Python polyclasses.DMP类代码示例

def chebyshevu_poly(n, x=None, polys=False):
    """Generates Chebyshev polynomial of the second kind of degree `n` in `x`.

    Parameters
    ==========

    n : int
        `n` decides the degree of polynomial
    x : optional
    polys : bool, optional
        ``polys=True`` returns an expression, otherwise
        (default) returns an expression.
    """
    if n < 0:
        raise ValueError(
            "can't generate 2nd kind Chebyshev polynomial of degree %s" % n)

    poly = DMP(dup_chebyshevu(int(n), ZZ), ZZ)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    return poly if polys else poly.as_expr()

def gegenbauer_poly(n, a, x=None, polys=False):
    """Generates Gegenbauer polynomial of degree `n` in `x`.

    Parameters
    ==========

    n : int
        `n` decides the degree of polynomial
    x : optional
    a
        Decides minimal domain for the list of
        coefficients.
    polys : bool, optional
        ``polys=True`` returns an expression, otherwise
        (default) returns an expression.
    """
    if n < 0:
        raise ValueError(
            "can't generate Gegenbauer polynomial of degree %s" % n)

    K, a = construct_domain(a, field=True)
    poly = DMP(dup_gegenbauer(int(n), a, K), K)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    return poly if polys else poly.as_expr()

def test_DUP_to_dict():
    f = DMP([[3],[],[2],[],[8]], ZZ)

    assert f.to_dict() == \
        {(4, 0): 3, (2, 0): 2, (0, 0): 8}
    assert f.to_sympy_dict() == \
        {(4, 0): ZZ.to_sympy(3), (2, 0): ZZ.to_sympy(2), (0, 0): ZZ.to_sympy(8)}

def jacobi_poly(n, a, b, x=None, polys=False):
    """Generates Jacobi polynomial of degree `n` in `x`.

    Parameters
    ==========

    n : int
        `n` decides the degree of polynomial
    a
        Lower limit of minimal domain for the list of
        coefficients.
    b
        Upper limit of minimal domain for the list of
        coefficients.
    x : optional
    polys : bool, optional
        ``polys=True`` returns an expression, otherwise
        (default) returns an expression.
    """
    if n < 0:
        raise ValueError("can't generate Jacobi polynomial of degree %s" % n)

    K, v = construct_domain([a, b], field=True)
    poly = DMP(dup_jacobi(int(n), v[0], v[1], K), K)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    return poly if polys else poly.as_expr()

def laguerre_poly(n, x=None, alpha=None, polys=False):
    """Generates Laguerre polynomial of degree `n` in `x`.

    Parameters
    ==========

    n : int
        `n` decides the degree of polynomial
    x : optional
    alpha
        Decides minimal domain for the list
        of coefficients.
    polys : bool, optional
        ``polys=True`` returns an expression, otherwise
        (default) returns an expression.
    """
    if n < 0:
        raise ValueError("can't generate Laguerre polynomial of degree %s" % n)

    if alpha is not None:
        K, alpha = construct_domain(
            alpha, field=True)  # XXX: ground_field=True
    else:
        K, alpha = QQ, QQ(0)

    poly = DMP(dup_laguerre(int(n), alpha, K), K)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    return poly if polys else poly.as_expr()

def spherical_bessel_fn(n, x=None, polys=False):
    """
    Coefficients for the spherical Bessel functions.

    Those are only needed in the jn() function.

    The coefficients are calculated from:

    fn(0, z) = 1/z
    fn(1, z) = 1/z**2
    fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)

    Parameters
    ==========

    n : int
        `n` decides the degree of polynomial
    x : optional
    polys : bool, optional
        ``polys=True`` returns an expression, otherwise
        (default) returns an expression.

    Examples
    ========

    >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn
    >>> from sympy import Symbol
    >>> z = Symbol("z")
    >>> fn(1, z)
    z**(-2)
    >>> fn(2, z)
    -1/z + 3/z**3
    >>> fn(3, z)
    -6/z**2 + 15/z**4
    >>> fn(4, z)
    1/z - 45/z**3 + 105/z**5

    """

    if n < 0:
        dup = dup_spherical_bessel_fn_minus(-int(n), ZZ)
    else:
        dup = dup_spherical_bessel_fn(int(n), ZZ)

    poly = DMP(dup, ZZ)

    if x is not None:
        poly = Poly.new(poly, 1/x)
    else:
        poly = PurePoly.new(poly, 1/Dummy('x'))

    return poly if polys else poly.as_expr()

    def __new__(cls, expr, coeffs=Tuple(), alias=None, **args):
        """Construct a new algebraic number. """
        expr = sympify(expr)

        if isinstance(expr, (tuple, Tuple)):
            minpoly, root = expr

            if not minpoly.is_Poly:
                minpoly = Poly(minpoly)
        elif expr.is_AlgebraicNumber:
            minpoly, root = expr.minpoly, expr.root
        else:
            minpoly, root = minimal_polynomial(
                expr, args.get('gen'), polys=True), expr

        dom = minpoly.get_domain()

