def SingularityAnalysis(var, zeta=1, alpha=0, beta=0, delta=0,
precision=None, normalized=True):
r"""
Return the asymptotic expansion of the coefficients of
an power series with specified pole and logarithmic singularity.
More precisely, this extracts the `n`-th coefficient
.. MATH::
[z^n] \left(\frac{1}{1-z/\zeta}\right)^\alpha
\left(\frac{1}{z/\zeta} \log \frac{1}{1-z/\zeta}\right)^\beta
\left(\frac{1}{z/\zeta} \log
\left(\frac{1}{z/\zeta} \log \frac{1}{1-z/\zeta}\right)\right)^\delta
(if ``normalized=True``, the default) or
.. MATH::
[z^n] \left(\frac{1}{1-z/\zeta}\right)^\alpha
\left(\log \frac{1}{1-z/\zeta}\right)^\beta
\left(\log
\left(\frac{1}{z/\zeta} \log \frac{1}{1-z/\zeta}\right)\right)^\delta
(if ``normalized=False``).
INPUT:
- ``var`` -- a string for the variable name.
- ``zeta`` -- (default: `1`) the location of the singularity.
- ``alpha`` -- (default: `0`) the pole order of the singularty.
- ``beta`` -- (default: `0`) the order of the logarithmic singularity.
- ``delta`` -- (default: `0`) the order of the log-log singularity.
Not yet implemented for ``delta != 0``.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``normalized`` -- (default: ``True``) a boolean, see above.
OUTPUT:
An asymptotic expansion.
EXAMPLES::
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=1)
1
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=2)
n + 1
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=3)
1/2*n^2 + 3/2*n + 1
sage: _.parent()
Asymptotic Ring <n^ZZ> over Rational Field
::
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=-3/2,
....: precision=3)
3/4/sqrt(pi)*n^(-5/2)
+ 45/32/sqrt(pi)*n^(-7/2)
+ 1155/512/sqrt(pi)*n^(-9/2)
+ O(n^(-11/2))
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=-1/2,
....: precision=3)
-1/2/sqrt(pi)*n^(-3/2)
- 3/16/sqrt(pi)*n^(-5/2)
- 25/256/sqrt(pi)*n^(-7/2)
+ O(n^(-9/2))
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=1/2,
....: precision=4)
1/sqrt(pi)*n^(-1/2)
- 1/8/sqrt(pi)*n^(-3/2)
+ 1/128/sqrt(pi)*n^(-5/2)
+ 5/1024/sqrt(pi)*n^(-7/2)
+ O(n^(-9/2))
sage: _.parent()
Asymptotic Ring <n^QQ> over Symbolic Constants Subring
::
sage: S = SR.subring(rejecting_variables=('n',))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=S.var('a'),
....: precision=4).map_coefficients(lambda c: c.factor())
1/gamma(a)*n^(a - 1)
+ (1/2*(a - 1)*a/gamma(a))*n^(a - 2)
+ (1/24*(3*a - 1)*(a - 1)*(a - 2)*a/gamma(a))*n^(a - 3)
+ (1/48*(a - 1)^2*(a - 2)*(a - 3)*a^2/gamma(a))*n^(a - 4)
+ O(n^(a - 5))
sage: _.parent()
Asymptotic Ring <n^(Symbolic Subring rejecting the variable n)>
over Symbolic Subring rejecting the variable n
::
sage: ae = asymptotic_expansions.SingularityAnalysis('n',
....: alpha=1/2, beta=1, precision=4); ae
1/sqrt(pi)*n^(-1/2)*log(n) + ((euler_gamma + 2*log(2))/sqrt(pi))*n^(-1/2)
- 5/8/sqrt(pi)*n^(-3/2)*log(n) + (1/8*(3*euler_gamma + 6*log(2) - 8)/sqrt(pi)
- (euler_gamma + 2*log(2) - 2)/sqrt(pi))*n^(-3/2) + O(n^(-5/2)*log(n))
sage: n = ae.parent().gen()
sage: ae.subs(n=n-1).map_coefficients(lambda x: x.canonicalize_radical())
1/sqrt(pi)*n^(-1/2)*log(n)
+ ((euler_gamma + 2*log(2))/sqrt(pi))*n^(-1/2)
- 1/8/sqrt(pi)*n^(-3/2)*log(n)
+ (-1/8*(euler_gamma + 2*log(2))/sqrt(pi))*n^(-3/2)
+ O(n^(-5/2)*log(n))
::
sage: asymptotic_expansions.SingularityAnalysis('n',
....: alpha=1, beta=1/2, precision=4)
log(n)^(1/2) + 1/2*euler_gamma*log(n)^(-1/2)
+ (-1/8*euler_gamma^2 + 1/48*pi^2)*log(n)^(-3/2)
+ (1/16*euler_gamma^3 - 1/32*euler_gamma*pi^2 + 1/8*zeta(3))*log(n)^(-5/2)
+ O(log(n)^(-7/2))
::
sage: ae = asymptotic_expansions.SingularityAnalysis('n',
....: alpha=0, beta=2, precision=14)
sage: n = ae.parent().gen()
sage: ae.subs(n=n-2)
2*n^(-1)*log(n) + 2*euler_gamma*n^(-1) - n^(-2) - 1/6*n^(-3) + O(n^(-5))
::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=-1/2, beta=1, precision=2, normalized=False)
-1/2/sqrt(pi)*n^(-3/2)*log(n)
+ (-1/2*(euler_gamma + 2*log(2) - 2)/sqrt(pi))*n^(-3/2)
+ O(n^(-5/2)*log(n))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1/2, alpha=0, beta=1, precision=3, normalized=False)
2^n*n^(-1) + O(2^n*n^(-2))
ALGORITHM:
See [FS2009]_ together with the
`errata list <http://algo.inria.fr/flajolet/Publications/AnaCombi/errata.pdf>`_.
REFERENCES:
.. [FS2009] Philippe Flajolet and Robert Sedgewick,
`Analytic combinatorics <http://algo.inria.fr/flajolet/Publications/AnaCombi/book.pdf>`_.
