def __init__(self, factors=None): # Factors
"""Initialize Factors from dict or expr.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x
>>> from sympy import I
>>> e = 2*x**3
>>> Factors(e)
Factors({2: 1, x: 3})
>>> Factors(e.as_powers_dict())
Factors({2: 1, x: 3})
>>> f = _
>>> f.factors # underlying dictionary
{2: 1, x: 3}
>>> f.gens # base of each factor
frozenset({2, x})
>>> Factors(0)
Factors({0: 1})
>>> Factors(I)
Factors({I: 1})
Notes
=====
Although a dictionary can be passed, only minimal checking is
performed: powers of -1 and I are made canonical.
"""
if isinstance(factors, (SYMPY_INTS, float)):
factors = S(factors)
if isinstance(factors, Factors):
factors = factors.factors.copy()
elif factors is None or factors is S.One:
factors = {}
elif factors is S.Zero or factors == 0:
factors = {S.Zero: S.One}
elif isinstance(factors, Number):
n = factors
factors = {}
if n < 0:
factors[S.NegativeOne] = S.One
n = -n
if n is not S.One:
if n.is_Float or n.is_Integer or n is S.Infinity:
factors[n] = S.One
elif n.is_Rational:
# since we're processing Numbers, the denominator is
# stored with a negative exponent; all other factors
# are left .
if n.p != 1:
factors[Integer(n.p)] = S.One
factors[Integer(n.q)] = S.NegativeOne
else:
raise ValueError(
'Expected Float|Rational|Integer, not %s' % n)
elif isinstance(factors, Basic) and not factors.args:
factors = {factors: S.One}
elif isinstance(factors, Expr):
c, nc = factors.args_cnc()
i = c.count(I)
for _ in range(i):
c.remove(I)
factors = dict(Mul._from_args(c).as_powers_dict())
if i:
factors[I] = S.One * i
if nc:
factors[Mul(*nc, evaluate=False)] = S.One
else:
factors = factors.copy() # /!\ should be dict-like
# tidy up -/+1 and I exponents if Rational
handle = []
for k in factors:
if k is I or k in (-1, 1):
handle.append(k)
if handle:
i1 = S.One
for k in handle:
if not _isnumber(factors[k]):
continue
i1 *= k**factors.pop(k)
if i1 is not S.One:
for a in i1.args if i1.is_Mul else [
i1
]: # at worst, -1.0*I*(-1)**e
if a is S.NegativeOne:
factors[a] = S.One
elif a is I:
factors[I] = S.One
elif a.is_Pow:
if S.NegativeOne not in factors:
factors[S.NegativeOne] = S.Zero
factors[S.NegativeOne] += a.exp
elif a == 1:
factors[a] = S.One
elif a == -1:
factors[-a] = S.One
factors[S.NegativeOne] = S.One
else:
raise ValueError('unexpected factor in i1: %s' % a)
self.factors = factors
try:
self.gens = frozenset(factors.keys())
except AttributeError:
raise TypeError('expecting Expr or dictionary')
def test_Integer():
sT(Integer(4), "Integer(4)")
def test_list():
sT([x, Integer(4)], "[Symbol('x'), Integer(4)]")
def _eval_anticommutator_YGate(self, other, **hints):
return Integer(2)*IdentityGate(self.targets[0])
def _eval_commutator_XGate(self, other, **hints):
return Integer(2)*I*YGate(self.targets[0])
def _eval_Mod(self, q):
n, k = self.args
if any(x.is_integer is False for x in (n, k, q)):
raise ValueError("Integers expected for binomial Mod")
if all(x.is_Integer for x in (n, k, q)):
n, k = map(int, (n, k))
aq, res = abs(q), 1
# handle negative integers k or n
if k < 0:
return 0
if n < 0:
n = -n + k - 1
res = -1 if k % 2 else 1
# non negative integers k and n
if k > n:
return 0
isprime = aq.is_prime
aq = int(aq)
if isprime:
if aq < n:
# use Lucas Theorem
N, K = n, k
while N or K:
res = res * binomial(N % aq, K % aq) % aq
N, K = N // aq, K // aq
else:
# use Factorial Modulo
d = n - k
if k > d:
k, d = d, k
kf = 1
for i in range(2, k + 1):
kf = kf * i % aq
df = kf
for i in range(k + 1, d + 1):
df = df * i % aq
res *= df
for i in range(d + 1, n + 1):
res = res * i % aq
res *= pow(kf * df % aq, aq - 2, aq)
res %= aq
else:
