def convert_factors(data, raw_data):
try:
return self._convert_factors_(data)
except ValueError as e:
from misc import combine_exceptions
raise combine_exceptions(
ValueError('%s is not in %s.' % (raw_data, self)), e)
def convert_factors(data, raw_data):
try:
return self._convert_factors_(data)
except ValueError as e:
from misc import combine_exceptions
raise combine_exceptions(ValueError("%s is not in %s." % (raw_data, self)), e)
def _log_factor_(self, base=None):
r"""
Helper method for calculating the logarithm of the factorization
of this element.
INPUT:
- ``base`` -- the base of the logarithm. If ``None``
(default value) is used, the natural logarithm is taken.
OUTPUT:
A tuple of pairs, where the first entry is either a growth
element or something out of which we can construct a growth element
and the second a multiplicative coefficient.
TESTS::
sage: from sage.rings.asymptotic.growth_group import GrowthGroup
sage: G = GrowthGroup('QQ^x * x^ZZ * log(x)^ZZ * y^ZZ * log(y)^ZZ')
sage: x, y = G.gens_monomial()
sage: (x * y).log_factor() # indirect doctest
((log(x), 1), (log(y), 1))
"""
if self.is_one():
return tuple()
def try_create_growth(g):
try:
return self.parent()(g)
except (TypeError, ValueError):
return g
try:
return sum(
iter(
tuple((try_create_growth(g), c) for g, c in factor._log_factor_(base=base))
for factor in self.cartesian_factors()
if factor != factor.parent().one()
),
tuple(),
)
except (ArithmeticError, TypeError, ValueError) as e:
from misc import combine_exceptions
raise combine_exceptions(ArithmeticError("Cannot build log(%s) in %s." % (self, self.parent())), e)
def _log_factor_(self, base=None):
r"""
Helper method for calculating the logarithm of the factorization
of this element.
INPUT:
- ``base`` -- the base of the logarithm. If ``None``
(default value) is used, the natural logarithm is taken.
OUTPUT:
A tuple of pairs, where the first entry is either a growth
element or something out of which we can construct a growth element
and the second a multiplicative coefficient.
TESTS::
sage: from sage.rings.asymptotic.growth_group import GrowthGroup
sage: G = GrowthGroup('QQ^x * x^ZZ * log(x)^ZZ * y^ZZ * log(y)^ZZ')
sage: x, y = G.gens_monomial()
sage: (x * y).log_factor() # indirect doctest
((log(x), 1), (log(y), 1))
"""
if self.is_one():
return tuple()
def try_create_growth(g):
try:
return self.parent()(g)
except (TypeError, ValueError):
return g
try:
return sum(
iter(
tuple((try_create_growth(g), c)
for g, c in factor._log_factor_(base=base))
for factor in self.cartesian_factors()
if factor != factor.parent().one()), tuple())
except (ArithmeticError, TypeError, ValueError) as e:
from misc import combine_exceptions
raise combine_exceptions(
ArithmeticError('Cannot build log(%s) in %s.' %
(self, self.parent())), e)
def Binomial_kn_over_n(var, k, precision=None, skip_constant_factor=False):
r"""
Return the asymptotic expansion of the binomial coefficient
`kn` choose `n`.
INPUT:
- ``var`` -- a string for the variable name.
- ``k`` -- a number or symbolic constant.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_factor`` -- (default: ``False``) a
boolean. If set, then the constant factor `\sqrt{k/(2\pi(k-1))}`
is left out.
As a consequence, the coefficient ring of the output changes
from ``Symbolic Constants Subring`` (if ``False``) to
``Rational Field`` (if ``True``).
OUTPUT:
An asymptotic expansion.