Cambridge University Press, Cambridge, 2009.
TESTS::
sage: ex = asymptotic_expansions.SingularityAnalysis('n', alpha=-1/2,
....: precision=4)
sage: n = ex.parent().gen()
sage: coefficients = ((1-x)^(1/2)).series(
....: x, 21).truncate().coefficients(x, sparse=False)
sage: ex.compare_with_values(n, # rel tol 1e-6
....: lambda k: coefficients[k], [5, 10, 20])
[(5, 0.015778873294?), (10, 0.01498952777?), (20, 0.0146264622?)]
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=3, precision=2)
1/2*n^2 + 3/2*n + O(1)
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=3, precision=3)
1/2*n^2 + 3/2*n + 1
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=3, precision=4)
1/2*n^2 + 3/2*n + 1
::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=0)
Traceback (most recent call last):
...
NotImplementedOZero: The error term in the result is O(0)
which means 0 for sufficiently large n.
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=-1)
Traceback (most recent call last):
...
NotImplementedOZero: The error term in the result is O(0)
which means 0 for sufficiently large n.
::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'm', alpha=-1/2, precision=3)
-1/2/sqrt(pi)*m^(-3/2)
- 3/16/sqrt(pi)*m^(-5/2)
- 25/256/sqrt(pi)*m^(-7/2)
+ O(m^(-9/2))
sage: _.parent()
Asymptotic Ring <m^QQ> over Symbolic Constants Subring
Location of the singularity::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=1, zeta=2, precision=3)
(1/2)^n
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=1, zeta=1/2, precision=3)
2^n
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=1, zeta=CyclotomicField(3).gen(),
....: precision=3)
(-zeta3 - 1)^n
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=4, zeta=2, precision=3)
1/6*(1/2)^n*n^3 + (1/2)^n*n^2 + 11/6*(1/2)^n*n + O((1/2)^n)
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=-1, zeta=2, precision=3)
Traceback (most recent call last):
...
NotImplementedOZero: The error term in the result is O(0)
which means 0 for sufficiently large n.
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=1/2, zeta=2, precision=3)
1/sqrt(pi)*(1/2)^n*n^(-1/2) - 1/8/sqrt(pi)*(1/2)^n*n^(-3/2)
+ 1/128/sqrt(pi)*(1/2)^n*n^(-5/2) + O((1/2)^n*n^(-7/2))
The following tests correspond to Table VI.5 in [FS2009]_. ::
sage: A.<n> = AsymptoticRing('n^QQ * log(n)^QQ', QQ)
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=-1/2, beta=1, precision=2,
....: normalized=False) * (- sqrt(pi*n^3))
1/2*log(n) + 1/2*euler_gamma + log(2) - 1 + O(n^(-1)*log(n))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=0, beta=1, precision=3,
....: normalized=False)
n^(-1) + O(n^(-2))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=0, beta=2, precision=14,
....: normalized=False) * n
2*log(n) + 2*euler_gamma - n^(-1) - 1/6*n^(-2) + O(n^(-4))
sage: (asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=1/2, beta=1, precision=4,
....: normalized=False) * sqrt(pi*n)).\
....: map_coefficients(lambda x: x.expand())
log(n) + euler_gamma + 2*log(2) - 1/8*n^(-1)*log(n) +
(-1/8*euler_gamma - 1/4*log(2))*n^(-1) + O(n^(-2)*log(n))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=1, beta=1, precision=13,
....: normalized=False)
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4)
+ O(n^(-6))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=1, beta=2, precision=4,
....: normalized=False)
log(n)^2 + 2*euler_gamma*log(n) + euler_gamma^2 - 1/6*pi^2
+ O(n^(-1)*log(n))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=3/2, beta=1, precision=3,
....: normalized=False) * sqrt(pi/n)
2*log(n) + 2*euler_gamma + 4*log(2) - 4 + 3/4*n^(-1)*log(n)
+ O(n^(-1))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=2, beta=1, precision=5,
....: normalized=False)
n*log(n) + (euler_gamma - 1)*n + log(n) + euler_gamma + 1/2
+ O(n^(-1))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=2, beta=2, precision=4,
....: normalized=False) / n
log(n)^2 + (2*euler_gamma - 2)*log(n)
- 2*euler_gamma + euler_gamma^2 - 1/6*pi^2 + 2
+ n^(-1)*log(n)^2 + O(n^(-1)*log(n))
Be aware that the last result does *not* coincide with [FS2009]_,
they do have a different error term.
Checking parameters::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, 1, 1/2, precision=0, normalized=False)
Traceback (most recent call last):
...
ValueError: beta and delta must be integers
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, 1, 1, 1/2, normalized=False)
Traceback (most recent call last):
...
ValueError: beta and delta must be integers
::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=0, beta=0, delta=1, precision=3)
Traceback (most recent call last):
...
NotImplementedError: not implemented for delta!=0
"""
from itertools import islice, count
from asymptotic_ring import AsymptoticRing
from growth_group import ExponentialGrowthGroup, \
MonomialGrowthGroup
from sage.arith.all import falling_factorial
from sage.categories.cartesian_product import cartesian_product
from sage.functions.other import binomial, gamma
from sage.calculus.calculus import limit
from sage.misc.cachefunc import cached_function
from sage.arith.srange import srange
from sage.rings.rational_field import QQ
from sage.rings.integer_ring import ZZ
from sage.symbolic.ring import SR
SCR = SR.subring(no_variables=True)
s = SR('s')
iga = 1/gamma(alpha)
if iga.parent() is SR:
try:
iga = SCR(iga)
except TypeError:
pass
coefficient_ring = iga.parent()
if beta != 0:
coefficient_ring = SCR
@cached_function
def inverse_gamma_derivative(shift, r):
"""
Return value of `r`-th derivative of 1/Gamma
at alpha-shift.
