def weak_normalizer(a, d, DE, z=None):
"""
Weak normalization.
Explanation
===========
Given a derivation D on k[t] and f == a/d in k(t), return q in k[t]
such that f - Dq/q is weakly normalized with respect to t.
f in k(t) is said to be "weakly normalized" with respect to t if
residue_p(f) is not a positive integer for any normal irreducible p
in k[t] such that f is in R_p (Definition 6.1.1). If f has an
elementary integral, this is equivalent to no logarithm of
integral(f) whose argument depends on t has a positive integer
coefficient, where the arguments of the logarithms not in k(t) are
in k[t].
Returns (q, f - Dq/q)
"""
z = z or Dummy('z')
dn, ds = splitfactor(d, DE)
# Compute d1, where dn == d1*d2**2*...*dn**n is a square-free
# factorization of d.
g = gcd(dn, dn.diff(DE.t))
d_sqf_part = dn.quo(g)
d1 = d_sqf_part.quo(gcd(d_sqf_part, g))
a1, b = gcdex_diophantine(
d.quo(d1).as_poly(DE.t), d1.as_poly(DE.t), a.as_poly(DE.t))
r = (a - Poly(z, DE.t) * derivation(d1, DE)).as_poly(DE.t).resultant(
d1.as_poly(DE.t))
r = Poly(r, z)
if not r.expr.has(z):
return (Poly(1, DE.t), (a, d))
N = [i for i in r.real_roots() if i in ZZ and i > 0]
q = reduce(mul,
[gcd(a - Poly(n, DE.t) * derivation(d1, DE), d1) for n in N],
Poly(1, DE.t))
dq = derivation(q, DE)
sn = q * a - d * dq
sd = q * d
sn, sd = sn.cancel(sd, include=True)
return (q, (sn, sd))
def keysgenerator(self):
"""
creates a public/private keys of 'keysize' bits in size:
calculates/generates public n, e for encryption and private d for decryption
:return: public and private keys (e, n) (d, n)
"""
p = self.generateprime()
q = self.generateprime()
n = p * q
phi = (p-1) * (q-1)
del p, q #for security purposes
e = randint(0, phi-1)
while gcd(e, phi) != 1:
e = randint(0, phi-1)
dt, k1, r = gcdex(e, phi)
d = dt % phi
public_key = (e, n)
private_key = (d, n)
return public_key, private_key
def weak_normalizer(a, d, DE, z=None):
"""
Weak normalization.
Given a derivation D on k[t] and f == a/d in k(t), return q in k[t]
such that f - Dq/q is weakly normalized with respect to t.
f in k(t) is said to be "weakly normalized" with respect to t if
residue_p(f) is not a positive integer for any normal irreducible p
in k[t] such that f is in R_p (Definition 6.1.1). If f has an
elementary integral, this is equivalent to no logarithm of
integral(f) whose argument depends on t has a positive integer
coefficient, where the arguments of the logarithms not in k(t) are
in k[t].
Returns (q, f - Dq/q)
"""
z = z or Dummy('z')
dn, ds = splitfactor(d, DE)
