def test(As,Bs,Cs,Ds,Us,sI_A,Js,Ys,B0,C0,C,Psi,Lamda):
D1=BlockMatrix([[Us.inv()*Psi,zeros(Us.rows,1)],[-Lamda,eye(Lamda.rows)]]).as_mutable()
D2=BlockMatrix([[sI_A,B0],[-C,Js]]).as_mutable()
D3=BlockMatrix([[As,Bs],[-Cs,Ds]]).as_mutable()
D4=BlockMatrix([[C0,-Ys],[zeros(Ys.cols,C0.cols),eye(Ys.cols)]]).as_mutable()
return expand(simplify(D1*D2))==expand(D3*D4)
def _eval_simplify(self, ratio=1.7, measure=None, rational=False, inverse=False):
from sympy.simplify.simplify import factor_sum, sum_combine
from sympy.core.function import expand
from sympy.core.mul import Mul
# split the function into adds
terms = Add.make_args(expand(self.function))
s_t = [] # Sum Terms
o_t = [] # Other Terms
for term in terms:
if term.has(Sum):
# if there is an embedded sum here
# it is of the form x * (Sum(whatever))
# hence we make a Mul out of it, and simplify all interior sum terms
subterms = Mul.make_args(expand(term))
out_terms = []
for subterm in subterms:
# go through each term
if isinstance(subterm, Sum):
# if it's a sum, simplify it
out_terms.append(subterm._eval_simplify())
else:
# otherwise, add it as is
out_terms.append(subterm)
# turn it back into a Mul
s_t.append(Mul(*out_terms))
else:
o_t.append(term)
# next try to combine any interior sums for further simplification
result = Add(sum_combine(s_t), *o_t)
return factor_sum(result, limits=self.limits)
def _eval_simplify(self, ratio=1.7, measure=None):
from sympy.simplify.simplify import factor_sum, sum_combine
from sympy.core.function import expand
from sympy.core.mul import Mul
# split the function into adds
terms = Add.make_args(expand(self.function))
s_t = [] # Sum Terms
o_t = [] # Other Terms
for term in terms:
if term.has(Sum):
# if there is an embedded sum here
# it is of the form x * (Sum(whatever))
# hence we make a Mul out of it, and simplify all interior sum terms
subterms = Mul.make_args(expand(term))
out_terms = []
for subterm in subterms:
# go through each term
if isinstance(subterm, Sum):
# if it's a sum, simplify it
out_terms.append(subterm._eval_simplify())
else:
# otherwise, add it as is
out_terms.append(subterm)
# turn it back into a Mul
s_t.append(Mul(*out_terms))
else:
o_t.append(term)
# next try to combine any interior sums for further simplification
result = Add(sum_combine(s_t), *o_t)
return factor_sum(result, limits=self.limits)
def test_order_noncommutative():
A = Symbol('A', commutative=False)
assert Order(A + A * x, x) == Order(1, x)
assert (A + A * x) * Order(x) == Order(x)
assert (A * x) * Order(x) == Order(x**2, x)
assert expand((1 + Order(x)) * A * A * x) == A * A * x + Order(x**2, x)
assert expand((A * A + Order(x)) * x) == A * A * x + Order(x**2, x)
assert expand((A + Order(x)) * A * x) == A * A * x + Order(x**2, x)
def test_expand_frac():
assert expand((x + y)*y/x/(x + 1), frac=True) == \
(x*y + y**2)/(x**2 + x)
assert expand((x + y)*y/x/(x + 1), numer=True) == \
(x*y + y**2)/(x*(x + 1))
assert expand((x + y)*y/x/(x + 1), denom=True) == \
y*(x + y)/(x**2 + x)
eq = (x + 1)**2/y
assert expand_numer(eq, multinomial=False) == eq
def test_expand_frac():
assert expand((x + y)*y/x/(x + 1), frac=True) == \
(x*y + y**2)/(x**2 + x)
assert expand((x + y)*y/x/(x + 1), numer=True) == \
(x*y + y**2)/(x*(x + 1))
assert expand((x + y)*y/x/(x + 1), denom=True) == \
y*(x + y)/(x**2 + x)
eq = (x + 1)**2/y
assert expand_numer(eq, multinomial=False) == eq
def __new__(cls, *args, **old_assumptions):
from sympy.physics.quantum.state import KetBase, BraBase
if len(args) != 2:
raise ValueError('2 parameters expected, got %d' % len(args))
ket_expr = expand(args[0])
bra_expr = expand(args[1])
if (isinstance(ket_expr, (KetBase, Mul)) and
isinstance(bra_expr, (BraBase, Mul))):
ket_c, kets = ket_expr.args_cnc()
bra_c, bras = bra_expr.args_cnc()
if len(kets) != 1 or not isinstance(kets[0], KetBase):
raise TypeError('KetBase subclass expected'
', got: %r' % Mul(*kets))
if len(bras) != 1 or not isinstance(bras[0], BraBase):
raise TypeError('BraBase subclass expected'
', got: %r' % Mul(*bras))
if not kets[0].dual_class() == bras[0].__class__:
raise TypeError(
'ket and bra are not dual classes: %r, %r' %
(kets[0].__class__, bras[0].__class__)
)
# TODO: make sure the hilbert spaces of the bra and ket are
# compatible
obj = Expr.__new__(cls, *(kets[0], bras[0]), **old_assumptions)
obj.hilbert_space = kets[0].hilbert_space
return Mul(*(ket_c + bra_c)) * obj
op_terms = []
if isinstance(ket_expr, Add) and isinstance(bra_expr, Add):
for ket_term in ket_expr.args:
for bra_term in bra_expr.args:
op_terms.append(OuterProduct(ket_term, bra_term,
**old_assumptions))
elif isinstance(ket_expr, Add):
for ket_term in ket_expr.args:
op_terms.append(OuterProduct(ket_term, bra_expr,
**old_assumptions))
elif isinstance(bra_expr, Add):
for bra_term in bra_expr.args:
op_terms.append(OuterProduct(ket_expr, bra_term,
**old_assumptions))
else:
raise TypeError(
'Expected ket and bra expression, got: %r, %r' %
(ket_expr, bra_expr)
)
return Add(*op_terms)
def _add_splines(c, b1, d, b2):
"""Construct c*b1 + d*b2."""
if b1 == S.Zero or c == S.Zero:
return expand(piecewise_fold(d * b2))
if b2 == S.Zero or d == S.Zero:
return expand(piecewise_fold(c * b1))
new_args = []
n_intervals = len(b1.args)
assert n_intervals == len(b2.args)
new_args.append((expand(c * b1.args[0].expr), b1.args[0].cond))
for i in range(1, n_intervals - 1):
new_args.append((expand(c * b1.args[i].expr + d * b2.args[i - 1].expr), b1.args[i].cond))
new_args.append((expand(d * b2.args[-2].expr), b2.args[-2].cond))
new_args.append(b2.args[-1])
return Piecewise(*new_args)
def _qsimplify_fermion_product_site(e):
"""
Assumes that all fermion operators have the same name.
"""
work = [e]
out = []
while len(work):
e = work.pop()
if not isinstance(e, Mul):
out.append(e)
continue
c, nc = e.args_cnc()
ann = [f.is_annihilation for f in nc]
# Find the first inverted pair of operators
idx = next((i for i in range(len(ann)) if ann[i:i+2] == [True,False]), None)
if idx is None:
out.append(e)
continue
# Substitute the inverted pair for a cannonical subexpression
e = Mul(*c) * (Mul(*nc[:idx])
* (Integer(1) - Mul(nc[idx+1],nc[idx]))
* Mul(*nc[idx+2:]))
# Expand the new as a sum of fermion operator strings
e = expand(e)
# Recursive simplify
if isinstance(e, Add):
work.extend(e.args)
else:
work.append(e)
return Add(*out)
def mysimp(expr):
from sympy.core.function import expand
from sympy.simplify.powsimp import powsimp
from sympy.simplify.simplify import logcombine
return expand(
powsimp(logcombine(expr, force=True), force=True, deep=True),
force=True).replace(exp_polar, exp)
def can_do_meijer(a1, a2, b1, b2, numeric=True):
"""
This helper function tries to hyperexpand() the meijer g-function
corresponding to the parameters a1, a2, b1, b2.
It returns False if this expansion still contains g-functions.
If numeric is True, it also tests the so-obtained formula numerically
(at random values) and returns False if the test fails.
Else it returns True.
