def test_expand_relational():
n = symbols('n', negative=True)
p, q = symbols('p q', positive=True)
r = ((n + q*(-n/q + 1))/(q*(-n/q + 1)) < 0)
assert r is not S.false
assert r.expand() is S.false
assert (q > 0).expand() is S.true
def test_relational_simplification_numerically():
def test_simplification_numerically_function(original, simplified):
symb = original.free_symbols
n = len(symb)
valuelist = list(set(list(combinations(list(range(-(n-1), n))*n, n))))
for values in valuelist:
sublist = dict(zip(symb, values))
originalvalue = original.subs(sublist)
simplifiedvalue = simplified.subs(sublist)
assert originalvalue == simplifiedvalue, "Original: {}\nand"\
" simplified: {}\ndo not evaluate to the same value for {}"\
"".format(original, simplified, sublist)
w, x, y, z = symbols('w x y z', real=True)
d, e = symbols('d e', real=False)
expressions = (And(Eq(x, y), x >= y, w < y, y >= z, z < y),
And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y),
Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y),
And(x >= y, Eq(y, x)),
Or(And(Eq(x, y), x >= y, w < y, Or(y >= z, z < y)),
And(Eq(x, y), x >= 1, 2 < y, y >= -1, z < y)),
(Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)),
)
for expression in expressions:
test_simplification_numerically_function(expression,
expression.simplify())
def test_rewrite():
x, y, z = symbols('x y z')
a, b = symbols('a b')
f1 = sin(x) + cos(x)
assert f1.rewrite(cos,exp) == exp(I*x)/2 + sin(x) + exp(-I*x)/2
assert f1.rewrite([cos],sin) == sin(x) + sin(x + pi/2, evaluate=False)
f2 = sin(x) + cos(y)/gamma(z)
assert f2.rewrite(sin,exp) == -I*(exp(I*x) - exp(-I*x))/2 + cos(y)/gamma(z)
def test_as_Boolean():
nz = symbols('nz', nonzero=True)
assert all(as_Boolean(i) is S.true for i in (True, S.true, 1, nz))
z = symbols('z', zero=True)
assert all(as_Boolean(i) is S.false for i in (False, S.false, 0, z))
assert all(as_Boolean(i) == i for i in (x, x < 0))
for i in (2, S(2), x + 1, []):
raises(TypeError, lambda: as_Boolean(i))
def test_relational_simplification():
w, x, y, z = symbols('w x y z', real=True)
d, e = symbols('d e', real=False)
# Test all combinations or sign and order
assert Or(x >= y, x < y).simplify() == S.true
assert Or(x >= y, y > x).simplify() == S.true
assert Or(x >= y, -x > -y).simplify() == S.true
assert Or(x >= y, -y < -x).simplify() == S.true
assert Or(-x <= -y, x < y).simplify() == S.true
assert Or(-x <= -y, -x > -y).simplify() == S.true
assert Or(-x <= -y, y > x).simplify() == S.true
assert Or(-x <= -y, -y < -x).simplify() == S.true
assert Or(y <= x, x < y).simplify() == S.true
assert Or(y <= x, y > x).simplify() == S.true
assert Or(y <= x, -x > -y).simplify() == S.true
assert Or(y <= x, -y < -x).simplify() == S.true
assert Or(-y >= -x, x < y).simplify() == S.true
assert Or(-y >= -x, y > x).simplify() == S.true
assert Or(-y >= -x, -x > -y).simplify() == S.true
assert Or(-y >= -x, -y < -x).simplify() == S.true
assert Or(x < y, x >= y).simplify() == S.true
assert Or(y > x, x >= y).simplify() == S.true
assert Or(-x > -y, x >= y).simplify() == S.true
assert Or(-y < -x, x >= y).simplify() == S.true
assert Or(x < y, -x <= -y).simplify() == S.true
assert Or(-x > -y, -x <= -y).simplify() == S.true
assert Or(y > x, -x <= -y).simplify() == S.true
assert Or(-y < -x, -x <= -y).simplify() == S.true
assert Or(x < y, y <= x).simplify() == S.true
assert Or(y > x, y <= x).simplify() == S.true
assert Or(-x > -y, y <= x).simplify() == S.true
assert Or(-y < -x, y <= x).simplify() == S.true
assert Or(x < y, -y >= -x).simplify() == S.true
assert Or(y > x, -y >= -x).simplify() == S.true
assert Or(-x > -y, -y >= -x).simplify() == S.true
assert Or(-y < -x, -y >= -x).simplify() == S.true
# Some other tests
assert Or(x >= y, w < z, x <= y).simplify() == S.true
assert And(x >= y, x < y).simplify() == S.false
assert Or(x >= y, Eq(y, x)).simplify() == (x >= y)
assert And(x >= y, Eq(y, x)).simplify() == Eq(x, y)
assert Or(Eq(x, y), x >= y, w < y, z < y).simplify() == \
Or(x >= y, y > Min(w, z))
assert And(Eq(x, y), x >= y, w < y, y >= z, z < y).simplify() == \
And(Eq(x, y), y > Max(w, z))
assert Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
(Eq(x, y) | (x >= 1) | (y > Min(2, z)))
assert And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
(Eq(x, y) & (x >= 1) & (y >= 5) & (y > z))
assert (Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)).simplify() == \
(Eq(x, y) & Eq(d, e) & (d >= e))
assert And(Eq(x, y), Eq(x, -y)).simplify() == And(Eq(x, 0), Eq(y, 0))
assert Xor(x >= y, x <= y).simplify() == Ne(x, y)
def execute(self, grp):
x = symbols('x')
x_val = self.x.get_data(Int32).values[0]
fun = self.fun.get_data(Text).values[0]
expr = parse_expr(fun)
res = expr.subs(x, x_val)
self.out.put_data(Int32(int(res)))
def test_multivariate_bool_as_set():
x, y = symbols("x,y")
assert And(x >= 0, y >= 0).as_set() == Interval(0, oo) * Interval(0, oo)
assert Or(x >= 0, y >= 0).as_set() == S.Reals * S.Reals - Interval(-oo, 0, True, True) * Interval(
-oo, 0, True, True
)
def test_is_cnf():
x, y, z = symbols("x,y,z")
assert is_cnf(x) is True
assert is_cnf(x | y | z) is True
assert is_cnf(x & y & z) is True
assert is_cnf((x | y) & z) is True
assert is_cnf((x & y) | z) is False
def test_is_dnf():
x, y, z = symbols('x,y,z')
assert is_dnf(x) is True
assert is_dnf(x | y | z) is True
assert is_dnf(x & y & z) is True
assert is_dnf((x & y) | z) is True
assert is_dnf((x | y) & z) is False
def test_issue_14700():
A, B, C, D, E, F, G, H = symbols('A B C D E F G H')
q = ((B & D & H & ~F) | (B & H & ~C & ~D) | (B & H & ~C & ~F) |
(B & H & ~D & ~G) | (B & H & ~F & ~G) | (C & G & ~B & ~D) |
(C & G & ~D & ~H) | (C & G & ~F & ~H) | (D & F & H & ~B) |
(D & F & ~G & ~H) | (B & D & F & ~C & ~H) | (D & E & F & ~B & ~C) |
(D & F & ~A & ~B & ~C) | (D & F & ~A & ~C & ~H) |
(A & B & D & F & ~E & ~H))
soldnf = ((B & D & H & ~F) | (D & F & H & ~B) | (B & H & ~C & ~D) |
(B & H & ~D & ~G) | (C & G & ~B & ~D) | (C & G & ~D & ~H) |
(C & G & ~F & ~H) | (D & F & ~G & ~H) | (D & E & F & ~C & ~H) |
(D & F & ~A & ~C & ~H) | (A & B & D & F & ~E & ~H))
solcnf = ((B | C | D) & (B | D | G) & (C | D | H) & (C | F | H) &
(D | G | H) & (F | G | H) & (B | F | ~D | ~H) &
(~B | ~D | ~F | ~H) & (D | ~B | ~C | ~G | ~H) &
(A | H | ~C | ~D | ~F | ~G) & (H | ~C | ~D | ~E | ~F | ~G) &
(B | E | H | ~A | ~D | ~F | ~G))
assert simplify_logic(q, "dnf") == soldnf
assert simplify_logic(q, "cnf") == solcnf
minterms = [[0, 1, 0, 0], [0, 1, 0, 1], [0, 1, 1, 0], [0, 1, 1, 1],
[0, 0, 1, 1], [1, 0, 1, 1]]
dontcares = [[1, 0, 0, 0], [1, 0, 0, 1], [1, 1, 0, 0], [1, 1, 0, 1]]
assert SOPform([w, x, y, z], minterms) == (x & ~w) | (y & z & ~x)
# Should not be more complicated with don't cares
assert SOPform([w, x, y, z], minterms, dontcares) == \
(x & ~w) | (y & z & ~x)
def test_tan():
R, x, y = ring('x, y', QQ)
assert rs_tan(x, x, 9)/x**5 == \
S(17)/315*x**2 + S(2)/15 + S(1)/3*x**(-2) + x**(-4)
assert rs_tan(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 + 17*x**8*y**9/45 + \
4*x**7*y**9/3 + 17*x**7*y**7/315 + x**6*y**9/3 + 2*x**6*y**7/3 + \
x**5*y**7 + 2*x**5*y**5/15 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[tan(a), a])
assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**S(3)/3 +
2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + S(1)/3)*x**3 + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
(tan(a)**4 + S(4)/3*tan(a)**2 + S(1)/3)*x**3 + (tan(a)**2 + 1)*x**2*y + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
R, x, y = ring('x, y', EX)
assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 +
2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 +
2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
EX(tan(a)**2 + 1)*x + EX(tan(a))
p = x + x**2 + 5
assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + S(67701870330562640) / \
668083460499)
def test_cos():
R, x, y = ring('x, y', QQ)
assert rs_cos(x, x, 9)/x**5 == \
S(1)/40320*x**3 - S(1)/720*x + S(1)/24*x**(-1) - S(1)/2*x**(-3) + x**(-5)
assert rs_cos(x*y + x**2*y**3, x, 9) == x**8*y**12/24 - \
x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 - \
x**7*y**8/120 + x**6*y**8/4 - x**6*y**6/720 + x**5*y**6/6 - \
x**4*y**6/2 + x**4*y**4/24 - x**3*y**4 - x**2*y**2/2 + 1
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
assert rs_cos(x + a, x, 5) == cos(a)*x**4/24 + sin(a)*x**3/6 - \
cos(a)*x**2/2 - sin(a)*x + cos(a)
assert rs_cos(x + x**2*y + a, x, 5) == -cos(a)*x**4*y**2/2 + \
sin(a)*x**4*y/2 + cos(a)*x**4/24 - cos(a)*x**3*y + sin(a)*x**3/6 - \
sin(a)*x**2*y - cos(a)*x**2/2 - sin(a)*x + cos(a)
R, x, y = ring('x, y', EX)
assert rs_cos(x + a, x, 5) == EX(cos(a)/24)*x**4 + EX(sin(a)/6)*x**3 - \
EX(cos(a)/2)*x**2 - EX(sin(a))*x + EX(cos(a))
assert rs_cos(x + x**2*y + a, x, 5) == -EX(cos(a)/2)*x**4*y**2 + \
EX(sin(a)/2)*x**4*y + EX(cos(a)/24)*x**4 - EX(cos(a))*x**3*y + \
EX(sin(a)/6)*x**3 - EX(sin(a))*x**2*y - EX(cos(a)/2)*x**2 - \
EX(sin(a))*x + EX(cos(a))
def test_sin():
R, x, y = ring('x, y', QQ)
assert rs_sin(x, x, 9)/x**5 == \
-S(1)/5040*x**2 + S(1)/120 - S(1)/6*x**(-2) + x**(-4)
assert rs_sin(x*y + x**2*y**3, x, 9) == x**8*y**11/12 - \
x**8*y**9/720 + x**7*y**9/12 - x**7*y**7/5040 - x**6*y**9/6 + \
x**6*y**7/24 - x**5*y**7/2 + x**5*y**5/120 - x**4*y**5/2 - \
x**3*y**3/6 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
assert rs_sin(x + a, x, 5) == sin(a)*x**4/24 - cos(a)*x**3/6 - \
sin(a)*x**2/2 + cos(a)*x + sin(a)
assert rs_sin(x + x**2*y + a, x, 5) == -sin(a)*x**4*y**2/2 - \
cos(a)*x**4*y/2 + sin(a)*x**4/24 - sin(a)*x**3*y - cos(a)*x**3/6 + \
cos(a)*x**2*y - sin(a)*x**2/2 + cos(a)*x + sin(a)
R, x, y = ring('x, y', EX)
assert rs_sin(x + a, x, 5) == EX(sin(a)/24)*x**4 - EX(cos(a)/6)*x**3 - \
EX(sin(a)/2)*x**2 + EX(cos(a))*x + EX(sin(a))
assert rs_sin(x + x**2*y + a, x, 5) == -EX(sin(a)/2)*x**4*y**2 - \
EX(cos(a)/2)*x**4*y + EX(sin(a)/24)*x**4 - EX(sin(a))*x**3*y - \
EX(cos(a)/6)*x**3 + EX(cos(a))*x**2*y - EX(sin(a)/2)*x**2 + \
EX(cos(a))*x + EX(sin(a))
def test_log():
R, x = ring('x', QQ)
p = 1 + x
p1 = rs_log(p, x, 4)/x**2
assert p1 == S(1)/3*x - S(1)/2 + x**(-1)
p = 1 + x +2*x**2/3
p1 = rs_log(p, x, 9)
assert p1 == -17*x**8/648 + 13*x**7/189 - 11*x**6/162 - x**5/45 + \
7*x**4/36 - x**3/3 + x**2/6 + x
p2 = rs_series_inversion(p, x, 9)
p3 = rs_log(p2, x, 9)
assert p3 == -p1
R, x, y = ring('x, y', QQ)
p = 1 + x + 2*y*x**2
p1 = rs_log(p, x, 6)
assert p1 == (4*x**5*y**2 - 2*x**5*y - 2*x**4*y**2 + x**5/5 + 2*x**4*y -
x**4/4 - 2*x**3*y + x**3/3 + 2*x**2*y - x**2/2 + x)
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', EX)
assert rs_log(x + a, x, 5) == -EX(1/(4*a**4))*x**4 + EX(1/(3*a**3))*x**3 \
- EX(1/(2*a**2))*x**2 + EX(1/a)*x + EX(log(a))
assert rs_log(x + x**2*y + a, x, 4) == -EX(a**(-2))*x**3*y + \
EX(1/(3*a**3))*x**3 + EX(1/a)*x**2*y - EX(1/(2*a**2))*x**2 + \
EX(1/a)*x + EX(log(a))
p = x + x**2 + 3
assert rs_log(p, x, 10).compose(x, 5) == EX(log(3) + S(19281291595)/9920232)
def test_exp():
R, x = ring('x', QQ)
p = x + x**4
for h in [10, 30]:
q = rs_series_inversion(1 + p, x, h) - 1
p1 = rs_exp(q, x, h)
q1 = rs_log(p1, x, h)
assert q1 == q
p1 = rs_exp(p, x, 30)
assert p1.coeff(x**29) == QQ(74274246775059676726972369, 353670479749588078181744640000)
prec = 21
p = rs_log(1 + x, x, prec)
p1 = rs_exp(p, x, prec)
assert p1 == x + 1
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[exp(a), a])
assert rs_exp(x + a, x, 5) == exp(a)*x**4/24 + exp(a)*x**3/6 + \
exp(a)*x**2/2 + exp(a)*x + exp(a)
assert rs_exp(x + x**2*y + a, x, 5) == exp(a)*x**4*y**2/2 + \
exp(a)*x**4*y/2 + exp(a)*x**4/24 + exp(a)*x**3*y + \
exp(a)*x**3/6 + exp(a)*x**2*y + exp(a)*x**2/2 + exp(a)*x + exp(a)
R, x, y = ring('x, y', EX)
assert rs_exp(x + a, x, 5) == EX(exp(a)/24)*x**4 + EX(exp(a)/6)*x**3 + \
EX(exp(a)/2)*x**2 + EX(exp(a))*x + EX(exp(a))
assert rs_exp(x + x**2*y + a, x, 5) == EX(exp(a)/2)*x**4*y**2 + \
EX(exp(a)/2)*x**4*y + EX(exp(a)/24)*x**4 + EX(exp(a))*x**3*y + \
EX(exp(a)/6)*x**3 + EX(exp(a))*x**2*y + EX(exp(a)/2)*x**2 + \
EX(exp(a))*x + EX(exp(a))
def test_sin():
R, x, y = ring('x, y', QQ)
assert rs_sin(x, x, 9) == \
x - x**3/6 + x**5/120 - x**7/5040
assert rs_sin(x*y + x**2*y**3, x, 9) == 1/12*x**8*y**11 - \
1/720*x**8*y**9 + 1/12*x**7*y**9 - 1/5040*x**7*y**7 - 1/6*x**6*y**9 \
+ 1/24*x**6*y**7 - 1/2*x**5*y**7 + 1/120*x**5*y**5 - 1/2*x**4*y**5 \
- 1/6*x**3*y**3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
assert rs_sin(x + a, x, 5) == sin(a)*x**4/24 - cos(a)*x**3/6 - \
sin(a)*x**2/2 + cos(a)*x + sin(a)
assert rs_sin(x + x**2*y + a, x, 5) == -sin(a)*x**4*y**2/2 - \
cos(a)*x**4*y/2 + sin(a)*x**4/24 - sin(a)*x**3*y - cos(a)*x**3/6 + \
cos(a)*x**2*y - sin(a)*x**2/2 + cos(a)*x + sin(a)
R, x, y = ring('x, y', EX)
assert rs_sin(x + a, x, 5) == EX(sin(a)/24)*x**4 - EX(cos(a)/6)*x**3 - \
EX(sin(a)/2)*x**2 + EX(cos(a))*x + EX(sin(a))
assert rs_sin(x + x**2*y + a, x, 5) == -EX(sin(a)/2)*x**4*y**2 - \
EX(cos(a)/2)*x**4*y + EX(sin(a)/24)*x**4 - EX(sin(a))*x**3*y - \
EX(cos(a)/6)*x**3 + EX(cos(a))*x**2*y - EX(sin(a)/2)*x**2 + \
EX(cos(a))*x + EX(sin(a))
def test_cos():
R, x, y = ring('x, y', QQ)
assert rs_cos(x, x, 9) == \
1/40320*x**8 - 1/720*x**6 + 1/24*x**4 - 1/2*x**2 + 1
assert rs_cos(x*y + x**2*y**3, x, 9) == 1/24*x**8*y**12 - \
1/48*x**8*y**10 + 1/40320*x**8*y**8 + 1/6*x**7*y**10 - \
1/120*x**7*y**8 + 1/4*x**6*y**8 - 1/720*x**6*y**6 + 1/6*x**5*y**6 \
- 1/2*x**4*y**6 + 1/24*x**4*y**4 - x**3*y**4 - 1/2*x**2*y**2 + 1
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
assert rs_cos(x + a, x, 5) == cos(a)*x**4/24 + sin(a)*x**3/6 - \
cos(a)*x**2/2 - sin(a)*x + cos(a)
assert rs_cos(x + x**2*y + a, x, 5) == -cos(a)*x**4*y**2/2 + \
sin(a)*x**4*y/2 + cos(a)*x**4/24 - cos(a)*x**3*y + sin(a)*x**3/6 - \
sin(a)*x**2*y - cos(a)*x**2/2 - sin(a)*x + cos(a)
R, x, y = ring('x, y', EX)
assert rs_cos(x + a, x, 5) == EX(cos(a)/24)*x**4 + EX(sin(a)/6)*x**3 - \
EX(cos(a)/2)*x**2 - EX(sin(a))*x + EX(cos(a))
assert rs_cos(x + x**2*y + a, x, 5) == -EX(cos(a)/2)*x**4*y**2 + \
EX(sin(a)/2)*x**4*y + EX(cos(a)/24)*x**4 - EX(cos(a))*x**3*y + \
EX(sin(a)/6)*x**3 - EX(sin(a))*x**2*y - EX(cos(a)/2)*x**2 - \
EX(sin(a))*x + EX(cos(a))
def test_literal_evalf_is_number_is_zero_is_comparable():
from sympy.integrals.integrals import Integral
from sympy.core.symbol import symbols
from sympy.core.function import Function
from sympy.functions.elementary.trigonometric import cos, sin
x = symbols('x')
f = Function('f')
