def test_MarginalDistribution():
a1, p1, p2 = symbols('a1 p1 p2', positive=True)
C = Multinomial('C', 2, p1, p2)
B = MultivariateBeta('B', a1, C[0])
MGR = MarginalDistribution(B, (C[0], ))
mgrc = Mul(
Symbol('B'),
Piecewise(
ExprCondPair(
Mul(
Integer(2),
Pow(Symbol('p1', positive=True),
Indexed(IndexedBase(Symbol('C')), Integer(0))),
Pow(Symbol('p2', positive=True),
Indexed(IndexedBase(Symbol('C')), Integer(1))),
Pow(
factorial(Indexed(IndexedBase(Symbol('C')),
Integer(0))), Integer(-1)),
Pow(
factorial(Indexed(IndexedBase(Symbol('C')),
Integer(1))), Integer(-1))),
Eq(
Add(Indexed(IndexedBase(Symbol('C')), Integer(0)),
Indexed(IndexedBase(Symbol('C')), Integer(1))),
Integer(2))), ExprCondPair(Integer(0), True)),
Pow(gamma(Symbol('a1', positive=True)), Integer(-1)),
gamma(
Add(Symbol('a1', positive=True),
Indexed(IndexedBase(Symbol('C')), Integer(0)))),
Pow(gamma(Indexed(IndexedBase(Symbol('C')), Integer(0))), Integer(-1)),
Pow(Indexed(IndexedBase(Symbol('B')), Integer(0)),
Add(Symbol('a1', positive=True), Integer(-1))),
Pow(Indexed(IndexedBase(Symbol('B')), Integer(1)),
Add(Indexed(IndexedBase(Symbol('C')), Integer(0)), Integer(-1))))
assert MGR(C) == mgrc
def test_postorder_traversal():
expr = z + w * (x + y)
expected = [z, w, x, y, x + y, w * (x + y), w * (x + y) + z]
assert list(postorder_traversal(expr, keys=default_sort_key)) == expected
assert list(postorder_traversal(expr, keys=True)) == expected
expr = Piecewise((x, x < 1), (x**2, True))
expected = [
x, 1, x, x < 1,
ExprCondPair(x, x < 1), 2, x, x**2, true,
ExprCondPair(x**2, True),
Piecewise((x, x < 1), (x**2, True))
]
assert list(postorder_traversal(expr, keys=default_sort_key)) == expected
assert list(postorder_traversal(
[expr], keys=default_sort_key)) == expected + [[expr]]
assert list(
postorder_traversal(Integral(x**2, (x, 0, 1)),
keys=default_sort_key)) == [
2, x, x**2, 0, 1, x,
Tuple(x, 0, 1),
Integral(x**2, Tuple(x, 0, 1))
]
assert list(postorder_traversal(
('abc',
('d',
'ef')))) == ['abc', 'd', 'ef', ('d', 'ef'), ('abc', ('d', 'ef'))]
def test_preorder_traversal():
expr = z + w * (x + y)
expected1 = [z + w * (x + y), z, w * (x + y), w, x + y, y, x]
expected2 = [z + w * (x + y), z, w * (x + y), w, x + y, x, y]
expected3 = [z + w * (x + y), w * (x + y), w, x + y, y, x, z]
assert list(preorder_traversal(expr)) in [expected1, expected2, expected3]
expr = Piecewise((x, x < 1), (x**2, True))
assert list(preorder_traversal(expr)) == [
Piecewise((x, x < 1), (x**2, True)),
ExprCondPair(x, x < 1), x, x < 1, x, 1,
ExprCondPair(x**2, True), x**2, x, 2, True
]
assert list(postorder_traversal(Integral(x**2, (x, 0, 1)))) == [
x, 2, x**2, x, 0, 1,
Tuple(x, 0, 1),
Integral(x**2, Tuple(x, 0, 1))
]
assert list(postorder_traversal(
('abc',
('d',
'ef')))) == ['abc', 'd', 'ef', ('d', 'ef'), ('abc', ('d', 'ef'))]
expr = (x**(y**z))**(x**(y**z))
expected = [(x**(y**z))**(x**(y**z)), x**(y**z), x**(y**z)]
result = []