"""
from sympy.core.function import expand
from sympy.functions.elementary.complexes import unpolarify
r = hyperexpand(meijerg(a1, a2, b1, b2, z))
if r.has(meijerg):
return False
# NOTE hyperexpand() returns a truly branched function, whereas numerical
# evaluation only works on the main branch. Since we are evaluating on
# the main branch, this should not be a problem, but expressions like
# exp_polar(I*pi/2*x)**a are evaluated incorrectly. We thus have to get
# rid of them. The expand heuristically does this...
r = unpolarify(expand(r, force=True, power_base=True, power_exp=False,
mul=False, log=False, multinomial=False, basic=False))
if not numeric:
return True
repl = {}
for n, ai in enumerate(meijerg(a1, a2, b1, b2, z).free_symbols - {z}):
repl[ai] = randcplx(n)
return tn(meijerg(a1, a2, b1, b2, z).subs(repl), r.subs(repl), z)
def _add_splines(c, b1, d, b2):
"""Construct c*b1 + d*b2."""
if b1 == S.Zero or c == S.Zero:
return expand(piecewise_fold(d * b2))
if b2 == S.Zero or d == S.Zero:
return expand(piecewise_fold(c * b1))
new_args = []
n_intervals = len(b1.args)
assert (n_intervals == len(b2.args))
new_args.append((expand(c * b1.args[0].expr), b1.args[0].cond))
for i in range(1, n_intervals - 1):
new_args.append((expand(c * b1.args[i].expr + d * b2.args[i - 1].expr),
b1.args[i].cond))
new_args.append((expand(d * b2.args[-2].expr), b2.args[-2].cond))
new_args.append(b2.args[-1])
return Piecewise(*new_args)
def test_complete_simple_double_pendulum():
q1, q2 = dynamicsymbols('q1 q2')
u1, u2 = dynamicsymbols('u1 u2')
m, l, g = symbols('m l g')
C = Body('C') # ceiling
PartP = Body('P', mass=m)
PartR = Body('R', mass=m)
J1 = PinJoint('J1', C, PartP, speeds=u1, coordinates=q1,
child_joint_pos=-l*PartP.x, parent_axis=C.z,
child_axis=PartP.z)
J2 = PinJoint('J2', PartP, PartR, speeds=u2, coordinates=q2,
child_joint_pos=-l*PartR.x, parent_axis=PartP.z,
child_axis=PartR.z)
PartP.apply_force(m*g*C.x)
PartR.apply_force(m*g*C.x)
method = JointsMethod(C, J1, J2)
method.form_eoms()
assert expand(method.mass_matrix_full) == Matrix([[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 2*l**2*m*cos(q2) + 3*l**2*m, l**2*m*cos(q2) + l**2*m],
[0, 0, l**2*m*cos(q2) + l**2*m, l**2*m]])
assert trigsimp(method.forcing_full) == trigsimp(Matrix([[u1], [u2], [-g*l*m*(sin(q1 + q2) + sin(q1)) -
g*l*m*sin(q1) + l**2*m*(2*u1 + u2)*u2*sin(q2)],
[-g*l*m*sin(q1 + q2) - l**2*m*u1**2*sin(q2)]]))
def test_issue_10847():
assert manualintegrate(x**2 / (x**2 - c),
x) == c * atan(x / sqrt(-c)) / sqrt(-c) + x
rc = Symbol('c', real=True)
assert manualintegrate(x**2 / (x**2 - rc), x) == \
rc*Piecewise((atan(x/sqrt(-rc))/sqrt(-rc), -rc > 0),
(-acoth(x/sqrt(rc))/sqrt(rc), And(-rc < 0, x**2 > rc)),
(-atanh(x/sqrt(rc))/sqrt(rc), And(-rc < 0, x**2 < rc))) + x
assert manualintegrate(sqrt(x - y) * log(z / x), x) == \
4*y**Rational(3, 2)*atan(sqrt(x - y)/sqrt(y))/3 - 4*y*sqrt(x - y)/3 +\
2*(x - y)**Rational(3, 2)*log(z/x)/3 + 4*(x - y)**Rational(3, 2)/9
ry = Symbol('y', real=True)
rz = Symbol('z', real=True)
assert manualintegrate(sqrt(x - ry) * log(rz / x), x) == \
4*ry**2*Piecewise((atan(sqrt(x - ry)/sqrt(ry))/sqrt(ry), ry > 0),
(-acoth(sqrt(x - ry)/sqrt(-ry))/sqrt(-ry), And(x - ry > -ry, ry < 0)),
(-atanh(sqrt(x - ry)/sqrt(-ry))/sqrt(-ry), And(x - ry < -ry, ry < 0)))/3 \
- 4*ry*sqrt(x - ry)/3 + 2*(x - ry)**Rational(3, 2)*log(rz/x)/3 \
+ 4*(x - ry)**Rational(3, 2)/9
assert manualintegrate(
sqrt(x) * log(x),
x) == 2 * x**Rational(3, 2) * log(x) / 3 - 4 * x**Rational(3, 2) / 9
assert manualintegrate(sqrt(a*x + b) / x, x) == \
2*b*atan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b) + 2*sqrt(a*x + b)
ra = Symbol('a', real=True)
rb = Symbol('b', real=True)
assert manualintegrate(sqrt(ra*x + rb) / x, x) == \
-2*rb*Piecewise((-atan(sqrt(ra*x + rb)/sqrt(-rb))/sqrt(-rb), -rb > 0),
(acoth(sqrt(ra*x + rb)/sqrt(rb))/sqrt(rb), And(-rb < 0, ra*x + rb > rb)),
(atanh(sqrt(ra*x + rb)/sqrt(rb))/sqrt(rb), And(-rb < 0, ra*x + rb < rb))) \
+ 2*sqrt(ra*x + rb)
assert expand(manualintegrate(sqrt(ra*x + rb) / (x + rc), x)) == -2*ra*rc*Piecewise((atan(sqrt(ra*x + rb)/sqrt(ra*rc - rb))/sqrt(ra*rc - rb), \
ra*rc - rb > 0), (-acoth(sqrt(ra*x + rb)/sqrt(-ra*rc + rb))/sqrt(-ra*rc + rb), And(ra*rc - rb < 0, ra*x + rb > -ra*rc + rb)), \
(-atanh(sqrt(ra*x + rb)/sqrt(-ra*rc + rb))/sqrt(-ra*rc + rb), And(ra*rc - rb < 0, ra*x + rb < -ra*rc + rb))) \
+ 2*rb*Piecewise((atan(sqrt(ra*x + rb)/sqrt(ra*rc - rb))/sqrt(ra*rc - rb), ra*rc - rb > 0), \
(-acoth(sqrt(ra*x + rb)/sqrt(-ra*rc + rb))/sqrt(-ra*rc + rb), And(ra*rc - rb < 0, ra*x + rb > -ra*rc + rb)), \
(-atanh(sqrt(ra*x + rb)/sqrt(-ra*rc + rb))/sqrt(-ra*rc + rb), And(ra*rc - rb < 0, ra*x + rb < -ra*rc + rb))) + 2*sqrt(ra*x + rb)
assert manualintegrate(
sqrt(2 * x + 3) / (x + 1),
x) == 2 * sqrt(2 * x + 3) - log(sqrt(2 * x + 3) +
1) + log(sqrt(2 * x + 3) - 1)
assert manualintegrate(
sqrt(2 * x + 3) / 2 * x, x
) == (2 * x + 3)**Rational(5, 2) / 20 - (2 * x + 3)**Rational(3, 2) / 4
assert manualintegrate(
x**Rational(3, 2) * log(x),
x) == 2 * x**Rational(5, 2) * log(x) / 5 - 4 * x**Rational(5, 2) / 25
assert manualintegrate(x**(-3) * log(x),
x) == -log(x) / (2 * x**2) - 1 / (4 * x**2)
assert manualintegrate(log(y)/(y**2*(1 - 1/y)), y) == \
log(y)*log(-1 + 1/y) - Integral(log(-1 + 1/y)/y, y)
def test_issues_5919_6830():
# issue 5919
n = -1 + 1 / x
z = n / x / (-n)**2 - 1 / n / x
assert expand(
z) == 1 / (x**2 - 2 * x + 1) - 1 / (x - 2 + 1 / x) - 1 / (-x + 1)
# issue 6830
p = (1 + x)**2
assert expand_multinomial(
(1 + x * p)**2) == (x**2 * (x**4 + 4 * x**3 + 6 * x**2 + 4 * x + 1) +
2 * x * (x**2 + 2 * x + 1) + 1)
assert expand_multinomial(
(1 + (y + x) * p)**2) == (2 * ((x + y) * (x**2 + 2 * x + 1)) +
(x**2 + 2 * x * y + y**2) *
(x**4 + 4 * x**3 + 6 * x**2 + 4 * x + 1) + 1)
A = Symbol('A', commutative=False)
p = (1 + A)**2
assert expand_multinomial(
(1 + x * p)**2) == (x**2 * (1 + 4 * A + 6 * A**2 + 4 * A**3 + A**4) +
2 * x * (1 + 2 * A + A**2) + 1)
assert expand_multinomial(
(1 + (y + x) * p)**2) == ((x + y) * (1 + 2 * A + A**2) * 2 +
(x**2 + 2 * x * y + y**2) *
(1 + 4 * A + 6 * A**2 + 4 * A**3 + A**4) + 1)
assert expand_multinomial((1 + (y + x) * p)**3) == (
(x + y) * (1 + 2 * A + A**2) * 3 + (x**2 + 2 * x * y + y**2) *