"""
if r == 0:
result = iga*falling_factorial(alpha-1, shift)
else:
result = limit((1/gamma(s)).diff(s, r), s=alpha-shift)
try:
return coefficient_ring(result)
except TypeError:
return result
if isinstance(alpha, int):
alpha = ZZ(alpha)
if isinstance(beta, int):
beta = ZZ(beta)
if isinstance(delta, int):
delta = ZZ(delta)
if precision is None:
precision = AsymptoticRing.__default_prec__
if not normalized and not (beta in ZZ and delta in ZZ):
raise ValueError("beta and delta must be integers")
if delta != 0:
raise NotImplementedError("not implemented for delta!=0")
groups = []
if zeta != 1:
groups.append(ExponentialGrowthGroup((1/zeta).parent(), var))
groups.append(MonomialGrowthGroup(alpha.parent(), var))
if beta != 0:
groups.append(MonomialGrowthGroup(beta.parent(), 'log({})'.format(var)))
group = cartesian_product(groups)
A = AsymptoticRing(growth_group=group, coefficient_ring=coefficient_ring,
default_prec=precision)
n = A.gen()
if zeta == 1:
exponential_factor = 1
else:
exponential_factor = n.rpow(1/zeta)
if beta in ZZ and beta >= 0:
it = ((k, r)
for k in count()
for r in srange(beta+1))
k_max = precision
else:
it = ((0, r)
for r in count())
k_max = 0
if beta != 0:
log_n = n.log()
else:
# avoid construction of log(n)
# because it does not exist in growth group.
log_n = 1
it = reversed(list(islice(it, precision+1)))
if normalized:
beta_denominator = beta
else:
beta_denominator = 0
L = _sa_coefficients_lambda_(max(1, k_max), beta=beta_denominator)
(k, r) = next(it)
result = (n**(-k) * log_n**(-r)).O()
if alpha in ZZ and beta == 0:
if alpha > 0 and alpha <= precision:
result = A(0)
elif alpha <= 0 and precision > 0:
from misc import NotImplementedOZero
raise NotImplementedOZero(A)
for (k, r) in it:
result += binomial(beta, r) * \
sum(L[(k, ell)] * (-1)**ell *
inverse_gamma_derivative(ell, r)
for ell in srange(k, 2*k+1)
if (k, ell) in L) * \
n**(-k) * log_n**(-r)
result *= exponential_factor * n**(alpha-1) * log_n**beta
return result
def HarmonicNumber(var, precision=None, skip_constant_summand=False):
r"""
Return the asymptotic expansion of a harmonic number.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_summand`` -- (default: ``False``) a
boolean. If set, then the constant summand ``euler_gamma`` 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.HarmonicNumber('n', precision=5)
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
TESTS::
sage: ex = asymptotic_expansions.HarmonicNumber('n', precision=5)
sage: n = ex.parent().gen()
sage: ex.compare_with_values(n, # rel tol 1e-6
....: lambda x: sum(1/k for k in srange(1, x+1)), [5, 10, 20])
[(5, 0.0038125360?), (10, 0.00392733?), (20, 0.0039579?)]
sage: asymptotic_expansions.HarmonicNumber('n')
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4)
- 1/252*n^(-6) + 1/240*n^(-8) - 1/132*n^(-10)
+ 691/32760*n^(-12) - 1/12*n^(-14) + 3617/8160*n^(-16)
- 43867/14364*n^(-18) + 174611/6600*n^(-20) - 77683/276*n^(-22)
+ 236364091/65520*n^(-24) - 657931/12*n^(-26)
+ 3392780147/3480*n^(-28) - 1723168255201/85932*n^(-30)
+ 7709321041217/16320*n^(-32)
- 151628697551/12*n^(-34) + O(n^(-36))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.HarmonicNumber(
....: 'n', precision=5, skip_constant_summand=True)
log(n) + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Rational Field
sage: for p in range(5):
....: print asymptotic_expansions.HarmonicNumber(
....: 'n', precision=p)
O(log(n))
log(n) + O(1)
log(n) + euler_gamma + O(n^(-1))
log(n) + euler_gamma + 1/2*n^(-1) + O(n^(-2))
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + O(n^(-4))
sage: asymptotic_expansions.HarmonicNumber('m', precision=5)
log(m) + euler_gamma + 1/2*m^(-1) - 1/12*m^(-2) + 1/120*m^(-4) + O(m^(-6))
"""
if not skip_constant_summand:
from sage.symbolic.ring import SR
coefficient_ring = SR.subring(no_variables=True)
else:
from sage.rings.rational_field import QQ
coefficient_ring = QQ
from asymptotic_ring import AsymptoticRing
A = AsymptoticRing(growth_group='{n}^ZZ * log({n})^ZZ'.format(n=var),
coefficient_ring=coefficient_ring)
n = A.gen()
if precision is None:
precision = A.default_prec
from sage.functions.log import log
result = A.zero()
if precision >= 1:
result += log(n)
if precision >= 2 and not skip_constant_summand:
from sage.symbolic.constants import euler_gamma
result += coefficient_ring(euler_gamma)
if precision >= 3:
result += 1 / (2 * n)
from sage.arith.srange import srange
from sage.arith.all import bernoulli
for k in srange(2, 2*precision - 4, 2):
result += -bernoulli(k) / k / n**k
if precision < 1:
result += (log(n)).O()
elif precision == 1:
result += A(1).O()
elif precision == 2:
result += (1 / n).O()
else:
result += (1 / n**(2*precision - 4)).O()
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
def Stirling(var, precision=None, skip_constant_factor=False):
r"""
Return Stirling's approximation formula for factorials.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer `\ge 3`. 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{2\pi}` 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.Stirling('n', precision=5)
sqrt(2)*sqrt(pi)*e^(n*log(n))*(e^n)^(-1)*n^(1/2) +
1/12*sqrt(2)*sqrt(pi)*e^(n*log(n))*(e^n)^(-1)*n^(-1/2) +
1/288*sqrt(2)*sqrt(pi)*e^(n*log(n))*(e^n)^(-1)*n^(-3/2) +
O(e^(n*log(n))*(e^n)^(-1)*n^(-5/2))
sage: _.parent()
Asymptotic Ring <(e^(n*log(n)))^QQ * (e^n)^QQ * n^QQ * log(n)^QQ>
over Symbolic Constants Subring
.. SEEALSO::
:meth:`log_Stirling`,
:meth:`~sage.rings.asymptotic.asymptotic_ring.AsymptoticExpansion.factorial`.