pt = preorder_traversal(expr)
for i in pt:
result.append(i)
if i == x**(y**z):
pt.skip()
assert result == expected
def _non_eval_xreplace(cls, expr, rule):
"""
Duplicate of sympy's xreplace but with non-evaluate statement included
"""
if expr in rule:
return rule[expr]
elif rule:
args = []
altered = False
for a in expr.args:
try:
new_a = cls._non_eval_xreplace(a, rule)
except AttributeError:
new_a = a
if new_a != a:
altered = True
args.append(new_a)
args = tuple(args)
if altered:
if isinstance(expr, ExprCondPair):
return ExprCondPair(
cls._non_eval_xreplace(expr.args[0], rule),
cls._non_eval_xreplace(expr.args[1], rule))
else:
return expr.func(*args, evaluate=False)
return expr
def test_postorder_traversal():
expr = z+w*(x+y)
expected1 = [z, w, y, x, x + y, w*(x + y), z + w*(x + y)]
expected2 = [z, w, x, y, x + y, w*(x + y), z + w*(x + y)]
expected3 = [w, y, x, x + y, w*(x + y), z, z + w*(x + y)]
assert list(postorder_traversal(expr)) in [expected1, expected2, expected3]
expr = Piecewise((x,x<1),(x**2,True))
assert list(postorder_traversal(expr)) == [
x, x, 1, x < 1, ExprCondPair(x, x < 1), x, 2, x**2, True,
ExprCondPair(x**2, True), Piecewise((x, x < 1), (x**2, True))
]
assert list(preorder_traversal(Integral(x**2, (x, 0, 1)))) == [
Integral(x**2, (x, 0, 1)), x**2, x, 2, Tuple(x, 0, 1), x, 0, 1
]
assert list(preorder_traversal(('abc', ('d', 'ef')))) == [
('abc', ('d', 'ef')), 'abc', ('d', 'ef'), 'd', 'ef']
def test_preorder_traversal():
expr = z + w * (x + y)
expected1 = [z + w * (x + y), z, w * (x + y), w, x + y, y, x]
expected2 = [z + w * (x + y), z, w * (x + y), w, x + y, x, y]
expected3 = [z + w * (x + y), w * (x + y), w, x + y, y, x, z]
assert list(preorder_traversal(expr)) in [expected1, expected2, expected3]
expr = Piecewise((x, x < 1), (x**2, True))
assert list(preorder_traversal(expr)) == [
Piecewise((x, x < 1), (x**2, True)),
ExprCondPair(x, x < 1), x, x < 1, x, 1,
ExprCondPair(x**2, True), x**2, x, 2, True
]
assert list(postorder_traversal(Integral(x**2, (x, 0, 1)))) == [
x, 2, x**2, x, 0, 1, (0, 1), (x, (0, 1)), ((x, (0, 1)), ),
Integral(x**2, (x, 0, 1))
]
assert list(postorder_traversal(
('abc',
('d',
'ef')))) == ['abc', 'd', 'ef', ('d', 'ef'), ('abc', ('d', 'ef'))]
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 sympy import bspline_basis
>>> from sympy.abc import x
>>> d = 0
>>> knots = range(5)
>>> bspline_basis(d, knots, 0, x)
Piecewise((1, [0, 1]), (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, [0, 1)), (2/3 - 2*x + 2*x**2 - x**3/2, [1, 2)), (-22/3 + 10*x - 4*x**2 + x**3/2, [2, 3)), (32/3 - 8*x + 2*x**2 - x**3/6, [3, 4]), (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((1 - x/2, [0, 2]), (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 sympy import lambdify
>>> d = 3
>>> knots = range(10)
>>> b0 = bspline_basis(d, knots, 0, x)
>>> f = lambdify(x, b0)
>>> y = f(0.5)
[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, True)), (0, True))