(1 + 4 * A + 6 * A**2 + 4 * A**3 + A**4) * 3 +
(x**3 + 3 * x**2 * y + 3 * x * y**2 + y**3) *
(1 + 6 * A + 15 * A**2 + 20 * A**3 + 15 * A**4 + 6 * A**5 + A**6) + 1)
# unevaluate powers
eq = (Pow((x + 1) * ((A + 1)**2), 2, evaluate=False))
# - in this case the base is not an Add so no further
# expansion is done
assert expand_multinomial(eq) == \
(x**2 + 2*x + 1)*(1 + 4*A + 6*A**2 + 4*A**3 + A**4)
# - but here, the expanded base *is* an Add so it gets expanded
eq = (Pow(((A + 1)**2), 2, evaluate=False))
assert expand_multinomial(eq) == 1 + 4 * A + 6 * A**2 + 4 * A**3 + A**4
# coverage
def ok(a, b, n):
e = (a + I * b)**n
return verify_numerically(e, expand_multinomial(e))
for a in [2, S.Half]:
for b in [3, R(1, 3)]:
for n in range(2, 6):
assert ok(a, b, n)
assert expand_multinomial((x + 1 + O(z))**2) == \
1 + 2*x + x**2 + O(z)
assert expand_multinomial((x + 1 + O(z))**3) == \
1 + 3*x + 3*x**2 + x**3 + O(z)
assert expand_multinomial(3**(x + y + 3)) == 27 * 3**(x + y)
def test__erfs():
assert _erfs(z).diff(z) == -2/sqrt(S.Pi) + 2*z*_erfs(z)
assert _erfs(1/z).series(z) == \
z/sqrt(pi) - z**3/(2*sqrt(pi)) + 3*z**5/(4*sqrt(pi)) + O(z**6)
assert expand(erf(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== erf(z).diff(z)
assert _erfs(z).rewrite("intractable") == (-erf(z) + 1)*exp(z**2)
raises(ArgumentIndexError, lambda: _erfs(z).fdiff(2))
def test_Piecewise():
e1 = x * (x + y) - y * (x + y)
e2 = sin(x)**2 + cos(x)**2
e3 = expand((x + y) * y / x)
# s1 = simplify(e1)
s2 = simplify(e2)
# s3 = simplify(e3)
# trigsimp tries not to touch non-trig containing args
assert trigsimp(Piecewise((e1, e3 < e2), (e3, True))) == \
Piecewise((e1, e3 < s2), (e3, True))
def test_two_dof_joints():
q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2')
m, c1, c2, k1, k2 = symbols('m c1 c2 k1 k2')
W = Body('W')
B1 = Body('B1', mass=m)
B2 = Body('B2', mass=m)
J1 = PrismaticJoint('J1', W, B1, coordinates=q1, speeds=u1)
J2 = PrismaticJoint('J2', B1, B2, coordinates=q2, speeds=u2)
W.apply_force(k1*q1*W.x, reaction_body=B1)
W.apply_force(c1*u1*W.x, reaction_body=B1)
B1.apply_force(k2*q2*W.x, reaction_body=B2)
B1.apply_force(c2*u2*W.x, reaction_body=B2)
method = JointsMethod(W, J1, J2)
method.form_eoms()
MM = method.mass_matrix
forcing = method.forcing
rhs = MM.LUsolve(forcing)
assert expand(rhs[0]) == expand((-k1 * q1 - c1 * u1 + k2 * q2 + c2 * u2)/m)
assert expand(rhs[1]) == expand((k1 * q1 + c1 * u1 - 2 * k2 * q2 - 2 *
c2 * u2) / m)
def test__eis():
assert _eis(z).diff(z) == -_eis(z) + 1/z
assert _eis(1/z).series(z) == \
z + z**2 + 2*z**3 + 6*z**4 + 24*z**5 + O(z**6)
assert Ei(z).rewrite('tractable') == exp(z)*_eis(z)
assert li(z).rewrite('tractable') == z*_eis(log(z))
assert _eis(z).rewrite('intractable') == exp(-z)*Ei(z)
assert expand(li(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== li(z).diff(z)
assert expand(Ei(z).rewrite('tractable').diff(z).rewrite('intractable')) \
== Ei(z).diff(z)
assert _eis(z).series(z, n=3) == EulerGamma + log(z) + z*(-log(z) - \
EulerGamma + 1) + z**2*(log(z)/2 - Rational(3, 4) + EulerGamma/2)\
+ O(z**3*log(z))
raises(ArgumentIndexError, lambda: _eis(z).fdiff(2))
def highest_row_degree_matrix(T,s):
"""
Returns a matrix containing with the coefficients of the a polynomial
including those which the coefficient is the Zero matrix
computes the highest row degree coefficient matrix of poynomial matrix A
A : polynomial Matrix
D :degree of matrix up to which we want to complete the list with zeroes
Example: TODO
"""
s=symbols('s')
RD=row_degrees(T,s)
return Matrix(T.rows,T.cols, lambda i,j: expand(T[i,j]).coeff(s,RD[i]))
def _equation(self, fx, x, order):
P, Q = self.wilds()
df = fx.diff(x)
eq = self.ode_problem.eq_expanded
c = Wild('c', exclude=[fx])
d = Wild('d', exclude=[df, fx.diff(x, 2)])
e = Wild('e', exclude=[df])
eq = expand(eq)
eq = eq.collect(fx)
r = eq.match(e * df + d)
self.fxx = None
self.gx = None
self.kx = None
if r:
r2 = r.copy()
r2[c] = S.Zero
if r2[d].is_Add:
# Separate the terms having f(x) to r[d] and
# remaining to r[c]
no_f, r2[d] = r2[d].as_independent(fx)
r2[c] += no_f
factor = simplify(r2[d].diff(fx) / r[e])
if factor and not factor.has(fx):
r2[d] = factor_terms(r2[d])
u = r2[d].as_independent(fx, as_Add=False)[1]
r2.update({'a': e, 'b': d, 'c': c, 'u': u})
r2[d] /= u
r2[e] /= u.diff(fx)
self.coeffs = r2
du = u.diff(x)
temp = du + (r2[r2['b']] /
r2[r2['a']]) * u - r2[r2['c']] / r2[r2['a']]
temp = expand(temp / temp.coeff(fx.diff(x)))
self.fxx = r2[e]
self.kx = r2[d]
self.gx = -r2[c]
self.substituting = u
return fx.diff(x) + P * fx - Q
return S.Zero
def test_factor_expand():
A = MatrixSymbol("A", n, n)
B = MatrixSymbol("B", n, n)
expr1 = (A + B)*(C + D)
expr2 = A*C + B*C + A*D + B*D
assert expr1 != expr2
assert expand(expr1) == expr2
assert factor(expr2) == expr1
expr = B**(-1)*(A**(-1)*B**(-1) - A**(-1)*C*B**(-1))**(-1)*A**(-1)
I = Identity(n)
# Ideally we get the first, but we at least don't want a wrong answer
assert factor(expr) in [I - C, B**-1*(A**-1*(I - C)*B**-1)**-1*A**-1]
def test_issues_5919_6830():
# issue 5919
n = -1 + 1/x
z = n/x/(-n)**2 - 1/n/x
assert expand(z) == 1/(x**2 - 2*x + 1) - 1/(x - 2 + 1/x) - 1/(-x + 1)
# issue 6830
p = (1 + x)**2
assert expand_multinomial((1 + x*p)**2) == (
x**2*(x**4 + 4*x**3 + 6*x**2 + 4*x + 1) + 2*x*(x**2 + 2*x + 1) + 1)
assert expand_multinomial((1 + (y + x)*p)**2) == (
2*((x + y)*(x**2 + 2*x + 1)) + (x**2 + 2*x*y + y**2)*
(x**4 + 4*x**3 + 6*x**2 + 4*x + 1) + 1)
A = Symbol('A', commutative=False)
p = (1 + A)**2
assert expand_multinomial((1 + x*p)**2) == (
x**2*(1 + 4*A + 6*A**2 + 4*A**3 + A**4) + 2*x*(1 + 2*A + A**2) + 1)
assert expand_multinomial((1 + (y + x)*p)**2) == (
(x + y)*(1 + 2*A + A**2)*2 + (x**2 + 2*x*y + y**2)*
(1 + 4*A + 6*A**2 + 4*A**3 + A**4) + 1)
assert expand_multinomial((1 + (y + x)*p)**3) == (
(x + y)*(1 + 2*A + A**2)*3 + (x**2 + 2*x*y + y**2)*(1 + 4*A +
6*A**2 + 4*A**3 + A**4)*3 + (x**3 + 3*x**2*y + 3*x*y**2 + y**3)*(1 + 6*A
+ 15*A**2 + 20*A**3 + 15*A**4 + 6*A**5 + A**6) + 1)
# unevaluate powers
eq = (Pow((x + 1)*((A + 1)**2), 2, evaluate=False))
# - in this case the base is not an Add so no further
# expansion is done
assert expand_multinomial(eq) == \