EXAMPLES::
sage: asymptotic_expansions.Binomial_kn_over_n('n', k=2, precision=3)
1/sqrt(pi)*4^n*n^(-1/2)
- 1/8/sqrt(pi)*4^n*n^(-3/2)
+ 1/128/sqrt(pi)*4^n*n^(-5/2)
+ O(4^n*n^(-7/2))
sage: _.parent()
Asymptotic Ring <QQ^n * n^QQ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.Binomial_kn_over_n('n', k=3, precision=3)
1/2*sqrt(3)/sqrt(pi)*(27/4)^n*n^(-1/2)
- 7/144*sqrt(3)/sqrt(pi)*(27/4)^n*n^(-3/2)
+ 49/20736*sqrt(3)/sqrt(pi)*(27/4)^n*n^(-5/2)
+ O((27/4)^n*n^(-7/2))
::
sage: asymptotic_expansions.Binomial_kn_over_n('n', k=7/5, precision=3)
1/2*sqrt(7)/sqrt(pi)*(7/10*7^(2/5)*2^(3/5))^n*n^(-1/2)
- 13/112*sqrt(7)/sqrt(pi)*(7/10*7^(2/5)*2^(3/5))^n*n^(-3/2)
+ 169/12544*sqrt(7)/sqrt(pi)*(7/10*7^(2/5)*2^(3/5))^n*n^(-5/2)
+ O((7/10*7^(2/5)*2^(3/5))^n*n^(-7/2))
sage: _.parent()
Asymptotic Ring <(Symbolic Constants Subring)^n * n^QQ>
over Symbolic Constants Subring
TESTS::
sage: expansion = asymptotic_expansions.Binomial_kn_over_n('n', k=7/5, precision=3)
sage: n = expansion.parent().gen()
sage: expansion.compare_with_values(n, lambda x: binomial(7/5*x, x), [5, 10, 20]) # rel tol 1e-6
[(5, -0.0287383845047?), (10, -0.030845971026?), (20, -0.03162833549?)]
sage: asymptotic_expansions.Binomial_kn_over_n(
....: 'n', k=5, precision=3, skip_constant_factor=True)
(3125/256)^n*n^(-1/2)
- 7/80*(3125/256)^n*n^(-3/2)
+ 49/12800*(3125/256)^n*n^(-5/2)
+ O((3125/256)^n*n^(-7/2))
sage: _.parent()
Asymptotic Ring <QQ^n * n^QQ> over Rational Field
sage: asymptotic_expansions.Binomial_kn_over_n(
....: 'n', k=4, precision=1, skip_constant_factor=True)
(256/27)^n*n^(-1/2) + O((256/27)^n*n^(-3/2))
::
sage: S = asymptotic_expansions.Stirling('n', precision=5)
sage: n = S.parent().gen()
sage: all( # long time
....: SR(asymptotic_expansions.Binomial_kn_over_n(
....: 'n', k=k, precision=3)).canonicalize_radical() ==
....: SR(S.subs(n=k*n) / (S.subs(n=(k-1)*n) * S)).canonicalize_radical()
....: for k in [2, 3, 4])
True
"""
from sage.symbolic.ring import SR
SCR = SR.subring(no_variables=True)
try:
SCR.coerce(k)
except TypeError as e:
from misc import combine_exceptions
raise combine_exceptions(
TypeError('Cannot use k={}.'.format(k)), e)
S = AsymptoticExpansionGenerators._log_StirlingNegativePowers_(
var, precision=max(precision - 2,0))
n = S.parent().gen()
result = (S.subs(n=k*n) - S.subs(n=(k-1)*n) - S).exp()
from sage.rings.rational_field import QQ
P = S.parent().change_parameter(
growth_group='QQ^{n} * {n}^QQ'.format(n=var),
coefficient_ring=QQ)
n = P.gen()
b = k**k / (k-1)**(k-1)
if b.parent() is SR:
b = SCR(b).canonicalize_radical()
result *= n.rpow(b)
result *= n**(-QQ(1)/QQ(2))
if not skip_constant_factor:
result *= (k/((k-1)*2*SCR('pi'))).sqrt()
return result
def Binomial_kn_over_n(var, k, precision=None, skip_constant_factor=False):
r"""
Return the asymptotic expansion of the binomial coefficient
`kn` choose `n`.
INPUT:
- ``var`` -- a string for the variable name.
- ``k`` -- a number or symbolic constant.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_factor`` -- (default: ``False``) a
boolean. If set, then the constant factor `\sqrt{k/(2\pi(k-1))}`
is left out.
As a consequence, the coefficient ring of the output changes
from ``Symbolic Constants Subring`` (if ``False``) to
``Rational Field`` (if ``True``).