# the following should not be changed without a lot of dicussion
# `foo.is_number` should be equivalent to `not foo.free_symbols`
# it should not attempt anything fancy; see is_zero, is_constant
# and equals for more rigorous tests.
assert f(1).is_number is True
i = Integral(0, (x, x, x))
# expressions that are symbolically 0 can be difficult to prove
# so in case there is some easy way to know if something is 0
# it should appear in the is_zero property for that object;
# if is_zero is true evalf should always be able to compute that
# zero
assert i.n() == 0
assert i.is_zero
assert i.is_number is False
assert i.evalf(2, strict=False) == 0
# issue 10268
n = sin(1)**2 + cos(1)**2 - 1
assert n.is_comparable is False
assert n.n(2).is_comparable is False
assert n.n(2).n(2).is_comparable
def test_tan():
R, x, y = ring('x, y', QQ)
assert rs_tan(x, x, 9) == \
x + x**3/3 + 2*x**5/15 + 17*x**7/315
assert rs_tan(x*y + x**2*y**3, x, 9) == 4/3*x**8*y**11 + 17/45*x**8*y**9 + \
4/3*x**7*y**9 + 17/315*x**7*y**7 + 1/3*x**6*y**9 + 2/3*x**6*y**7 + \
x**5*y**7 + 2/15*x**5*y**5 + x**4*y**5 + 1/3*x**3*y**3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[tan(a), a])
assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**3/3 + \
2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + 1/3)*x**3 + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
(tan(a)**4 + 4/3*tan(a)**2 + 1/3)*x**3 + (tan(a)**2 + 1)*x**2*y + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
R, x, y = ring('x, y', EX)
assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 + \
2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 + \
2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
EX(tan(a)**2 + 1)*x + EX(tan(a))
p = x + x**2 + 5
assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + 67701870330562640/ \
668083460499)
def test_literal_evalf_is_number_is_zero_is_comparable():
from sympy.integrals.integrals import Integral
from sympy.core.symbol import symbols
from sympy.core.function import Function
from sympy.functions.elementary.trigonometric import cos, sin
x = symbols('x')
f = Function('f')
# issue 5033
assert f.is_number is False
# issue 6646
assert f(1).is_number is False
i = Integral(0, (x, x, x))
# expressions that are symbolically 0 can be difficult to prove
# so in case there is some easy way to know if something is 0
# it should appear in the is_zero property for that object;
# if is_zero is true evalf should always be able to compute that
# zero
assert i.n() == 0
assert i.is_zero
assert i.is_number is False
assert i.evalf(2, strict=False) == 0
# issue 10268
n = sin(1)**2 + cos(1)**2 - 1
assert n.is_comparable is False
assert n.n(2).is_comparable is False
assert n.n(2).n(2).is_comparable
def test_preorder_traversal():
expr = Basic(b21, b3)
assert list(preorder_traversal(expr)) == [expr, b21, b2, b1, b1, b3, b2, b1]
assert list(preorder_traversal(("abc", ("d", "ef")))) == [("abc", ("d", "ef")), "abc", ("d", "ef"), "d", "ef"]
result = []
pt = preorder_traversal(expr)
for i in pt:
result.append(i)
if i == b2:
pt.skip()
assert result == [expr, b21, b2, b1, b3, b2]
w, x, y, z = symbols("w:z")
expr = z + w * (x + y)
assert list(preorder_traversal([expr], key=default_sort_key)) == [
[w * (x + y) + z],
w * (x + y) + z,
z,
w * (x + y),
w,
x + y,
x,
y,
]
def test_rewrite():
x, y, z = symbols('x y z')
a, b = symbols('a b')
f1 = sin(x) + cos(x)
assert f1.rewrite(cos,exp) == exp(I*x)/2 + sin(x) + exp(-I*x)/2
assert f1.rewrite([cos],sin) == sin(x) + sin(x + pi/2, evaluate=False)
f2 = sin(x) + cos(y)/gamma(z)
assert f2.rewrite(sin,exp) == -I*(exp(I*x) - exp(-I*x))/2 + cos(y)/gamma(z)
assert Max(a, b).rewrite(Piecewise) == Piecewise((a, a >= b), (b, True))
assert Max(x, y, z).rewrite(Piecewise) == Piecewise((x, (x >= y) & (x >= z)), (y, y >= z), (z, True))
assert Max(x, y, a, b).rewrite(Piecewise) == Piecewise((a, (a >= b) & (a >= x) & (a >= y)),
(b, (b >= x) & (b >= y)), (x, x >= y), (y, True))
assert Min(a, b).rewrite(Piecewise) == Piecewise((a, a <= b), (b, True))
assert Min(x, y, z).rewrite(Piecewise) == Piecewise((x, (x <= y) & (x <= z)), (y, y <= z), (z, True))
assert Min(x, y, a, b).rewrite(Piecewise) == Piecewise((a, (a <= b) & (a <= x) & (a <= y)),
(b, (b <= x) & (b <= y)), (x, x <= y), (y, True))
def test_replace_map():
F, G = symbols('F, G', cls=Function)
K = OperationsOnlyMatrix(2, 2, [(G(0), {F(0): G(0)}), (G(1), {F(1): G(1)}), (G(1), {F(1) \
: G(1)}), (G(2), {F(2): G(2)})])
M = OperationsOnlyMatrix(2, 2, lambda i, j: F(i+j))
N = M.replace(F, G, True)
assert N == K
def as_real_imag(self, deep=True, **hints):
from sympy.core.symbol import symbols
from sympy.polys.polytools import poly
from sympy.core.function import expand_multinomial
if self.exp.is_Integer:
exp = self.exp
re, im = self.base.as_real_imag(deep=deep)
if re.func == C.re or im.func == C.im:
return self, S.Zero
a, b = symbols('a b', cls=Dummy)
if exp >= 0:
if re.is_Number and im.is_Number:
# We can be more efficient in this case
expr = expand_multinomial(self.base**exp)
return expr.as_real_imag()
expr = poly((a + b)**exp) # a = re, b = im; expr = (a + b*I)**exp
else:
mag = re**2 + im**2
re, im = re/mag, -im/mag
if re.is_Number and im.is_Number:
# We can be more efficient in this case
expr = expand_multinomial((re + im*S.ImaginaryUnit)**-exp)
return expr.as_real_imag()
expr = poly((a + b)**-exp)
# Terms with even b powers will be real
r = [i for i in expr.terms() if not i[0][1] % 2]
re_part = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
# Terms with odd b powers will be imaginary
r = [i for i in expr.terms() if i[0][1] % 4 == 1]
im_part1 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
r = [i for i in expr.terms() if i[0][1] % 4 == 3]
im_part3 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
return (re_part.subs({a: re, b: S.ImaginaryUnit*im}),
im_part1.subs({a: re, b: im}) + im_part3.subs({a: re, b: -im}))
elif self.exp.is_Rational:
# NOTE: This is not totally correct since for x**(p/q) with
# x being imaginary there are actually q roots, but
# only a single one is returned from here.
re, im = self.base.as_real_imag(deep=deep)
if re.func == C.re or im.func == C.im:
return self, S.Zero
r = Pow(Pow(re, 2) + Pow(im, 2), S.Half)
t = C.atan2(im, re)
rp, tp = Pow(r, self.exp), t*self.exp
return (rp*C.cos(tp), rp*C.sin(tp))
else:
if deep:
hints['complex'] = False
return (C.re(self.expand(deep, **hints)),
C.im(self.expand(deep, **hints)))
else:
return (C.re(self), C.im(self))
def test_rs_series():
x, a, b, c = symbols('x, a, b, c')
assert rs_series(a, a, 5).as_expr() == a
assert rs_series(sin(1/a), a, 5).as_expr() == sin(1/a)
assert rs_series(sin(a), a, 5).as_expr() == (sin(a).series(a, 0,
5)).removeO()
assert rs_series(sin(a) + cos(a), a, 5).as_expr() == ((sin(a) +
cos(a)).series(a, 0, 5)).removeO()
assert rs_series(sin(a)*cos(a), a, 5).as_expr() == ((sin(a)*
cos(a)).series(a, 0, 5)).removeO()
p = (sin(a) - a)*(cos(a**2) + a**4/2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
10).removeO())
p = sin(a**2/2 + a/3) + cos(a/5)*sin(a/2)**3
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = sin(x**2 + a)*(cos(x**3 - 1) - a - a**2)
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = sin(a**2 - a/3 + 2)**5*exp(a**3 - a/2)
assert expand(rs_series(p, a, 10).as_expr()) == expand(p.series(a, 0,
10).removeO())
p = sin(a + b + c)
assert expand(rs_series(p, a, 5).as_expr()) == expand(p.series(a, 0,
5).removeO())
p = tan(sin(a**2 + 4) + b + c)
assert expand(rs_series(p, a, 6).as_expr()) == expand(p.series(a, 0,
6).removeO())
def test_atomic():
g, h = map(Function, 'gh')
x = symbols('x')
assert _atomic(g(x + h(x))) == {g(x + h(x))}
assert _atomic(g(x + h(x)), recursive=True) == {h(x), x, g(x + h(x))}
assert _atomic(1) == set()
assert _atomic(Basic(1,2)) == {Basic(1, 2)}
def test_replace_exceptions():
from sympy import Wild
x, y = symbols('x y')
e = (x**2 + x*y)
raises(TypeError, lambda: e.replace(sin, 2))
b = Wild('b')
c = Wild('c')
raises(TypeError, lambda: e.replace(b*c, c.is_real))
raises(TypeError, lambda: e.replace(b.is_real, 1))
raises(TypeError, lambda: e.replace(lambda d: d.is_Number, 1))
def test_is_diagonalizable():
a, b, c = symbols('a b c')
m = EigenOnlyMatrix(2, 2, [a, c, c, b])
assert m.is_symmetric()
assert m.is_diagonalizable()
assert not EigenOnlyMatrix(2, 2, [1, 1, 0, 1]).is_diagonalizable()
m = EigenOnlyMatrix(2, 2, [0, -1, 1, 0])
assert m.is_diagonalizable()
assert not m.is_diagonalizable(reals_only=True)
def test_bool_as_set():
x = symbols("x")
assert And(x <= 2, x >= -2).as_set() == Interval(-2, 2)
assert Or(x >= 2, x <= -2).as_set() == Interval(-oo, -2) + Interval(2, oo)
assert Not(x > 2).as_set() == Interval(-oo, 2)
# issue 10240
assert Not(And(x > 2, x < 3)).as_set() == Union(Interval(-oo, 2), Interval(3, oo))
assert true.as_set() == S.UniversalSet
assert false.as_set() == EmptySet()
def test_ITE():
A, B, C = map(Boolean, symbols('A,B,C'))
assert ITE(True, False, True) is false
assert ITE(True, True, False) is true
assert ITE(False, True, False) is false
assert ITE(False, False, True) is true
assert isinstance(ITE(A, B, C), ITE)
A = True
assert ITE(A, B, C) == B
A = False
assert ITE(A, B, C) == C
B = True
assert ITE(And(A, B), B, C) == C
assert ITE(Or(A, False), And(B, True), False) is false
x = symbols('x')
assert ITE(x, A, B) == Not(x)
assert ITE(x, B, A) == x
def _trigpats():
global _trigpat
a, b, c = symbols('a b c', cls=Wild)
d = Wild('d', commutative=False)