(x**2 + 2*x + 1)*(1 + 4*A + 6*A**2 + 4*A**3 + A**4)
# - but here, the expanded base *is* an Add so it gets expanded
eq = (Pow(((A + 1)**2), 2, evaluate=False))
assert expand_multinomial(eq) == 1 + 4*A + 6*A**2 + 4*A**3 + A**4
# coverage
def ok(a, b, n):
e = (a + I*b)**n
return verify_numerically(e, expand_multinomial(e))
for a in [2, S.Half]:
for b in [3, S(1)/3]:
for n in range(2, 6):
assert ok(a, b, n)
assert expand_multinomial((x + 1 + O(z))**2) == \
1 + 2*x + x**2 + O(z)
assert expand_multinomial((x + 1 + O(z))**3) == \
1 + 3*x + 3*x**2 + x**3 + O(z)
assert expand_multinomial(3**(x + y + 3)) == 27*3**(x + y)
def test_expand_non_commutative():
A = Symbol("A", commutative=False)
B = Symbol("B", commutative=False)
C = Symbol("C", commutative=False)
a = Symbol("a")
b = Symbol("b")
i = Symbol("i", integer=True)
n = Symbol("n", negative=True)
m = Symbol("m", negative=True)
p = Symbol("p", polar=True)
np = Symbol("p", polar=False)
assert (C * (A + B)).expand() == C * A + C * B
assert (C * (A + B)).expand() != A * C + B * C
assert ((A + B)**2).expand() == A**2 + A * B + B * A + B**2
assert ((A +
B)**3).expand() == (A**2 * B + B**2 * A + A * B**2 + B * A**2 +
A**3 + B**3 + A * B * A + B * A * B)
# issue 6219
assert ((a * A * B * A**-1)**2).expand() == a**2 * A * B**2 / A
# Note that (a*A*B*A**-1)**2 is automatically converted to a**2*(A*B*A**-1)**2
assert ((a * A * B *
A**-1)**2).expand(deep=False) == a**2 * (A * B * A**-1)**2
assert ((a * A * B * A**-1)**2).expand() == a**2 * (A * B**2 * A**-1)
assert ((a * A * B *
A**-1)**2).expand(force=True) == a**2 * A * B**2 * A**(-1)
assert ((a * A * B)**2).expand() == a**2 * A * B * A * B
assert ((a * A)**2).expand() == a**2 * A**2
assert ((a * A * B)**i).expand() == a**i * (A * B)**i
assert ((a * A *
(B *
(A * B / A)**2))**i).expand() == a**i * (A * B * A * B**2 / A)**i
# issue 6558
assert (A * B * (A * B)**-1).expand() == 1
assert ((a * A)**i).expand() == a**i * A**i
assert ((a * A * B * A**-1)**3).expand() == a**3 * A * B**3 / A
assert ((a * A * B * A * B / A)**3).expand() == a**3 * A * B * (
A * B**2) * (A * B**2) * A * B * A**(-1)
assert ((a * A * B * A * B / A)**-2).expand(
) == A * B**-1 * A**-1 * B**-2 * A**-1 * B**-1 * A**-1 / a**2
assert ((a * b * A * B *
A**-1)**i).expand() == a**i * b**i * (A * B / A)**i
assert ((a * (a * b)**i)**i).expand() == a**i * a**(i**2) * b**(i**2)
e = Pow(Mul(a, 1 / a, A, B, evaluate=False), S(2), evaluate=False)
assert e.expand() == A * B * A * B
assert sqrt(a * (A * b)**i).expand() == sqrt(a * b**i * A**i)
assert (sqrt(-a)**a).expand() == sqrt(-a)**a
assert expand((-2 * n)**(i / 3)) == 2**(i / 3) * (-n)**(i / 3)
assert expand(
(-2 * n * m)**(i / a)) == (-2)**(i / a) * (-n)**(i / a) * (-m)**(i / a)
assert expand((-2 * a * p)**b) == 2**b * p**b * (-a)**b
assert expand((-2 * a * np)**b) == 2**b * (-a * np)**b
assert expand(sqrt(A * B)) == sqrt(A * B)
assert expand(sqrt(-2 * a * b)) == sqrt(2) * sqrt(-a * b)
def _matches(self):
eq = self.ode_problem.eq_expanded
f = self.ode_problem.func.func
x = self.ode_problem.sym
order = self.ode_problem.order
df = f(x).diff(x)
if order != 1:
return False
eq = expand(eq / eq.coeff(df))
eq = eq.collect(f(x), func=cancel)
pattern = self._equation(f(x), x, 1)
self._wilds_match = match = eq.match(pattern)
return match is not None
def _cg_simp_add(e):
#TODO: Improve simplification method
"""Takes a sum of terms involving Clebsch-Gordan coefficients and
simplifies the terms.
Explanation
===========
First, we create two lists, cg_part, which is all the terms involving CG
coefficients, and other_part, which is all other terms. The cg_part list
is then passed to the simplification methods, which return the new cg_part
and any additional terms that are added to other_part
"""
cg_part = []
other_part = []
e = expand(e)
for arg in e.args:
if arg.has(CG):
if isinstance(arg, Sum):
other_part.append(_cg_simp_sum(arg))
elif isinstance(arg, Mul):
terms = 1
for term in arg.args:
if isinstance(term, Sum):
terms *= _cg_simp_sum(term)
else:
terms *= term
if terms.has(CG):
cg_part.append(terms)
else:
other_part.append(terms)
else:
cg_part.append(arg)
else:
other_part.append(arg)
cg_part, other = _check_varsh_871_1(cg_part)
other_part.append(other)
cg_part, other = _check_varsh_871_2(cg_part)
other_part.append(other)
cg_part, other = _check_varsh_872_9(cg_part)
other_part.append(other)
return Add(*cg_part) + Add(*other_part)
def sum_simplify(s):
"""Main function for Sum simplification"""
from sympy.concrete.summations import Sum
from sympy.core.function import expand
terms = Add.make_args(expand(s))
s_t = [] # Sum Terms
o_t = [] # Other Terms
for term in terms:
if isinstance(term, Mul):
other = 1
sum_terms = []
if not term.has(Sum):
o_t.append(term)
continue
mul_terms = Mul.make_args(term)
for mul_term in mul_terms:
if isinstance(mul_term, Sum):
r = mul_term._eval_simplify()
sum_terms.extend(Add.make_args(r))
else:
other = other * mul_term
if len(sum_terms):
#some simplification may have happened
#use if so
s_t.append(Mul(*sum_terms) * other)
else:
o_t.append(other)
elif isinstance(term, Sum):
#as above, we need to turn this into an add list
r = term._eval_simplify()
s_t.extend(Add.make_args(r))
else:
o_t.append(term)
result = Add(sum_combine(s_t), *o_t)
return result
def sum_simplify(s):
"""Main function for Sum simplification"""
from sympy.concrete.summations import Sum
from sympy.core.function import expand
terms = Add.make_args(expand(s))
s_t = [] # Sum Terms
o_t = [] # Other Terms
for term in terms:
if isinstance(term, Mul):
other = 1
sum_terms = []
if not term.has(Sum):
o_t.append(term)
continue
mul_terms = Mul.make_args(term)
for mul_term in mul_terms:
if isinstance(mul_term, Sum):
r = mul_term._eval_simplify()
sum_terms.extend(Add.make_args(r))
else:
other = other * mul_term
if len(sum_terms):
#some simplification may have happened
#use if so
s_t.append(Mul(*sum_terms) * other)
else:
o_t.append(other)
elif isinstance(term, Sum):
#as above, we need to turn this into an add list
r = term._eval_simplify()
s_t.extend(Add.make_args(r))
else:
o_t.append(term)
result = Add(sum_combine(s_t), *o_t)
return result
def entropy(density):
"""Compute the entropy of a matrix/density object.