OUTPUT:
An asymptotic expansion.
EXAMPLES::
sage: asymptotic_expansions.Binomial_kn_over_n('n', k=2, precision=3)
1/sqrt(pi)*4^n*n^(-1/2)
- 1/8/sqrt(pi)*4^n*n^(-3/2)
+ 1/128/sqrt(pi)*4^n*n^(-5/2)
+ O(4^n*n^(-7/2))
sage: _.parent()
Asymptotic Ring <QQ^n * n^QQ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.Binomial_kn_over_n('n', k=3, precision=3)
1/2*sqrt(3)/sqrt(pi)*(27/4)^n*n^(-1/2)
- 7/144*sqrt(3)/sqrt(pi)*(27/4)^n*n^(-3/2)
+ 49/20736*sqrt(3)/sqrt(pi)*(27/4)^n*n^(-5/2)
+ O((27/4)^n*n^(-7/2))
::
sage: asymptotic_expansions.Binomial_kn_over_n('n', k=7/5, precision=3)
1/2*sqrt(7)/sqrt(pi)*(7/10*7^(2/5)*2^(3/5))^n*n^(-1/2)
- 13/112*sqrt(7)/sqrt(pi)*(7/10*7^(2/5)*2^(3/5))^n*n^(-3/2)
+ 169/12544*sqrt(7)/sqrt(pi)*(7/10*7^(2/5)*2^(3/5))^n*n^(-5/2)
+ O((7/10*7^(2/5)*2^(3/5))^n*n^(-7/2))
sage: _.parent()
Asymptotic Ring <(Symbolic Constants Subring)^n * n^QQ>
over Symbolic Constants Subring
TESTS::
sage: expansion = asymptotic_expansions.Binomial_kn_over_n('n', k=7/5, precision=3)
sage: n = expansion.parent().gen()
sage: expansion.compare_with_values(n, lambda x: binomial(7/5*x, x), [5, 10, 20]) # rel tol 1e-6
[(5, -0.0287383845047?), (10, -0.030845971026?), (20, -0.03162833549?)]
sage: asymptotic_expansions.Binomial_kn_over_n(
....: 'n', k=5, precision=3, skip_constant_factor=True)
(3125/256)^n*n^(-1/2)
- 7/80*(3125/256)^n*n^(-3/2)
+ 49/12800*(3125/256)^n*n^(-5/2)
+ O((3125/256)^n*n^(-7/2))
sage: _.parent()
Asymptotic Ring <QQ^n * n^QQ> over Rational Field
sage: asymptotic_expansions.Binomial_kn_over_n(
....: 'n', k=4, precision=1, skip_constant_factor=True)
(256/27)^n*n^(-1/2) + O((256/27)^n*n^(-3/2))
::
sage: S = asymptotic_expansions.Stirling('n', precision=5)
sage: n = S.parent().gen()
sage: all( # long time
....: SR(asymptotic_expansions.Binomial_kn_over_n(
....: 'n', k=k, precision=3)).canonicalize_radical() ==
....: SR(S.subs(n=k*n) / (S.subs(n=(k-1)*n) * S)).canonicalize_radical()
....: for k in [2, 3, 4])
True
"""
from sage.symbolic.ring import SR
SCR = SR.subring(no_variables=True)
try:
SCR.coerce(k)
except TypeError as e:
from misc import combine_exceptions
raise combine_exceptions(TypeError('Cannot use k={}.'.format(k)),
e)
S = AsymptoticExpansionGenerators._log_StirlingNegativePowers_(
var, precision=max(precision - 2, 0))
n = S.parent().gen()
result = (S.subs(n=k * n) - S.subs(n=(k - 1) * n) - S).exp()
from sage.rings.rational_field import QQ
P = S.parent().change_parameter(
growth_group='QQ^{n} * {n}^QQ'.format(n=var), coefficient_ring=QQ)
n = P.gen()
b = k**k / (k - 1)**(k - 1)
if b.parent() is SR:
b = SCR(b).canonicalize_radical()
result *= n.rpow(b)
result *= n**(-QQ(1) / QQ(2))
if not skip_constant_factor:
result *= (k / ((k - 1) * 2 * SCR('pi'))).sqrt()
return result
