Step 2

Author: anutosh491

integration_tests/gruntz_demo.py

@@ -343,7 +343,7 @@ def limitinf(e, x):
     c0, e0 = mrv_leadterm(e, x)
     sig = signinf(e0, x)
     if sig == 1:
-        return Integer(0)
+        return S(0)
     if sig == -1:
         return signinf(c0, x)*oo
     if sig == 0:

integration_tests/gruntz_demo2.py

@@ -0,0 +1,334 @@
+"""
+Limits
+======
+
+Implemented according to the PhD thesis
+https://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
+do not 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 functools import reduce
+
+from sympy.core import Basic, S, Mul, PoleError, expand_mul, evaluate
+from sympy.core.cache import cacheit
+from sympy.core.numbers import I, oo
+from sympy.core.symbol import Dummy, Wild, Symbol
+from sympy.core.traversal import bottom_up
+from sympy.core.sorting import ordered
+
+from sympy.functions import log, exp, sign, sin
+from sympy.series.order import Order
+from sympy.utilities.exceptions import SymPyDeprecationWarning
+from sympy.utilities.misc import debug_decorator as debug
+from sympy.utilities.timeutils import timethis
+
+def mrv(e, x):
+    """
+    Calculate the MRV set of the expression.
+
+    Examples
+    ========
+
+    >>> mrv(log(x - log(x))/log(x), x)
+    {x}
+
+    """
+
+    if e == x:
+        return {x}
+    if e.is_Mul or e.is_Add:
+        a, b = e.as_two_terms()
+        ans1 = mrv(a, x)
+        ans2 = mrv(b, x)
+        return mrv_max(mrv(a, x), mrv(b, x), x)
+    if e.is_Pow:
+        return mrv(e.base, x)
+    if e.is_Function:
+        return reduce(lambda a, b: mrv_max(a, b, x), (mrv(a, x) for a in e.args))
+    raise NotImplementedError(f"Can't calculate the MRV of {e}.")
+
+def mrv_max(f, g, x):
+    """Compute the maximum of two MRV sets.
+
+    Examples
+    ========
+
+    >>> mrv_max({log(x)}, {x**5}, x)
+    {x**5}
+
+    """
+
+    if not f:
+        return g
+    if not g:
+        return f
+    if f & g:
+        return f | g
+
+def rewrite(e, x, w):
+    r"""
+    Rewrites the expression in terms of the MRV 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 MRV in `x` subexpression.
+
+    Returns
+    =======
+
+    The rewritten expression
+
+    Examples
+    ========
+
+    >>> rewrite(exp(x)*log(x), x, y)
+    (log(x)/y, -x)
+
+    """
+
+    Omega = mrv(e, x)
+
+    if x in Omega:
+        # Moving up in the asymptotical scale:
+        with evaluate(False):
+            e = e.subs(x, exp(x))
+            Omega = {s.subs(x, exp(x)) for s in Omega}
+
+    Omega = list(ordered(Omega, keys=lambda a: -len(mrv(a, x))))
+
+    for g in Omega:
+        sig = signinf(g.exp, x)
+        if sig not in (1, -1):
+            raise NotImplementedError(f'Result depends on the sign of {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.subs(a, b)
+
+    return e
+
+@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
+    ========
+
+    >>> leadterm(1/exp(-x + exp(-x)) - exp(x), x)
+    (-1, 0)
+
+    """
+
+    w = Dummy('w', real=True, positive=True)
+    e = rewrite(e, x, w)
+    return e.leadterm(w)
+
+@cacheit
+def signinf(e, x):
+    r"""
+    Determine sign of the expression at the 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`.
+
+    """
+
+    if not e.has(x):
+        return sign(e)
+    if e == x or (e.is_Pow and signinf(e.base, x) == 1):
+        return S(1)
+
+@cacheit
+def limitinf(e, x):
+    """
+    Compute the limit of the expression at the infinity.
+
+    Examples
+    ========
+
+    >>> limitinf(exp(x)*(exp(1/x - exp(-x)) - exp(1/x)), x)
+    -1
+
+    """
+
+    if not e.has(x):
+        return e
+
+    c0, e0 = mrv_leadterm(e, x)
+    sig = signinf(e0, x)
+    if sig == 1:
+        return Integer(0)
+    if sig == -1:
+        return signinf(c0, x)*oo
+    if sig == 0:
+        return limitinf(c0, x)
+    raise NotImplementedError(f'Result depends on the sign of {sig}.')
+
+
+def gruntz(e, z, z0, dir="+"):
+    """
+    Compute the limit of e(z) at the point z0 using the Gruntz algorithm.
+
+    Explanation
+    ===========
+
+    ``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 does not 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 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)
+    return r
+
+# tests
+x = Symbol('x')
+ans = gruntz(sin(x)/x, x, 0)
+print(ans)