Python polynomial_element.Polynomial类代码示例

    def __init__(self, parent, x=None, check=True, is_gen=False, construct=False):
        """
        TESTS::

            sage: PolynomialRing(RIF, 'z', sparse=True)([RIF(-1, 1), RIF(-1,1)])
            0.?*z + 0.?
            sage: PolynomialRing(CIF, 'z', sparse=True)([CIF(RIF(-1,1), RIF(-1,1)), RIF(-1,1)])
            0.?*z + 0.? + 0.?*I
        """
        Polynomial.__init__(self, parent, is_gen=is_gen)
        if x is None:
            self.__coeffs = {}
            return
        R = parent.base_ring()
        if isinstance(x, Polynomial):
            if x.parent() == self.parent():
                x = dict(x.dict())
            elif x.parent() == R:
                x = {0:x}
            else:
                w = {}
                for n, c in x.dict().iteritems():
                    w[n] = R(c)
                # The following line has been added in trac ticket #9944.
                # Apparently, the "else" case has never occured before.
                x = w
        elif isinstance(x, list):
            x = dict((i, c) for (i, c) in enumerate(x) if c)
        elif isinstance(x, pari_gen):
            y = {}
            for i in range(len(x)):
                y[i] = R(x[i])
            x = y
            check = True
        elif not isinstance(x, dict):
            x = {0:x}   # constant polynomials
        if check:
            self.__coeffs = {}
            for i, z in x.iteritems():
                self.__coeffs[i] = R(z)
        else:
            self.__coeffs = x
        if check:
            self.__normalize()

 def __init__(self, parent, x=None, check=True, is_gen=False, construct=False):
     Polynomial.__init__(self, parent, is_gen=is_gen)
     if x is None:
         self.__coeffs = {}
         return
     R = parent.base_ring()
     if isinstance(x, Polynomial):
         if x.parent() == self.parent():
             x = dict(x.dict())
         elif x.parent() == R:
             x = {0:x}
         else:
             w = {}
             for n, c in x.dict().iteritems():
                 w[n] = R(c)
             # The following line has been added in trac ticket #9944.
             # Apparently, the "else" case has never occured before.
             x = w  
     elif isinstance(x, list):
         y = {}
         for i in xrange(len(x)):
             if x[i] != 0:
                 y[i] = x[i]
         x = y
     elif isinstance(x, pari_gen):
         y = {}
         for i in range(len(x)):
             y[i] = R(x[i])
         x = y
         check = True
     elif not isinstance(x, dict):
         x = {0:x}   # constant polynomials
     if check:
         self.__coeffs = {}
         for i, z in x.iteritems():
             self.__coeffs[i] = R(z)
     else:
         self.__coeffs = x
     if check:
         self.__normalize()

    def __init__(self, parent, x=None, check=True, is_gen=False, construct = False, absprec = infinity, relprec = infinity):
        """
        TESTS:
            sage: K = Qp(13,7)
            sage: R.<t> = K[]
            sage: R([K(13), K(1)])
            (1 + O(13^7))*t + (13 + O(13^8))
            sage: T.<t> = ZZ[]
            sage: R(t + 2)
            (1 + O(13^7))*t + (2 + O(13^7))
        """
        Polynomial.__init__(self, parent, is_gen=is_gen)
        parentbr = parent.base_ring()
        from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
        if construct:
            (self._poly, self._valbase, self._relprecs, self._normalized, self._valaddeds, self._list) = x #the last two of these may be None
            return
        elif is_gen:
            self._poly = PolynomialRing(ZZ, parent.variable_name()).gen()
            self._valbase = 0
            self._valaddeds = [infinity, 0]
            self._relprecs = [infinity, parentbr.precision_cap()]
            self._normalized = True
            self._list = None
            return

        #First we list the types that are turned into Polynomials
        if isinstance(x, ZZX):
            x = Polynomial_integer_dense(PolynomialRing(ZZ, parent.variable_name()), x, construct = True)
        elif isinstance(x, fraction_field_element.FractionFieldElement) and \
               x.denominator() == 1:
            #Currently we ignore precision information in the denominator.  This should be changed eventually
            x = x.numerator()

