```
"""
Limits
======
Implemented according to the PhD thesis
http://www.cybertester.com/data/gruntz.pdf, which contains very thorough
descriptions of the algorithm including many examples. We summarize here
the gist of it.
All functions are sorted according to how rapidly varying they are at
infinity using the following rules. Any two functions f and g can be
compared using the properties of L:
L=lim log|f(x)| / log|g(x)| (for x -> oo)
We define >, < ~ according to::
1. f > g .... L=+-oo
we say that:
- f is greater than any power of g
- f is more rapidly varying than g
- f goes to infinity/zero faster than g
2. f < g .... L=0
we say that:
- f is lower than any power of g
3. f ~ g .... L!=0, +-oo
we say that:
- both f and g are bounded from above and below by suitable integral
powers of the other
Examples
========
::
2 < x < exp(x) < exp(x**2) < exp(exp(x))
2 ~ 3 ~ -5
x ~ x**2 ~ x**3 ~ 1/x ~ x**m ~ -x
exp(x) ~ exp(-x) ~ exp(2x) ~ exp(x)**2 ~ exp(x+exp(-x))
f ~ 1/f
So we can divide all the functions into comparability classes (x and x^2
belong to one class, exp(x) and exp(-x) belong to some other class). In
principle, we could compare any two functions, but in our algorithm, we
don't compare anything below the class 2~3~-5 (for example log(x) is
below this), so we set 2~3~-5 as the lowest comparability class.
Given the function f, we find the list of most rapidly varying (mrv set)
subexpressions of it. This list belongs to the same comparability class.
Let's say it is {exp(x), exp(2x)}. Using the rule f ~ 1/f we find an
element "w" (either from the list or a new one) from the same
comparability class which goes to zero at infinity. In our example we
set w=exp(-x) (but we could also set w=exp(-2x) or w=exp(-3x) ...). We
rewrite the mrv set using w, in our case {1/w, 1/w^2}, and substitute it
into f. Then we expand f into a series in w::
f = c0*w^e0 + c1*w^e1 + ... + O(w^en), where e0<e1<...<en, c0!=0
but for x->oo, lim f = lim c0*w^e0, because all the other terms go to zero,
because w goes to zero faster than the ci and ei. So::
for e0>0, lim f = 0
for e0<0, lim f = +-oo (the sign depends on the sign of c0)
for e0=0, lim f = lim c0
We need to recursively compute limits at several places of the algorithm, but
as is shown in the PhD thesis, it always finishes.
Important functions from the implementation:
compare(a, b, x) compares "a" and "b" by computing the limit L.
mrv(e, x) returns list of most rapidly varying (mrv) subexpressions of "e"
rewrite(e, Omega, x, wsym) rewrites "e" in terms of w
leadterm(f, x) returns the lowest power term in the series of f
mrv_leadterm(e, x) returns the lead term (c0, e0) for e
limitinf(e, x) computes lim e (for x->oo)
limit(e, z, z0) computes any limit by converting it to the case x->oo
All the functions are really simple and straightforward except
rewrite(), which is the most difficult/complex part of the algorithm.
When the algorithm fails, the bugs are usually in the series expansion
(i.e. in SymPy) or in rewrite.
This code is almost exact rewrite of the Maple code inside the Gruntz
thesis.
Debugging
---------
Because the gruntz algorithm is highly recursive, it's difficult to
figure out what went wrong inside a debugger. Instead, turn on nice
debug prints by defining the environment variable SYMPY_DEBUG. For
example:
[user@localhost]: SYMPY_DEBUG=True ./bin/isympy
In [1]: limit(sin(x)/x, x, 0)
limitinf(_x*sin(1/_x), _x) = 1
+-mrv_leadterm(_x*sin(1/_x), _x) = (1, 0)
| +-mrv(_x*sin(1/_x), _x) = set([_x])
| | +-mrv(_x, _x) = set([_x])
| | +-mrv(sin(1/_x), _x) = set([_x])
| | +-mrv(1/_x, _x) = set([_x])
| | +-mrv(_x, _x) = set([_x])
| +-mrv_leadterm(exp(_x)*sin(exp(-_x)), _x, set([exp(_x)])) = (1, 0)
| +-rewrite(exp(_x)*sin(exp(-_x)), set([exp(_x)]), _x, _w) = (1/_w*sin(_w), -_x)
| +-sign(_x, _x) = 1
| +-mrv_leadterm(1, _x) = (1, 0)
+-sign(0, _x) = 0
+-limitinf(1, _x) = 1
And check manually which line is wrong. Then go to the source code and
debug this function to figure out the exact problem.