This computes -Tr(density*ln(density)) using the eigenvalue decomposition
of density, which is given as either a Density instance or a matrix
(numpy.ndarray, sympy.Matrix or scipy.sparse).
Parameters
==========
density : density matrix of type Density, SymPy matrix,
scipy.sparse or numpy.ndarray
Examples
========
>>> from sympy.physics.quantum.density import Density, entropy
>>> from sympy.physics.quantum.spin import JzKet
>>> from sympy import S
>>> up = JzKet(S(1)/2,S(1)/2)
>>> down = JzKet(S(1)/2,-S(1)/2)
>>> d = Density((up,S(1)/2),(down,S(1)/2))
>>> entropy(d)
log(2)/2
"""
if isinstance(density, Density):
density = represent(density) # represent in Matrix
if isinstance(density, scipy_sparse_matrix):
density = to_numpy(density)
if isinstance(density, Matrix):
eigvals = density.eigenvals().keys()
return expand(-sum(e * log(e) for e in eigvals))
elif isinstance(density, numpy_ndarray):
import numpy as np
eigvals = np.linalg.eigvals(density)
return -np.sum(eigvals * np.log(eigvals))
else:
raise ValueError(
"numpy.ndarray, scipy.sparse or SymPy matrix expected")
def _matches(self):
eq = self.ode_problem.eq_expanded
f = self.ode_problem.func.func
x = self.ode_problem.sym
order = self.ode_problem.order
df = f(x).diff(x)
if order != 1:
return False
pattern = self._equation(f(x), x, 1)
if not pattern.coeff(df).has(Wild):
eq = expand(eq / eq.coeff(df))
eq = eq.collect([f(x).diff(x)], func = cancel)
self._wilds_match = match = eq.match(pattern)
if match is not None:
return self._verify(f(x))
return False
def match_2nd_hypergeometric(eq, func):
x = func.args[0]
df = func.diff(x)
a3 = Wild('a3', exclude=[func, func.diff(x), func.diff(x, 2)])
b3 = Wild('b3', exclude=[func, func.diff(x), func.diff(x, 2)])
c3 = Wild('c3', exclude=[func, func.diff(x), func.diff(x, 2)])
deq = a3 * (func.diff(x, 2)) + b3 * df + c3 * func
r = collect(eq, [func.diff(x, 2), func.diff(x), func]).match(deq)
if r:
if not all(val.is_polynomial() for val in r.values()):
n, d = eq.as_numer_denom()
eq = expand(n)
r = collect(eq, [func.diff(x, 2), func.diff(x), func]).match(deq)
if r and r[a3] != 0:
A = cancel(r[b3] / r[a3])
B = cancel(r[c3] / r[a3])
return [A, B]
else:
return []
def test_expand_non_commutative():
A = Symbol('A', commutative=False)
B = Symbol('B', commutative=False)
C = Symbol('C', commutative=False)
a = Symbol('a')
b = Symbol('b')
i = Symbol('i', integer=True)
n = Symbol('n', negative=True)
m = Symbol('m', negative=True)
p = Symbol('p', polar=True)
np = Symbol('p', polar=False)
assert (C*(A + B)).expand() == C*A + C*B
assert (C*(A + B)).expand() != A*C + B*C
assert ((A + B)**2).expand() == A**2 + A*B + B*A + B**2
assert ((A + B)**3).expand() == (A**2*B + B**2*A + A*B**2 + B*A**2 +
A**3 + B**3 + A*B*A + B*A*B)
# issue 6219
assert ((a*A*B*A**-1)**2).expand() == a**2*A*B**2/A
# Note that (a*A*B*A**-1)**2 is automatically converted to a**2*(A*B*A**-1)**2
assert ((a*A*B*A**-1)**2).expand(deep=False) == a**2*(A*B*A**-1)**2
assert ((a*A*B*A**-1)**2).expand() == a**2*(A*B**2*A**-1)
assert ((a*A*B*A**-1)**2).expand(force=True) == a**2*A*B**2*A**(-1)
assert ((a*A*B)**2).expand() == a**2*A*B*A*B
assert ((a*A)**2).expand() == a**2*A**2
assert ((a*A*B)**i).expand() == a**i*(A*B)**i
assert ((a*A*(B*(A*B/A)**2))**i).expand() == a**i*(A*B*A*B**2/A)**i
# issue 6558
assert (A*B*(A*B)**-1).expand() == A*B*(A*B)**-1
assert ((a*A)**i).expand() == a**i*A**i
assert ((a*A*B*A**-1)**3).expand() == a**3*A*B**3/A
assert ((a*A*B*A*B/A)**3).expand() == \
a**3*A*B*(A*B**2)*(A*B**2)*A*B*A**(-1)
assert ((a*A*B*A*B/A)**-3).expand() == \
a**-3*(A*B*(A*B**2)*(A*B**2)*A*B*A**(-1))**-1
assert ((a*b*A*B*A**-1)**i).expand() == a**i*b**i*(A*B/A)**i
assert ((a*(a*b)**i)**i).expand() == a**i*a**(i**2)*b**(i**2)
e = Pow(Mul(a, 1/a, A, B, evaluate=False), S(2), evaluate=False)
assert e.expand() == A*B*A*B
assert sqrt(a*(A*b)**i).expand() == sqrt(a*b**i*A**i)
assert (sqrt(-a)**a).expand() == sqrt(-a)**a
assert expand((-2*n)**(i/3)) == 2**(i/3)*(-n)**(i/3)
assert expand((-2*n*m)**(i/a)) == (-2)**(i/a)*(-n)**(i/a)*(-m)**(i/a)
assert expand((-2*a*p)**b) == 2**b*p**b*(-a)**b
assert expand((-2*a*np)**b) == 2**b*(-a*np)**b
assert expand(sqrt(A*B)) == sqrt(A*B)
assert expand(sqrt(-2*a*b)) == sqrt(2)*sqrt(-a*b)
def classify_pde(eq, func=None, dict=False, **kwargs):
"""
Returns a tuple of possible pdsolve() classifications for a PDE.
The tuple is ordered so that first item is the classification that
pdsolve() uses to solve the PDE by default. In general,
classifications at the near the beginning of the list will produce
better solutions faster than those near the end, thought there are
always exceptions. To make pdsolve use a different classification,
use pdsolve(PDE, func, hint=<classification>). See also the pdsolve()
docstring for different meta-hints you can use.
If ``dict`` is true, classify_pde() will return a dictionary of
hint:match expression terms. This is intended for internal use by
pdsolve(). Note that because dictionaries are ordered arbitrarily,
this will most likely not be in the same order as the tuple.
You can get help on different hints by doing help(pde.pde_hintname),
where hintname is the name of the hint without "_Integral".
See sympy.pde.allhints or the sympy.pde docstring for a list of all
supported hints that can be returned from classify_pde.