# Binomial Factorization is performed by calculating the
# exponents of primes <= n in `n! /(k! (n - k)!)`,
# for non-negative integers n and k. As the exponent of
# prime in n! is e_p(n) = [n/p] + [n/p**2] + ...
# the exponent of prime in binomial(n, k) would be
# e_p(n) - e_p(k) - e_p(n - k)
M = int(_sqrt(n))
for prime in sieve.primerange(2, n + 1):
if prime > n - k:
res = res * prime % aq
elif prime > n // 2:
continue
elif prime > M:
if n % prime < k % prime:
res = res * prime % aq
else:
N, K = n, k
exp = a = 0
while N > 0:
a = int((N % prime) < (K % prime + a))
N, K = N // prime, K // prime
exp += a
if exp > 0:
res *= pow(prime, exp, aq)
res %= aq
return Integer(res % q)
def _eval_anticommutator(self, other, **hints):
return Integer(2)*other
def gate_simp(circuit):
"""Simplifies gates symbolically
It first sorts gates using gate_sort. It then applies basic
simplification rules to the circuit, e.g., XGate**2 = Identity
"""
# Bubble sort out gates that commute.
circuit = gate_sort(circuit)
# Do simplifications by subing a simplification into the first element
# which can be simplified. We recursively call gate_simp with new circuit
# as input more simplifications exist.
if isinstance(circuit, Add):
return sum(gate_simp(t) for t in circuit.args)
elif isinstance(circuit, Mul):
circuit_args = circuit.args
elif isinstance(circuit, Pow):
b, e = circuit.as_base_exp()
circuit_args = (gate_simp(b)**e,)
else:
return circuit
# Iterate through each element in circuit, simplify if possible.
for i in range(len(circuit_args)):
# H,X,Y or Z squared is 1.
# T**2 = S, S**2 = Z
if isinstance(circuit_args[i], Pow):
if isinstance(circuit_args[i].base,
(HadamardGate, XGate, YGate, ZGate)) \
and isinstance(circuit_args[i].exp, Number):
# Build a new circuit taking replacing the
# H,X,Y,Z squared with one.
newargs = (circuit_args[:i] +
(circuit_args[i].base**(circuit_args[i].exp % 2),) +
circuit_args[i + 1:])
# Recursively simplify the new circuit.
circuit = gate_simp(Mul(*newargs))
break
elif isinstance(circuit_args[i].base, PhaseGate):
# Build a new circuit taking old circuit but splicing
# in simplification.
newargs = circuit_args[:i]
# Replace PhaseGate**2 with ZGate.
newargs = newargs + (ZGate(circuit_args[i].base.args[0])**
(Integer(circuit_args[i].exp/2)), circuit_args[i].base**
(circuit_args[i].exp % 2))
# Append the last elements.
newargs = newargs + circuit_args[i + 1:]
# Recursively simplify the new circuit.
circuit = gate_simp(Mul(*newargs))
break
elif isinstance(circuit_args[i].base, TGate):
# Build a new circuit taking all the old elements.
newargs = circuit_args[:i]
# Put an Phasegate in place of any TGate**2.
newargs = newargs + (PhaseGate(circuit_args[i].base.args[0])**
Integer(circuit_args[i].exp/2), circuit_args[i].base**
(circuit_args[i].exp % 2))
# Append the last elements.
newargs = newargs + circuit_args[i + 1:]
# Recursively simplify the new circuit.
circuit = gate_simp(Mul(*newargs))
break
return circuit
def _eval_anticommutator(self, other, **hints):
if isinstance(other, OneQubitGate):
if self.targets != other.targets or self.__class__ == other.__class__:
return Integer(2)*self*other
return Operator._eval_anticommutator(self, other, **hints)
def continued_fraction(a):
"""Return the continued fraction representation of a Rational or
quadratic irrational.
