Coverage for sympy/series/gruntz.py : 61%

""" 

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 

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 "=" 

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 

def meets(self, s2): 

"""Tell whether or not self and s2 have non-empty intersection""" 

return set(self.keys()).intersection(list(s2.keys())) != set() 

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 

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) 

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 

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 

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 

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 

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 

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) 

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 

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 

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 z.is_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)