        if coeffs != Tuple():
            if not isinstance(coeffs, ANP):
                rep = DMP.from_sympy_list(sympify(coeffs), 0, dom)
                scoeffs = Tuple(*coeffs)
            else:
                rep = DMP.from_list(coeffs.to_list(), 0, dom)
                scoeffs = Tuple(*coeffs.to_list())

            if rep.degree() >= minpoly.degree():
                rep = rep.rem(minpoly.rep)

            sargs = (root, scoeffs)

        else:
            rep = DMP.from_list([1, 0], 0, dom)

            if ask(Q.negative(root)):
                rep = -rep

            sargs = (root, coeffs)

        if alias is not None:
            if not isinstance(alias, Symbol):
                alias = Symbol(alias)
            sargs = sargs + (alias,)

        obj = Expr.__new__(cls, *sargs)

        obj.rep = rep
        obj.root = root
        obj.alias = alias
        obj.minpoly = minpoly

        return obj

def legendre_poly(n, x=None, **args):
    """Generates Legendre polynomial of degree `n` in `x`. """
    if n < 0:
        raise ValueError("can't generate Legendre polynomial of degree %s" % n)

    poly = DMP(dup_legendre(int(n), QQ), QQ)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    if not args.get('polys', False):
        return poly.as_expr()
    else:
        return poly

def chebyshevu_poly(n, x=None, **args):
    """Generates Chebyshev polynomial of the second kind of degree `n` in `x`. """
    if n < 0:
        raise ValueError("can't generate 2nd kind Chebyshev polynomial of degree %s" % n)

    poly = DMP(dup_chebyshevu(int(n), ZZ), ZZ)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    if not args.get('polys', False):
        return poly.as_expr()
    else:
        return poly

def cyclotomic_poly(n, x=None, **args):
    """Generates cyclotomic polynomial of order `n` in `x`. """
    if n <= 0:
        raise ValueError("can't generate cyclotomic polynomial of order %s" % n)

    poly = DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    if not args.get('polys', False):
        return poly.as_expr()
    else:
        return poly

def gegenbauer_poly(n, a, x=None, **args):
    """Generates Gegenbauer polynomial of degree `n` in `x`. """
    if n < 0:
        raise ValueError("can't generate Gegenbauer polynomial of degree %s" % n)

    K, a = construct_domain(a, field=True)
    poly = DMP(dup_gegenbauer(int(n), a, K), K)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    if not args.get('polys', False):
        return poly.as_expr()
    else:
        return poly

def jacobi_poly(n, a, b, x=None, **args):
    """Generates Jacobi polynomial of degree `n` in `x`. """
    if n < 0:
        raise ValueError("can't generate Jacobi polynomial of degree %s" % n)

    K, v = construct_domain([a, b], field=True)
    poly = DMP(dup_jacobi(int(n), v[0], v[1], K), K)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    if not args.get('polys', False):
        return poly.as_expr()
    else:
        return poly

def spherical_bessel_fn(n, x=None, **args):
    """
    Coefficients for the spherical Bessel functions.

    Those are only needed in the jn() function.

    The coefficients are calculated from:

    fn(0, z) = 1/z
    fn(1, z) = 1/z**2
    fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)

    Examples
    ========

    >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn
    >>> from sympy import Symbol
    >>> z = Symbol("z")
    >>> fn(1, z)
    z**(-2)
    >>> fn(2, z)
    -1/z + 3/z**3
    >>> fn(3, z)
    -6/z**2 + 15/z**4
    >>> fn(4, z)
    1/z - 45/z**3 + 105/z**5

    """
    from sympy import sympify

    if n < 0:
        dup = dup_spherical_bessel_fn_minus(-int(n), ZZ)
    else:
        dup = dup_spherical_bessel_fn(int(n), ZZ)

    poly = DMP(dup, ZZ)

    if x is not None:
        poly = Poly.new(poly, 1/x)
    else:
        poly = PurePoly.new(poly, 1/Dummy('x'))

    if not args.get('polys', False):
        return poly.as_expr()
    else:
        return poly

def laguerre_poly(n, x=None, alpha=None, **args):
    """Generates Laguerre polynomial of degree `n` in `x`. """
    if n < 0:
        raise ValueError("can't generate Laguerre polynomial of degree %s" % n)

    if alpha is not None:
        K, alpha = construct_domain(alpha, field=True) # XXX: ground_field=True
    else:
        K, alpha = QQ, QQ(0)

    poly = DMP(dup_laguerre(int(n), alpha, K), K)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    if not args.get('polys', False):
        return poly.as_expr()
    else:
        return poly

def legendre_poly(n, x=None, polys=False):
    """Generates Legendre polynomial of degree `n` in `x`.

    Parameters
    ----------
    n : int
        `n` decides the degree of polynomial
    x : optional
    polys : bool, optional
        ``polys=True`` returns an expression, otherwise
        (default) returns an expression.
    """
    if n < 0:
        raise ValueError("can't generate Legendre polynomial of degree %s" % n)

    poly = DMP(dup_legendre(int(n), QQ), QQ)

    if x is not None:
        poly = Poly.new(poly, x)
    else:
        poly = PurePoly.new(poly, Dummy('x'))

    return poly if polys else poly.as_expr()