# for the simplifications like sinh/cosh -> tanh:
# DO NOT REORDER THE FIRST 14 since these are assumed to be in this
# order in _match_div_rewrite.
matchers_division = (
(a * sin(b)**c / cos(b)**c, a * tan(b)**c, sin(b), cos(b)),
(a * tan(b)**c * cos(b)**c, a * sin(b)**c, sin(b), cos(b)),
(a * cot(b)**c * sin(b)**c, a * cos(b)**c, sin(b), cos(b)),
(a * tan(b)**c / sin(b)**c, a / cos(b)**c, sin(b), cos(b)),
(a * cot(b)**c / cos(b)**c, a / sin(b)**c, sin(b), cos(b)),
(a * cot(b)**c * tan(b)**c, a, sin(b), cos(b)),
(a * (cos(b) + 1)**c * (cos(b) - 1)**c, a * (-sin(b)**2)**c,
cos(b) + 1, cos(b) - 1),
(a * (sin(b) + 1)**c * (sin(b) - 1)**c, a * (-cos(b)**2)**c,
sin(b) + 1, sin(b) - 1),
(a * sinh(b)**c / cosh(b)**c, a * tanh(b)**c, S.One, S.One),
(a * tanh(b)**c * cosh(b)**c, a * sinh(b)**c, S.One, S.One),
(a * coth(b)**c * sinh(b)**c, a * cosh(b)**c, S.One, S.One),
(a * tanh(b)**c / sinh(b)**c, a / cosh(b)**c, S.One, S.One),
(a * coth(b)**c / cosh(b)**c, a / sinh(b)**c, S.One, S.One),
(a * coth(b)**c * tanh(b)**c, a, S.One, S.One),
(c * (tanh(a) + tanh(b)) / (1 + tanh(a) * tanh(b)), tanh(a + b) * c,
S.One, S.One),
)
matchers_add = (
(c * sin(a) * cos(b) + c * cos(a) * sin(b) + d, sin(a + b) * c + d),
(c * cos(a) * cos(b) - c * sin(a) * sin(b) + d, cos(a + b) * c + d),
(c * sin(a) * cos(b) - c * cos(a) * sin(b) + d, sin(a - b) * c + d),
(c * cos(a) * cos(b) + c * sin(a) * sin(b) + d, cos(a - b) * c + d),
(c * sinh(a) * cosh(b) + c * sinh(b) * cosh(a) + d,
sinh(a + b) * c + d),
(c * cosh(a) * cosh(b) + c * sinh(a) * sinh(b) + d,
cosh(a + b) * c + d),
)
# for cos(x)**2 + sin(x)**2 -> 1
matchers_identity = (
(a * sin(b)**2, a - a * cos(b)**2),
(a * tan(b)**2, a * (1 / cos(b))**2 - a),
(a * cot(b)**2, a * (1 / sin(b))**2 - a),
(a * sin(b + c), a * (sin(b) * cos(c) + sin(c) * cos(b))),
(a * cos(b + c), a * (cos(b) * cos(c) - sin(b) * sin(c))),
(a * tan(b + c), a * ((tan(b) + tan(c)) / (1 - tan(b) * tan(c)))),
(a * sinh(b)**2, a * cosh(b)**2 - a),
(a * tanh(b)**2, a - a * (1 / cosh(b))**2),
(a * coth(b)**2, a + a * (1 / sinh(b))**2),
(a * sinh(b + c), a * (sinh(b) * cosh(c) + sinh(c) * cosh(b))),
(a * cosh(b + c), a * (cosh(b) * cosh(c) + sinh(b) * sinh(c))),
(a * tanh(b + c), a * ((tanh(b) + tanh(c)) / (1 + tanh(b) * tanh(c)))),
)
# Reduce any lingering artifacts, such as sin(x)**2 changing
# to 1-cos(x)**2 when sin(x)**2 was "simpler"
artifacts = (
(a - a * cos(b)**2 + c, a * sin(b)**2 + c, cos),
(a - a * (1 / cos(b))**2 + c, -a * tan(b)**2 + c, cos),
(a - a * (1 / sin(b))**2 + c, -a * cot(b)**2 + c, sin),
(a - a * cosh(b)**2 + c, -a * sinh(b)**2 + c, cosh),
(a - a * (1 / cosh(b))**2 + c, a * tanh(b)**2 + c, cosh),
(a + a * (1 / sinh(b))**2 + c, a * coth(b)**2 + c, sinh),
# same as above but with noncommutative prefactor
(a * d - a * d * cos(b)**2 + c, a * d * sin(b)**2 + c, cos),
(a * d - a * d * (1 / cos(b))**2 + c, -a * d * tan(b)**2 + c, cos),
(a * d - a * d * (1 / sin(b))**2 + c, -a * d * cot(b)**2 + c, sin),
(a * d - a * d * cosh(b)**2 + c, -a * d * sinh(b)**2 + c, cosh),
(a * d - a * d * (1 / cosh(b))**2 + c, a * d * tanh(b)**2 + c, cosh),
(a * d + a * d * (1 / sinh(b))**2 + c, a * d * coth(b)**2 + c, sinh),
)
_trigpat = (a, b, c, d, matchers_division, matchers_add, matchers_identity,
artifacts)
return _trigpat
def pde_1st_linear_constant_coeff(eq, func, order, match, solvefun):
r"""
Solves a first order linear partial differential equation
with constant coefficients.
The general form of this partial differential equation is
.. math:: a \frac{df(x,y)}{dx} + b \frac{df(x,y)}{dy} + c f(x,y) = G(x,y)
where `a`, `b` and `c` are constants and `G(x, y)` can be an arbitrary
function in `x` and `y`.
The general solution of the PDE is::
>>> from sympy.solvers import pdsolve
>>> from sympy.abc import x, y, a, b, c
>>> from sympy import Function, pprint
>>> f = Function('f')
>>> G = Function('G')
>>> u = f(x,y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> genform = a*u + b*ux + c*uy - G(x,y)
>>> pprint(genform)
d d
a*f(x, y) + b*--(f(x, y)) + c*--(f(x, y)) - G(x, y)
dx dy
>>> pprint(pdsolve(genform, hint='1st_linear_constant_coeff_Integral'))
// b*x + c*y \
|| / |
|| | |
|| | a*xi |
|| | ------- |
|| | 2 2 |
|| | /b*xi + c*eta -b*eta + c*xi\ b + c |
|| | G|------------, -------------|*e d(xi)|
|| | | 2 2 2 2 | |
|| | \ b + c b + c / |
|| | |
|| / |
|| |
f(x, y) = ||F(eta) + -------------------------------------------------------|*
|| 2 2 |
\\ b + c /
<BLANKLINE>
\|
||
||
||
||
||
||
||
||
-a*xi ||
-------||
2 2||
b + c ||
e ||
||
/|eta=-b*y + c*x, xi=b*x + c*y
Examples
========
>>> from sympy.solvers.pde import pdsolve
>>> from sympy import Function, diff, pprint, exp
>>> from sympy.abc import x,y
>>> f = Function('f')
>>> eq = -2*f(x,y).diff(x) + 4*f(x,y).diff(y) + 5*f(x,y) - exp(x + 3*y)
>>> pdsolve(eq)
Eq(f(x, y), (F(4*x + 2*y) + exp(x/2 + 4*y)/15)*exp(x/2 - y))
References
==========
- Viktor Grigoryan, "Partial Differential Equations"
Math 124A - Fall 2010, pp.7
"""
# TODO : For now homogeneous first order linear PDE's having
# two variables are implemented. Once there is support for
# solving systems of ODE's, this can be extended to n variables.
xi, eta = symbols("xi eta")
f = func.func
x = func.args[0]
y = func.args[1]
b = match[match['b']]
c = match[match['c']]
d = match[match['d']]
e = -match[match['e']]
expterm = exp(-S(d) / (b**2 + c**2) * xi)
functerm = solvefun(eta)
solvedict = solve((b * x + c * y - xi, c * x - b * y - eta), x, y)
# Integral should remain as it is in terms of xi,
# doit() should be done in _handle_Integral.
genterm = (1 / S(b**2 + c**2)) * Integral(
(1 / expterm * e).subs(solvedict), (xi, b * x + c * y))
return Eq(
f(x, y),
Subs(expterm * (functerm + genterm), (eta, xi),
(c * x - b * y, b * x + c * y)))
def pde_1st_linear_variable_coeff(eq, func, order, match, solvefun):
r"""
Solves a first order linear partial differential equation
with variable coefficients. The general form of this partial differential equation is
.. math:: a(x, y) \frac{df(x, y)}{dx} + a(x, y) \frac{df(x, y)}{dy}
+ c(x, y) f(x, y) - G(x, y)
where `a(x, y)`, `b(x, y)`, `c(x, y)` and `G(x, y)` are arbitrary functions
in `x` and `y`. This PDE is converted into an ODE by making the following transformation.
1] `\xi` as `x`
2] `\eta` as the constant in the solution to the differential equation
`\frac{dy}{dx} = -\frac{b}{a}`
Making the following substitutions reduces it to the linear ODE
.. math:: a(\xi, \eta)\frac{du}{d\xi} + c(\xi, \eta)u - d(\xi, \eta) = 0
which can be solved using dsolve.