Examples
========
>>> from sympy.ntheory.continued_fraction import continued_fraction
>>> from sympy import sqrt
>>> continued_fraction((1 + 2*sqrt(3))/5)
[0, 1, [8, 3, 34, 3]]
See Also
========
continued_fraction_periodic, continued_fraction_reduce, continued_fraction_convergents
"""
e = _sympify(a)
if all(i.is_Rational for i in e.atoms()):
if e.is_Integer:
return continued_fraction_periodic(e, 1, 0)
elif e.is_Rational:
return continued_fraction_periodic(e.p, e.q, 0)
elif e.is_Pow and e.exp is S.Half and e.base.is_Integer:
return continued_fraction_periodic(0, 1, e.base)
elif e.is_Mul and len(
e.args) == 2 and (e.args[0].is_Rational and e.args[1].is_Pow
and e.args[1].base.is_Integer
and e.args[1].exp is S.Half):
a, b = e.args
return continued_fraction_periodic(0, a.q, b.base, a.p)
else:
# this should not have to work very hard- no
# simplification, cancel, etc... which should be
# done by the user. e.g. This is a fancy 1 but
# the user should simplify it first:
# sqrt(2)*(1 + sqrt(2))/(sqrt(2) + 2)
p, d = e.expand().as_numer_denom()
if d.is_Integer:
if p.is_Rational:
return continued_fraction_periodic(p, d)
# look for a + b*c
# with c = sqrt(s)
if p.is_Add and len(p.args) == 2:
a, bc = p.args
else:
a = S.Zero
bc = p
if a.is_Integer:
b = S.NaN
if bc.is_Mul and len(bc.args) == 2:
b, c = bc.args
elif bc.is_Pow:
b = Integer(1)
c = bc
if b.is_Integer and (c.is_Pow and c.exp is S.Half
and c.base.is_Integer):
# (a + b*sqrt(c))/d
c = c.base
return continued_fraction_periodic(a, d, c, b)
raise ValueError('expecting a rational or quadratic irrational, not %s' %
e)
def _real_to_rational(expr, tolerance=None, rational_conversion='base10'):
"""
Replace all reals in expr with rationals.
>>> from sympy import Rational
>>> from sympy.simplify.simplify import _real_to_rational
>>> from sympy.abc import x
>>> _real_to_rational(.76 + .1*x**.5)
sqrt(x)/10 + 19/25
If rational_conversion='base10', this uses the base-10 string. If
rational_conversion='exact', the exact, base-2 representation is used.
>>> _real_to_rational(0.333333333333333, rational_conversion='exact')
6004799503160655/18014398509481984
>>> _real_to_rational(0.333333333333333)
1/3
"""
expr = _sympify(expr)
inf = Float('inf')
p = expr
reps = {}
reduce_num = None
if tolerance is not None and tolerance < 1:
reduce_num = ceiling(1/tolerance)
for fl in p.atoms(Float):
key = fl
if reduce_num is not None:
r = Rational(fl).limit_denominator(reduce_num)
elif (tolerance is not None and tolerance >= 1 and
fl.is_Integer is False):
r = Rational(tolerance*round(fl/tolerance)
).limit_denominator(int(tolerance))
else:
if rational_conversion == 'exact':
r = Rational(fl)
reps[key] = r
continue
elif rational_conversion != 'base10':
raise ValueError("rational_conversion must be 'base10' or 'exact'")
r = nsimplify(fl, rational=False)
# e.g. log(3).n() -> log(3) instead of a Rational
if fl and not r:
r = Rational(fl)
elif not r.is_Rational:
if fl == inf or fl == -inf:
r = S.ComplexInfinity
elif fl < 0:
fl = -fl
d = Pow(10, int((mpmath.log(fl)/mpmath.log(10))))
r = -Rational(str(fl/d))*d
elif fl > 0:
d = Pow(10, int((mpmath.log(fl)/mpmath.log(10))))
r = Rational(str(fl/d))*d
else:
r = Integer(0)
reps[key] = r
return p.subs(reps, simultaneous=True)
def continued_fraction_reduce(cf):
"""
Reduce a continued fraction to a rational or quadratic irrational.