"""
from __future__ import print_function, division
from sympy.core import Basic, S, oo, Symbol, I, Dummy, Wild, Mul
from sympy.functions import log, exp
from sympy.series.order import Order
from sympy.simplify.powsimp import powsimp
from sympy import cacheit
from sympy.core.compatibility import reduce
from sympy.utilities.timeutils import timethis
timeit = timethis('gruntz')
from sympy.utilities.misc import debug_decorator as debug
[docs]def compare(a, b, x):
"""Returns "<" if a<b, "=" for a == b, ">" for a>b"""
# log(exp(...)) must always be simplified here for termination
la, lb = log(a), log(b)
if isinstance(a, Basic) and a.func is exp:
la = a.args[0]
if isinstance(b, Basic) and b.func is exp:
lb = b.args[0]
c = limitinf(la/lb, x)
if c == 0:
return "<"
elif c.is_infinite:
return ">"
else:
return "="
[docs]class SubsSet(dict):
"""
Stores (expr, dummy) pairs, and how to rewrite expr-s.
The gruntz algorithm needs to rewrite certain expressions in term of a new
variable w. We cannot use subs, because it is just too smart for us. For
example::
> Omega=[exp(exp(_p - exp(-_p))/(1 - 1/_p)), exp(exp(_p))]
> O2=[exp(-exp(_p) + exp(-exp(-_p))*exp(_p)/(1 - 1/_p))/_w, 1/_w]
> e = exp(exp(_p - exp(-_p))/(1 - 1/_p)) - exp(exp(_p))
> e.subs(Omega[0],O2[0]).subs(Omega[1],O2[1])
-1/w + exp(exp(p)*exp(-exp(-p))/(1 - 1/p))
is really not what we want!
So we do it the hard way and keep track of all the things we potentially
want to substitute by dummy variables. Consider the expression::
exp(x - exp(-x)) + exp(x) + x.
The mrv set is {exp(x), exp(-x), exp(x - exp(-x))}.
We introduce corresponding dummy variables d1, d2, d3 and rewrite::
d3 + d1 + x.
This class first of all keeps track of the mapping expr->variable, i.e.
will at this stage be a dictionary::
{exp(x): d1, exp(-x): d2, exp(x - exp(-x)): d3}.
[It turns out to be more convenient this way round.]
But sometimes expressions in the mrv set have other expressions from the
mrv set as subexpressions, and we need to keep track of that as well. In
this case, d3 is really exp(x - d2), so rewrites at this stage is::
{d3: exp(x-d2)}.
The function rewrite uses all this information to correctly rewrite our
expression in terms of w. In this case w can be choosen to be exp(-x),
i.e. d2. The correct rewriting then is::
exp(-w)/w + 1/w + x.
"""
def __init__(self):
self.rewrites = {}
def __repr__(self):
return super(SubsSet, self).__repr__() + ', ' + self.rewrites.__repr__()
def __getitem__(self, key):
if not key in self:
self[key] = Dummy()
return dict.__getitem__(self, key)
def do_subs(self, e):
for expr, var in self.items():
e = e.subs(var, expr)
return e
[docs] def meets(self, s2):
"""Tell whether or not self and s2 have non-empty intersection"""
return set(self.keys()).intersection(list(s2.keys())) != set()
[docs] def union(self, s2, exps=None):
"""Compute the union of self and s2, adjusting exps"""
res = self.copy()
tr = {}
for expr, var in s2.items():
if expr in self:
if exps:
exps = exps.subs(var, res[expr])
tr[var] = res[expr]
else:
res[expr] = var
for var, rewr in s2.rewrites.items():
res.rewrites[var] = rewr.subs(tr)
return res, exps
def copy(self):
r = SubsSet()
r.rewrites = self.rewrites.copy()
for expr, var in self.items():
r[expr] = var
return r
@debug
[docs]def mrv(e, x):
"""Returns a SubsSet of most rapidly varying (mrv) subexpressions of 'e',
and e rewritten in terms of these"""
e = powsimp(e, deep=True, combine='exp')
if not isinstance(e, Basic):
raise TypeError("e should be an instance of Basic")
if not e.has(x):
return SubsSet(), e
elif e == x:
s = SubsSet()
return s, s[x]
elif e.is_Mul or e.is_Add:
i, d = e.as_independent(x) # throw away x-independent terms
if d.func != e.func:
s, expr = mrv(d, x)
return s, e.func(i, expr)
a, b = d.as_two_terms()
s1, e1 = mrv(a, x)
s2, e2 = mrv(b, x)
return mrv_max1(s1, s2, e.func(i, e1, e2), x)
elif e.is_Pow:
b, e = e.as_base_exp()
if e.has(x):
return mrv(exp(e * log(b)), x)
else:
s, expr = mrv(b, x)
return s, expr**e
elif e.func is log:
s, expr = mrv(e.args[0], x)
return s, log(expr)
elif e.func is exp:
