Source code for diofant.series.gruntz

The Gruntz Algorithm

This section explains the basics of the algorithm :cite:`Gruntz1996limits` used for computing
limits.  Most of the time the :py:func:`~diofant.series.limits.limit` function
should just work.  However it is still useful to keep in mind how it is
implemented in case something does not work as expected.

First we define an ordering on functions of single variable `x` according
to how rapidly varying they at infinity.  Any two functions `f(x)` and
`g(x)` can be compared using the properties of:

    .. math::
        L = \lim\limits_{x\to\infty}\frac{\log|f(x)|}{\log|g(x)|}

We shall say that `f(x)` *dominates* `g(x)`, written `f(x) \succ
g(x)`, iff `L=\pm\infty`.  We also say that `f(x)` and `g(x)` are *of the
same comparability class* if neither `f(x) \succ g(x)` nor `g(x) \succ
f(x)` and shall denote it as `f(x) \asymp g(x)`.

It is easy to show the following examples:

* `e^{e^x} \succ e^{x^2} \succ e^x \succ x \succ 42`
* `2 \asymp 3 \asymp -5`
* `x \asymp x^2 \asymp x^3 \asymp -x`
* `e^x \asymp e^{-x} \asymp e^{2x} \asymp e^{x + e^{-x}}`
* `f(x) \asymp 1/f(x)`

Using these definitions yields the following strategy for
computing `\lim_{x \to \infty} f(x)`:

1. Given the function `f(x)`, we find the set of *most rapidly varying
   subexpressions* (MRV set) of it.  All items of this set belongs to the
   same comparability class.  Let's say it is `\{e^x, e^{2x}\}`.

2. Choose an expression `\omega` which is positive and tends to zero and
   which is in the same comparability class as any element of the MRV set.
   Such element always exists.  Then we rewrite the MRV set using `\omega`,
   in our case `\{\omega^{-1}, \omega^{-2}\}`, and substitute it into `f(x)`.

3. Let `f(\omega)` be the function which is obtained from `f(x)` after the
   rewrite step above.  Consider all expressions independent of `\omega` as
   constants and compute the leading term of the power series of `f(\omega)`
   around `\omega = 0^+`:

       .. math:: f(\omega) = c_0 \omega^{e_0} + c_1 \omega^{e_1} + \ldots

   where `e_0 < e_1 < e_2 \ldots`

4. If the leading exponent `e_0 > 0` then the limit is `0`.  If `e_0 < 0`,
   then the answer is `\pm\infty` (depends on sign of `c_0`).  Finally,
   if `e_0 = 0`, the limit is the limit of the leading coefficient `c_0`.

This exposition glossed over several details.  For example, limits could be
computed recursively (steps 1 and 4).  Please address to the Gruntz thesis :cite:`Gruntz1996limits`
for proof of the termination (pp. 52-60).


from functools import reduce

from ..core import Add, Dummy, E, Float, Integer, Mul, cacheit, evaluate, oo
from ..core.compatibility import ordered
from ..functions import Abs, exp, log
from ..functions import sign as sgn