Compute the rational or quadratic irrational number from its
terminating or periodic continued fraction expansion. The
continued fraction expansion (cf) should be supplied as a
terminating iterator supplying the terms of the expansion. For
terminating continued fractions, this is equivalent to
``list(continued_fraction_convergents(cf))[-1]``, only a little more
efficient. If the expansion has a repeating part, a list of the
repeating terms should be returned as the last element from the
iterator. This is the format returned by
continued_fraction_periodic.
For quadratic irrationals, returns the largest solution found,
which is generally the one sought, if the fraction is in canonical
form (all terms positive except possibly the first).
Examples
========
>>> from sympy.ntheory.continued_fraction import continued_fraction_reduce
>>> continued_fraction_reduce([1, 2, 3, 4, 5])
225/157
>>> continued_fraction_reduce([-2, 1, 9, 7, 1, 2])
-256/233
>>> continued_fraction_reduce([2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8]).n(10)
2.718281835
>>> continued_fraction_reduce([1, 4, 2, [3, 1]])
(sqrt(21) + 287)/238
>>> continued_fraction_reduce([[1]])
(1 + sqrt(5))/2
>>> from sympy.ntheory.continued_fraction import continued_fraction_periodic
>>> continued_fraction_reduce(continued_fraction_periodic(8, 5, 13))
(sqrt(13) + 8)/5
See Also
========
continued_fraction_periodic
"""
from sympy.core.exprtools import factor_terms
from sympy.core.symbol import Dummy
from sympy.solvers import solve
period = []
x = Dummy('x')
def untillist(cf):
for nxt in cf:
if isinstance(nxt, list):
period.extend(nxt)
yield x
break
yield nxt
a = Integer(0)
for a in continued_fraction_convergents(untillist(cf)):
pass
if period:
y = Dummy('y')
solns = solve(continued_fraction_reduce(period + [y]) - y, y)
solns.sort()
pure = solns[-1]
rv = a.subs(x, pure).radsimp()
else:
rv = a
if rv.is_Add:
rv = factor_terms(rv)
if rv.is_Mul and rv.args[0] == -1:
rv = rv.func(*rv.args)
return rv
def test_MarginalDistribution():
a1, p1, p2 = symbols('a1 p1 p2', positive=True)
C = Multinomial('C', 2, p1, p2)
B = MultivariateBeta('B', a1, C[0])
MGR = MarginalDistribution(B, (C[0], ))
mgrc = Mul(
Symbol('B'),
Piecewise(
ExprCondPair(
Mul(
Integer(2),
Pow(Symbol('p1', positive=True),
Indexed(IndexedBase(Symbol('C')), Integer(0))),
Pow(Symbol('p2', positive=True),
Indexed(IndexedBase(Symbol('C')), Integer(1))),
Pow(
factorial(Indexed(IndexedBase(Symbol('C')),
Integer(0))), Integer(-1)),
Pow(
factorial(Indexed(IndexedBase(Symbol('C')),
Integer(1))), Integer(-1))),
Eq(
Add(Indexed(IndexedBase(Symbol('C')), Integer(0)),
Indexed(IndexedBase(Symbol('C')), Integer(1))),
Integer(2))), ExprCondPair(Integer(0), True)),
Pow(gamma(Symbol('a1', positive=True)), Integer(-1)),
gamma(
Add(Symbol('a1', positive=True),
Indexed(IndexedBase(Symbol('C')), Integer(0)))),
Pow(gamma(Indexed(IndexedBase(Symbol('C')), Integer(0))), Integer(-1)),
Pow(Indexed(IndexedBase(Symbol('B')), Integer(0)),
Add(Symbol('a1', positive=True), Integer(-1))),
Pow(Indexed(IndexedBase(Symbol('B')), Integer(1)),
Add(Indexed(IndexedBase(Symbol('C')), Integer(0)), Integer(-1))))
assert MGR(C) == mgrc
def test_core_numbers():
for c in (Integer(2), Rational(2, 3), Float("1.2")):
check(c)
def wigner_3j(j_1, j_2, j_3, m_1, m_2, m_3):
r"""
Calculate the Wigner 3j symbol `\operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)`.