# We know from the theory of this algorithm that exp(log(...)) may always
# be simplified here, and doing so is vital for termination.
if e.args[0].func is log:
return mrv(e.args[0].args[0], x)
# if a product has an infinite factor the result will be
# infinite if there is no zero, otherwise NaN; here, we
# consider the result infinite if any factor is infinite
li = limitinf(e.args[0], x)
if any(_.is_infinite for _ in Mul.make_args(li)):
s1 = SubsSet()
e1 = s1[e]
s2, e2 = mrv(e.args[0], x)
su = s1.union(s2)[0]
su.rewrites[e1] = exp(e2)
return mrv_max3(s1, e1, s2, exp(e2), su, e1, x)
else:
s, expr = mrv(e.args[0], x)
return s, exp(expr)
elif e.is_Function:
l = [mrv(a, x) for a in e.args]
l2 = [s for (s, _) in l if s != SubsSet()]
if len(l2) != 1:
# e.g. something like BesselJ(x, x)
raise NotImplementedError("MRV set computation for functions in"
" several variables not implemented.")
s, ss = l2[0], SubsSet()
args = [ss.do_subs(x[1]) for x in l]
return s, e.func(*args)
elif e.is_Derivative:
raise NotImplementedError("MRV set computation for derviatives"
" not implemented yet.")
return mrv(e.args[0], x)
raise NotImplementedError(
"Don't know how to calculate the mrv of '%s'" % e)
[docs]def mrv_max3(f, expsf, g, expsg, union, expsboth, x):
"""Computes the maximum of two sets of expressions f and g, which
are in the same comparability class, i.e. max() compares (two elements of)
f and g and returns either (f, expsf) [if f is larger], (g, expsg)
[if g is larger] or (union, expsboth) [if f, g are of the same class].
"""
if not isinstance(f, SubsSet):
raise TypeError("f should be an instance of SubsSet")
if not isinstance(g, SubsSet):
raise TypeError("g should be an instance of SubsSet")
if f == SubsSet():
return g, expsg
elif g == SubsSet():
return f, expsf
elif f.meets(g):
return union, expsboth
c = compare(list(f.keys())[0], list(g.keys())[0], x)
if c == ">":
return f, expsf
elif c == "<":
return g, expsg
else:
if c != "=":
raise ValueError("c should be =")
return union, expsboth
[docs]def mrv_max1(f, g, exps, x):
"""Computes the maximum of two sets of expressions f and g, which
are in the same comparability class, i.e. mrv_max1() compares (two elements of)
f and g and returns the set, which is in the higher comparability class
of the union of both, if they have the same order of variation.
Also returns exps, with the appropriate substitutions made.
"""
u, b = f.union(g, exps)
return mrv_max3(f, g.do_subs(exps), g, f.do_subs(exps),
u, b, x)
@debug
@cacheit
@timeit
[docs]def sign(e, x):
"""
Returns a sign of an expression e(x) for x->oo.
::
e > 0 for x sufficiently large ... 1
e == 0 for x sufficiently large ... 0
e < 0 for x sufficiently large ... -1
The result of this function is currently undefined if e changes sign
arbitarily often for arbitrarily large x (e.g. sin(x)).