Examples
========
>>> from sympy.solvers.pde import classify_pde
>>> from sympy import Function, diff, Eq
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> u = f(x, y)
>>> ux = u.diff(x)
>>> uy = u.diff(y)
>>> eq = Eq(1 + (2*(ux/u)) + (3*(uy/u)))
>>> classify_pde(eq)
('1st_linear_constant_coeff_homogeneous',)
"""
prep = kwargs.pop('prep', True)
if func and len(func.args) != 2:
raise NotImplementedError("Right now only partial "
"differential equations of two variables are supported")
if prep or func is None:
prep, func_ = _preprocess(eq, func)
if func is None:
func = func_
if isinstance(eq, Equality):
if eq.rhs != 0:
return classify_pde(eq.lhs - eq.rhs, func)
eq = eq.lhs
f = func.func
x = func.args[0]
y = func.args[1]
fx = f(x,y).diff(x)
fy = f(x,y).diff(y)
# TODO : For now pde.py uses support offered by the ode_order function
# to find the order with respect to a multi-variable function. An
# improvement could be to classify the order of the PDE on the basis of
# individual variables.
order = ode_order(eq, f(x,y))
# hint:matchdict or hint:(tuple of matchdicts)
# Also will contain "default":<default hint> and "order":order items.
matching_hints = {'order': order}
if not order:
if dict:
matching_hints["default"] = None
return matching_hints
else:
return ()
eq = expand(eq)
a = Wild('a', exclude = [f(x,y)])
b = Wild('b', exclude = [f(x,y), fx, fy, x, y])
c = Wild('c', exclude = [f(x,y), fx, fy, x, y])
d = Wild('d', exclude = [f(x,y), fx, fy, x, y])
e = Wild('e', exclude = [f(x,y), fx, fy])
n = Wild('n', exclude = [x, y])
# Try removing the smallest power of f(x,y)
# from the highest partial derivatives of f(x,y)
reduced_eq = None
if eq.is_Add:
var = set(combinations_with_replacement((x,y), order))
dummyvar = deepcopy(var)
power = None
for i in var:
coeff = eq.coeff(f(x,y).diff(*i))
if coeff != 1:
match = coeff.match(a*f(x,y)**n)
if match and match[a]:
power = match[n]
dummyvar.remove(i)
break
dummyvar.remove(i)
for i in dummyvar:
coeff = eq.coeff(f(x,y).diff(*i))
if coeff != 1:
match = coeff.match(a*f(x,y)**n)
if match and match[a] and match[n] < power:
power = match[n]
if power:
den = f(x,y)**power
reduced_eq = Add(*[arg/den for arg in eq.args])
if not reduced_eq:
reduced_eq = eq
if order == 1:
reduced_eq = collect(reduced_eq, f(x, y))
r = reduced_eq.match(b*fx + c*fy + d*f(x,y) + e)
if r:
if not r[e]:
## Linear first-order homogeneous partial-differential
## equation with constant coefficients
r.update({'b': b, 'c': c, 'd': d})
matching_hints["1st_linear_constant_coeff_homogeneous"] = r
else:
if r[b]**2 + r[c]**2 != 0:
## Linear first-order general partial-differential
## equation with constant coefficients
r.update({'b': b, 'c': c, 'd': d, 'e': e})
matching_hints["1st_linear_constant_coeff"] = r
matching_hints[
"1st_linear_constant_coeff_Integral"] = r
else:
b = Wild('b', exclude=[f(x, y), fx, fy])
c = Wild('c', exclude=[f(x, y), fx, fy])
d = Wild('d', exclude=[f(x, y), fx, fy])
r = reduced_eq.match(b*fx + c*fy + d*f(x,y) + e)
if r:
r.update({'b': b, 'c': c, 'd': d, 'e': e})
matching_hints["1st_linear_variable_coeff"] = r
# Order keys based on allhints.
retlist = []
for i in allhints:
if i in matching_hints:
retlist.append(i)
if dict:
# Dictionaries are ordered arbitrarily, so make note of which
# hint would come first for pdsolve(). Use an ordered dict in Py 3.
matching_hints["default"] = None
matching_hints["ordered_hints"] = tuple(retlist)
for i in allhints:
if i in matching_hints:
matching_hints["default"] = i
break
return matching_hints
else:
return tuple(retlist)
def find_linear_recurrence(self, n, d=None, gfvar=None):
r"""
Finds the shortest linear recurrence that satisfies the first n
terms of sequence of order `\leq` ``n/2`` if possible.
If ``d`` is specified, find shortest linear recurrence of order
`\leq` min(d, n/2) if possible.
Returns list of coefficients ``[b(1), b(2), ...]`` corresponding to the
recurrence relation ``x(n) = b(1)*x(n-1) + b(2)*x(n-2) + ...``
Returns ``[]`` if no recurrence is found.
If gfvar is specified, also returns ordinary generating function as a
function of gfvar.
Examples
========
>>> from sympy import sequence, sqrt, oo, lucas
>>> from sympy.abc import n, x, y
>>> sequence(n**2).find_linear_recurrence(10, 2)
[]
>>> sequence(n**2).find_linear_recurrence(10)
[3, -3, 1]
>>> sequence(2**n).find_linear_recurrence(10)
[2]
>>> sequence(23*n**4+91*n**2).find_linear_recurrence(10)
[5, -10, 10, -5, 1]
>>> sequence(sqrt(5)*(((1 + sqrt(5))/2)**n - (-(1 + sqrt(5))/2)**(-n))/5).find_linear_recurrence(10)
[1, 1]
>>> sequence(x+y*(-2)**(-n), (n, 0, oo)).find_linear_recurrence(30)
[1/2, 1/2]
>>> sequence(3*5**n + 12).find_linear_recurrence(20,gfvar=x)
([6, -5], 3*(5 - 21*x)/((x - 1)*(5*x - 1)))
>>> sequence(lucas(n)).find_linear_recurrence(15,gfvar=x)
([1, 1], (x - 2)/(x**2 + x - 1))
"""
from sympy.matrices import Matrix
x = [simplify(expand(t)) for t in self[:n]]
lx = len(x)
if d is None:
r = lx // 2
else:
r = min(d, lx // 2)
coeffs = []
for l in range(1, r + 1):
l2 = 2 * l
mlist = []
for k in range(l):
mlist.append(x[k:k + l])
m = Matrix(mlist)
if m.det() != 0:
y = simplify(m.LUsolve(Matrix(x[l:l2])))
if lx == l2:
coeffs = flatten(y[::-1])
break
mlist = []
for k in range(l, lx - l):
mlist.append(x[k:k + l])
m = Matrix(mlist)
if m * y == Matrix(x[l2:]):
coeffs = flatten(y[::-1])
break
if gfvar is None:
return coeffs
else:
l = len(coeffs)
if l == 0:
return [], None
else:
n, d = x[l - 1] * gfvar**(l - 1), 1 - coeffs[l - 1] * gfvar**l
for i in range(l - 1):
n += x[i] * gfvar**i
for j in range(l - i - 1):
n -= coeffs[i] * x[j] * gfvar**(i + j + 1)
d -= coeffs[i] * gfvar**(i + 1)
return coeffs, simplify(factor(n) / factor(d))
def test_meijerint():
from sympy.core.function import expand
from sympy.core.symbol import symbols
from sympy.functions.elementary.complexes import arg
s, t, mu = symbols('s t mu', real=True)
assert integrate(
meijerg([], [], [0], [], s * t) *
meijerg([], [], [mu / 2], [-mu / 2], t**2 / 4),
(t, 0, oo)).is_Piecewise
s = symbols('s', positive=True)
assert integrate(x**s*meijerg([[], []], [[0], []], x), (x, 0, oo)) == \
gamma(s + 1)
assert integrate(x**s * meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=True) == gamma(s + 1)
assert isinstance(
integrate(x**s * meijerg([[], []], [[0], []], x), (x, 0, oo),
meijerg=False), Integral)
assert meijerint_indefinite(exp(x), x) == exp(x)
# TODO what simplifications should be done automatically?
# This tests "extra case" for antecedents_1.
a, b = symbols('a b', positive=True)
assert simplify(meijerint_definite(x**a, x, 0, b)[0]) == \
b**(a + 1)/(a + 1)
# This tests various conditions and expansions:
assert meijerint_definite((x + 1)**3 * exp(-x), x, 0, oo) == (16, True)
# Again, how about simplifications?
sigma, mu = symbols('sigma mu', positive=True)
i, c = meijerint_definite(exp(-((x - mu) / (2 * sigma))**2), x, 0, oo)
assert simplify(i) == sqrt(pi) * sigma * (2 - erfc(mu / (2 * sigma)))
assert c == True
i, _ = meijerint_definite(exp(-mu * x) * exp(sigma * x), x, 0, oo)
# TODO it would be nice to test the condition
assert simplify(i) == 1 / (mu - sigma)
# Test substitutions to change limits
assert meijerint_definite(exp(x), x, -oo, 2) == (exp(2), True)
# Note: causes a NaN in _check_antecedents
assert expand(meijerint_definite(exp(x), x, 0, I)[0]) == exp(I) - 1
assert expand(meijerint_definite(exp(-x), x, 0, x)[0]) == \
1 - exp(-exp(I*arg(x))*abs(x))
# Test -oo to oo
assert meijerint_definite(exp(-x**2), x, -oo, oo) == (sqrt(pi), True)
assert meijerint_definite(exp(-abs(x)), x, -oo, oo) == (2, True)
assert meijerint_definite(exp(-(2*x - 3)**2), x, -oo, oo) == \
(sqrt(pi)/2, True)
assert meijerint_definite(exp(-abs(2 * x - 3)), x, -oo, oo) == (1, True)
assert meijerint_definite(
exp(-((x - mu) / sigma)**2 / 2) / sqrt(2 * pi * sigma**2), x, -oo,
oo) == (1, True)
assert meijerint_definite(sinc(x)**2, x, -oo, oo) == (pi, True)
# Test one of the extra conditions for 2 g-functinos
assert meijerint_definite(exp(-x) * sin(x), x, 0, oo) == (S.Half, True)
# Test a bug
def res(n):
return (1 / (1 + x**2)).diff(x, n).subs(x, 1) * (-1)**n
for n in range(6):
assert integrate(exp(-x)*sin(x)*x**n, (x, 0, oo), meijerg=True) == \
res(n)