Parameters
==========
j_1, j_2, j_3, m_1, m_2, m_3 :
Integer or half integer.
Returns
=======
Rational number times the square root of a rational number.
Examples
========
>>> from sympy.physics.wigner import wigner_3j
>>> wigner_3j(2, 6, 4, 0, 0, 0)
sqrt(715)/143
>>> wigner_3j(2, 6, 4, 0, 0, 1)
0
It is an error to have arguments that are not integer or half
integer values::
sage: wigner_3j(2.1, 6, 4, 0, 0, 0)
Traceback (most recent call last):
...
ValueError: j values must be integer or half integer
sage: wigner_3j(2, 6, 4, 1, 0, -1.1)
Traceback (most recent call last):
...
ValueError: m values must be integer or half integer
Notes
=====
The Wigner 3j symbol obeys the following symmetry rules:
- invariant under any permutation of the columns (with the
exception of a sign change where `J:=j_1+j_2+j_3`):
.. math::
\begin{aligned}
\operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)
&=\operatorname{Wigner3j}(j_3,j_1,j_2,m_3,m_1,m_2) \\
&=\operatorname{Wigner3j}(j_2,j_3,j_1,m_2,m_3,m_1) \\
&=(-1)^J \operatorname{Wigner3j}(j_3,j_2,j_1,m_3,m_2,m_1) \\
&=(-1)^J \operatorname{Wigner3j}(j_1,j_3,j_2,m_1,m_3,m_2) \\
&=(-1)^J \operatorname{Wigner3j}(j_2,j_1,j_3,m_2,m_1,m_3)
\end{aligned}
- invariant under space inflection, i.e.
.. math::
\operatorname{Wigner3j}(j_1,j_2,j_3,m_1,m_2,m_3)
=(-1)^J \operatorname{Wigner3j}(j_1,j_2,j_3,-m_1,-m_2,-m_3)
- symmetric with respect to the 72 additional symmetries based on
the work by [Regge58]_
- zero for `j_1`, `j_2`, `j_3` not fulfilling triangle relation
- zero for `m_1 + m_2 + m_3 \neq 0`
- zero for violating any one of the conditions
`j_1 \ge |m_1|`, `j_2 \ge |m_2|`, `j_3 \ge |m_3|`
Algorithm
=========
This function uses the algorithm of [Edmonds74]_ to calculate the
value of the 3j symbol exactly. Note that the formula contains
alternating sums over large factorials and is therefore unsuitable
for finite precision arithmetic and only useful for a computer
algebra system [Rasch03]_.