Note that this returns zero only if e is *constantly* zero
for x sufficiently large. [If e is constant, of course, this is just
the same thing as the sign of e.]
"""
from sympy import sign as _sign
if not isinstance(e, Basic):
raise TypeError("e should be an instance of Basic")
if e.is_positive:
return 1
elif e.is_negative:
return -1
elif e.is_zero:
return 0
elif not e.has(x):
return _sign(e)
elif e == x:
return 1
elif e.is_Mul:
a, b = e.as_two_terms()
sa = sign(a, x)
if not sa:
return 0
return sa * sign(b, x)
elif e.func is exp:
return 1
elif e.is_Pow:
s = sign(e.base, x)
if s == 1:
return 1
if e.exp.is_Integer:
return s**e.exp
elif e.func is log:
return sign(e.args[0] - 1, x)
# if all else fails, do it the hard way
c0, e0 = mrv_leadterm(e, x)
return sign(c0, x)
@debug
@timeit
@cacheit
[docs]def limitinf(e, x):
"""Limit e(x) for x-> oo"""
#rewrite e in terms of tractable functions only
e = e.rewrite('tractable', deep=True)
if not e.has(x):
return e # e is a constant
if e.has(Order):
e = e.expand().removeO()
if not x.is_positive:
# We make sure that x.is_positive is True so we
# get all the correct mathematical behavior from the expression.
# We need a fresh variable.
p = Dummy('p', positive=True, finite=True)
e = e.subs(x, p)
x = p
c0, e0 = mrv_leadterm(e, x)
sig = sign(e0, x)
if sig == 1:
return S.Zero # e0>0: lim f = 0
elif sig == -1: # e0<0: lim f = +-oo (the sign depends on the sign of c0)
if c0.match(I*Wild("a", exclude=[I])):
return c0*oo
s = sign(c0, x)
#the leading term shouldn't be 0:
if s == 0:
raise ValueError("Leading term should not be 0")
return s*oo
elif sig == 0:
return limitinf(c0, x) # e0=0: lim f = lim c0
def moveup2(s, x):
r = SubsSet()
for expr, var in s.items():
r[expr.subs(x, exp(x))] = var
for var, expr in s.rewrites.items():
r.rewrites[var] = s.rewrites[var].subs(x, exp(x))
return r
def moveup(l, x):
return [e.subs(x, exp(x)) for e in l]
@debug
@timeit
[docs]def calculate_series(e, x, logx=None):
""" Calculates at least one term of the series of "e" in "x".
This is a place that fails most often, so it is in its own function.
"""
from sympy.polys import cancel
for t in e.lseries(x, logx=logx):
t = cancel(t)
if t.simplify():
break
return t
@debug
@timeit
@cacheit
[docs]def mrv_leadterm(e, x):
"""Returns (c0, e0) for e."""
Omega = SubsSet()
if not e.has(x):
return (e, S.Zero)
if Omega == SubsSet():
Omega, exps = mrv(e, x)
if not Omega:
# e really does not depend on x after simplification
series = calculate_series(e, x)
c0, e0 = series.leadterm(x)
if e0 != 0:
raise ValueError("e0 should be 0")
return c0, e0
if x in Omega:
#move the whole omega up (exponentiate each term):
Omega_up = moveup2(Omega, x)
e_up = moveup([e], x)[0]
exps_up = moveup([exps], x)[0]
# NOTE: there is no need to move this down!
e = e_up
Omega = Omega_up
exps = exps_up
#
# The positive dummy, w, is used here so log(w*2) etc. will expand;
# a unique dummy is needed in this algorithm
#
# 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, finite=True)
f, logw = rewrite(exps, Omega, x, w)
series = calculate_series(f, w, logx=logw)
return series.leadterm(w)
[docs]def build_expression_tree(Omega, rewrites):
r""" Helper function for rewrite.
We need to sort Omega (mrv set) so that we replace an expression before
we replace any expression in terms of which it has to be rewritten::
e1 ---> e2 ---> e3
\
-> e4
Here we can do e1, e2, e3, e4 or e1, e2, e4, e3.
To do this we assemble the nodes into a tree, and sort them by height.
This function builds the tree, rewrites then sorts the nodes.