elif d > 0:
denom = knots[n + d] - knots[n]
if denom != S.Zero:
A = (x - knots[n]) / denom
b1 = bspline_basis(d - 1, knots, n, x, close=False)
else:
b1 = A = S.Zero
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=False)
else:
b2 = B = S.Zero
result = _add_splines(A, b1, B, b2)
else:
raise ValueError('degree must be non-negative: %r' % n)
if close:
final_ec_pair = result.args[-2]
final_cond = final_ec_pair.cond
final_expr = final_ec_pair.expr
new_args = final_cond.args[:3] + (False, )
new_ec_pair = ExprCondPair(final_expr, Interval(*new_args))
new_args = result.args[:-2] + (new_ec_pair, result.args[-1])
result = Piecewise(*new_args)
return result
def test_sympy__functions__elementary__piecewise__ExprCondPair():
from sympy.functions.elementary.piecewise import ExprCondPair
assert _test_args(ExprCondPair(1, True))
def test_piecewise1():
# Test canonicalization
assert unchanged(Piecewise, ExprCondPair(x, x < 1), ExprCondPair(0, True))
assert Piecewise((x, x < 1),
(0, True)) == Piecewise(ExprCondPair(x, x < 1),
ExprCondPair(0, True))
assert Piecewise((x, x < 1), (0, True), (1, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, False), (-1, 1 > 2)) == \
Piecewise((x, x < 1))
assert Piecewise((x, x < 1), (0, x < 1), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, x < 2), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (0, True)) == \
Piecewise((x, Or(x < 1, x < 2)), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (x, True)) == x
assert Piecewise((x, True)) == x
# Explicitly constructed empty Piecewise not accepted
raises(TypeError, lambda: Piecewise())
# False condition is never retained
assert Piecewise((2*x, x < 0), (x, False)) == \
Piecewise((2*x, x < 0), (x, False), evaluate=False) == \
Piecewise((2*x, x < 0))
assert Piecewise((x, False)) == Undefined
raises(TypeError, lambda: Piecewise(x))
assert Piecewise((x, 1)) == x # 1 and 0 are accepted as True/False
raises(TypeError, lambda: Piecewise((x, 2)))
raises(TypeError, lambda: Piecewise((x, x**2)))
raises(TypeError, lambda: Piecewise(([1], True)))
assert Piecewise(((1, 2), True)) == Tuple(1, 2)
cond = (Piecewise((1, x < 0), (2, True)) < y)
assert Piecewise((1, cond)) == Piecewise((1, ITE(x < 0, y > 1, y > 2)))
assert Piecewise((1, x > 0), (2, And(x <= 0, x > -1))) == Piecewise(
(1, x > 0), (2, x > -1))
# Test subs
p = Piecewise((-1, x < -1), (x**2, x < 0), (log(x), x >= 0))
p_x2 = Piecewise((-1, x**2 < -1), (x**4, x**2 < 0), (log(x**2), x**2 >= 0))
assert p.subs(x, x**2) == p_x2
assert p.subs(x, -5) == -1
assert p.subs(x, -1) == 1
assert p.subs(x, 1) == log(1)
# More subs tests
p2 = Piecewise((1, x < pi), (-1, x < 2 * pi), (0, x > 2 * pi))
p3 = Piecewise((1, Eq(x, 0)), (1 / x, True))
p4 = Piecewise((1, Eq(x, 0)), (2, 1 / x > 2))
assert p2.subs(x, 2) == 1
assert p2.subs(x, 4) == -1
assert p2.subs(x, 10) == 0
assert p3.subs(x, 0.0) == 1
assert p4.subs(x, 0.0) == 1