# Compute d1, where dn == d1*d2**2*...*dn**n is a square-free
# factorization of d.
g = gcd(dn, dn.diff(DE.t))
d_sqf_part = dn.quo(g)
d1 = d_sqf_part.quo(gcd(d_sqf_part, g))
a1, b = gcdex_diophantine(d.quo(d1).as_poly(DE.t), d1.as_poly(DE.t),
a.as_poly(DE.t))
r = (a - Poly(z, DE.t)*derivation(d1, DE)).as_poly(DE.t).resultant(
d1.as_poly(DE.t))
r = Poly(r, z)
if not r.has(z):
return (Poly(1, DE.t), (a, d))
N = [i for i in r.real_roots() if i in ZZ and i > 0]
q = reduce(mul, [gcd(a - Poly(n, DE.t)*derivation(d1, DE), d1) for n in N],
Poly(1, DE.t))
dq = derivation(q, DE)
sn = q*a - d*dq
sd = q*d
sn, sd = sn.cancel(sd, include=True)
return (q, (sn, sd))
def rrs_simple(n):
"""
@type n: integer > 0
@rtype: list
@return: a reduced residue system modulo n
"""
return [x for x in range(n) if gcd(x, n) == 1]
def extract(f):
p = f.args[0]
for q in f.args[1:]:
p = gcd(p, q, *symbols)
if p.is_number:
return S.One, f
return p, Add(*[quo(g, p, *symbols) for g in f.args])
def deflation(p):
for y in V:
if not p.has_any_symbols(y):
continue
if derivation(p) is not S.Zero:
c, q = p.as_poly(y).as_primitive()
return deflation(c)*gcd(q, q.diff(y))
else:
return p
def deflation(p):
for y in V:
if not p.has(y):
continue
if derivation(p) is not S.Zero:
c, q = p.as_poly(y).primitive()
return deflation(c)*gcd(q, q.diff(y)).as_basic()
else:
return p
def _deflation(p):
for y in V:
if not p.has(y):
continue
if _derivation(p) is not S.Zero:
c, q = p.as_poly(y).primitive()
return _deflation(c)*gcd(q, q.diff(y)).as_expr()
else:
return p
def _deflation(p):
for y in V:
if not p.has(y):
continue
if _derivation(p) is not S.Zero:
c, q = p.as_poly(y).primitive()
return _deflation(c)*gcd(q, q.diff(y)).as_expr()
return p
def rsa(bits):
p = nextprime(randrange(2**(bits // 2 + 1)))
q = nextprime(randrange(2**(bits // 2 + 1)))
n = p * q
phi_n = (p - 1) * (q - 1)
while True:
e = randrange(1, phi_n)
if gcd(e, phi_n) == 1:
break
d = inv_mod(e, phi_n)
return e, d, n
def splitter(p):
for y in V:
if not p.has_any_symbols(y):
continue
if derivation(y) is not S.Zero:
c, q = p.as_poly(y).as_primitive()
q = q.as_basic()
h = gcd(q, derivation(q), y)
s = quo(h, gcd(q, q.diff(y), y), y)
c_split = splitter(c)
if s.as_poly(y).degree == 0:
return (c_split[0], q * c_split[1])
q_split = splitter(Poly.cancel((q, s), *V))
return (c_split[0] * q_split[0] * s, c_split[1] * q_split[1])
else:
return (S.One, p)
def _splitter(p):
for y in V:
if not p.has(y):
continue
if _derivation(y) is not S.Zero:
c, q = p.as_poly(y).primitive()
q = q.as_expr()
h = gcd(q, _derivation(q), y)
s = quo(h, gcd(q, q.diff(y), y), y)
c_split = _splitter(c)
if s.as_poly(y).degree() == 0:
return (c_split[0], q * c_split[1])
q_split = _splitter(cancel(q / s))
return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1])
else:
return (S.One, p)
def splitter(p):
for y in V:
if not p.has_any_symbols(y):
continue
if derivation(y) is not S.Zero:
c, q = p.as_poly(y).as_primitive()
q = q.as_basic()
h = gcd(q, derivation(q), y)
s = quo(h, gcd(q, q.diff(y), y), y)
c_split = splitter(c)
if s.as_poly(y).degree == 0:
return (c_split[0], q * c_split[1])
q_split = splitter(Poly.cancel((q, s), *V))
return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1])
else:
return (S.One, p)
def _splitter(p):
for y in V:
if not p.has(y):
continue
if _derivation(y) is not S.Zero:
c, q = p.as_poly(y).primitive()
q = q.as_expr()
h = gcd(q, _derivation(q), y)
s = quo(h, gcd(q, q.diff(y), y), y)
c_split = _splitter(c)
if s.as_poly(y).degree() == 0:
return (c_split[0], q * c_split[1])
q_split = _splitter(cancel(q / s))
return (c_split[0]*q_split[0]*s, c_split[1]*q_split[1])
return (S.One, p)
def crack_given_decrypt(n, m):
n = int(n)
m = int(m)
while True:
if m % 2 != 0:
break
divide_out = True
for _ in range(5):
a = randrange(1, n)
if gcd(a, n) == 1:
if pow(a, m // 2, n) != 1:
divide_out = False
break
if divide_out:
m //= 2
else:
break
while True:
a = randrange(1, n)
g = gcd(pow(a, m // 2, n) - 1, n)
if g != 1 and g != n:
return g
def _split_gcd(*a):
"""
split the list of integers ``a`` into a list of integers, ``a1`` having
``g = gcd(a1)``, and a list ``a2`` whose elements are not divisible by
``g``. Returns ``g, a1, a2``
Examples
========
>>> from sympy.simplify.radsimp import _split_gcd
>>> _split_gcd(55, 35, 22, 14, 77, 10)
(5, [55, 35, 10], [22, 14, 77])
"""
g = a[0]
b1 = [g]
b2 = []
for x in a[1:]:
g1 = gcd(g, x)
if g1 == 1:
b2.append(x)
else:
g = g1
b1.append(x)
return g, b1, b2
def _split_gcd(*a):
"""
Split the list of integers ``a`` into a list of integers, ``a1`` having
``g = gcd(a1)``, and a list ``a2`` whose elements are not divisible by
``g``. Returns ``g, a1, a2``.