# This used to test trigexpand... now it is done by linear substitution
assert simplify(integrate(exp(-x) * sin(x + a), (x, 0, oo),
meijerg=True)) == sqrt(2) * sin(a + pi / 4) / 2
# Test the condition 14 from prudnikov.
# (This is besselj*besselj in disguise, to stop the product from being
# recognised in the tables.)
a, b, s = symbols('a b s')
from sympy.functions.elementary.complexes import re
assert meijerint_definite(
meijerg([], [], [a / 2], [-a / 2], x / 4) *
meijerg([], [], [b / 2], [-b / 2], x / 4) * x**(s - 1), x, 0,
oo) == ((4 * 2**(2 * s - 2) * gamma(-2 * s + 1) *
gamma(a / 2 + b / 2 + s) /
(gamma(-a / 2 + b / 2 - s + 1) *
gamma(a / 2 - b / 2 - s + 1) * gamma(a / 2 + b / 2 - s + 1)),
(re(s) < 1) & (re(s) < S(1) / 2) &
(re(a) / 2 + re(b) / 2 + re(s) > 0)))
# test a bug
assert integrate(sin(x**a)*sin(x**b), (x, 0, oo), meijerg=True) == \
Integral(sin(x**a)*sin(x**b), (x, 0, oo))
# test better hyperexpand
assert integrate(exp(-x**2)*log(x), (x, 0, oo), meijerg=True) == \
(sqrt(pi)*polygamma(0, S.Half)/4).expand()
# Test hyperexpand bug.
from sympy.functions.special.gamma_functions import lowergamma
n = symbols('n', integer=True)
assert simplify(integrate(exp(-x)*x**n, x, meijerg=True)) == \
lowergamma(n + 1, x)
# Test a bug with argument 1/x
alpha = symbols('alpha', positive=True)
assert meijerint_definite((2 - x)**alpha*sin(alpha/x), x, 0, 2) == \
(sqrt(pi)*alpha*gamma(alpha + 1)*meijerg(((), (alpha/2 + S.Half,
alpha/2 + 1)), ((0, 0, S.Half), (Rational(-1, 2),)), alpha**2/16)/4, True)
# test a bug related to 3016
a, s = symbols('a s', positive=True)
assert simplify(integrate(x**s*exp(-a*x**2), (x, -oo, oo))) == \
a**(-s/2 - S.Half)*((-1)**s + 1)*gamma(s/2 + S.Half)/2
def test_expand_log():
t = Symbol('t', positive=True)
# after first expansion, -2*log(2) + log(4); then 0 after second
assert expand(log(t**2) - log(t**2/4) - 2*log(2)) == 0
def eq_expanded(self) -> Expr:
return expand(self.eq_preprocessed)
def solveset_complex(f, symbol):
""" Solve a complex valued equation.
Parameters
==========
f : Expr
The target equation
symbol : Symbol
The variable for which the equation is solved
Returns
=======
Set
A set of values for `symbol` for which `f` equal to
zero. An `EmptySet` is returned if no solution is found.
A `ConditionSet` is returned as an unsolved object if algorithms
to evaluate complete solutions are not yet implemented.
`solveset_complex` claims to be complete in the solution set that
it returns.
Raises
======
NotImplementedError
The algorithms to solve inequalities in complex domain are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
See Also
========
solveset_real: solver for real domain
Examples
========
>>> from sympy import Symbol, exp
>>> from sympy.solvers.solveset import solveset_complex
>>> from sympy.abc import x, a, b, c
>>> solveset_complex(a*x**2 + b*x +c, x)
{-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a)}
* Due to the fact that complex extension of my real valued functions are
multivariate even some simple equations can have infinitely many
solution.
>>> solveset_complex(exp(x) - 1, x)
ImageSet(Lambda(_n, 2*_n*I*pi), Integers())
"""
if not symbol.is_Symbol:
raise ValueError(" %s is not a symbol" % (symbol))
f = sympify(f)
original_eq = f
if not isinstance(f, (Expr, Number)):
raise ValueError(" %s is not a valid sympy expression" % (f))
f = together(f)
# Without this equations like a + 4*x**2 - E keep oscillating
# into form a/4 + x**2 - E/4 and (a + 4*x**2 - E)/4
if not fraction(f)[1].has(symbol):
f = expand(f)
if f.is_zero:
return S.Complexes
elif not f.has(symbol):
result = EmptySet()
elif f.is_Mul and all([_is_finite_with_finite_vars(m) for m in f.args]):
result = Union(*[solveset_complex(m, symbol) for m in f.args])
else:
lhs, rhs_s = invert_complex(f, 0, symbol)
if lhs == symbol:
result = rhs_s
elif isinstance(rhs_s, FiniteSet):
equations = [lhs - rhs for rhs in rhs_s]
result = EmptySet()
for equation in equations:
if equation == f:
if any(
_has_rational_power(g, symbol)[0]
for g in equation.args):
result += _solve_radical(equation, symbol,
solveset_complex)
else:
result += _solve_as_rational(
equation,
symbol,
solveset_solver=solveset_complex,
as_poly_solver=_solve_as_poly_complex)
else:
result += solveset_complex(equation, symbol)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Complexes)
if isinstance(result, FiniteSet):
result = [
s for s in result
if isinstance(s, RootOf) or domain_check(original_eq, symbol, s)
]
return FiniteSet(*result)
else:
return result
def solveset_real(f, symbol):
""" Solves a real valued equation.
Parameters
==========
f : Expr
The target equation
symbol : Symbol
The variable for which the equation is solved
Returns
=======
Set
A set of values for `symbol` for which `f` is equal to
zero. An `EmptySet` is returned if no solution is found.
A `ConditionSet` is returned as unsolved object if algorithms
to evaluate complete solutions are not yet implemented.
`solveset_real` claims to be complete in the set of the solution it
returns.
Raises
======
NotImplementedError
Algorithms to solve inequalities in complex domain are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
See Also
=======
solveset_complex : solver for complex domain
Examples
========
>>> from sympy import Symbol, exp, sin, sqrt, I
>>> from sympy.solvers.solveset import solveset_real
>>> x = Symbol('x', real=True)
>>> a = Symbol('a', real=True, finite=True, positive=True)
>>> solveset_real(x**2 - 1, x)
{-1, 1}
>>> solveset_real(sqrt(5*x + 6) - 2 - x, x)
{-1, 2}
>>> solveset_real(x - I, x)
EmptySet()
>>> solveset_real(x - a, x)
{a}
>>> solveset_real(exp(x) - a, x)
{log(a)}
* In case the equation has infinitely many solutions an infinitely indexed
`ImageSet` is returned.
>>> solveset_real(sin(x) - 1, x)
ImageSet(Lambda(_n, 2*_n*pi + pi/2), Integers())
* If the equation is true for any arbitrary value of the symbol a `S.Reals`
set is returned.