The general form of this PDE is::
>>> from sympy.solvers.pde import pdsolve
>>> from sympy.abc import x, y
>>> from sympy import Function, pprint
>>> a, b, c, G, f= [Function(i) for i in ['a', 'b', 'c', 'G', 'f']]
>>> u = f(x,y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> genform = a(x, y)*u + b(x, y)*ux + c(x, y)*uy - G(x,y)
>>> pprint(genform)
d d
-G(x, y) + a(x, y)*f(x, y) + b(x, y)*--(f(x, y)) + c(x, y)*--(f(x, y))
dx dy
Examples
========
>>> from sympy.solvers.pde import pdsolve
>>> from sympy import Function, diff, pprint, exp
>>> from sympy.abc import x,y
>>> f = Function('f')
>>> eq = x*(u.diff(x)) - y*(u.diff(y)) + y**2*u - y**2
>>> pdsolve(eq)
Eq(f(x, y), F(x*y)*exp(y**2/2) + 1)
References
==========
- Viktor Grigoryan, "Partial Differential Equations"
Math 124A - Fall 2010, pp.7
"""
from sympy.integrals.integrals import integrate
from sympy.solvers.ode import dsolve
xi, eta = symbols("xi eta")
f = func.func
x = func.args[0]
y = func.args[1]
b = match[match['b']]
c = match[match['c']]
d = match[match['d']]
e = -match[match['e']]
if not d:
# To deal with cases like b*ux = e or c*uy = e
if not (b and c):
if c:
try:
tsol = integrate(e / c, y)
except NotImplementedError:
raise NotImplementedError("Unable to find a solution"
" due to inability of integrate")
else:
return Eq(f(x, y), solvefun(x) + tsol)
if b:
try:
tsol = integrate(e / b, x)
except NotImplementedError:
raise NotImplementedError("Unable to find a solution"
" due to inability of integrate")
else:
return Eq(f(x, y), solvefun(y) + tsol)
if not c:
# To deal with cases when c is 0, a simpler method is used.
# The PDE reduces to b*(u.diff(x)) + d*u = e, which is a linear ODE in x
plode = f(x).diff(x) * b + d * f(x) - e
sol = dsolve(plode, f(x))
syms = sol.free_symbols - plode.free_symbols - {x, y}
rhs = _simplify_variable_coeff(sol.rhs, syms, solvefun, y)
return Eq(f(x, y), rhs)
if not b:
# To deal with cases when b is 0, a simpler method is used.
# The PDE reduces to c*(u.diff(y)) + d*u = e, which is a linear ODE in y
plode = f(y).diff(y) * c + d * f(y) - e
sol = dsolve(plode, f(y))
syms = sol.free_symbols - plode.free_symbols - {x, y}
rhs = _simplify_variable_coeff(sol.rhs, syms, solvefun, x)
return Eq(f(x, y), rhs)
dummy = Function('d')
h = (c / b).subs(y, dummy(x))
sol = dsolve(dummy(x).diff(x) - h, dummy(x))
if isinstance(sol, list):
sol = sol[0]
solsym = sol.free_symbols - h.free_symbols - {x, y}
if len(solsym) == 1:
solsym = solsym.pop()
etat = (solve(sol, solsym)[0]).subs(dummy(x), y)
ysub = solve(eta - etat, y)[0]
deq = (b * (f(x).diff(x)) + d * f(x) - e).subs(y, ysub)
final = (dsolve(deq, f(x), hint='1st_linear')).rhs
if isinstance(final, list):
final = final[0]
finsyms = final.free_symbols - deq.free_symbols - {x, y}
rhs = _simplify_variable_coeff(final, finsyms, solvefun, etat)
return Eq(f(x, y), rhs)
else:
raise NotImplementedError(
"Cannot solve the partial differential equation due"
" to inability of constantsimp")
def test_matrix():
x = symbols('x')
m = Matrix([[I, x * I], [2, 4]])
assert Dagger(m) == m.H
def test_issue_19869():
t = symbols('t')
assert (maximum(sqrt(3)*(t - 1)/(3*sqrt(t**2 + 1)), t)
) == sqrt(3)/3
def test_symbols():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
assert symbols('x') == x
assert symbols('x ') == x
assert symbols(' x ') == x
assert symbols('x,') == (x, )
assert symbols('x, ') == (x, )
assert symbols('x ,') == (x, )
assert symbols('x , y') == (x, y)
assert symbols('x,y,z') == (x, y, z)
assert symbols('x y z') == (x, y, z)
assert symbols('x,y,z,') == (x, y, z)
assert symbols('x y z ') == (x, y, z)
xyz = Symbol('xyz')
abc = Symbol('abc')
assert symbols('xyz') == xyz
assert symbols('xyz,') == (xyz, )
assert symbols('xyz,abc') == (xyz, abc)
assert symbols(('xyz', )) == (xyz, )
assert symbols(('xyz,', )) == ((xyz, ), )
assert symbols(('x,y,z,', )) == ((x, y, z), )
assert symbols(('xyz', 'abc')) == (xyz, abc)
assert symbols(('xyz,abc', )) == ((xyz, abc), )
assert symbols(('xyz,abc', 'x,y,z')) == ((xyz, abc), (x, y, z))
assert symbols(('x', 'y', 'z')) == (x, y, z)
assert symbols(['x', 'y', 'z']) == [x, y, z]
assert symbols({'x', 'y', 'z'}) == {x, y, z}
raises(ValueError, lambda: symbols(''))
raises(ValueError, lambda: symbols(','))
raises(ValueError, lambda: symbols('x,,y,,z'))
raises(ValueError, lambda: symbols(('x', '', 'y', '', 'z')))
a, b = symbols('x,y', real=True)
assert a.is_real and b.is_real
x0 = Symbol('x0')
x1 = Symbol('x1')
x2 = Symbol('x2')
y0 = Symbol('y0')
y1 = Symbol('y1')
assert symbols('x0:0') == ()
assert symbols('x0:1') == (x0, )
assert symbols('x0:2') == (x0, x1)
assert symbols('x0:3') == (x0, x1, x2)
assert symbols('x:0') == ()
assert symbols('x:1') == (x0, )
assert symbols('x:2') == (x0, x1)
assert symbols('x:3') == (x0, x1, x2)
assert symbols('x1:1') == ()
assert symbols('x1:2') == (x1, )
assert symbols('x1:3') == (x1, x2)
assert symbols('x1:3,x,y,z') == (x1, x2, x, y, z)
assert symbols('x:3,y:2') == (x0, x1, x2, y0, y1)
assert symbols(('x:3', 'y:2')) == ((x0, x1, x2), (y0, y1))
a = Symbol('a')
b = Symbol('b')
c = Symbol('c')
d = Symbol('d')
assert symbols('x:z') == (x, y, z)
assert symbols('a:d,x:z') == (a, b, c, d, x, y, z)
assert symbols(('a:d', 'x:z')) == ((a, b, c, d), (x, y, z))
aa = Symbol('aa')
ab = Symbol('ab')
ac = Symbol('ac')
ad = Symbol('ad')
assert symbols('aa:d') == (aa, ab, ac, ad)
assert symbols('aa:d,x:z') == (aa, ab, ac, ad, x, y, z)
assert symbols(('aa:d', 'x:z')) == ((aa, ab, ac, ad), (x, y, z))
assert type(symbols(
('q:2', 'u:2'),
cls=Function)[0][0]) == UndefinedFunction # issue 23532
# issue 6675
def sym(s):
return str(symbols(s))
assert sym('a0:4') == '(a0, a1, a2, a3)'
assert sym('a2:4,b1:3') == '(a2, a3, b1, b2)'
assert sym('a1(2:4)') == '(a12, a13)'
assert sym('a0:2.0:2') == '(a0.0, a0.1, a1.0, a1.1)'
assert sym('aa:cz') == '(aaz, abz, acz)'
assert sym('aa:c0:2') == '(aa0, aa1, ab0, ab1, ac0, ac1)'
assert sym('aa:ba:b') == '(aaa, aab, aba, abb)'
assert sym('a:3b') == '(a0b, a1b, a2b)'
assert sym('a-1:3b') == '(a-1b, a-2b)'
assert sym(r'a:2\,:2' +
chr(0)) == '(a0,0%s, a0,1%s, a1,0%s, a1,1%s)' % ((chr(0), ) * 4)
assert sym('x(:a:3)') == '(x(a0), x(a1), x(a2))'
assert sym('x(:c):1') == '(xa0, xb0, xc0)'
assert sym('x((:a)):3') == '(x(a)0, x(a)1, x(a)2)'
assert sym('x(:a:3') == '(x(a0, x(a1, x(a2)'
assert sym(':2') == '(0, 1)'
assert sym(':b') == '(a, b)'
assert sym(':b:2') == '(a0, a1, b0, b1)'
assert sym(':2:2') == '(00, 01, 10, 11)'
assert sym(':b:b') == '(aa, ab, ba, bb)'
raises(ValueError, lambda: symbols(':'))
raises(ValueError, lambda: symbols('a:'))
raises(ValueError, lambda: symbols('::'))
raises(ValueError, lambda: symbols('a::'))
raises(ValueError, lambda: symbols(':a:'))
raises(ValueError, lambda: symbols('::a'))
def _regular_point_ellipse(self, a, b, c, d, e, f):
D = 4 * a * c - b**2
ok = D
if not ok:
raise ValueError("Rational Point on the conic does not exist")
if a == 0 and c == 0:
K = -1
L = 4 * (d * e - b * f)
elif c != 0:
K = D
L = 4 * c**2 * d**2 - 4 * b * c * d * e + 4 * a * c * e**2 + 4 * b**2 * c * f - 16 * a * c**2 * f
else:
K = D
L = 4 * a**2 * e**2 - 4 * b * a * d * e + 4 * b**2 * a * f
ok = L != 0 and not (K > 0 and L < 0)
if not ok:
raise ValueError("Rational Point on the conic does not exist")
K = Rational(K).limit_denominator(10**12)
L = Rational(L).limit_denominator(10**12)
k1, k2 = K.p, K.q
l1, l2 = L.p, L.q
g = gcd(k2, l2)
a1 = (l2 * k2) / g
b1 = (k1 * l2) / g
c1 = -(l1 * k2) / g
a2 = sign(a1) * core(abs(a1), 2)
r1 = sqrt(a1 / a2)
b2 = sign(b1) * core(abs(b1), 2)
r2 = sqrt(b1 / b2)
c2 = sign(c1) * core(abs(c1), 2)
r3 = sqrt(c1 / c2)
g = gcd(gcd(a2, b2), c2)
a2 = a2 / g
b2 = b2 / g
c2 = c2 / g
g1 = gcd(a2, b2)
a2 = a2 / g1
b2 = b2 / g1
c2 = c2 * g1
g2 = gcd(a2, c2)
a2 = a2 / g2
b2 = b2 * g2
c2 = c2 / g2
g3 = gcd(b2, c2)
a2 = a2 * g3
b2 = b2 / g3
c2 = c2 / g3
x, y, z = symbols("x y z")
eq = a2 * x**2 + b2 * y**2 + c2 * z**2
solutions = diophantine(eq)
if len(solutions) == 0:
raise ValueError("Rational Point on the conic does not exist")
flag = False
for sol in solutions:
syms = Tuple(*sol).free_symbols
rep = {s: 3 for s in syms}
sol_z = sol[2]
if sol_z == 0:
flag = True
continue
if not isinstance(sol_z, (int, Integer)):
syms_z = sol_z.free_symbols
if len(syms_z) == 1:
p = next(iter(syms_z))
p_values = Complement(
S.Integers, solveset(Eq(sol_z, 0), p, S.Integers))
rep[p] = next(iter(p_values))
if len(syms_z) == 2:
p, q = list(ordered(syms_z))
for i in S.Integers:
subs_sol_z = sol_z.subs(p, i)
q_values = Complement(
S.Integers,
solveset(Eq(subs_sol_z, 0), q, S.Integers))
if not q_values.is_empty:
rep[p] = i
rep[q] = next(iter(q_values))
break
if len(syms) != 0:
x, y, z = tuple(s.subs(rep) for s in sol)
else:
x, y, z = sol
flag = False
break
if flag:
raise ValueError("Rational Point on the conic does not exist")
x = (x * g3) / r1
y = (y * g2) / r2
z = (z * g1) / r3
x = x / z
y = y / z
if a == 0 and c == 0:
x_reg = (x + y - 2 * e) / (2 * b)
y_reg = (x - y - 2 * d) / (2 * b)
elif c != 0:
x_reg = (x - 2 * d * c + b * e) / K
y_reg = (y - b * x_reg - e) / (2 * c)
else:
y_reg = (x - 2 * e * a + b * d) / K
x_reg = (y - b * y_reg - d) / (2 * a)
return x_reg, y_reg
def cse_release_variables(r, e):
"""
Return tuples giving ``(a, b)`` where ``a`` is a symbol and ``b`` is
either an expression or None. The value of None is used when a
symbol is no longer needed for subsequent expressions.
