Python SR.subring方法代码示例

# 需要导入模块: from sage.symbolic.ring import SR [as 别名]
# 或者: from sage.symbolic.ring.SR import subring [as 别名]
    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

        ::

#.........这里部分代码省略.........

# 需要导入模块: from sage.symbolic.ring import SR [as 别名]
# 或者: from sage.symbolic.ring.SR import subring [as 别名]
    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(
#.........这里部分代码省略.........

# 需要导入模块: from sage.symbolic.ring import SR [as 别名]
# 或者: from sage.symbolic.ring.SR import subring [as 别名]
    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:
#.........这里部分代码省略.........

# 需要导入模块: from sage.symbolic.ring import SR [as 别名]
# 或者: from sage.symbolic.ring.SR import subring [as 别名]
    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:
#.........这里部分代码省略.........

# 需要导入模块: from sage.symbolic.ring import SR [as 别名]
# 或者: from sage.symbolic.ring.SR import subring [as 别名]
    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