        #We now coerce various types into lists of coefficients.  There are fast pathways for some types of polynomials
        if isinstance(x, Polynomial):
            if x.parent() is self.parent():
                if not absprec is infinity or not relprec is infinity:
                    x._normalize()
                self._poly = x._poly
                self._valbase = x._valbase
                self._valaddeds = x._valaddeds
                self._relprecs = x._relprecs
                self._normalized = x._normalized
                self._list = x._list
                if not absprec is infinity or not relprec is infinity:
                    self._adjust_prec_info(absprec, relprec)
                return
            elif x.base_ring() is ZZ:
                self._poly = x
                self._valbase = Integer(0)
                p = parentbr.prime()
                self._relprecs = [c.valuation(p) + parentbr.precision_cap() for c in x.list()]
                self._comp_valaddeds()
                self._normalized = len(self._valaddeds) == 0 or (min(self._valaddeds) == 0)
                self._list = None
                if not absprec is infinity or not relprec is infinity:
                    self._adjust_prec_info(absprec, relprec)
                return
            else:
                x = [parentbr(a) for a in x.list()]
                check = False
        elif isinstance(x, dict):
            zero = parentbr.zero_element()
            n = max(x.keys())
            v = [zero for _ in xrange(n + 1)]
            for i, z in x.iteritems():
                v[i] = z
            x = v
        elif isinstance(x, pari_gen):
            x = [parentbr(w) for w in x.list()]
            check = False
        #The default behavior if we haven't already figured out what the type is is to assume it coerces into the base_ring as a constant polynomial
        elif not isinstance(x, list):
            x = [x] # constant polynomial

        # In contrast to other polynomials, the zero element is not distinguished
        # by having its list empty. Instead, it has list [0]
        if not x:
            x = [parentbr.zero_element()]
        if check:
            x = [parentbr(z) for z in x]

        # Remove this -- for p-adics this is terrible, since it kills any non exact zero.
        #if len(x) == 1 and not x[0]:
        #    x = []

        self._list = x
        self._valaddeds = [a.valuation() for a in x]
        self._valbase = sage.rings.padics.misc.min(self._valaddeds)
        if self._valbase is infinity:
            self._valaddeds = []
            self._relprecs = []
            self._poly = PolynomialRing(ZZ, parent.variable_name())()
            self._normalized = True
            if not absprec is infinity or not relprec is infinity:
                self._adjust_prec_info(absprec, relprec)
        else:
            self._valaddeds = [c - self._valbase for c in self._valaddeds]
            self._relprecs = [a.precision_absolute() - self._valbase for a in x]
#.........这里部分代码省略.........

 def __init__(self, parent, x=None, check=True, is_gen=False, construct=False):
     Polynomial.__init__(self, parent, is_gen, construct)

    def gcd(self,other,algorithm=None):
        """
        Return the gcd of this polynomial and ``other``

        INPUT:

        - ``other`` -- a polynomial defined over the same ring as this
          polynomial.

        ALGORITHM:

        Two algorithms are provided:

        - ``generic``: Uses the generic implementation, which depends on the
          base ring being a UFD or a field.
        - ``dense``: The polynomials are converted to the dense representation,
          their gcd is computed and is converted back to the sparse
          representation.

        Default is ``dense`` for polynomials over ZZ and ``generic`` in the
        other cases.

        EXAMPLES::

            sage: R.<x> = PolynomialRing(ZZ,sparse=True)
            sage: p = x^6 + 7*x^5 + 8*x^4 + 6*x^3 + 2*x^2 + x + 2
            sage: q = 2*x^4 - x^3 - 2*x^2 - 4*x - 1
            sage: gcd(p,q)
            x^2 + x + 1
            sage: gcd(p, q, algorithm = "dense")
            x^2 + x + 1
            sage: gcd(p, q, algorithm = "generic")
            x^2 + x + 1
            sage: gcd(p, q, algorithm = "foobar")
            Traceback (most recent call last):
            ...
            ValueError: Unknown algorithm 'foobar'

        TESTS:

        Check that :trac:`19676` is fixed::

            sage: S.<y> = R[]
            sage: x.gcd(y)
            1
            sage: (6*x).gcd(9)
            3
        """

        from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
        from sage.arith.all import lcm

        if algorithm is None:
            if self.base_ring() == ZZ:
                algorithm = "dense"
            else:
                algorithm = "generic"
        if algorithm=="dense":
            S = self.parent()
            # FLINT is faster but a bug makes the conversion extremely slow,
            # so NTL is used in those cases where the conversion is too slow. Cf
            # <https://groups.google.com/d/msg/sage-devel/6qhW90dgd1k/Hoq3N7fWe4QJ>
            sd = self.degree()
            od = other.degree()
            if max(sd,od)<100 or \
               min(len(self.__coeffs)/sd, len(other.__coeffs)/od)>.06:
                implementation="FLINT"
            else:
                implementation="NTL"
            D = PolynomialRing(S.base_ring(),'x',implementation=implementation)
            g = D(self).gcd(D(other))
            return S(g)
        elif algorithm=="generic":
            return Polynomial.gcd(self,other)
        else:
            raise ValueError("Unknown algorithm '%s'" % algorithm)