Use of such output can reduce the memory footprint of lambdified
expressions that contain large, repeated subexpressions.
Examples
========
>>> from sympy import cse
>>> from sympy.simplify.cse_main import cse_release_variables
>>> from sympy.abc import x, y
>>> eqs = [(x + y - 1)**2, x, x + y, (x + y)/(2*x + 1) + (x + y - 1)**2, (2*x + 1)**(x + y)]
>>> defs, rvs = cse_release_variables(*cse(eqs))
>>> for i in defs:
... print(i)
...
(x0, x + y)
(x1, (x0 - 1)**2)
(x2, 2*x + 1)
(_3, x0/x2 + x1)
(_4, x2**x0)
(x2, None)
(_0, x1)
(x1, None)
(_2, x0)
(x0, None)
(_1, x)
>>> print(rvs)
(_0, _1, _2, _3, _4)
"""
if not r:
return r, e
s, p = zip(*r)
esyms = symbols('_:%d' % len(e))
syms = list(esyms)
s = list(s)
in_use = set(s)
p = list(p)
# sort e so those with most sub-expressions appear first
e = [(e[i], syms[i]) for i in range(len(e))]
e, syms = zip(*sorted(e,
key=lambda x: -sum([p[s.index(i)].count_ops()
for i in x[0].free_symbols & in_use])))
syms = list(syms)
p += e
rv = []
i = len(p) - 1
while i >= 0:
_p = p.pop()
c = in_use & _p.free_symbols
if c: # sorting for canonical results
rv.extend([(s, None) for s in sorted(c, key=str)])
if i >= len(r):
rv.append((syms.pop(), _p))
else:
rv.append((s[i], _p))
in_use -= c
i -= 1
rv.reverse()
return rv, esyms
def test_integrate():
x, y = symbols('x y')
m = CalculusOnlyMatrix(2, 1, [x, y])
assert m.integrate(x) == Matrix(2, 1, [x**2/2, y*x])
def test_replace():
F, G = symbols('F, G', cls=Function)
K = OperationsOnlyMatrix(2, 2, lambda i, j: G(i+j))
M = OperationsOnlyMatrix(2, 2, lambda i, j: F(i+j))
N = M.replace(F, G)
assert N == K
def analyse_gens(gens, hints):
"""
Analyse the generators ``gens``, using the hints ``hints``.
The meaning of ``hints`` is described in the main docstring.
Return a new list of generators, and also the ideal we should
work with.
"""
# First parse the hints
n, funcs, iterables, extragens = parse_hints(hints)
debug('n=%s' % n, 'funcs:', funcs, 'iterables:', iterables,
'extragens:', extragens)
# We just add the extragens to gens and analyse them as before
gens = list(gens)
gens.extend(extragens)
# remove duplicates
funcs = list(set(funcs))
iterables = list(set(iterables))
gens = list(set(gens))
# all the functions we can do anything with
allfuncs = {sin, cos, tan, sinh, cosh, tanh}
# sin(3*x) -> ((3, x), sin)
trigterms = [(g.args[0].as_coeff_mul(), g.func) for g in gens
if g.func in allfuncs]
# Our list of new generators - start with anything that we cannot
# work with (i.e. is not a trigonometric term)
freegens = [g for g in gens if g.func not in allfuncs]
newgens = []
trigdict = {}
for (coeff, var), fn in trigterms:
trigdict.setdefault(var, []).append((coeff, fn))
res = [] # the ideal
for key, val in trigdict.items():
# We have now assembeled a dictionary. Its keys are common
# arguments in trigonometric expressions, and values are lists of
# pairs (fn, coeff). x0, (fn, coeff) in trigdict means that we
# need to deal with fn(coeff*x0). We take the rational gcd of the
# coeffs, call it ``gcd``. We then use x = x0/gcd as "base symbol",
# all other arguments are integral multiples thereof.
# We will build an ideal which works with sin(x), cos(x).
# If hint tan is provided, also work with tan(x). Moreover, if
# n > 1, also work with sin(k*x) for k <= n, and similarly for cos
# (and tan if the hint is provided). Finally, any generators which
# the ideal does not work with but we need to accommodate (either
# because it was in expr or because it was provided as a hint)
# we also build into the ideal.
# This selection process is expressed in the list ``terms``.
# build_ideal then generates the actual relations in our ideal,
# from this list.
fns = [x[1] for x in val]
val = [x[0] for x in val]
gcd = reduce(igcd, val)
terms = [(fn, v / gcd) for (fn, v) in zip(fns, val)]
fs = set(funcs + fns)
for c, s, t in ([cos, sin, tan], [cosh, sinh, tanh]):
if any(x in fs for x in (c, s, t)):
fs.add(c)
fs.add(s)
for fn in fs:
for k in range(1, n + 1):
terms.append((fn, k))
extra = []
for fn, v in terms:
if fn == tan:
extra.append((sin, v))
extra.append((cos, v))
if fn in [sin, cos] and tan in fs:
extra.append((tan, v))
if fn == tanh:
extra.append((sinh, v))
extra.append((cosh, v))
if fn in [sinh, cosh] and tanh in fs:
extra.append((tanh, v))
terms.extend(extra)
x = gcd * Mul(*key)
r = build_ideal(x, terms)
res.extend(r)
newgens.extend({fn(v * x) for fn, v in terms})
# Add generators for compound expressions from iterables
for fn, args in iterables:
if fn == tan:
# Tan expressions are recovered from sin and cos.
iterables.extend([(sin, args), (cos, args)])
elif fn == tanh:
# Tanh expressions are recovered from sihn and cosh.
iterables.extend([(sinh, args), (cosh, args)])
else:
dummys = symbols('d:%i' % len(args), cls=Dummy)
expr = fn(Add(*dummys)).expand(trig=True).subs(
list(zip(dummys, args)))
res.append(fn(Add(*args)) - expr)
if myI in gens:
res.append(myI**2 + 1)
freegens.remove(myI)
newgens.append(myI)
return res, freegens, newgens
def test_exp_taylor_term():
x = symbols('x')
assert exp(x).taylor_term(1, x) == x
assert exp(x).taylor_term(3, x) == x**3 / 6
assert exp(x).taylor_term(4, x) == x**4 / 24
assert exp(x).taylor_term(-1, x) is S.Zero
def pde_1st_linear_constant_coeff(eq, func, order, match, solvefun):
r"""
Solves a first order linear partial differential equation
with constant coefficients.
The general form of this partial differential equation is
.. math:: a \frac{\partial f(x,y)}{\partial x}
+ b \frac{\partial f(x,y)}{\partial y}
+ c f(x,y) = G(x,y)
where `a`, `b` and `c` are constants and `G(x, y)` can be an arbitrary
function in `x` and `y`.
The general solution of the PDE is:
.. math::
f(x, y) = \left. \left[F(\eta) + \frac{1}{a^2 + b^2}
\int\limits^{a x + b y} G\left(\frac{a \xi + b \eta}{a^2 + b^2},
\frac{- a \eta + b \xi}{a^2 + b^2} \right)
e^{\frac{c \xi}{a^2 + b^2}}\, d\xi\right]
e^{- \frac{c \xi}{a^2 + b^2}}
\right|_{\substack{\eta=- a y + b x\\ \xi=a x + b y }}\, ,
where `F(\eta)` is an arbitrary single-valued function. The solution
can be found in SymPy with ``pdsolve``::
>>> from sympy.solvers import pdsolve
>>> from sympy.abc import x, y, a, b, c
>>> from sympy import Function, pprint
>>> f = Function('f')
>>> G = Function('G')
>>> u = f(x,y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> genform = a*ux + b*uy + c*u - G(x,y)
>>> pprint(genform)
d d
a*--(f(x, y)) + b*--(f(x, y)) + c*f(x, y) - G(x, y)
dx dy
>>> pprint(pdsolve(genform, hint='1st_linear_constant_coeff_Integral'))
// a*x + b*y \
|| / |
|| | |
|| | c*xi |
|| | ------- |
|| | 2 2 |
|| | /a*xi + b*eta -a*eta + b*xi\ a + b |
|| | G|------------, -------------|*e d(xi)|
|| | | 2 2 2 2 | |
|| | \ a + b a + b / |
|| | |
|| / |
|| |
f(x, y) = ||F(eta) + -------------------------------------------------------|*
|| 2 2 |
\\ a + b /
<BLANKLINE>
\|
||
||
||
||
||
||
||
||
-c*xi ||
-------||
2 2||
a + b ||
e ||
||
/|eta=-a*y + b*x, xi=a*x + b*y
Examples
========
>>> from sympy.solvers.pde import pdsolve
>>> from sympy import Function, pprint, exp
>>> from sympy.abc import x,y
>>> f = Function('f')
>>> eq = -2*f(x,y).diff(x) + 4*f(x,y).diff(y) + 5*f(x,y) - exp(x + 3*y)
>>> pdsolve(eq)
Eq(f(x, y), (F(4*x + 2*y) + exp(x/2 + 4*y)/15)*exp(x/2 - y))
References
==========
- Viktor Grigoryan, "Partial Differential Equations"
Math 124A - Fall 2010, pp.7
"""