Examples
========
>>> from sympy.simplify.radsimp import _split_gcd
>>> _split_gcd(55, 35, 22, 14, 77, 10)
(5, [55, 35, 10], [22, 14, 77])
"""
g = a[0]
b1 = [g]
b2 = []
for x in a[1:]:
g1 = gcd(g, x)
if g1 == 1:
b2.append(x)
else:
g = g1
b1.append(x)
return g, b1, b2
def nc_gcd(aa, bb):
a, b = [i.as_coeff_Mul() for i in [aa, bb]]
c = gcd(a[0], b[0]).as_numer_denom()[0]
g = Mul(*(a[1].args_cnc(cset=True)[0] & b[1].args_cnc(cset=True)[0]))
return _keep_coeff(c, g)
def relsimp(func, args, **kwargs):
lhs, rhs = args
r = func(lhs.simplify(**kwargs), rhs.simplify(**kwargs))
if not isinstance(r.lhs, Expr) or not isinstance(r.rhs, Expr):
return r.canonical
dif = r.lhs - r.rhs
r = r.canonical
# If there is only one symbol in the expression,
# try to write it on a simplified form
free = list(filter(lambda x: x.is_real is not False, r.free_symbols))
if len(free) == 1:
try:
x = free.pop()
dif = r.lhs - r.rhs
m, b = linear_coeffs(dif, x)
if m.is_zero is False:
if m.is_negative:
# Dividing with a negative number, so change order of arguments
# canonical will put the symbol back on the lhs later
r = r.function(-b / m, x)
else:
r = r.function(x, -b / m)
else:
r = r.function(b, S.Zero)
except ValueError:
# maybe not a linear function, try polynomial
try:
p = poly(dif, x)
c = p.all_coeffs()
constant = c[-1]
c[-1] = 0
scale = gcd(c)
c = [ctmp / scale for ctmp in c]
r = r.function(Poly.from_list(c, x).as_expr(), -constant / scale)
except PolynomialError:
pass
elif len(free) >= 2:
try:
free = list(ordered(free))
dif = r.lhs - r.rhs
m = linear_coeffs(dif, *free)
constant = m[-1]
del m[-1]
scale = gcd(m)
m = [mtmp / scale for mtmp in m]
nzm = list(filter(lambda f: f[0] != 0, list(zip(m, free))))
if scale.is_zero is False:
if constant != 0:
# lhs: expression, rhs: constant
newexpr = Add(*[i * j for i, j in nzm])
r = r.function(newexpr, -constant / scale)
else:
# keep first term on lhs
lhsterm = nzm[0][0] * nzm[0][1]
del nzm[0]
newexpr = Add(*[i * j for i, j in nzm])
r = r.function(lhsterm, -newexpr)
else:
r = r.function(constant, S.Zero)
except ValueError:
