def test_intersections():
assert S.Integers.intersect(S.Reals) == S.Integers
assert 5 in S.Integers.intersect(S.Reals)
assert 5 in S.Integers.intersect(S.Reals)
assert -5 not in S.Naturals.intersect(S.Reals)
assert 5.5 not in S.Integers.intersect(S.Reals)
assert 5 in S.Integers.intersect(Interval(3, oo))
assert -5 in S.Integers.intersect(Interval(-oo, 3))
assert all(x.is_Integer
for x in take(10, S.Integers.intersect(Interval(3, oo))))
def test_intersections():
assert S.Integers.intersection(S.Reals) == S.Integers
assert 5 in S.Integers.intersection(S.Reals)
assert 5 in S.Integers.intersection(S.Reals)
assert -5 not in S.Naturals.intersection(S.Reals)
assert 5.5 not in S.Integers.intersection(S.Reals)
assert 5 in S.Integers.intersection(Interval(3, oo))
assert -5 in S.Integers.intersection(Interval(-oo, 3))
assert all(x.is_Integer
for x in itertools.islice(S.Integers.intersection(Interval(3, oo)), 10))
def test_halfcircle():
r, th = symbols('r, theta', extended_real=True)
L = Lambda((r, th), (r*cos(th), r*sin(th)))
halfcircle = ImageSet(L, Interval(0, 1)*Interval(0, pi))
assert (1, 0) in halfcircle
assert (0, -1) not in halfcircle
assert (0, 0) in halfcircle
assert not halfcircle.is_iterable
def test_halfcircle():
# This test sometimes works and sometimes doesn't.
# It may be an issue with solve? Maybe with using Lambdas/dummys?
# I believe the code within fancysets is correct
r, th = symbols('r, theta', extended_real=True)
L = Lambda((r, th), (r*cos(th), r*sin(th)))
halfcircle = ImageSet(L, Interval(0, 1)*Interval(0, pi))
assert (1, 0) in halfcircle
assert (0, -1) not in halfcircle
assert (0, 0) in halfcircle
assert not halfcircle.is_iterable
def _intersect(self, other):
from diofant.functions.elementary.integers import floor, ceiling
from diofant.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, False, True))
if other == S.Integers:
return self
return
def test_naturals():
N = S.Naturals
assert 5 in N
assert -5 not in N
assert 5.5 not in N
ni = iter(N)
a, b, c, d = next(ni), next(ni), next(ni), next(ni)
assert (a, b, c, d) == (1, 2, 3, 4)
assert isinstance(a, Basic)
assert N.intersect(Interval(-5, 5)) == Range(1, 6)
assert N.intersect(Interval(-5, 5, True, True)) == Range(1, 5)
assert N.boundary == N
assert N.inf == 1
assert N.sup == oo
def test_integers():
Z = S.Integers
assert 5 in Z
assert -5 in Z
assert 5.5 not in Z
zi = iter(Z)
a, b, c, d = next(zi), next(zi), next(zi), next(zi)
assert (a, b, c, d) == (0, 1, -1, 2)
assert isinstance(a, Basic)
assert Z.intersect(Interval(-5, 5)) == Range(-5, 6)
assert Z.intersect(Interval(-5, 5, True, True)) == Range(-4, 5)
assert Z.inf == -oo
assert Z.sup == oo
assert Z.boundary == Z
def _intersect(self, other):
from diofant.functions.elementary.integers import floor, ceiling
if other is Interval(S.NegativeInfinity, S.Infinity, True,
True) 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
def test_univariate_relational_as_set():
assert (x > 0).as_set() == Interval(0, oo, True, True)
assert (x >= 0).as_set() == Interval(0, oo, False, True)
assert (x < 0).as_set() == Interval(-oo, 0, True, True)
assert (x <= 0).as_set() == Interval(-oo, 0, True)
assert Eq(x, 0).as_set() == FiniteSet(0)
assert Ne(x, 0).as_set() == Interval(-oo, 0, True, True) + \
Interval(0, oo, True, True)
assert (x**2 >= 4).as_set() == (Interval(-oo, -2, True) +
Interval(2, oo, False, True))
def test_range_interval_intersection():
# Intersection with intervals
assert FiniteSet(*Range(0, 10, 1).intersect(Interval(2, 6))) == \
FiniteSet(2, 3, 4, 5, 6)
# Open Intervals are removed
assert (FiniteSet(*Range(0, 10, 1).intersect(Interval(2, 6,
True, True))) ==
FiniteSet(3, 4, 5))
# Try this with large steps
assert (FiniteSet(*Range(0, 100, 10).intersect(Interval(15, 55))) ==
FiniteSet(20, 30, 40, 50))
# Going backwards
assert (FiniteSet(*Range(10, -9, -3).intersect(Interval(-5, 6))) ==
FiniteSet(-5, -2, 1, 4))
assert (FiniteSet(*Range(10, -9, -3).intersect(Interval(-5, 6, True))) ==
FiniteSet(-2, 1, 4))