TESTS::
sage: expansion = asymptotic_expansions.Stirling('n', precision=5)
sage: n = expansion.parent().gen()
sage: expansion.compare_with_values(n, lambda x: x.factorial(), [5, 10, 20]) # rel tol 1e-6
[(5, 0.00675841118?), (10, 0.0067589306?), (20, 0.006744925?)]
sage: asymptotic_expansions.Stirling('n', precision=5,
....: skip_constant_factor=True)
e^(n*log(n))*(e^n)^(-1)*n^(1/2) +
1/12*e^(n*log(n))*(e^n)^(-1)*n^(-1/2) +
1/288*e^(n*log(n))*(e^n)^(-1)*n^(-3/2) +
O(e^(n*log(n))*(e^n)^(-1)*n^(-5/2))
sage: _.parent()
Asymptotic Ring <(e^(n*log(n)))^QQ * (e^n)^QQ * n^QQ * log(n)^QQ>
over Rational Field
sage: asymptotic_expansions.Stirling('m', precision=4)
sqrt(2)*sqrt(pi)*e^(m*log(m))*(e^m)^(-1)*m^(1/2) +
O(e^(m*log(m))*(e^m)^(-1)*m^(-1/2))
sage: asymptotic_expansions.Stirling('m', precision=3)
O(e^(m*log(m))*(e^m)^(-1)*m^(1/2))
sage: asymptotic_expansions.Stirling('m', precision=2)
Traceback (most recent call last):
...
ValueError: precision must be at least 3
"""
if precision < 3:
raise ValueError("precision must be at least 3")
log_Stirling = AsymptoticExpansionGenerators.log_Stirling(
var, precision=precision, skip_constant_summand=True)
P = log_Stirling.parent().change_parameter(
growth_group='(e^({n}*log({n})))^QQ * (e^{n})^QQ * {n}^QQ * log({n})^QQ'.format(n=var))
from sage.functions.log import exp
result = exp(P(log_Stirling))
if not skip_constant_factor:
from sage.symbolic.ring import SR
SCR = SR.subring(no_variables=True)
result *= (2*SCR('pi')).sqrt()
return result
def log_Stirling(var, precision=None, skip_constant_summand=False):
r"""
Return the logarithm of Stirling's approximation formula
for factorials.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_summand`` -- (default: ``False``) a
boolean. If set, then the constant summand `\log(2\pi)/2` 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.log_Stirling('n', precision=7)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
- 1/360*n^(-3) + 1/1260*n^(-5) + O(n^(-7))
.. SEEALSO::
:meth:`Stirling`,
:meth:`~sage.rings.asymptotic.asymptotic_ring.AsymptoticExpansion.factorial`.
TESTS::
sage: expansion = asymptotic_expansions.log_Stirling('n', precision=7)
sage: n = expansion.parent().gen()
sage: expansion.compare_with_values(n, lambda x: x.factorial().log(), [5, 10, 20]) # rel tol 1e-6
[(5, 0.000564287?), (10, 0.0005870?), (20, 0.0006?)]
sage: asymptotic_expansions.log_Stirling('n')
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
- 1/360*n^(-3) + 1/1260*n^(-5) - 1/1680*n^(-7) + 1/1188*n^(-9)
- 691/360360*n^(-11) + 1/156*n^(-13) - 3617/122400*n^(-15)
+ 43867/244188*n^(-17) - 174611/125400*n^(-19) + 77683/5796*n^(-21)
- 236364091/1506960*n^(-23) + 657931/300*n^(-25)
- 3392780147/93960*n^(-27) + 1723168255201/2492028*n^(-29)
- 7709321041217/505920*n^(-31) + O(n^(-33))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=7, skip_constant_summand=True)
n*log(n) - n + 1/2*log(n) + 1/12*n^(-1) - 1/360*n^(-3) +
1/1260*n^(-5) + O(n^(-7))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Rational Field
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=0)
O(n*log(n))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=1)
n*log(n) + O(n)
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=2)
n*log(n) - n + O(log(n))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=3)
n*log(n) - n + 1/2*log(n) + O(1)
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=4)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + O(n^(-1))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=5)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
+ O(n^(-3))
sage: asymptotic_expansions.log_Stirling(
....: 'm', precision=7, skip_constant_summand=True)
m*log(m) - m + 1/2*log(m) + 1/12*m^(-1) - 1/360*m^(-3) +
1/1260*m^(-5) + O(m^(-7))
"""
if not skip_constant_summand:
from sage.symbolic.ring import SR
coefficient_ring = SR.subring(no_variables=True)
else:
from sage.rings.rational_field import QQ
coefficient_ring = QQ
from asymptotic_ring import AsymptoticRing
A = AsymptoticRing(growth_group='{n}^ZZ * log({n})^ZZ'.format(n=var),
coefficient_ring=coefficient_ring)
n = A.gen()
if precision is None:
precision = AsymptoticRing.__default_prec__
from sage.functions.log import log
result = A.zero()
if precision >= 1:
result += n * log(n)
if precision >= 2:
result += -n
if precision >= 3:
result += log(n) / 2
if precision >= 4 and not skip_constant_summand:
result += log(2*coefficient_ring('pi')) / 2
result += AsymptoticExpansionGenerators._log_StirlingNegativePowers_(
var, precision - 4)
if precision < 1:
result += (n * log(n)).O()
elif precision == 1:
result += n.O()
elif precision == 2:
result += log(n).O()
elif precision == 3:
result += A(1).O()
return result
def SingularityAnalysis(var,
zeta=1,
alpha=0,
beta=0,
delta=0,
precision=None,
normalized=True):
r"""
Return the asymptotic expansion of the coefficients of
an power series with specified pole and logarithmic singularity.