pass
# Did we get a simplified result?
r = r.canonical
rel = func(*args)
measure = kwargs['measure']
if measure(r) < kwargs['ratio'] * measure(rel):
return r
else:
return rel
def _eval_simplify(self, **kwargs):
from .add import Add
from sympy.core.expr import Expr
r = self
r = r.func(*[i.simplify(**kwargs) for i in r.args])
if r.is_Relational:
if not isinstance(r.lhs, Expr) or not isinstance(r.rhs, Expr):
return r
dif = r.lhs - r.rhs
# replace dif with a valid Number that will
# allow a definitive comparison with 0
v = None
if dif.is_comparable:
v = dif.n(2)
elif dif.equals(0): # XXX this is expensive
v = S.Zero
if v is not None:
r = r.func._eval_relation(v, S.Zero)
r = r.canonical
# If there is only one symbol in the expression,
# try to write it on a simplified form
free = list(
filter(lambda x: x.is_real is not False, r.free_symbols))
if len(free) == 1:
try:
from sympy.solvers.solveset import linear_coeffs
x = free.pop()
dif = r.lhs - r.rhs
m, b = linear_coeffs(dif, x)
if m.is_zero is False:
if m.is_negative:
# Dividing with a negative number, so change order of arguments
# canonical will put the symbol back on the lhs later
r = r.func(-b / m, x)
else:
r = r.func(x, -b / m)
else:
r = r.func(b, S.zero)
except ValueError:
# maybe not a linear function, try polynomial
from sympy.polys import Poly, poly, PolynomialError, gcd
try:
p = poly(dif, x)
c = p.all_coeffs()
constant = c[-1]
c[-1] = 0
scale = gcd(c)
c = [ctmp / scale for ctmp in c]
r = r.func(
Poly.from_list(c, x).as_expr(), -constant / scale)
except PolynomialError:
pass
elif len(free) >= 2:
try:
from sympy.solvers.solveset import linear_coeffs
from sympy.polys import gcd
free = list(ordered(free))
dif = r.lhs - r.rhs
m = linear_coeffs(dif, *free)
constant = m[-1]
del m[-1]
scale = gcd(m)
m = [mtmp / scale for mtmp in m]
nzm = list(filter(lambda f: f[0] != 0, list(zip(m, free))))
if scale.is_zero is False:
if constant != 0:
# lhs: expression, rhs: constant
newexpr = Add(*[i * j for i, j in nzm])
r = r.func(newexpr, -constant / scale)
else:
# keep first term on lhs
lhsterm = nzm[0][0] * nzm[0][1]
del nzm[0]
newexpr = Add(*[i * j for i, j in nzm])
r = r.func(lhsterm, -newexpr)
else:
r = r.func(constant, S.zero)
except ValueError:
pass
# Did we get a simplified result?
r = r.canonical
measure = kwargs['measure']
if measure(r) < kwargs['ratio'] * measure(self):
return r
else:
return self
def equivalence_hypergeometric(A, B, func):
# This method for finding the equivalence is only for 2F1 type.
# We can extend it for 1F1 and 0F1 type also.
x = func.args[0]
# making given equation in normal form
I1 = factor(cancel(A.diff(x)/2 + A**2/4 - B))
# computing shifted invariant(J1) of the equation
J1 = factor(cancel(x**2*I1 + S(1)/4))
num, dem = J1.as_numer_denom()
num = powdenest(expand(num))
dem = powdenest(expand(dem))
# this function will compute the different powers of variable(x) in J1.
# then it will help in finding value of k. k is power of x such that we can express
# J1 = x**k * J0(x**k) then all the powers in J0 become integers.
def _power_counting(num):
_pow = {0}
for val in num:
if val.has(x):
if isinstance(val, Pow) and val.as_base_exp()[0] == x:
_pow.add(val.as_base_exp()[1])
elif val == x:
_pow.add(val.as_base_exp()[1])
else:
_pow.update(_power_counting(val.args))
return _pow
pow_num = _power_counting((num, ))
pow_dem = _power_counting((dem, ))
pow_dem.update(pow_num)
_pow = pow_dem
k = gcd(_pow)
# computing I0 of the given equation
I0 = powdenest(simplify(factor(((J1/k**2) - S(1)/4)/((x**k)**2))), force=True)
I0 = factor(cancel(powdenest(I0.subs(x, x**(S(1)/k)), force=True)))
num, dem = I0.as_numer_denom()
max_num_pow = max(_power_counting((num, )))
dem_args = dem.args
sing_point = []
dem_pow = []