[docs]def compare(a, b, x): r""" Determine order relation between two functons. Returns ======= {1, 0, -1} Respectively, if `a(x) \succ b(x)`, `a(x) \asymp b(x)` or `b(x) \succ a(x)`. Examples ======== >>> x = Symbol('x', real=True, positive=True) >>> m = Symbol('m', real=True, positive=True) >>> compare(x, x**2, x) 0 >>> compare(1/x, x**m, x) 0 >>> compare(exp(x), exp(x**2), x) -1 >>> compare(exp(x), x**5, x) 1 """ # The log(exp(...)) must always be simplified here for termination. la = a.exp if a.is_Pow and a.base is E else log(a) lb = b.exp if b.is_Pow and b.base is E else log(b) c = limitinf(la/lb, x) if c.is_zero: return -1 elif c.is_infinite: return 1 else: return 0
[docs]def mrv(e, x): """ Calculate the MRV set of expression. Examples ======== >>> x = Symbol('x', real=True, positive=True) >>> mrv(log(x - log(x))/log(x), x) {x} >>> mrv(exp(x + exp(-x)), x) {E**(-x), E**(x + E**(-x))} """ if not e.has(x): return set() elif e == x: return {x} elif e.is_Mul or e.is_Add: a, b = e.as_two_terms() return mrv_max(mrv(a, x), mrv(b, x), x) elif e.is_Pow and e.base is E: if e.exp == x: return {e} elif any(a.is_infinite for a in Mul.make_args(limitinf(e.exp, x))): return mrv_max({e}, mrv(e.exp, x), x) else: return mrv(e.exp, x) elif e.is_Pow: assert not e.exp.has(x) return mrv(e.base, x) elif isinstance(e, log): return mrv(e.args[0], x) elif e.is_Function: return reduce(lambda a, b: mrv_max(a, b, x), [mrv(a, x) for a in e.args]) else: raise NotImplementedError("Don't know how to calculate the mrv of '%s'" % e)
[docs]def mrv_max(f, g, x): """Computes the maximum of two MRV sets.""" if not f or not g or f & g: return f | g c = compare(list(f)[0], list(g)[0], x) if c: return f if c > 0 else g else: return f | g
[docs]@cacheit def sign(e, x): r""" Determine a sign of an expression at infinity. Returns ======= {1, 0, -1} One or minus one, if `e > 0` or `e < 0` for `x` sufficiently large and zero if `e` is *constantly* zero for `x\to\infty`. The result of this function is currently undefined if `e` changes sign arbitrarily often at infinity (e.g. `\sin(x)`). """ if not e.has(x): return sgn(e).simplify() elif e == x: return 1 elif e.is_Mul: a, b = e.as_two_terms() return sign(a, x)*sign(b, x) elif e.is_Pow: s = sign(e.base, x) if s == 1: return 1 c0, e0 = mrv_leadterm(e, x) return sign(c0, x)
[docs]@cacheit def limitinf(e, x): """ Compute limit of the expression at the infinity. Examples ======== >>> x = Symbol('x', real=True, positive=True) >>> limitinf(exp(x)*(exp(1/x - exp(-x)) - exp(1/x)), x) -1 >>> limitinf(x/log(x**(log(x**(log(2)/log(x))))), x) oo """ assert x.is_real and x.is_positive assert not e.has(Float) # Rewrite e in terms of tractable functions only: e = e.rewrite('tractable', deep=True) def transform_abs(f): s = sgn(limitinf(f.args[0], x)) return s*f.args[0] if s in (1, -1) else f e = e.replace(lambda f: isinstance(f, Abs) and f.has(x), transform_abs) if not e.has(x): # This is a bit of a heuristic for nice results. We always rewrite # tractable functions in terms of familiar intractable ones. # TODO: It might be nicer to rewrite the exactly to what they were # initially, but that would take some work to implement. return e.rewrite('intractable', deep=True) c0, e0 = mrv_leadterm(e, x) sig = sign(e0, x) if sig == 1: return Integer(0) elif sig == -1: s = sign(c0, x) assert s != 0 return s*oo elif sig == 0: return limitinf(c0, x) else: raise NotImplementedError('Result depends on the sign of %s' % sig)
[docs]@cacheit def mrv_leadterm(e, x): """ Compute the leading term of the series. Returns ======= tuple The leading term `c_0 w^{e_0}` of the series of `e` in terms of the most rapidly varying subexpression `w` in form of the pair ``(c0, e0)`` of Expr. Examples ======== >>> x = Symbol('x', real=True, positive=True) >>> mrv_leadterm(1/exp(-x + exp(-x)) - exp(x), x) (-1, 0) """ if not e.has(x): return e, Integer(0) e = e.replace(lambda f: f.is_Pow and f.exp.has(x), lambda f: exp(log(f.base)*f.exp)) e = e.replace(lambda f: f.is_Mul and sum(a.is_Pow for a in f.args) > 1, lambda f: Mul(exp(Add(*[a.exp for a in f.args if a.is_Pow and a.base is E])), *[a for a in f.args if not a.is_Pow or a.base is not E])) # The positive dummy, w, is used here so log(w*2) etc. will expand. # TODO: For limits of complex functions, the algorithm would have to # be improved, or just find limits of Re and Im components separately. w = Dummy("w", real=True, positive=True) e, logw = rewrite(e, x, w) lt = e.compute_leading_term(w, logx=logw) c0, e0 = lt.as_coeff_exponent(w) if c0.has(w): raise NotImplementedError("Cannot compute mrv_leadterm(%s, %s). " "The coefficient should have been free of " "%s, but got %s." % (e, x, w, c0)) return c0, e0
[docs]def rewrite(e, x, w): r""" Rewrites expression in terms of the most rapidly varying subexpression. Parameters ========== e : Expr an expression x : Symbol variable of the `e` w : Symbol The symbol which is going to be used for substitution in place of the most rapidly varying in `x` subexpression. Returns ======= tuple A pair: rewritten (in `w`) expression and `\log(w)`. Examples ======== >>> x = Symbol('x', real=True, positive=True) >>> m = Symbol('m', real=True, positive=True) >>> rewrite(exp(x), x, m) (1/m, -x) >>> rewrite(exp(x)*log(log(exp(x))), x, m) (log(x)/m, -x) """ Omega = mrv(e, x) if not Omega: return e, None # e really does not depend on x assert all(e.has(t) for t in Omega) if x in Omega: # Moving up in the asymptotical scale: with evaluate(False): e = e.xreplace({x: exp(x)}) Omega = {s.xreplace({x: exp(x)}) for s in Omega} Omega = list(ordered(Omega, keys=lambda a: -len(mrv(a, x)))) for g in Omega: sig = sign(g.exp, x) if sig not in (1, -1): raise NotImplementedError('Result depends on the sign of %s' % sig) if sig == 1: w = 1/w # if g goes to oo, substitute 1/w # Rewrite and substitute subexpressions in the Omega. for a in Omega: c = limitinf(a.exp/g.exp, x) b = exp(a.exp - c*g.exp)*w**c # exponential must never be expanded here with evaluate(False): e = e.xreplace({a: b}) return e, -sig*g.exp