Authors
=======
- Jens Rasch (2009-03-24): initial version
"""
if int(j_1 * 2) != j_1 * 2 or int(j_2 * 2) != j_2 * 2 or \
int(j_3 * 2) != j_3 * 2:
raise ValueError("j values must be integer or half integer")
if int(m_1 * 2) != m_1 * 2 or int(m_2 * 2) != m_2 * 2 or \
int(m_3 * 2) != m_3 * 2:
raise ValueError("m values must be integer or half integer")
if m_1 + m_2 + m_3 != 0:
return S.Zero
prefid = Integer((-1)**int(j_1 - j_2 - m_3))
m_3 = -m_3
a1 = j_1 + j_2 - j_3
if a1 < 0:
return S.Zero
a2 = j_1 - j_2 + j_3
if a2 < 0:
return S.Zero
a3 = -j_1 + j_2 + j_3
if a3 < 0:
return S.Zero
if (abs(m_1) > j_1) or (abs(m_2) > j_2) or (abs(m_3) > j_3):
return S.Zero
maxfact = max(j_1 + j_2 + j_3 + 1, j_1 + abs(m_1), j_2 + abs(m_2),
j_3 + abs(m_3))
_calc_factlist(int(maxfact))
argsqrt = Integer(_Factlist[int(j_1 + j_2 - j_3)] *
_Factlist[int(j_1 - j_2 + j_3)] *
_Factlist[int(-j_1 + j_2 + j_3)] *
_Factlist[int(j_1 - m_1)] *
_Factlist[int(j_1 + m_1)] *
_Factlist[int(j_2 - m_2)] *
_Factlist[int(j_2 + m_2)] *
_Factlist[int(j_3 - m_3)] *
_Factlist[int(j_3 + m_3)]) / \
_Factlist[int(j_1 + j_2 + j_3 + 1)]
ressqrt = sqrt(argsqrt)
if ressqrt.is_complex or ressqrt.is_infinite:
ressqrt = ressqrt.as_real_imag()[0]
imin = max(-j_3 + j_1 + m_2, -j_3 + j_2 - m_1, 0)
imax = min(j_2 + m_2, j_1 - m_1, j_1 + j_2 - j_3)
sumres = 0
for ii in range(int(imin), int(imax) + 1):
den = _Factlist[ii] * \
_Factlist[int(ii + j_3 - j_1 - m_2)] * \
_Factlist[int(j_2 + m_2 - ii)] * \
_Factlist[int(j_1 - ii - m_1)] * \
_Factlist[int(ii + j_3 - j_2 + m_1)] * \
_Factlist[int(j_1 + j_2 - j_3 - ii)]
sumres = sumres + Integer((-1)**ii) / den
res = ressqrt * sumres * prefid
return res
class TwoQubitGate(Gate):
"""A two qubit unitary gate base class."""
nqubits = Integer(2)
def continued_fraction_reduce(cf):
"""
Reduce a continued fraction to a rational or quadratic irrational.
Compute the rational or quadratic irrational number from its
terminating or periodic continued fraction expansion. The
continued fraction expansion (cf) should be supplied as a
terminating iterator supplying the terms of the expansion. For
terminating continued fractions, this is equivalent to
``list(continued_fraction_convergents(cf))[-1]``, only a little more
efficient. If the expansion has a repeating part, a list of the
repeating terms should be returned as the last element from the
iterator. This is the format returned by
continued_fraction_periodic.
For quadratic irrationals, returns the largest solution found,
which is generally the one sought, if the fraction is in canonical
form (all terms positive except possibly the first).
Examples
========
>>> from sympy.ntheory.continued_fraction import continued_fraction_reduce
>>> continued_fraction_reduce([1, 2, 3, 4, 5])
225/157
>>> continued_fraction_reduce([-2, 1, 9, 7, 1, 2])
-256/233
>>> continued_fraction_reduce([2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8]).n(10)
2.718281835
>>> continued_fraction_reduce([1, 4, 2, [3, 1]])
(sqrt(21) + 287)/238
>>> continued_fraction_reduce([[1]])
1/2 + sqrt(5)/2
>>> from sympy.ntheory.continued_fraction import continued_fraction_periodic
>>> continued_fraction_reduce(continued_fraction_periodic(8, 5, 13))
(sqrt(13) + 8)/5
See Also
========
continued_fraction_periodic
"""
from sympy.core.symbol import Dummy
from sympy.solvers import solve
period = []
x = Dummy('x')
def untillist(cf):
for nxt in cf:
if isinstance(nxt, list):
period.extend(nxt)
yield x
break
yield nxt
a = Integer(0)
for a in continued_fraction_convergents(untillist(cf)):
pass
if period:
y = Dummy('y')
solns = solve(continued_fraction_reduce(period + [y]) - y, y)
solns.sort()
pure = solns[-1]
return a.subs(x, pure).radsimp()
else:
return a
def test_issue_6981():
S = set(divisors(4)).union(set(divisors(Integer(2))))
assert S == {1, 2, 4}