# calculating singular point of I0.
for arg in dem_args:
if arg.has(x):
if isinstance(arg, Pow):
# (x-a)**n
dem_pow.append(arg.as_base_exp()[1])
sing_point.append(list(roots(arg.as_base_exp()[0], x).keys())[0])
else:
# (x-a) type
dem_pow.append(arg.as_base_exp()[1])
sing_point.append(list(roots(arg, x).keys())[0])
dem_pow.sort()
# checking if equivalence is exists or not.
if equivalence(max_num_pow, dem_pow) == "2F1":
return {'I0':I0, 'k':k, 'sing_point':sing_point, 'type':"2F1"}
else:
return None
def normal(f, g, n=None):
"""Given relatively prime univariate polynomials 'f' and 'g',
rewrite their quotient to a normal form defined as follows:
f(n) A(n) C(n+1)
---- = Z -----------
g(n) B(n) C(n)
where Z is arbitrary constant and A, B, C are monic
polynomials in 'n' with following properties:
(1) gcd(A(n), B(n+h)) = 1 for all 'h' in N
(2) gcd(B(n), C(n+1)) = 1
(3) gcd(A(n), C(n)) = 1
This normal form, or rational factorization in other words,
is crucial step in Gosper's algorithm and in difference
equations solving. It can be also used to decide if two
hypergeometric are similar or not.
This procedure will return a tuple containing elements
of this factorization in the form (Z*A, B, C). For example:
>>> from sympy import Symbol, normal
>>> n = Symbol('n', integer=True)
>>> normal(4*n+5, 2*(4*n+1)*(2*n+3), n)
(1/4, 3/2 + n, 1/4 + n)
"""
f, g = map(sympify, (f, g))
p = f.as_poly(n, field=True)
q = g.as_poly(n, field=True)
a, p = p.LC(), p.monic()
b, q = q.LC(), q.monic()
A = p.as_basic()
B = q.as_basic()
C, Z = S.One, a / b
h = Dummy('h')
res = resultant(A, B.subs(n, n+h), n)
nni_roots = roots(res, h, filter='Z',
predicate=lambda r: r >= 0).keys()
if not nni_roots:
return (f, g, S.One)
else:
for i in sorted(nni_roots):
d = gcd(A, B.subs(n, n+i), n)
A = quo(A, d, n)
B = quo(B, d.subs(n, n-i), n)
C *= Mul(*[ d.subs(n, n-j) for j in xrange(1, i+1) ])
return (Z*A, B, C)
def factor_nc(expr):
"""Return the factored form of ``expr`` while handling non-commutative
expressions.
**examples**
>>> from sympy.core.exprtools import factor_nc
>>> from sympy import Symbol
>>> from sympy.abc import x
>>> A = Symbol('A', commutative=False)
>>> B = Symbol('B', commutative=False)
>>> factor_nc((x**2 + 2*A*x + A**2).expand())
(x + A)**2
>>> factor_nc(((x + A)*(x + B)).expand())
(x + A)*(x + B)
"""
from sympy.simplify.simplify import _mexpand
from sympy.polys import gcd, factor
expr = sympify(expr)
if not isinstance(expr, Expr) or not expr.args:
return expr
if not expr.is_Add:
return expr.func(*[factor_nc(a) for a in expr.args])
expr, rep, nc_symbols = _mask_nc(expr)
if rep:
return factor(expr).subs(rep)
else:
args = [a.args_cnc() for a in Add.make_args(expr)]
c = g = l = r = S.One
hit = False
# find any commutative gcd term
for i, a in enumerate(args):
if i == 0:
c = Mul._from_args(a[0])
elif a[0]:
c = gcd(c, Mul._from_args(a[0]))
else:
c = S.One
if c is not S.One:
hit = True
c, g = c.as_coeff_Mul()
if g is not S.One:
for i, (cc, _) in enumerate(args):
cc = list(Mul.make_args(Mul._from_args(list(cc))/g))
args[i][0] = cc
else:
for i, (cc, _) in enumerate(args):
cc[0] = cc[0]/c
args[i][0] = cc
# find any noncommutative common prefix
for i, a in enumerate(args):
if i == 0:
n = a[1][:]
else:
n = common_prefix(n, a[1])
if not n:
