def integral(self, var=None):
"""
Return the integral of this polynomial.
By default, the integration variable is the variable of the
polynomial.
Otherwise, the integration variable is the optional parameter ``var``
.. NOTE::
The integral is always chosen so that the constant term is 0.
EXAMPLES::
sage: R.<x> = PolynomialRing(ZZ, sparse=True)
sage: (1 + 3*x^10 - 2*x^100).integral()
-2/101*x^101 + 3/11*x^11 + x
TESTS:
Check that :trac:`18600` is fixed::
sage: R.<x> = PolynomialRing(ZZ, sparse=True)
sage: (x^2^100).integral()
1/1267650600228229401496703205377*x^1267650600228229401496703205377
Check the correctness when the base ring is a polynomial ring::
sage: R.<x> = PolynomialRing(ZZ, sparse=True)
sage: S.<t> = PolynomialRing(R, sparse=True)
sage: (x*t+1).integral()
1/2*x*t^2 + t
sage: (x*t+1).integral(x)
1/2*x^2*t + x
Check the correctness when the base ring is not an integral domain::
sage: R.<x> = PolynomialRing(Zmod(4), sparse=True)
sage: (x^4 + 2*x^2 + 3).integral()
x^5 + 2*x^3 + 3*x
sage: x.integral()
Traceback (most recent call last):
...
ZeroDivisionError: Inverse does not exist.
"""
R = self.parent()
# TODO:
# calling the coercion model bin_op is much more accurate than using the
# true division (which is bypassed by polynomials). But it does not work
# in all cases!!
from sage.structure.element import coercion_model as cm
import operator
try:
Q = cm.bin_op(R.one(), ZZ.one(), operator.truediv).parent()
except TypeError:
F = (R.base_ring().one()/ZZ.one()).parent()
Q = R.change_ring(F)
if var is not None and var != R.gen():
return Q({k:v.integral(var) for k,v in six.iteritems(self.__coeffs)}, check=False)
return Q({ k+1:v/(k+1) for k,v in six.iteritems(self.__coeffs)}, check=False)
def integral(self, var=None):
"""
Return the integral of this polynomial.
By default, the integration variable is the variable of the
polynomial.
Otherwise, the integration variable is the optional parameter ``var``
.. NOTE::
The integral is always chosen so that the constant term is 0.
EXAMPLES::
sage: R.<x> = PolynomialRing(ZZ, sparse=True)
sage: (1 + 3*x^10 - 2*x^100).integral()
-2/101*x^101 + 3/11*x^11 + x
TESTS:
Check that :trac:`18600` is fixed::
sage: R.<x> = PolynomialRing(ZZ, sparse=True)
sage: (x^2^100).integral()
1/1267650600228229401496703205377*x^1267650600228229401496703205377
Check the correctness when the base ring is a polynomial ring::
sage: R.<x> = PolynomialRing(ZZ, sparse=True)
sage: S.<t> = PolynomialRing(R, sparse=True)
sage: (x*t+1).integral()
1/2*x*t^2 + t
sage: (x*t+1).integral(x)
1/2*x^2*t + x
Check the correctness when the base ring is not an integral domain::
sage: R.<x> = PolynomialRing(Zmod(4), sparse=True)
sage: (x^4 + 2*x^2 + 3).integral()
x^5 + 2*x^3 + 3*x
sage: x.integral()
Traceback (most recent call last):
...
ZeroDivisionError: Inverse does not exist.
"""
R = self.parent()
# TODO:
# calling the coercion model bin_op is much more accurate than using the
# true division (which is bypassed by polynomials). But it does not work
# in all cases!!
from sage.structure.element import coercion_model as cm
import operator
try:
Q = cm.bin_op(R.one(), ZZ.one(), operator.div).parent()
except TypeError:
F = (R.base_ring().one()/ZZ.one()).parent()
Q = R.change_ring(F)
if var is not None and var != R.gen():
return Q({k:v.integral(var) for k,v in six.iteritems(self.__coeffs)}, check=False)
return Q({ k+1:v/(k+1) for k,v in six.iteritems(self.__coeffs)}, check=False)