def asymptotic_simplify_term(term, x):
"""
Recursively drop additive terms which are dominated asymptotically by other terms.
:param term: term to process
:param x: variable going to infinity
:returns: a simpler term with the same asymptotic behaviour
.. note::
This should be replaced by Sage's asymptotic ring.
"""
from sage.symbolic.operators import add_vararg, mul_vararg
if term.operator() == add_vararg:
ret = []
for op1 in term.operands():
for op2 in term.operands():
# akward notation to make limit(foo, kappa=infinity) work
if limit(abs(op1)/abs(op2), **{str(x):Infinity}) < 1:
break
else:
ret.append(asymptotic_simplify_term(op1, x))
ret = add_vararg(*ret)
elif term.operator() == mul_vararg:
ret = mul_vararg(*[asymptotic_simplify_term(op, x) for op in term.operands()])
else:
ret = term
return ret
def asymptotic_simplify_term(term, x):
"""
Recursively drop additive terms which are dominated asymptotically by other terms.
:param term: term to process
:param x: variable going to infinity
:returns: a simpler term with the same asymptotic behaviour
.. note::
This should be replaced by Sage's asymptotic ring.
"""
from sage.symbolic.operators import add_vararg, mul_vararg
if term.operator() == add_vararg:
ret = []
for op1 in term.operands():
for op2 in term.operands():
# akward notation to make limit(foo, kappa=infinity) work
if limit(abs(op1) / abs(op2), **{str(x): Infinity}) < 1:
break
else:
ret.append(asymptotic_simplify_term(op1, x))
ret = add_vararg(*ret)
elif term.operator() == mul_vararg:
ret = mul_vararg(
*[asymptotic_simplify_term(op, x) for op in term.operands()])
else:
ret = term
return ret
def _substitute_(self, rules):
r"""
Substitute the given ``rules`` in this
Cartesian product growth element.
INPUT:
- ``rules`` -- a dictionary.
The neutral element of the group is replaced by the value
to key ``'_one_'``.
OUTPUT:
An object.
TESTS::
sage: from sage.rings.asymptotic.growth_group import GrowthGroup
sage: G = GrowthGroup('x^QQ * log(x)^QQ')
sage: G(x^3 * log(x)^5)._substitute_({'x': SR.var('z')})
z^3*log(z)^5
sage: _.parent()
Symbolic Ring
sage: G(x^3 * log(x)^5)._substitute_({'x': 2.2}) # rel tol 1e-6
3.24458458945
sage: _.parent()
Real Field with 53 bits of precision
sage: G(1 / x)._substitute_({'x': 0})
Traceback (most recent call last):
...
ZeroDivisionError: Cannot substitute in x^(-1) in
Growth Group x^QQ * log(x)^QQ.
> *previous* ZeroDivisionError: Cannot substitute in x^(-1) in
Growth Group x^QQ.
>> *previous* ZeroDivisionError: rational division by zero
sage: G(1)._substitute_({'_one_': 'one'})
'one'
"""
if self.is_one():
return rules["_one_"]
from sage.symbolic.operators import mul_vararg
try:
return mul_vararg(*tuple(x._substitute_(rules) for x in self.cartesian_factors()))
except (ArithmeticError, TypeError, ValueError) as e:
from misc import substitute_raise_exception
substitute_raise_exception(self, e)
def _substitute_(self, rules):
r"""
Substitute the given ``rules`` in this
Cartesian product growth element.
INPUT:
- ``rules`` -- a dictionary.
The neutral element of the group is replaced by the value
to key ``'_one_'``.
OUTPUT:
An object.
TESTS::
sage: from sage.rings.asymptotic.growth_group import GrowthGroup
sage: G = GrowthGroup('x^QQ * log(x)^QQ')
sage: G(x^3 * log(x)^5)._substitute_({'x': SR.var('z')})
z^3*log(z)^5
sage: _.parent()
Symbolic Ring
sage: G(x^3 * log(x)^5)._substitute_({'x': 2.2}) # rel tol 1e-6
3.24458458945
sage: _.parent()
Real Field with 53 bits of precision
sage: G(1 / x)._substitute_({'x': 0})
Traceback (most recent call last):
...
ZeroDivisionError: Cannot substitute in x^(-1) in
Growth Group x^QQ * log(x)^QQ.
> *previous* ZeroDivisionError: Cannot substitute in x^(-1) in
Growth Group x^QQ.
>> *previous* ZeroDivisionError: rational division by zero
sage: G(1)._substitute_({'_one_': 'one'})
'one'
"""
if self.is_one():
return rules['_one_']
from sage.symbolic.operators import mul_vararg
try:
return mul_vararg(
*tuple(x._substitute_(rules)
for x in self.cartesian_factors()))
except (ArithmeticError, TypeError, ValueError) as e:
from .misc import substitute_raise_exception
substitute_raise_exception(self, e)