# is there a power that can be extracted?
if not args[0][1]:
break
b, e = args[0][1][0].as_base_exp()
ok = False
if e.is_Integer:
for t in args:
if not t[1]:
break
bt, et = t[1][0].as_base_exp()
if et.is_Integer and bt == b:
e = min(e, et)
else:
break
else:
ok = hit = True
l = b**e
il = b**-e
for i, a in enumerate(args):
args[i][1][0] = il*args[i][1][0]
break
if not ok:
break
else:
hit = True
lenn = len(n)
l = Mul(*n)
for i, a in enumerate(args):
args[i][1] = args[i][1][lenn:]
# find any noncommutative common suffix
for i, a in enumerate(args):
if i == 0:
n = a[1][:]
else:
n = common_suffix(n, a[1])
if not n:
# is there a power that can be extracted?
if not args[0][1]:
break
b, e = args[0][1][-1].as_base_exp()
ok = False
if e.is_Integer:
for t in args:
if not t[1]:
break
bt, et = t[1][-1].as_base_exp()
if et.is_Integer and bt == b:
e = min(e, et)
else:
break
else:
ok = hit = True
r = b**e
il = b**-e
for i, a in enumerate(args):
args[i][1][-1] = args[i][1][-1]*il
break
if not ok:
break
else:
hit = True
lenn = len(n)
r = Mul(*n)
for i, a in enumerate(args):
args[i][1] = a[1][:len(a[1]) - lenn]
if hit:
mid = Add(*[Mul(*cc)*Mul(*nc) for cc, nc in args])
else:
mid = expr
# sort the symbols so the Dummys would appear in the same
# order as the original symbols, otherwise you may introduce
# a factor of -1, e.g. A**2 - B**2) -- {A:y, B:x} --> y**2 - x**2
# and the former factors into two terms, (A - B)*(A + B) while the
# latter factors into 3 terms, (-1)*(x - y)*(x + y)
rep1 = [(n, Dummy()) for n in sorted(nc_symbols, key=default_sort_key)]
unrep1 = [(v, k) for k, v in rep1]
unrep1.reverse()
new_mid, r2, _ = _mask_nc(mid.subs(rep1))
new_mid = factor(new_mid)
new_mid = new_mid.subs(r2).subs(unrep1)
if new_mid.is_Pow:
return _keep_coeff(c, g*l*new_mid*r)
if new_mid.is_Mul:
# XXX TODO there should be a way to inspect what order the terms
# must be in and just select the plausible ordering without
# checking permutations
cfac = []
ncfac = []
for f in new_mid.args:
if f.is_commutative:
cfac.append(f)
else:
b, e = f.as_base_exp()
assert e.is_Integer
ncfac.extend([b]*e)
pre_mid = g*Mul(*cfac)*l
target = _mexpand(expr/c)
for s in variations(ncfac, len(ncfac)):
ok = pre_mid*Mul(*s)*r
if _mexpand(ok) == target:
return _keep_coeff(c, ok)
# mid was an Add that didn't factor successfully
return _keep_coeff(c, g*l*mid*r)
def rsolve_ratio(coeffs, f, n, **hints):
"""Given linear recurrence operator L of order 'k' with polynomial
coefficients and inhomogeneous equation Ly = f, where 'f' is a
polynomial, we seek for all rational solutions over field K of
characteristic zero.
This procedure accepts only polynomials, however if you are
interested in solving recurrence with rational coefficients
then use rsolve() which will pre-process the given equation
and run this procedure with polynomial arguments.
The algorithm performs two basic steps:
(1) Compute polynomial v(n) which can be used as universal
denominator of any rational solution of equation Ly = f.