More precisely, this extracts the `n`-th coefficient
.. MATH::
[z^n] \left(\frac{1}{1-z/\zeta}\right)^\alpha
\left(\frac{1}{z/\zeta} \log \frac{1}{1-z/\zeta}\right)^\beta
\left(\frac{1}{z/\zeta} \log
\left(\frac{1}{z/\zeta} \log \frac{1}{1-z/\zeta}\right)\right)^\delta
(if ``normalized=True``, the default) or
.. MATH::
[z^n] \left(\frac{1}{1-z/\zeta}\right)^\alpha
\left(\log \frac{1}{1-z/\zeta}\right)^\beta
\left(\log
\left(\frac{1}{z/\zeta} \log \frac{1}{1-z/\zeta}\right)\right)^\delta
(if ``normalized=False``).
INPUT:
- ``var`` -- a string for the variable name.
- ``zeta`` -- (default: `1`) the location of the singularity.
- ``alpha`` -- (default: `0`) the pole order of the singularty.
- ``beta`` -- (default: `0`) the order of the logarithmic singularity.
- ``delta`` -- (default: `0`) the order of the log-log singularity.
Not yet implemented for ``delta != 0``.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``normalized`` -- (default: ``True``) a boolean, see above.
OUTPUT:
An asymptotic expansion.
EXAMPLES::
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=1)
1
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=2)
n + 1
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=3)
1/2*n^2 + 3/2*n + 1
sage: _.parent()
Asymptotic Ring <n^ZZ> over Rational Field
::
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=-3/2,
....: precision=3)
3/4/sqrt(pi)*n^(-5/2)
+ 45/32/sqrt(pi)*n^(-7/2)
+ 1155/512/sqrt(pi)*n^(-9/2)
+ O(n^(-11/2))
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=-1/2,
....: precision=3)
-1/2/sqrt(pi)*n^(-3/2)
- 3/16/sqrt(pi)*n^(-5/2)
- 25/256/sqrt(pi)*n^(-7/2)
+ O(n^(-9/2))
sage: asymptotic_expansions.SingularityAnalysis('n', alpha=1/2,
....: precision=4)
1/sqrt(pi)*n^(-1/2)
- 1/8/sqrt(pi)*n^(-3/2)
+ 1/128/sqrt(pi)*n^(-5/2)
+ 5/1024/sqrt(pi)*n^(-7/2)
+ O(n^(-9/2))
sage: _.parent()
Asymptotic Ring <n^QQ> over Symbolic Constants Subring
::
sage: S = SR.subring(rejecting_variables=('n',))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=S.var('a'),
....: precision=4).map_coefficients(lambda c: c.factor())
1/gamma(a)*n^(a - 1)
+ (1/2*(a - 1)*a/gamma(a))*n^(a - 2)
+ (1/24*(3*a - 1)*(a - 1)*(a - 2)*a/gamma(a))*n^(a - 3)
+ (1/48*(a - 1)^2*(a - 2)*(a - 3)*a^2/gamma(a))*n^(a - 4)
+ O(n^(a - 5))
sage: _.parent()
Asymptotic Ring <n^(Symbolic Subring rejecting the variable n)>
over Symbolic Subring rejecting the variable n
::
sage: ae = asymptotic_expansions.SingularityAnalysis('n',
....: alpha=1/2, beta=1, precision=4); ae
1/sqrt(pi)*n^(-1/2)*log(n) + ((euler_gamma + 2*log(2))/sqrt(pi))*n^(-1/2)
- 5/8/sqrt(pi)*n^(-3/2)*log(n) + (1/8*(3*euler_gamma + 6*log(2) - 8)/sqrt(pi)
- (euler_gamma + 2*log(2) - 2)/sqrt(pi))*n^(-3/2) + O(n^(-5/2)*log(n))
sage: n = ae.parent().gen()
sage: ae.subs(n=n-1).map_coefficients(lambda x: x.canonicalize_radical())
1/sqrt(pi)*n^(-1/2)*log(n)
+ ((euler_gamma + 2*log(2))/sqrt(pi))*n^(-1/2)
- 1/8/sqrt(pi)*n^(-3/2)*log(n)
+ (-1/8*(euler_gamma + 2*log(2))/sqrt(pi))*n^(-3/2)
+ O(n^(-5/2)*log(n))
::
sage: asymptotic_expansions.SingularityAnalysis('n',
....: alpha=1, beta=1/2, precision=4)
log(n)^(1/2) + 1/2*euler_gamma*log(n)^(-1/2)
+ (-1/8*euler_gamma^2 + 1/48*pi^2)*log(n)^(-3/2)
+ (1/16*euler_gamma^3 - 1/32*euler_gamma*pi^2 + 1/8*zeta(3))*log(n)^(-5/2)
+ O(log(n)^(-7/2))
::
sage: ae = asymptotic_expansions.SingularityAnalysis('n',
....: alpha=0, beta=2, precision=14)
sage: n = ae.parent().gen()
sage: ae.subs(n=n-2)
2*n^(-1)*log(n) + 2*euler_gamma*n^(-1) - n^(-2) - 1/6*n^(-3) + O(n^(-5))
::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=-1/2, beta=1, precision=2, normalized=False)
-1/2/sqrt(pi)*n^(-3/2)*log(n)
+ (-1/2*(euler_gamma + 2*log(2) - 2)/sqrt(pi))*n^(-3/2)
+ O(n^(-5/2)*log(n))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1/2, alpha=0, beta=1, precision=3, normalized=False)
2^n*n^(-1) + O(2^n*n^(-2))
ALGORITHM:
See [FS2009]_ together with the
`errata list <http://algo.inria.fr/flajolet/Publications/AnaCombi/errata.pdf>`_.
REFERENCES:
.. [FS2009] Philippe Flajolet and Robert Sedgewick,
`Analytic combinatorics <http://algo.inria.fr/flajolet/Publications/AnaCombi/book.pdf>`_.
Cambridge University Press, Cambridge, 2009.
TESTS::
sage: ex = asymptotic_expansions.SingularityAnalysis('n', alpha=-1/2,
....: precision=4)
sage: n = ex.parent().gen()
sage: coefficients = ((1-x)^(1/2)).series(
....: x, 21).truncate().coefficients(x, sparse=False)
sage: ex.compare_with_values(n, # rel tol 1e-6
....: lambda k: coefficients[k], [5, 10, 20])
[(5, 0.015778873294?), (10, 0.01498952777?), (20, 0.0146264622?)]