"""
class Node:
def ht(self):
return reduce(lambda x, y: x + y,
[x.ht() for x in self.before], 1)
nodes = {}
for expr, v in Omega:
n = Node()
n.before = []
n.var = v
n.expr = expr
nodes[v] = n
for _, v in Omega:
if v in rewrites:
n = nodes[v]
r = rewrites[v]
for _, v2 in Omega:
if r.has(v2):
n.before.append(nodes[v2])
return nodes
@debug
@timeit
[docs]def rewrite(e, Omega, x, wsym):
"""e(x) ... the function
Omega ... the mrv set
wsym ... the symbol which is going to be used for w
Returns the rewritten e in terms of w and log(w). See test_rewrite1()
for examples and correct results.
"""
from sympy import ilcm
if not isinstance(Omega, SubsSet):
raise TypeError("Omega should be an instance of SubsSet")
if len(Omega) == 0:
raise ValueError("Length can not be 0")
#all items in Omega must be exponentials
for t in Omega.keys():
if not t.func is exp:
raise ValueError("Value should be exp")
rewrites = Omega.rewrites
Omega = list(Omega.items())
nodes = build_expression_tree(Omega, rewrites)
Omega.sort(key=lambda x: nodes[x[1]].ht(), reverse=True)
# make sure we know the sign of each exp() term; after the loop,
# g is going to be the "w" - the simplest one in the mrv set
for g, _ in Omega:
sig = sign(g.args[0], x)
if sig != 1 and sig != -1:
raise NotImplementedError('Result depends on the sign of %s' % sig)
if sig == 1:
wsym = 1/wsym # if g goes to oo, substitute 1/w
#O2 is a list, which results by rewriting each item in Omega using "w"
O2 = []
denominators = []
for f, var in Omega:
c = limitinf(f.args[0]/g.args[0], x)
if c.is_Rational:
denominators.append(c.q)
arg = f.args[0]
if var in rewrites:
if not rewrites[var].func is exp:
raise ValueError("Value should be exp")
arg = rewrites[var].args[0]
O2.append((var, exp((arg - c*g.args[0]).expand())*wsym**c))
#Remember that Omega contains subexpressions of "e". So now we find
#them in "e" and substitute them for our rewriting, stored in O2
# the following powsimp is necessary to automatically combine exponentials,
# so that the .subs() below succeeds:
# TODO this should not be necessary
f = powsimp(e, deep=True, combine='exp')
for a, b in O2:
f = f.subs(a, b)
for _, var in Omega:
assert not f.has(var)
#finally compute the logarithm of w (logw).
logw = g.args[0]
if sig == 1:
logw = -logw # log(w)->log(1/w)=-log(w)
# Some parts of sympy have difficulty computing series expansions with
# non-integral exponents. The following heuristic improves the situation:
exponent = reduce(ilcm, denominators, 1)
f = f.subs(wsym, wsym**exponent)
logw /= exponent
return f, logw
[docs]def gruntz(e, z, z0, dir="+"):
"""
Compute the limit of e(z) at the point z0 using the Gruntz algorithm.
z0 can be any expression, including oo and -oo.
For dir="+" (default) it calculates the limit from the right
(z->z0+) and for dir="-" the limit from the left (z->z0-). For infinite z0
(oo or -oo), the dir argument doesn't matter.
This algorithm is fully described in the module docstring in the gruntz.py
file. It relies heavily on the series expansion. Most frequently, gruntz()
is only used if the faster limit() function (which uses heuristics) fails.
"""
if not isinstance(z, Symbol):
raise NotImplementedError("Second argument must be a Symbol")
#convert all limits to the limit z->oo; sign of z is handled in limitinf
r = None
if z0 == oo:
r = limitinf(e, z)
elif z0 == -oo:
r = limitinf(e.subs(z, -z), z)
else:
if str(dir) == "-":
e0 = e.subs(z, z0 - 1/z)
elif str(dir) == "+":
e0 = e.subs(z, z0 + 1/z)
else:
raise NotImplementedError("dir must be '+' or '-'")
r = limitinf(e0, z)
# This is a bit of a heuristic for nice results... we always rewrite
# tractable functions in terms of familiar intractable ones.
# It might be nicer to rewrite the exactly to what they were initially,
# but that would take some work to implement.
return r.rewrite('intractable', deep=True)
```