(2) Construct new linear difference equation by substitution
y(n) = u(n)/v(n) and solve it for u(n) finding all its
polynomial solutions. Return None if none were found.
Algorithm implemented here is a revised version of the original
Abramov's algorithm, developed in 1989. The new approach is much
simpler to implement and has better overall efficiency. This
method can be easily adapted to q-difference equations case.
Besides finding rational solutions alone, this functions is
an important part of Hyper algorithm were it is used to find
particular solution of inhomogeneous part of a recurrence.
For more information on the implemented algorithm refer to:
[1] S. A. Abramov, Rational solutions of linear difference
and q-difference equations with polynomial coefficients,
in: T. Levelt, ed., Proc. ISSAC '95, ACM Press, New York,
1995, 285-289
"""
f = sympify(f)
if not f.is_polynomial(n):
return None
coeffs = map(sympify, coeffs)
r = len(coeffs)-1
A, B = coeffs[r], coeffs[0]
A = A.subs(n, n-r).expand()
h = Symbol('h', dummy=True)
res = resultant(A, B.subs(n, n+h), n)
if not res.is_polynomial(h):
p, q = res.as_numer_denom()
res = exquo(p, q, h)
nni_roots = roots(res, h, filter='Z',
predicate=lambda r: r >= 0).keys()
if not nni_roots:
return rsolve_poly(coeffs, f, n, **hints)
else:
C, numers = S.One, [S.Zero]*(r+1)
for i in xrange(int(max(nni_roots)), -1, -1):
d = gcd(A, B.subs(n, n+i), n)
A = exquo(A, d, n)
B = exquo(B, d.subs(n, n-i), n)
C *= Mul(*[ d.subs(n, n-j) for j in xrange(0, i+1) ])
denoms = [ C.subs(n, n+i) for i in range(0, r+1) ]
for i in range(0, r+1):
g = gcd(coeffs[i], denoms[i], n)
numers[i] = exquo(coeffs[i], g, n)
denoms[i] = exquo(denoms[i], g, n)
for i in xrange(0, r+1):
numers[i] *= Mul(*(denoms[:i] + denoms[i+1:]))
result = rsolve_poly(numers, f * Mul(*denoms), n, **hints)
if result is not None:
if hints.get('symbols', False):
return (simplify(result[0] / C), result[1])
else:
return simplify(result / C)
else:
return None
def rsolve_ratio(coeffs, f, n, **hints):
"""
Given linear recurrence operator `\operatorname{L}` of order `k`
with polynomial coefficients and inhomogeneous equation
`\operatorname{L} y = f`, where `f` is a polynomial, we seek
for all rational solutions over field `K` of characteristic zero.
This procedure accepts only polynomials, however if you are
interested in solving recurrence with rational coefficients
then use ``rsolve`` which will pre-process the given equation
and run this procedure with polynomial arguments.
The algorithm performs two basic steps:
(1) Compute polynomial `v(n)` which can be used as universal
denominator of any rational solution of equation
`\operatorname{L} y = f`.
(2) Construct new linear difference equation by substitution
`y(n) = u(n)/v(n)` and solve it for `u(n)` finding all its
polynomial solutions. Return ``None`` if none were found.
Algorithm implemented here is a revised version of the original
Abramov's algorithm, developed in 1989. The new approach is much
simpler to implement and has better overall efficiency. This
method can be easily adapted to q-difference equations case.
Besides finding rational solutions alone, this functions is
an important part of Hyper algorithm were it is used to find
particular solution of inhomogeneous part of a recurrence.
