normalize_theta_set(FiniteSet(0, pi, 3*pi))
{0, pi}
"""
from sympy.functions.elementary.trigonometric import _pi_coeff as coeff
if theta.is_Interval:
interval_len = theta.measure
# one complete circle
if interval_len >= 2*S.Pi:
if interval_len == 2*S.Pi and theta.left_open and theta.right_open:
k = coeff(theta.start)
return Union(Interval(0, k*S.Pi, False, True),
Interval(k*S.Pi, 2*S.Pi, True, True))
return Interval(0, 2*S.Pi, False, True)
k_start, k_end = coeff(theta.start), coeff(theta.end)
if k_start is None or k_end is None:
raise NotImplementedError("Normalizing theta without pi as coefficient is "
"not yet implemented")
new_start = k_start*S.Pi
new_end = k_end*S.Pi
if new_start > new_end:
return Union(Interval(S.Zero, new_end, False, theta.right_open),
Interval(new_start, 2*S.Pi, theta.left_open, True))
else:
return Interval(new_start, new_end, theta.left_open, theta.right_open)
elif theta.is_FiniteSet:
new_theta = []
for element in theta:
k = coeff(element)
if k is None:
raise NotImplementedError('Normalizing theta without pi as '
'coefficient, is not Implemented.')
else:
new_theta.append(k*S.Pi)
return FiniteSet(*new_theta)
elif theta.is_Union:
return Union(*[normalize_theta_set(interval) for interval in theta.args])
elif theta.is_subset(S.Reals):
raise NotImplementedError("Normalizing theta when, it is of type %s is not "
"implemented" % type(theta))
else:
raise ValueError(" %s is not a real set" % (theta))
[docs]class ComplexRegion(Set):
"""
Represents the Set of all Complex Numbers. It can represent a
region of Complex Plane in both the standard forms Polar and
Rectangular coordinates.
* Polar Form
Input is in the form of the ProductSet or Union of ProductSets
of the intervals of r and theta, & use the flag polar=True.
Z = {z in C | z = r*[cos(theta) + I*sin(theta)], r in [r], theta in [theta]}
* Rectangular Form
Input is in the form of the ProductSet or Union of ProductSets
of interval of x and y the of the Complex numbers in a Plane.
Default input type is in rectangular form.
Z = {z in C | z = x + I*y, x in [Re(z)], y in [Im(z)]}
Examples
========
>>> from sympy.sets.fancysets import ComplexRegion
>>> from sympy.sets import Interval
>>> from sympy import S, I, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 6)
>>> c = Interval(1, 8)
>>> c1 = ComplexRegion(a*b) # Rectangular Form
>>> c1
ComplexRegion([2, 3] x [4, 6], False)
* c1 represents the rectangular region in complex plane
surrounded by the coordinates (2, 4), (3, 4), (3, 6) and
(2, 6), of the four vertices.
>>> c2 = ComplexRegion(Union(a*b, b*c))
>>> c2
ComplexRegion([2, 3] x [4, 6] U [4, 6] x [1, 8], False)
* c2 represents the Union of two rectangular regions in complex
plane. One of them surrounded by the coordinates of c1 and
other surrounded by the coordinates (4, 1), (6, 1), (6, 8) and
(4, 8).
>>> 2.5 + 4.5*I in c1
True
>>> 2.5 + 6.5*I in c1
False
>>> r = Interval(0, 1)
>>> theta = Interval(0, 2*S.Pi)
>>> c2 = ComplexRegion(r*theta, polar=True) # Polar Form
>>> c2 # unit Disk
ComplexRegion([0, 1] x [0, 2*pi), True)
* c2 represents the region in complex plane inside the
Unit Disk centered at the origin.
>>> 0.5 + 0.5*I in c2
True
>>> 1 + 2*I in c2
False
>>> unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True)
>>> upper_half_unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, S.Pi), polar=True)
>>> intersection = unit_disk.intersect(upper_half_unit_disk)
>>> intersection
ComplexRegion([0, 1] x [0, pi], True)
>>> intersection == upper_half_unit_disk
True
See Also
========
Reals
"""
is_ComplexRegion = True
def __new__(cls, sets, polar=False):
from sympy import symbols, Dummy, sympify, sin, cos
x, y, r, theta = symbols('x, y, r, theta', cls=Dummy)
I = S.ImaginaryUnit
polar = sympify(polar)
# Rectangular Form
if polar == False:
if all(_a.is_FiniteSet for _a in sets.args) and (len(sets.args) == 2):
# ** ProductSet of FiniteSets in the Complex Plane. **
# For Cases like ComplexRegion({2, 4}*{3}), It
# would return {2 + 3*I, 4 + 3*I}
complex_num = []
for x in sets.args[0]:
for y in sets.args[1]:
complex_num.append(x + I*y)
obj = FiniteSet(*complex_num)
else:
obj = ImageSet.__new__(cls, Lambda((x, y), x + I*y), sets)
obj._variables = (x, y)
obj._expr = x + I*y
# Polar Form
elif polar == True:
new_sets = []
# sets is Union of ProductSets
if not sets.is_ProductSet:
for k in sets.args:
new_sets.append(k)
# sets is ProductSets
else:
new_sets.append(sets)
# Normalize input theta
for k, v in enumerate(new_sets):
from sympy.sets import ProductSet
new_sets[k] = ProductSet(v.args[0],
normalize_theta_set(v.args[1]))
sets = Union(*new_sets)
obj = ImageSet.__new__(cls, Lambda((r, theta),
r*(cos(theta) + I*sin(theta))),
sets)
obj._variables = (r, theta)
obj._expr = r*(cos(theta) + I*sin(theta))
else:
raise ValueError("polar should be either True or False")
obj._sets = sets
obj._polar = polar
return obj
@property
[docs] def sets(self):
"""
Return raw input sets to the self.