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=3, precision=2)
1/2*n^2 + 3/2*n + O(1)
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=3, precision=3)
1/2*n^2 + 3/2*n + 1
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=3, precision=4)
1/2*n^2 + 3/2*n + 1
::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=0)
Traceback (most recent call last):
...
NotImplementedOZero: The error term in the result is O(0)
which means 0 for sufficiently large n.
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=-1)
Traceback (most recent call last):
...
NotImplementedOZero: The error term in the result is O(0)
which means 0 for sufficiently large n.
::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'm', alpha=-1/2, precision=3)
-1/2/sqrt(pi)*m^(-3/2)
- 3/16/sqrt(pi)*m^(-5/2)
- 25/256/sqrt(pi)*m^(-7/2)
+ O(m^(-9/2))
sage: _.parent()
Asymptotic Ring <m^QQ> over Symbolic Constants Subring
Location of the singularity::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=1, zeta=2, precision=3)
(1/2)^n
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=1, zeta=1/2, precision=3)
2^n
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=1, zeta=CyclotomicField(3).gen(),
....: precision=3)
(-zeta3 - 1)^n
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=4, zeta=2, precision=3)
1/6*(1/2)^n*n^3 + (1/2)^n*n^2 + 11/6*(1/2)^n*n + O((1/2)^n)
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=-1, zeta=2, precision=3)
Traceback (most recent call last):
...
NotImplementedOZero: The error term in the result is O(0)
which means 0 for sufficiently large n.
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=1/2, zeta=2, precision=3)
1/sqrt(pi)*(1/2)^n*n^(-1/2) - 1/8/sqrt(pi)*(1/2)^n*n^(-3/2)
+ 1/128/sqrt(pi)*(1/2)^n*n^(-5/2) + O((1/2)^n*n^(-7/2))
The following tests correspond to Table VI.5 in [FS2009]_. ::
sage: A.<n> = AsymptoticRing('n^QQ * log(n)^QQ', QQ)
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=-1/2, beta=1, precision=2,
....: normalized=False) * (- sqrt(pi*n^3))
1/2*log(n) + 1/2*euler_gamma + log(2) - 1 + O(n^(-1)*log(n))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=0, beta=1, precision=3,
....: normalized=False)
n^(-1) + O(n^(-2))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=0, beta=2, precision=14,
....: normalized=False) * n
2*log(n) + 2*euler_gamma - n^(-1) - 1/6*n^(-2) + O(n^(-4))
sage: (asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=1/2, beta=1, precision=4,
....: normalized=False) * sqrt(pi*n)).\
....: map_coefficients(lambda x: x.expand())
log(n) + euler_gamma + 2*log(2) - 1/8*n^(-1)*log(n) +
(-1/8*euler_gamma - 1/4*log(2))*n^(-1) + O(n^(-2)*log(n))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=1, beta=1, precision=13,
....: normalized=False)
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4)
+ O(n^(-6))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=1, beta=2, precision=4,
....: normalized=False)
log(n)^2 + 2*euler_gamma*log(n) + euler_gamma^2 - 1/6*pi^2
+ O(n^(-1)*log(n))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=3/2, beta=1, precision=3,
....: normalized=False) * sqrt(pi/n)
2*log(n) + 2*euler_gamma + 4*log(2) - 4 + 3/4*n^(-1)*log(n)
+ O(n^(-1))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=2, beta=1, precision=5,
....: normalized=False)
n*log(n) + (euler_gamma - 1)*n + log(n) + euler_gamma + 1/2
+ O(n^(-1))
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, alpha=2, beta=2, precision=4,
....: normalized=False) / n
log(n)^2 + (2*euler_gamma - 2)*log(n)
- 2*euler_gamma + euler_gamma^2 - 1/6*pi^2 + 2
+ n^(-1)*log(n)^2 + O(n^(-1)*log(n))
Be aware that the last result does *not* coincide with [FS2009]_,
they do have a different error term.
Checking parameters::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, 1, 1/2, precision=0, normalized=False)
Traceback (most recent call last):
...
ValueError: beta and delta must be integers
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', 1, 1, 1, 1/2, normalized=False)
Traceback (most recent call last):
...
ValueError: beta and delta must be integers
::
sage: asymptotic_expansions.SingularityAnalysis(
....: 'n', alpha=0, beta=0, delta=1, precision=3)
Traceback (most recent call last):
...
NotImplementedError: not implemented for delta!=0
"""
from itertools import islice, count
from asymptotic_ring import AsymptoticRing
from growth_group import ExponentialGrowthGroup, \
MonomialGrowthGroup
from sage.arith.all import falling_factorial
from sage.categories.cartesian_product import cartesian_product
from sage.functions.other import binomial, gamma
from sage.calculus.calculus import limit
from sage.misc.cachefunc import cached_function
from sage.arith.srange import srange
from sage.rings.rational_field import QQ
from sage.rings.integer_ring import ZZ
from sage.symbolic.ring import SR
SCR = SR.subring(no_variables=True)
s = SR('s')
iga = 1 / gamma(alpha)
if iga.parent() is SR:
try:
iga = SCR(iga)
except TypeError:
pass
coefficient_ring = iga.parent()
if beta != 0:
coefficient_ring = SCR
@cached_function
def inverse_gamma_derivative(shift, r):
"""
Return value of `r`-th derivative of 1/Gamma
at alpha-shift.