# TODO : For now homogeneous first order linear PDE's having
# two variables are implemented. Once there is support for
# solving systems of ODE's, this can be extended to n variables.
xi, eta = symbols("xi eta")
f = func.func
x = func.args[0]
y = func.args[1]
b = match[match['b']]
c = match[match['c']]
d = match[match['d']]
e = -match[match['e']]
expterm = exp(-S(d) / (b**2 + c**2) * xi)
functerm = solvefun(eta)
solvedict = solve((b * x + c * y - xi, c * x - b * y - eta), x, y)
# Integral should remain as it is in terms of xi,
# doit() should be done in _handle_Integral.
genterm = (1 / S(b**2 + c**2)) * Integral(
(1 / expterm * e).subs(solvedict), (xi, b * x + c * y))
return Eq(
f(x, y),
Subs(expterm * (functerm + genterm), (eta, xi),
(c * x - b * y, b * x + c * y)))
from sympy.vector.vector import Vector
from sympy.vector.coordsysrect import CoordSysCartesian
from sympy.simplify import simplify
from sympy.core.symbol import symbols
from sympy.core import S
from sympy import sin, cos
C = CoordSysCartesian('C')
i, j, k = C.base_vectors()
x, y, z = C.base_scalars()
delop = C.delop
a, b, c = symbols('a b c')
def test_del_operator():
#Tests for curl
assert delop ^ Vector.zero == Vector.zero
assert delop.cross(Vector.zero) == Vector.zero
assert delop ^ i == Vector.zero
assert delop.cross(2 * y**2 * j) == Vector.zero
v = x * y * z * (i + j + k)
assert delop ^ v == \
(-x*y + x*z)*i + (x*y - y*z)*j + (-x*z + y*z)*k
assert delop ^ v == delop.cross(v)
assert delop.cross(2 * x**2 * j) == 4 * x * k
#Tests for divergence
assert delop & Vector.zero == S(0)
assert delop.dot(Vector.zero) == S(0)
assert delop & i == S(0)
def test_commutative():
"""Test for commutativity of And and Or"""
A, B = map(Boolean, symbols('A,B'))
assert A & B == B & A
assert A | B == B | A
def test_multivariate_bool_as_set():
x, y = symbols('x,y')
assert And(x >= 0, y >= 0).as_set() == Interval(0, oo)*Interval(0, oo)
assert Or(x >= 0, y >= 0).as_set() == S.Reals*S.Reals - \
Interval(-oo, 0, True, True)*Interval(-oo, 0, True, True)
def test_bool_monomial():
x, y = symbols('x,y')
assert bool_monomial(1, [x, y]) == y
assert bool_monomial([1, 1], [x, y]) == And(x, y)
def test_simplification():
"""
Test working of simplification methods.
"""
set1 = [[0, 0, 1], [0, 1, 1], [1, 0, 0], [1, 1, 0]]
set2 = [[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]]
assert SOPform([x, y, z], set1) == Or(And(Not(x), z), And(Not(z), x))
assert Not(SOPform([x, y, z], set2)) == \
Not(Or(And(Not(x), Not(z)), And(x, z)))
assert POSform([x, y, z], set1 + set2) is true
assert SOPform([x, y, z], set1 + set2) is true
assert SOPform([Dummy(), Dummy(), Dummy()], set1 + set2) is true
minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
[1, 1, 1, 1]]
dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, 3, 7, 11, 15]
dontcares = [0, 2, 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, [0, 0, 1, 1], 7, [1, 0, 1, 1],
[1, 1, 1, 1]]
dontcares = [0, [0, 0, 1, 0], 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, {y: 1, z: 1}]
dontcares = [0, [0, 0, 1, 0], 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [{y: 1, z: 1}, 1]
dontcares = [[0, 0, 0, 0]]
minterms = [[0, 0, 0]]
raises(ValueError, lambda: SOPform([w, x, y, z], minterms))
raises(ValueError, lambda: POSform([w, x, y, z], minterms))
raises(TypeError, lambda: POSform([w, x, y, z], ["abcdefg"]))
# test simplification
ans = And(A, Or(B, C))
assert simplify_logic(A & (B | C)) == ans
assert simplify_logic((A & B) | (A & C)) == ans
assert simplify_logic(Implies(A, B)) == Or(Not(A), B)
assert simplify_logic(Equivalent(A, B)) == \
Or(And(A, B), And(Not(A), Not(B)))
assert simplify_logic(And(Equality(A, 2), C)) == And(Equality(A, 2), C)
assert simplify_logic(And(Equality(A, 2), A)) is S.false
assert simplify_logic(And(Equality(A, 2), A)) == And(Equality(A, 2), A)
assert simplify_logic(And(Equality(A, B), C)) == And(Equality(A, B), C)
assert simplify_logic(Or(And(Equality(A, 3), B), And(Equality(A, 3), C))) \
== And(Equality(A, 3), Or(B, C))
b = (~x & ~y & ~z) | (~x & ~y & z)
e = And(A, b)
assert simplify_logic(e) == A & ~x & ~y
raises(ValueError, lambda: simplify_logic(A & (B | C), form='blabla'))
# Check that expressions with nine variables or more are not simplified
# (without the force-flag)
a, b, c, d, e, f, g, h, j = symbols('a b c d e f g h j')
expr = a & b & c & d & e & f & g & h & j | \
a & b & c & d & e & f & g & h & ~j
# This expression can be simplified to get rid of the j variables
assert simplify_logic(expr) == expr
# check input
ans = SOPform([x, y], [[1, 0]])
assert SOPform([x, y], [[1, 0]]) == ans
assert POSform([x, y], [[1, 0]]) == ans
raises(ValueError, lambda: SOPform([x], [[1]], [[1]]))
assert SOPform([x], [[1]], [[0]]) is true
assert SOPform([x], [[0]], [[1]]) is true
assert SOPform([x], [], []) is false
raises(ValueError, lambda: POSform([x], [[1]], [[1]]))
assert POSform([x], [[1]], [[0]]) is true
assert POSform([x], [[0]], [[1]]) is true
assert POSform([x], [], []) is false
# check working of simplify
assert simplify((A & B) | (A & C)) == And(A, Or(B, C))
assert simplify(And(x, Not(x))) == False
assert simplify(Or(x, Not(x))) == True
assert simplify(And(Eq(x, 0), Eq(x, y))) == And(Eq(x, 0), Eq(y, 0))
assert And(Eq(x - 1, 0), Eq(x, y)).simplify() == And(Eq(x, 1), Eq(y, 1))
assert And(Ne(x - 1, 0), Ne(x, y)).simplify() == And(Ne(x, 1), Ne(x, y))
assert And(Eq(x - 1, 0), Ne(x, y)).simplify() == And(Eq(x, 1), Ne(y, 1))
assert And(Eq(x - 1, 0), Eq(x, z + y), Eq(y + x, 0)).simplify(
) == And(Eq(x, 1), Eq(y, -1), Eq(z, 2))
assert And(Eq(x - 1, 0), Eq(x + 2, 3)).simplify() == Eq(x, 1)
assert And(Ne(x - 1, 0), Ne(x + 2, 3)).simplify() == Ne(x, 1)
assert And(Eq(x - 1, 0), Eq(x + 2, 2)).simplify() == False
assert And(Ne(x - 1, 0), Ne(x + 2, 2)).simplify(
) == And(Ne(x, 1), Ne(x, 0))
def test_bool_minterm():
x, y = symbols('x,y')
assert bool_minterm(3, [x, y]) == And(x, y)
assert bool_minterm([1, 0], [x, y]) == And(Not(y), x)
def test_bool_maxterm():
x, y = symbols('x,y')
assert bool_maxterm(2, [x, y]) == Or(Not(x), y)
assert bool_maxterm([0, 1], [x, y]) == Or(Not(y), x)
POSform, SOPform, Xor, Xnor, conjuncts, disjuncts,
distribute_or_over_and, distribute_and_over_or,
eliminate_implications, is_nnf, is_cnf, is_dnf, simplify_logic,
to_nnf, to_cnf, to_dnf, to_int_repr, bool_map, true, false,
BooleanAtom, is_literal, term_to_integer, integer_to_term,
truth_table, as_Boolean, to_anf, is_anf, distribute_xor_over_and,
anf_coeffs, ANFform, bool_minterm, bool_maxterm, bool_monomial,
_check_pair, _convert_to_varsSOP, _convert_to_varsPOS, Exclusive,)
from sympy.assumptions.cnf import CNF
from sympy.testing.pytest import raises, XFAIL, slow
from sympy.utilities.iterables import cartes
from itertools import combinations, permutations
A, B, C, D = symbols('A:D')
a, b, c, d, e, w, x, y, z = symbols('a:e w:z')
def test_overloading():
"""Test that |, & are overloaded as expected"""
assert A & B == And(A, B)
assert A | B == Or(A, B)
assert (A & B) | C == Or(And(A, B), C)
assert A >> B == Implies(A, B)
assert A << B == Implies(B, A)
assert ~A == Not(A)
assert A ^ B == Xor(A, B)
def test_issue_16803():
n = symbols('n')
# No simplification done, but should not raise an exception
assert ((n > 3) | (n < 0) | ((n > 0) & (n < 3))).simplify() == \
((n > 3) | (n < 0) | ((n > 0) & (n < 3)))
def test_limit():
x, y = symbols('x y')
m = CalculusOnlyMatrix(2, 1, [1/x, y])
assert m.limit(x, 5) == Matrix(2, 1, [Rational(1, 5), y])
def sym(s):
return str(symbols(s))
def test_pinjoint_arbitrary_axis():
theta, omega = dynamicsymbols('theta_J, omega_J')