Examples
========
>>> from sympy import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.sets
[2, 3] x [4, 5]
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.sets
[2, 3] x [4, 5] U [4, 5] x [1, 7]
"""
return self._sets
@property
def args(self):
return (self._sets, self._polar)
@property
def variables(self):
return self._variables
@property
def expr(self):
return self._expr
@property
[docs] def psets(self):
"""
Return a tuple of sets (ProductSets) input of the self.
Examples
========
>>> from sympy import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.psets
([2, 3] x [4, 5],)
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.psets
([2, 3] x [4, 5], [4, 5] x [1, 7])
"""
if self.sets.is_ProductSet:
psets = ()
psets = psets + (self.sets, )
else:
psets = self.sets.args
return psets
@property
[docs] def a_interval(self):
"""
Return the union of intervals of `x` when, self is in
rectangular form, or the union of intervals of `r` when
self is in polar form.
Examples
========
>>> from sympy import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.a_interval
[2, 3]
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.a_interval
[2, 3] U [4, 5]
"""
a_interval = []
for element in self.psets:
a_interval.append(element.args[0])
a_interval = Union(*a_interval)
return a_interval
@property
[docs] def b_interval(self):
"""
Return the union of intervals of `y` when, self is in
rectangular form, or the union of intervals of `theta`
when self is in polar form.
Examples
========
>>> from sympy import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.b_interval
[4, 5]
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.b_interval
[1, 7]
"""
b_interval = []
for element in self.psets:
b_interval.append(element.args[1])
b_interval = Union(*b_interval)
return b_interval
@property
[docs] def polar(self):
"""
Returns True if self is in polar form.
Examples
========
>>> from sympy import Interval, ComplexRegion, Union, S
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> theta = Interval(0, 2*S.Pi)
>>> C1 = ComplexRegion(a*b)
>>> C1.polar
False
>>> C2 = ComplexRegion(a*theta, polar=True)
>>> C2.polar
True
"""
return self._polar
@property
def _measure(self):
"""
The measure of self.sets.
Examples
========
>>> from sympy import Interval, ComplexRegion, S
>>> a, b = Interval(2, 5), Interval(4, 8)
>>> c = Interval(0, 2*S.Pi)
>>> c1 = ComplexRegion(a*b)
>>> c1.measure
12
>>> c2 = ComplexRegion(a*c, polar=True)
>>> c2.measure
6*pi
"""
return self.sets._measure
def _contains(self, other):
from sympy.functions import arg, Abs
# self in rectangular form
if not self.polar:
re, im = other.as_real_imag()
for element in self.psets:
if And(element.args[0]._contains(re),
element.args[1]._contains(im)):
return True
return False
# self in polar form
elif self.polar:
if sympify(other).is_zero:
r, theta = S.Zero, S.Zero
else:
r, theta = Abs(other), arg(other)
for element in self.psets:
if And(element.args[0]._contains(r),
element.args[1]._contains(theta)):
return True
return False
def _intersect(self, other):
if other.is_ComplexRegion:
# self in rectangular form
if (not self.polar) and (not other.polar):
return ComplexRegion(Intersection(self.sets, other.sets))
# self in polar form
elif self.polar and other.polar:
r1, theta1 = self.a_interval, self.b_interval
r2, theta2 = other.a_interval, other.b_interval
new_r_interval = Intersection(r1, r2)
new_theta_interval = Intersection(theta1, theta2)
# 0 and 2*Pi means the same
if ((2*S.Pi in theta1 and S.Zero in theta2) or
(2*S.Pi in theta2 and S.Zero in theta1)):
new_theta_interval = Union(new_theta_interval,
FiniteSet(0))
return ComplexRegion(new_r_interval*new_theta_interval,
polar=True)
if other is S.Reals:
return other
if other.is_subset(S.Reals):
new_interval = []
# self in rectangular form
if not self.polar:
for element in self.psets:
if S.Zero in element.args[0]:
new_interval.append(element.args[0])
new_interval = Union(*new_interval)
return Intersection(new_interval, other)
# self in polar form
elif self.polar:
for element in self.psets:
if (0 in element.args[1]) or (S.Pi in element.args[1]):
new_interval.append(element.args[0])
new_interval = Union(*new_interval)
return Intersection(new_interval, other)
def _union(self, other):
if other.is_ComplexRegion:
# self in rectangular form
if (not self.polar) and (not other.polar):
return ComplexRegion(Union(self.sets, other.sets))
# self in polar form
elif self.polar and other.polar:
return ComplexRegion(Union(self.sets, other.sets), polar=True)
if self == S.Complexes:
return self
return None
class Complexes(with_metaclass(Singleton, ComplexRegion)):
def __new__(cls):
return ComplexRegion.__new__(cls, S.Reals*S.Reals)
def __eq__(self, other):
return other == ComplexRegion(S.Reals*S.Reals)
def __hash__(self):
return hash(ComplexRegion(S.Reals*S.Reals))
def __str__(self):
return "S.Complexes"
def __repr__(self):
return "S.Complexes"