"""
if r == 0:
result = iga * falling_factorial(alpha - 1, shift)
else:
result = limit((1 / gamma(s)).diff(s, r), s=alpha - shift)
try:
return coefficient_ring(result)
except TypeError:
return result
if isinstance(alpha, int):
alpha = ZZ(alpha)
if isinstance(beta, int):
beta = ZZ(beta)
if isinstance(delta, int):
delta = ZZ(delta)
if precision is None:
precision = AsymptoticRing.__default_prec__
if not normalized and not (beta in ZZ and delta in ZZ):
raise ValueError("beta and delta must be integers")
if delta != 0:
raise NotImplementedError("not implemented for delta!=0")
groups = []
if zeta != 1:
groups.append(ExponentialGrowthGroup((1 / zeta).parent(), var))
groups.append(MonomialGrowthGroup(alpha.parent(), var))
if beta != 0:
groups.append(
MonomialGrowthGroup(beta.parent(), 'log({})'.format(var)))
group = cartesian_product(groups)
A = AsymptoticRing(growth_group=group,
coefficient_ring=coefficient_ring,
default_prec=precision)
n = A.gen()
if zeta == 1:
exponential_factor = 1
else:
exponential_factor = n.rpow(1 / zeta)
if beta in ZZ and beta >= 0:
it = ((k, r) for k in count() for r in srange(beta + 1))
k_max = precision
else:
it = ((0, r) for r in count())
k_max = 0
if beta != 0:
log_n = n.log()
else:
# avoid construction of log(n)
# because it does not exist in growth group.
log_n = 1
it = reversed(list(islice(it, precision + 1)))
if normalized:
beta_denominator = beta
else:
beta_denominator = 0
L = _sa_coefficients_lambda_(max(1, k_max), beta=beta_denominator)
(k, r) = next(it)
result = (n**(-k) * log_n**(-r)).O()
if alpha in ZZ and beta == 0:
if alpha > 0 and alpha <= precision:
result = A(0)
elif alpha <= 0 and precision > 0:
from misc import NotImplementedOZero
raise NotImplementedOZero(A)
for (k, r) in it:
result += binomial(beta, r) * \
sum(L[(k, ell)] * (-1)**ell *
inverse_gamma_derivative(ell, r)
for ell in srange(k, 2*k+1)
if (k, ell) in L) * \
n**(-k) * log_n**(-r)
result *= exponential_factor * n**(alpha - 1) * log_n**beta
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
def HarmonicNumber(var, precision=None, skip_constant_summand=False):
r"""
Return the asymptotic expansion of a harmonic number.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_summand`` -- (default: ``False``) a
boolean. If set, then the constant summand ``euler_gamma`` 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.HarmonicNumber('n', precision=5)
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
TESTS::
sage: ex = asymptotic_expansions.HarmonicNumber('n', precision=5)
sage: n = ex.parent().gen()
sage: ex.compare_with_values(n, # rel tol 1e-6
....: lambda x: sum(1/k for k in srange(1, x+1)), [5, 10, 20])
[(5, 0.0038125360?), (10, 0.00392733?), (20, 0.0039579?)]
sage: asymptotic_expansions.HarmonicNumber('n')
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4)
- 1/252*n^(-6) + 1/240*n^(-8) - 1/132*n^(-10)
+ 691/32760*n^(-12) - 1/12*n^(-14) + 3617/8160*n^(-16)
- 43867/14364*n^(-18) + 174611/6600*n^(-20) - 77683/276*n^(-22)
+ 236364091/65520*n^(-24) - 657931/12*n^(-26)
+ 3392780147/3480*n^(-28) - 1723168255201/85932*n^(-30)
+ 7709321041217/16320*n^(-32)
- 151628697551/12*n^(-34) + O(n^(-36))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.HarmonicNumber(
....: 'n', precision=5, skip_constant_summand=True)
log(n) + 1/2*n^(-1) - 1/12*n^(-2) + 1/120*n^(-4) + O(n^(-6))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Rational Field
sage: for p in range(5):
....: print asymptotic_expansions.HarmonicNumber(
....: 'n', precision=p)
O(log(n))
log(n) + O(1)
log(n) + euler_gamma + O(n^(-1))
log(n) + euler_gamma + 1/2*n^(-1) + O(n^(-2))
log(n) + euler_gamma + 1/2*n^(-1) - 1/12*n^(-2) + O(n^(-4))
sage: asymptotic_expansions.HarmonicNumber('m', precision=5)
log(m) + euler_gamma + 1/2*m^(-1) - 1/12*m^(-2) + 1/120*m^(-4) + O(m^(-6))
"""
if not skip_constant_summand:
from sage.symbolic.ring import SR
coefficient_ring = SR.subring(no_variables=True)
else:
from sage.rings.rational_field import QQ
coefficient_ring = QQ
from asymptotic_ring import AsymptoticRing
A = AsymptoticRing(growth_group='{n}^ZZ * log({n})^ZZ'.format(n=var),
coefficient_ring=coefficient_ring)
n = A.gen()
if precision is None:
precision = A.default_prec
from sage.functions.log import log
result = A.zero()
if precision >= 1:
result += log(n)
if precision >= 2 and not skip_constant_summand:
from sage.symbolic.constants import euler_gamma
result += coefficient_ring(euler_gamma)
if precision >= 3:
result += 1 / (2 * n)
from sage.arith.srange import srange
from sage.arith.all import bernoulli
for k in srange(2, 2 * precision - 4, 2):
result += -bernoulli(k) / k / n**k
if precision < 1:
result += (log(n)).O()
elif precision == 1:
result += A(1).O()
elif precision == 2:
result += (1 / n).O()
else:
result += (1 / n**(2 * precision - 4)).O()
return result
def log_Stirling(var, precision=None, skip_constant_summand=False):
r"""
Return the logarithm of Stirling's approximation formula
for factorials.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer. If ``None``, then
the default precision of the asymptotic ring is used.
- ``skip_constant_summand`` -- (default: ``False``) a
boolean. If set, then the constant summand `\log(2\pi)/2` 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.log_Stirling('n', precision=7)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
- 1/360*n^(-3) + 1/1260*n^(-5) + O(n^(-7))
.. SEEALSO::
:meth:`Stirling`,
:meth:`~sage.rings.asymptotic.asymptotic_ring.AsymptoticExpansion.factorial`.