>>> solveset_real(x - x, x)
(-oo, oo)
"""
if not symbol.is_Symbol:
raise ValueError(" %s is not a symbol" % (symbol))
f = sympify(f)
if not isinstance(f, (Expr, Number)):
raise ValueError(" %s is not a valid sympy expression" % (f))
original_eq = f
f = together(f)
# In this, unlike in solveset_complex, expression should only
# be expanded when fraction(f)[1] does not contain the symbol
# for which we are solving
if not symbol in fraction(f)[1].free_symbols and f.is_rational_function():
f = expand(f)
if f.has(Piecewise):
f = piecewise_fold(f)
result = EmptySet()
if f.expand().is_zero:
return S.Reals
elif not f.has(symbol):
return EmptySet()
elif f.is_Mul and all([_is_finite_with_finite_vars(m) for m in f.args]):
# if f(x) and g(x) are both finite we can say that the solution of
# f(x)*g(x) == 0 is same as Union(f(x) == 0, g(x) == 0) is not true in
# general. g(x) can grow to infinitely large for the values where
# f(x) == 0. To be sure that we are not silently allowing any
# wrong solutions we are using this technique only if both f and g are
# finite for a finite input.
result = Union(*[solveset_real(m, symbol) for m in f.args])
elif _is_function_class_equation(TrigonometricFunction, f, symbol) or \
_is_function_class_equation(HyperbolicFunction, f, symbol):
result = _solve_real_trig(f, symbol)
elif f.is_Piecewise:
result = EmptySet()
expr_set_pairs = f.as_expr_set_pairs()
for (expr, in_set) in expr_set_pairs:
solns = solveset_real(expr, symbol).intersect(in_set)
result = result + solns
else:
lhs, rhs_s = invert_real(f, 0, symbol)
if lhs == symbol:
result = rhs_s
elif isinstance(rhs_s, FiniteSet):
equations = [lhs - rhs for rhs in rhs_s]
for equation in equations:
if equation == f:
if any(_has_rational_power(g, symbol)[0]
for g in equation.args):
result += _solve_radical(equation,
symbol,
solveset_real)
elif equation.has(Abs):
result += _solve_abs(f, symbol)
else:
result += _solve_as_rational(equation, symbol,
solveset_solver=solveset_real,
as_poly_solver=_solve_as_poly_real)
else:
result += solveset_real(equation, symbol)
else:
result = ConditionSet(symbol, Eq(f, 0), S.Reals)
if isinstance(result, FiniteSet):
result = [s for s in result
if isinstance(s, RootOf)
or domain_check(original_eq, symbol, s)]
return FiniteSet(*result).intersect(S.Reals)
else:
return result.intersect(S.Reals)
def solveset_complex(f, symbol):
""" Solve a complex valued equation.
Parameters
==========
f : Expr
The target equation
symbol : Symbol
The variable for which the equation is solved
Returns
=======
Set
A set of values for `symbol` for which `f` equal to
zero. An `EmptySet` is returned if no solution is found.
`solveset_complex` claims to be complete in the solution set that
it returns.
Raises
======
NotImplementedError
The algorithms for to find the solution of the given equation are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
See Also
========
solveset_real: solver for real domain
Examples
========
>>> from sympy import Symbol, exp
>>> from sympy.solvers.solveset import solveset_complex
>>> from sympy.abc import x, a, b, c
>>> solveset_complex(a*x**2 + b*x +c, x)
{-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a), -b/(2*a) + sqrt(-4*a*c + b**2)/(2*a)}
Due to the fact that complex extension of my real valued functions are
multivariate even some simple equations can have infinitely many solution.
>>> solveset_complex(exp(x) - 1, x)
ImageSet(Lambda(_n, 2*_n*I*pi), Integers())
"""
if not symbol.is_Symbol:
raise ValueError(" %s is not a symbol" % (symbol))
f = sympify(f)
original_eq = f
if not isinstance(f, (Expr, Number)):
raise ValueError(" %s is not a valid sympy expression" % (f))
f = together(f)
# Without this equations like a + 4*x**2 - E keep oscillating
# into form a/4 + x**2 - E/4 and (a + 4*x**2 - E)/4
if not fraction(f)[1].has(symbol):
f = expand(f)
if f.is_zero:
raise NotImplementedError("S.Complex set is not yet implemented")
elif not f.has(symbol):
result = EmptySet()
elif f.is_Mul and all([_is_finite_with_finite_vars(m) for m in f.args]):
result = Union(*[solveset_complex(m, symbol) for m in f.args])
else:
lhs, rhs_s = invert_complex(f, 0, symbol)
if lhs == symbol:
result = rhs_s
elif isinstance(rhs_s, FiniteSet):
equations = [lhs - rhs for rhs in rhs_s]
result = EmptySet()
for equation in equations:
if equation == f:
result += _solve_as_rational(equation, symbol,
solveset_solver=solveset_complex,
as_poly_solver=_solve_as_poly_complex)
else:
result += solveset_complex(equation, symbol)
else:
raise NotImplementedError
if isinstance(result, FiniteSet):
result = [s for s in result
if isinstance(s, RootOf)
or domain_check(original_eq, symbol, s)]
return FiniteSet(*result)
else:
return result
def test_expand_log():
t = Symbol('t', positive=True)
# after first expansion, -2*log(2) + log(4); then 0 after second
assert expand(log(t**2) - log(t**2/4) - 2*log(2)) == 0
def expand(self, *args, **kwargs):
return SeqFormula(expand(self.formula, *args, **kwargs), self.args[1])
def ALGO4(As,Bs,Cs,Ds,do_test):
#-------------------STEP 1------------------------------
if not is_row_proper(As):
Us,Ast=row_proper(As)
else:
Us,Ast=eye(As.rows),As
Bst=Us*Bs
Bst=expand(Bst)
r=Ast.cols
#-------------------STEP 2------------------------------
K=simplify(Ast.inv()*Bst) #very important
Ys=zeros(K.shape)
for i,j in product(range(K.rows),range(K.cols)):
Ys[i,j],q=div(numer(K[i,j]),denom(K[i,j]))
B_hat=Bst-Ast*Ys
#-------------------END STEP 2------------------------------
#-------------------STEP 3------------------------------
Psi=diag(*[[s**( mc.row_degrees(Ast,s)[j] -i -1) for i in range( mc.row_degrees(Ast,s)[j])] for j in range(r)]).T
S=diag(*[s**(rho) for rho in mc.row_degrees(Ast,s)])
Ahr=mc.highest_row_degree_matrix(Ast,s)
Help=Ast-S*Ahr
SOL={}
numvar=Psi.rows*Psi.cols
alr=symbols('a0:%d'%numvar)
Alr=Matrix(Psi.cols,Psi.rows,alr)
RHS=Psi*Alr
for i,j in product(range(Help.rows),range(Help.cols)): #diagonal explain later
SOL.update(solve_undetermined_coeffs(Eq(Help[i,j],RHS[i,j]),alr,s))
Alr=Alr.subs(SOL) #substitute(SOL)
Aoc=Matrix(BlockDiagMatrix(*[Matrix(rho, rho, lambda i,j: KroneckerDelta(i+1,j))for rho in mc.row_degrees(Ast,s)]))
Boc=eye(sum(mc.row_degrees(Ast,s)))
Coc=Matrix(BlockDiagMatrix(*[SparseMatrix(1,rho,{(x,0):1 if x==0 else 0 for x in range(rho)}) for rho in mc.row_degrees(Ast,s)]))
A0=Aoc-Alr*Ahr.inv()*Matrix(Coc)
C0=Ahr.inv()*Coc
SOL={}
numvar=Psi.cols*Bst.cols
b0=symbols('b0:%d'%numvar)
B0=Matrix(Psi.cols,Bst.cols,b0)
RHS=Psi*B0
for i,j in product(range(B_hat.rows),range(B_hat.cols)): #diagonal explain later
SOL.update(solve_undetermined_coeffs(Eq(B_hat[i,j],RHS[i,j]),b0,s))
B0=B0.subs(SOL) #substitute(SOL)
LHS_matrix=simplify(Cs*C0) #left hand side of the equation (1)
sI_A=s*eye(A0.cols)- A0
max_degree=mc.find_degree(LHS_matrix,s) #get the degree of the matrix at the LHS
#which is also the maximum degree for the coefficients of Λ(s)
#---------------------------Creating Matrices Λ(s) and C -------------------------------------
Lamda=[]
numvar=((max_degree))*A0.cols
a=symbols('a0:%d'%numvar)
for i in range(A0.cols): # paratirisi den douleuei to prin giat;i otra oxi diagonios
p=sum(a[n +i*(max_degree)]*s**n for n in range(max_degree)) # we want variables one degree lower because we are multiplying by first order monomials
Lamda.append(p)
Lamda=Matrix(Cs.rows,A0.cols,Lamda) #convert the list to Matrix
c=symbols('c0:%d'%(Lamda.rows*Lamda.cols))
C=Matrix(Lamda.rows,Lamda.cols,c)
#-----------------------------------------
RHS_matrix=Lamda*sI_A +C #right hand side of the equation (1)
'''
-----------Converting equation (1) to a system of linear -----------
-----------equations, comparing the coefficients of the -----------
-----------polynomials in both sides of the equation (1) -----------
'''
EQ=[Eq(LHS_matrix[i,j],expand(RHS_matrix[i,j])) for i,j in product(range(LHS_matrix.rows),range(LHS_matrix.cols)) ]