def test_integers():
Z = S.Integers
assert 5 in Z
assert -5 in Z
assert 5.5 not in Z
zi = iter(Z)
a, b, c, d = next(zi), next(zi), next(zi), next(zi)
assert (a, b, c, d) == (0, 1, -1, 2)
assert isinstance(a, Basic)
assert Z.intersection(Interval(-5, 5)) == Range(-5, 6)
assert Z.intersection(Interval(-5, 5, True, True)) == Range(-4, 5)
assert Z.intersection(Set(x)) == Intersection(Z, Set(x), evaluate=False)
assert Z.inf == -oo
assert Z.sup == oo
assert Z.boundary == Z
assert imageset(Lambda((x, y), x * y),
Z) == ImageSet(Lambda((x, y), x * y), Z)
def test_sympyissue_11732():
interval12 = Interval(1, 2)
finiteset1234 = FiniteSet(1, 2, 3, 4)
pointComplex = (1, 2)
assert (interval12 in S.Naturals) is False
assert (interval12 in S.Naturals0) is False
assert (interval12 in S.Integers) is False
assert (finiteset1234 in S.Naturals) is False
assert (finiteset1234 in S.Naturals0) is False
assert (finiteset1234 in S.Integers) is False
assert (pointComplex in S.Naturals) is False
assert (pointComplex in S.Naturals0) is False
assert (pointComplex in S.Integers) is False
def test_ImageSet():
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1/x), S.Naturals)
assert Rational(1, 5) in harmonics
assert Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.3) not in harmonics
assert harmonics.is_iterable
def _intersect(self, other):
if other.is_Interval:
return Intersection(S.Integers, other,
Interval(self._inf, S.Infinity, False, True))
return
def bspline_basis(d, knots, n, x, close=True):
"""The `n`-th B-spline at `x` of degree `d` with knots.
B-Splines are piecewise polynomials of degree `d` [1]_. They are defined on
a set of knots, which is a sequence of integers or floats.
The 0th degree splines have a value of one on a single interval:
>>> from diofant import bspline_basis
>>> from diofant.abc import x
>>> d = 0
>>> knots = range(5)
>>> bspline_basis(d, knots, 0, x)
Piecewise((1, And(x <= 1, x >= 0)), (0, true))
For a given ``(d, knots)`` there are ``len(knots)-d-1`` B-splines defined, that
are indexed by ``n`` (starting at 0).
Here is an example of a cubic B-spline:
>>> bspline_basis(3, range(5), 0, x)
Piecewise((x**3/6, And(x < 1, x >= 0)),
(-x**3/2 + 2*x**2 - 2*x + 2/3, And(x < 2, x >= 1)),
(x**3/2 - 4*x**2 + 10*x - 22/3, And(x < 3, x >= 2)),
(-x**3/6 + 2*x**2 - 8*x + 32/3, And(x <= 4, x >= 3)),
(0, true))
By repeating knot points, you can introduce discontinuities in the
B-splines and their derivatives:
>>> d = 1
>>> knots = [0,0,2,3,4]
>>> bspline_basis(d, knots, 0, x)
Piecewise((-x/2 + 1, And(x <= 2, x >= 0)), (0, true))
It is quite time consuming to construct and evaluate B-splines. If you
need to evaluate a B-splines many times, it is best to lambdify them
first:
>>> from diofant import lambdify
>>> d = 3
>>> knots = range(10)
>>> b0 = bspline_basis(d, knots, 0, x)
>>> f = lambdify(x, b0)
>>> y = f(0.5)
See Also
========
diofant.functions.special.bsplines.bspline_basis_set
References
==========
.. [1] http://en.wikipedia.org/wiki/B-spline
"""
knots = [sympify(k) for k in knots]
d = int(d)
n = int(n)
n_knots = len(knots)
n_intervals = n_knots - 1
if n + d + 1 > n_intervals:
raise ValueError('n + d + 1 must not exceed len(knots) - 1')
if d == 0:
result = Piecewise(
(S.One, Interval(knots[n], knots[n + 1], False,
not close).contains(x)), (0, True))
elif d > 0:
denom = knots[n + d + 1] - knots[n + 1]
if denom != S.Zero:
B = (knots[n + d + 1] - x) / denom
b2 = bspline_basis(d - 1, knots, n + 1, x, close)
else:
b2 = B = S.Zero
denom = knots[n + d] - knots[n]
if denom != S.Zero:
A = (x - knots[n]) / denom
b1 = bspline_basis(d - 1, knots, n, x, close
and (B == S.Zero or b2 == S.Zero))
else:
b1 = A = S.Zero
result = _add_splines(A, b1, B, b2)
else:
raise ValueError('degree must be non-negative: %r' % n)
return result
def test_core_interval():
for c in (Interval, Interval(0, 2)):
check(c)
def test_multivariate_relational_as_set():
assert (x*y >= 0).as_set() == Interval(0, oo)*Interval(0, oo) + \
Interval(-oo, 0)*Interval(-oo, 0)