f, g, h = symbols('f,g,h', cls=Function)
pf = Piecewise((f(x), x < -1), (f(x) + h(x) + 2, x <= 1))
pg = Piecewise((g(x), x < -1), (g(x) + h(x) + 2, x <= 1))
assert pg.subs(g, f) == pf
assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 0) == 1
assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 1) == 0
assert Piecewise((1, Eq(x, y)), (0, True)).subs(x, y) == 1
assert Piecewise((1, Eq(x, z)), (0, True)).subs(x, z) == 1
assert Piecewise((1, Eq(exp(x), cos(z))), (0, True)).subs(x, z) == \
Piecewise((1, Eq(exp(z), cos(z))), (0, True))
p5 = Piecewise((0, Eq(cos(x) + y, 0)), (1, True))
assert p5.subs(y, 0) == Piecewise((0, Eq(cos(x), 0)), (1, True))
assert Piecewise((-1, y < 1), (0, x < 0), (1, Eq(x, 0)),
(2, True)).subs(x, 1) == Piecewise((-1, y < 1), (2, True))
assert Piecewise((1, Eq(x**2, -1)), (2, x < 0)).subs(x, I) == 1
p6 = Piecewise((x, x > 0))
n = symbols('n', negative=True)
assert p6.subs(x, n) == Undefined
# Test evalf
assert p.evalf() == p
assert p.evalf(subs={x: -2}) == -1
assert p.evalf(subs={x: -1}) == 1
assert p.evalf(subs={x: 1}) == log(1)
assert p6.evalf(subs={x: -5}) == Undefined
# Test doit
f_int = Piecewise((Integral(x, (x, 0, 1)), x < 1))
assert f_int.doit() == Piecewise((S(1) / 2, x < 1))
# Test differentiation
f = x
fp = x * p
dp = Piecewise((0, x < -1), (2 * x, x < 0), (1 / x, x >= 0))
fp_dx = x * dp + p
assert diff(p, x) == dp
assert diff(f * p, x) == fp_dx
# Test simple arithmetic
assert x * p == fp
assert x * p + p == p + x * p
assert p + f == f + p
assert p + dp == dp + p
assert p - dp == -(dp - p)
# Test power
dp2 = Piecewise((0, x < -1), (4 * x**2, x < 0), (1 / x**2, x >= 0))
assert dp**2 == dp2
# Test _eval_interval
f1 = x * y + 2
f2 = x * y**2 + 3
peval = Piecewise((f1, x < 0), (f2, x > 0))
peval_interval = f1.subs(x, 0) - f1.subs(x, -1) + f2.subs(x, 1) - f2.subs(
x, 0)
assert peval._eval_interval(x, 0, 0) == 0
assert peval._eval_interval(x, -1, 1) == peval_interval
peval2 = Piecewise((f1, x < 0), (f2, True))
assert peval2._eval_interval(x, 0, 0) == 0
assert peval2._eval_interval(x, 1, -1) == -peval_interval
assert peval2._eval_interval(x, -1, -2) == f1.subs(x, -2) - f1.subs(x, -1)
assert peval2._eval_interval(x, -1, 1) == peval_interval
assert peval2._eval_interval(x, None, 0) == peval2.subs(x, 0)
assert peval2._eval_interval(x, -1, None) == -peval2.subs(x, -1)
# Test integration
assert p.integrate() == Piecewise((-x, x < -1),
(x**3 / 3 + S(4) / 3, x < 0),
(x * log(x) - x + S(4) / 3, True))
p = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
assert integrate(p, (x, -2, 2)) == S(5) / 6
assert integrate(p, (x, 2, -2)) == -S(5) / 6
p = Piecewise((0, x < 0), (1, x < 1), (0, x < 2), (1, x < 3), (0, True))
assert integrate(p, (x, -oo, oo)) == 2
p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
assert integrate(p, (x, -2, 2)) == Undefined
# Test commutativity
assert isinstance(p, Piecewise) and p.is_commutative is True