Examples
========
>>> from sympy.abc import x
>>> from sympy.solvers.recurr import rsolve_ratio
>>> rsolve_ratio([-2*x**3 + x**2 + 2*x - 1, 2*x**3 + x**2 - 6*x,
... - 2*x**3 - 11*x**2 - 18*x - 9, 2*x**3 + 13*x**2 + 22*x + 8], 0, x)
C2*(2*x - 3)/(2*(x**2 - 1))
References
==========
.. [1] S. A. Abramov, Rational solutions of linear difference
and q-difference equations with polynomial coefficients,
in: T. Levelt, ed., Proc. ISSAC '95, ACM Press, New York,
1995, 285-289
See Also
========
rsolve_hyper
"""
f = sympify(f)
if not f.is_polynomial(n):
return None
coeffs = map(sympify, coeffs)
r = len(coeffs) - 1
A, B = coeffs[r], coeffs[0]
A = A.subs(n, n - r).expand()
h = Dummy('h')
res = resultant(A, B.subs(n, n + h), n)
if not res.is_polynomial(h):
p, q = res.as_numer_denom()
res = quo(p, q, h)
nni_roots = roots(res, h, filter='Z',
predicate=lambda r: r >= 0).keys()
if not nni_roots:
return rsolve_poly(coeffs, f, n, **hints)
else:
C, numers = S.One, [S.Zero]*(r + 1)
for i in xrange(int(max(nni_roots)), -1, -1):
d = gcd(A, B.subs(n, n + i), n)
A = quo(A, d, n)
B = quo(B, d.subs(n, n - i), n)
C *= Mul(*[ d.subs(n, n - j) for j in xrange(0, i + 1) ])
denoms = [ C.subs(n, n + i) for i in range(0, r + 1) ]
for i in range(0, r + 1):
g = gcd(coeffs[i], denoms[i], n)
numers[i] = quo(coeffs[i], g, n)
denoms[i] = quo(denoms[i], g, n)
for i in xrange(0, r + 1):
numers[i] *= Mul(*(denoms[:i] + denoms[i + 1:]))
result = rsolve_poly(numers, f * Mul(*denoms), n, **hints)
if result is not None:
if hints.get('symbols', False):
return (simplify(result[0] / C), result[1])
else:
return simplify(result / C)
else:
return None
def gcd(f, g):
from sympy.polys import gcd
return f.__class__(gcd(f.ex, f.__class__(g).ex))
def nc_gcd(aa, bb):
a, b = [i.as_coeff_Mul() for i in [aa, bb]]
c = gcd(a[0], b[0]).as_numer_denom()[0]
g = Mul(*(a[1].args_cnc(cset=True)[0] & b[1].args_cnc(cset=True)[0]))
return _keep_coeff(c, g)
def normal(f, g, n=None):
"""Given relatively prime univariate polynomials 'f' and 'g',
rewrite their quotient to a normal form defined as follows:
f(n) A(n) C(n+1)
---- = Z -----------
g(n) B(n) C(n)
where Z is arbitrary constant and A, B, C are monic
polynomials in 'n' with following properties:
(1) gcd(A(n), B(n+h)) = 1 for all 'h' in N
(2) gcd(B(n), C(n+1)) = 1
(3) gcd(A(n), C(n)) = 1
This normal form, or rational factorization in other words,
is crucial step in Gosper's algorithm and in difference
equations solving. It can be also used to decide if two
hypergeometric are similar or not.
This procedure will return a tuple containing elements
of this factorization in the form (Z*A, B, C). For example:
>>> from sympy import Symbol, normal
>>> n = Symbol('n', integer=True)
>>> normal(4*n+5, 2*(4*n+1)*(2*n+3), n)
(1/4, 3/2 + n, 1/4 + n)
"""
f, g = map(sympify, (f, g))
p = f.as_poly(n, field=True)
q = g.as_poly(n, field=True)
a, p = p.LC(), p.monic()
b, q = q.LC(), q.monic()
A = p.as_basic()
B = q.as_basic()
C, Z = S.One, a / b
h = Symbol('h', dummy=True)
res = resultant(A, B.subs(n, n + h), n)
nni_roots = roots(res, h, filter='Z', predicate=lambda r: r >= 0).keys()
if not nni_roots:
return (f, g, S.One)
else:
for i in sorted(nni_roots):
d = gcd(A, B.subs(n, n + i), n)
A = quo(A, d, n)
B = quo(B, d.subs(n, n - i), n)
C *= Mul(*[d.subs(n, n - j) for j in xrange(1, i + 1)])
return (Z * A, B, C)
def gcd(f, g):
from sympy.polys import gcd
return f.__class__(gcd(f.ex, f.__class__(g).ex))