# When the bodies are attached though masscenters but axess are opposite.
N, A, P, C = _generate_body()
PinJoint('J', P, C, child_axis=-A.x)
assert -A.x.angle_between(N.x) == -pi
assert -A.x.express(N) == N.x
assert A.dcm(N) == Matrix([[-1, 0, 0], [0, -cos(theta), -sin(theta)],
[0, -sin(theta), cos(theta)]])
assert A.ang_vel_in(N) == omega * N.x
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
assert C.masscenter.pos_from(P.masscenter) == 0
assert C.masscenter.pos_from(P.masscenter).express(N).simplify() == 0
assert C.masscenter.vel(N) == 0
# When axes are different and parent joint is at masscenter but child joint
# is at a unit vector from child masscenter.
N, A, P, C = _generate_body()
PinJoint('J', P, C, child_axis=A.y, child_joint_pos=A.x)
assert A.y.angle_between(N.x) == 0 # Axis are aligned
assert A.y.express(N) == N.x
assert A.dcm(N) == Matrix([[0, -cos(theta), -sin(theta)], [1, 0, 0],
[0, -sin(theta), cos(theta)]])
assert A.ang_vel_in(N) == omega * N.x
assert A.ang_vel_in(N).express(A) == omega * A.y
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
angle = A.ang_vel_in(N).angle_between(A.y)
assert expand_mul(angle).xreplace({omega: 1}) == 0
assert C.masscenter.vel(N) == omega * A.z
assert C.masscenter.pos_from(P.masscenter) == -A.x
assert (C.masscenter.pos_from(
P.masscenter).express(N).simplify() == cos(theta) * N.y +
sin(theta) * N.z)
assert C.masscenter.vel(N).angle_between(A.x) == pi / 2
# Similar to previous case but wrt parent body
N, A, P, C = _generate_body()
PinJoint('J', P, C, parent_axis=N.y, parent_joint_pos=N.x)
assert N.y.angle_between(A.x) == 0 # Axis are aligned
assert N.y.express(A) == A.x
assert A.dcm(N) == Matrix([[0, 1, 0], [-cos(theta), 0,
sin(theta)],
[sin(theta), 0, cos(theta)]])
assert A.ang_vel_in(N) == omega * N.y
assert A.ang_vel_in(N).express(A) == omega * A.x
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
angle = A.ang_vel_in(N).angle_between(A.x)
assert expand_mul(angle).xreplace({omega: 1}) == 0
assert C.masscenter.vel(N).simplify() == -omega * N.z
assert C.masscenter.pos_from(P.masscenter) == N.x
# Both joint pos id defined but different axes
N, A, P, C = _generate_body()
PinJoint('J',
P,
C,
parent_joint_pos=N.x,
child_joint_pos=A.x,
child_axis=A.x + A.y)
assert expand_mul(N.x.angle_between(A.x + A.y)) == 0 # Axis are aligned
assert (A.x + A.y).express(N).simplify() == sqrt(2) * N.x
assert A.dcm(N).simplify() == Matrix(
[[sqrt(2) / 2, -sqrt(2) * cos(theta) / 2, -sqrt(2) * sin(theta) / 2],
[sqrt(2) / 2,
sqrt(2) * cos(theta) / 2,
sqrt(2) * sin(theta) / 2], [0, -sin(theta),
cos(theta)]])
assert A.ang_vel_in(N) == omega * N.x
assert (
A.ang_vel_in(N).express(A).simplify() == (omega * A.x + omega * A.y) /
sqrt(2))
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
angle = A.ang_vel_in(N).angle_between(A.x + A.y)
assert expand_mul(angle).xreplace({omega: 1}) == 0
assert C.masscenter.vel(N).simplify() == (omega * A.z) / sqrt(2)
assert C.masscenter.pos_from(P.masscenter) == N.x - A.x
assert (C.masscenter.pos_from(
P.masscenter).express(N).simplify() == (1 - sqrt(2) / 2) * N.x +
sqrt(2) * cos(theta) / 2 * N.y + sqrt(2) * sin(theta) / 2 * N.z)
assert (C.masscenter.vel(N).express(N).simplify() == -sqrt(2) * omega *
sin(theta) / 2 * N.y + sqrt(2) * omega * cos(theta) / 2 * N.z)
assert C.masscenter.vel(N).angle_between(A.x) == pi / 2
N, A, P, C = _generate_body()
PinJoint('J',
P,
C,
parent_joint_pos=N.x,
child_joint_pos=A.x,
child_axis=A.x + A.y - A.z)
assert expand_mul(N.x.angle_between(A.x + A.y - A.z)) == 0 # Axis aligned
assert (A.x + A.y - A.z).express(N).simplify() == sqrt(3) * N.x
assert A.dcm(N).simplify() == Matrix(
[[
sqrt(3) / 3, -sqrt(6) * sin(theta + pi / 4) / 3,
sqrt(6) * cos(theta + pi / 4) / 3
],
[
sqrt(3) / 3,
sqrt(6) * cos(theta + pi / 12) / 3,
sqrt(6) * sin(theta + pi / 12) / 3
],
[
-sqrt(3) / 3,
sqrt(6) * cos(theta + 5 * pi / 12) / 3,
sqrt(6) * sin(theta + 5 * pi / 12) / 3
]])
assert A.ang_vel_in(N) == omega * N.x
assert A.ang_vel_in(N).express(
A).simplify() == (omega * A.x + omega * A.y - omega * A.z) / sqrt(3)
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
angle = A.ang_vel_in(N).angle_between(A.x + A.y - A.z)
assert expand_mul(angle).xreplace({omega: 1}) == 0
assert C.masscenter.vel(N).simplify() == (omega * A.y +
omega * A.z) / sqrt(3)
assert C.masscenter.pos_from(P.masscenter) == N.x - A.x
assert (C.masscenter.pos_from(
P.masscenter).express(N).simplify() == (1 - sqrt(3) / 3) * N.x +
sqrt(6) * sin(theta + pi / 4) / 3 * N.y -
sqrt(6) * cos(theta + pi / 4) / 3 * N.z)
assert (C.masscenter.vel(N).express(N).simplify() == sqrt(6) * omega *
cos(theta + pi / 4) / 3 * N.y +
sqrt(6) * omega * sin(theta + pi / 4) / 3 * N.z)
assert C.masscenter.vel(N).angle_between(A.x) == pi / 2
N, A, P, C = _generate_body()
m, n = symbols('m n')
PinJoint('J',
P,
C,
parent_joint_pos=m * N.x,
child_joint_pos=n * A.x,
child_axis=A.x + A.y - A.z,
parent_axis=N.x - N.y + N.z)
angle = (N.x - N.y + N.z).angle_between(A.x + A.y - A.z)
assert expand_mul(angle) == 0 # Axis are aligned
assert ((
A.x - A.y +
A.z).express(N).simplify() == (-4 * cos(theta) / 3 - 1 / 3) * N.x +
(1 / 3 - 4 * sin(theta + pi / 6) / 3) * N.y +
(4 * cos(theta + pi / 3) / 3 - 1 / 3) * N.z)
assert A.dcm(N).simplify() == Matrix(
[[
S(1) / 3 - 2 * cos(theta) / 3,
-2 * sin(theta + pi / 6) / 3 - S(1) / 3,
2 * cos(theta + pi / 3) / 3 + S(1) / 3
],
[
2 * cos(theta + pi / 3) / 3 + S(1) / 3,
2 * cos(theta) / 3 - S(1) / 3,
2 * sin(theta + pi / 6) / 3 + S(1) / 3
],
[
-2 * sin(theta + pi / 6) / 3 - S(1) / 3,
2 * cos(theta + pi / 3) / 3 + S(1) / 3,
2 * cos(theta) / 3 - S(1) / 3
]])
assert A.ang_vel_in(N) == (omega * N.x - omega * N.y +
omega * N.z) / sqrt(3)
assert A.ang_vel_in(N).express(
A).simplify() == (omega * A.x + omega * A.y - omega * A.z) / sqrt(3)
assert A.ang_vel_in(N).magnitude() == sqrt(omega**2)
angle = A.ang_vel_in(N).angle_between(A.x + A.y - A.z)
assert expand_mul(angle).xreplace({omega: 1}) == 0
assert (
C.masscenter.vel(N).simplify() == (m * omega * N.y + m * omega * N.z +
n * omega * A.y + n * omega * A.z) /
sqrt(3))
assert C.masscenter.pos_from(P.masscenter) == m * N.x - n * A.x
assert (C.masscenter.pos_from(
P.masscenter).express(N).simplify() == (m + n *
(2 * cos(theta) - 1) / 3) * N.x
+ n * (2 * sin(theta + pi / 6) + 1) / 3 * N.y - n *
(2 * cos(theta + pi / 3) + 1) / 3 * N.z)
assert (C.masscenter.vel(N).express(N).simplify() == -2 * n * omega *
sin(theta) / 3 * N.x +
(sqrt(3) * m + 2 * n * cos(theta + pi / 6)) * omega / 3 * N.y +
(sqrt(3) * m + 2 * n * sin(theta + pi / 3)) * omega / 3 * N.z)
assert expand_mul(C.masscenter.vel(N).angle_between(m * N.x -
n * A.x)) == pi / 2
def test_log_taylor_term():
x = symbols('x')
assert log(x).taylor_term(0, x) == x
assert log(x).taylor_term(1, x) == -x**2 / 2
assert log(x).taylor_term(4, x) == x**5 / 5
assert log(x).taylor_term(-1, x) is S.Zero
def test_sorted_args():
x = symbols('x')
assert b21._sorted_args == b21.args
raises(AttributeError, lambda: x._sorted_args)
def test_exp_summation():
w = symbols("w")
m, n, i, j = symbols("m n i j")
expr = exp(Sum(w * i, (i, 0, n), (j, 0, m)))
assert expr.expand() == Product(exp(w * i), (i, 0, n), (j, 0, m))
def test_diff():
x, y = symbols('x y')
m = CalculusOnlyMatrix(2, 1, [x, y])
# TODO: currently not working as ``_MinimalMatrix`` cannot be sympified:
assert m.diff(x) == Matrix(2, 1, [1, 0])
def test_log_expand_fail():
x, y, z = symbols('x,y,z', positive=True)
assert (log(x * (y + z)) * (x + y)).expand(
mul=True,
log=True) == y * log(x) + y * log(y + z) + z * log(x) + z * log(y + z)