TESTS::
sage: expansion = asymptotic_expansions.log_Stirling('n', precision=7)
sage: n = expansion.parent().gen()
sage: expansion.compare_with_values(n, lambda x: x.factorial().log(), [5, 10, 20]) # rel tol 1e-6
[(5, 0.000564287?), (10, 0.0005870?), (20, 0.0006?)]
sage: asymptotic_expansions.log_Stirling('n')
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
- 1/360*n^(-3) + 1/1260*n^(-5) - 1/1680*n^(-7) + 1/1188*n^(-9)
- 691/360360*n^(-11) + 1/156*n^(-13) - 3617/122400*n^(-15)
+ 43867/244188*n^(-17) - 174611/125400*n^(-19) + 77683/5796*n^(-21)
- 236364091/1506960*n^(-23) + 657931/300*n^(-25)
- 3392780147/93960*n^(-27) + 1723168255201/2492028*n^(-29)
- 7709321041217/505920*n^(-31) + O(n^(-33))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Symbolic Constants Subring
::
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=7, skip_constant_summand=True)
n*log(n) - n + 1/2*log(n) + 1/12*n^(-1) - 1/360*n^(-3) +
1/1260*n^(-5) + O(n^(-7))
sage: _.parent()
Asymptotic Ring <n^ZZ * log(n)^ZZ> over Rational Field
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=0)
O(n*log(n))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=1)
n*log(n) + O(n)
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=2)
n*log(n) - n + O(log(n))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=3)
n*log(n) - n + 1/2*log(n) + O(1)
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=4)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + O(n^(-1))
sage: asymptotic_expansions.log_Stirling(
....: 'n', precision=5)
n*log(n) - n + 1/2*log(n) + 1/2*log(2*pi) + 1/12*n^(-1)
+ O(n^(-3))
sage: asymptotic_expansions.log_Stirling(
....: 'm', precision=7, skip_constant_summand=True)
m*log(m) - m + 1/2*log(m) + 1/12*m^(-1) - 1/360*m^(-3) +
1/1260*m^(-5) + O(m^(-7))
"""
if not skip_constant_summand:
from sage.symbolic.ring import SR
coefficient_ring = SR.subring(no_variables=True)
else:
from sage.rings.rational_field import QQ
coefficient_ring = QQ
from asymptotic_ring import AsymptoticRing
A = AsymptoticRing(growth_group='{n}^ZZ * log({n})^ZZ'.format(n=var),
coefficient_ring=coefficient_ring)
n = A.gen()
if precision is None:
precision = AsymptoticRing.__default_prec__
from sage.functions.log import log
result = A.zero()
if precision >= 1:
result += n * log(n)
if precision >= 2:
result += -n
if precision >= 3:
result += log(n) / 2
if precision >= 4 and not skip_constant_summand:
result += log(2 * coefficient_ring('pi')) / 2
result += AsymptoticExpansionGenerators._log_StirlingNegativePowers_(
var, precision - 4)
if precision < 1:
result += (n * log(n)).O()
elif precision == 1:
result += n.O()
elif precision == 2:
result += log(n).O()
elif precision == 3:
result += A(1).O()
return result
def Stirling(var, precision=None, skip_constant_factor=False):
r"""
Return Stirling's approximation formula for factorials.
INPUT:
- ``var`` -- a string for the variable name.
- ``precision`` -- (default: ``None``) an integer `\ge 3`. 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{2\pi}` 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.Stirling('n', precision=5)
sqrt(2)*sqrt(pi)*e^(n*log(n))*(e^n)^(-1)*n^(1/2) +
1/12*sqrt(2)*sqrt(pi)*e^(n*log(n))*(e^n)^(-1)*n^(-1/2) +
1/288*sqrt(2)*sqrt(pi)*e^(n*log(n))*(e^n)^(-1)*n^(-3/2) +
O(e^(n*log(n))*(e^n)^(-1)*n^(-5/2))
sage: _.parent()
Asymptotic Ring <(e^(n*log(n)))^QQ * (e^n)^QQ * n^QQ * log(n)^QQ>
over Symbolic Constants Subring
.. SEEALSO::
:meth:`log_Stirling`,
:meth:`~sage.rings.asymptotic.asymptotic_ring.AsymptoticExpansion.factorial`.
TESTS::
sage: expansion = asymptotic_expansions.Stirling('n', precision=5)
sage: n = expansion.parent().gen()
sage: expansion.compare_with_values(n, lambda x: x.factorial(), [5, 10, 20]) # rel tol 1e-6
[(5, 0.00675841118?), (10, 0.0067589306?), (20, 0.006744925?)]
sage: asymptotic_expansions.Stirling('n', precision=5,
....: skip_constant_factor=True)
e^(n*log(n))*(e^n)^(-1)*n^(1/2) +
1/12*e^(n*log(n))*(e^n)^(-1)*n^(-1/2) +
1/288*e^(n*log(n))*(e^n)^(-1)*n^(-3/2) +
O(e^(n*log(n))*(e^n)^(-1)*n^(-5/2))
sage: _.parent()
Asymptotic Ring <(e^(n*log(n)))^QQ * (e^n)^QQ * n^QQ * log(n)^QQ>
over Rational Field
sage: asymptotic_expansions.Stirling('m', precision=4)
sqrt(2)*sqrt(pi)*e^(m*log(m))*(e^m)^(-1)*m^(1/2) +
O(e^(m*log(m))*(e^m)^(-1)*m^(-1/2))
sage: asymptotic_expansions.Stirling('m', precision=3)
O(e^(m*log(m))*(e^m)^(-1)*m^(1/2))
sage: asymptotic_expansions.Stirling('m', precision=2)
Traceback (most recent call last):
...
ValueError: precision must be at least 3
"""
if precision < 3:
raise ValueError("precision must be at least 3")
log_Stirling = AsymptoticExpansionGenerators.log_Stirling(
var, precision=precision, skip_constant_summand=True)
P = log_Stirling.parent().change_parameter(
growth_group=
'(e^({n}*log({n})))^QQ * (e^{n})^QQ * {n}^QQ * log({n})^QQ'.format(
n=var))
from sage.functions.log import exp
result = exp(P(log_Stirling))
if not skip_constant_factor:
from sage.symbolic.ring import SR
SCR = SR.subring(no_variables=True)
result *= (2 * SCR('pi')).sqrt()
return result