Python Poly.length方法代码示例

# 需要导入模块: from sympy.polys.polytools import Poly [as 别名]
# 或者: from sympy.polys.polytools.Poly import length [as 别名]

#.........这里部分代码省略.........
            raise PolynomialError('multivariate polynomials are not supported')

    def _update_dict(result, root, k):
        if root in result:
            result[root] += k
        else:
            result[root] = k

    def _try_decompose(f):
        """Find roots using functional decomposition. """
        factors, roots = f.decompose(), []

        for root in _try_heuristics(factors[0]):
            roots.append(root)

        for factor in factors[1:]:
            previous, roots = list(roots), []

            for root in previous:
                g = factor - Poly(root, f.gen)

                for root in _try_heuristics(g):
                    roots.append(root)

        return roots

    def _try_heuristics(f):
        """Find roots using formulas and some tricks. """
        if f.is_ground:
            return []
        if f.is_monomial:
            return [S(0)]*f.degree()

        if f.length() == 2:
            if f.degree() == 1:
                return map(cancel, roots_linear(f))
            else:
                return roots_binomial(f)

        result = []

        for i in [-1, 1]:
            if not f.eval(i):
                f = f.quo(Poly(f.gen - i, f.gen))
                result.append(i)
                break

        n = f.degree()

        if n == 1:
            result += map(cancel, roots_linear(f))
        elif n == 2:
            result += map(cancel, roots_quadratic(f))
        elif f.is_cyclotomic:
            result += roots_cyclotomic(f)
        elif n == 3 and cubics:
            result += roots_cubic(f)
        elif n == 4 and quartics:
            result += roots_quartic(f)
        elif n == 5 and quintics:
            result += roots_quintic(f)

        return result

    (k,), f = f.terms_gcd()

# 需要导入模块: from sympy.polys.polytools import Poly [as 别名]
# 或者: from sympy.polys.polytools.Poly import length [as 别名]
def roots(f, *gens, **flags):
    """
    Computes symbolic roots of a univariate polynomial.

    Given a univariate polynomial f with symbolic coefficients (or
    a list of the polynomial's coefficients), returns a dictionary
    with its roots and their multiplicities.

    Only roots expressible via radicals will be returned.  To get
    a complete set of roots use RootOf class or numerical methods
    instead. By default cubic and quartic formulas are used in
    the algorithm. To disable them because of unreadable output
    set ``cubics=False`` or ``quartics=False`` respectively. If cubic
    roots are real but are expressed in terms of complex numbers
    (casus irreducibilis [1]) the ``trig`` flag can be set to True to
    have the solutions returned in terms of cosine and inverse cosine
    functions.

    To get roots from a specific domain set the ``filter`` flag with
    one of the following specifiers: Z, Q, R, I, C. By default all
    roots are returned (this is equivalent to setting ``filter='C'``).

    By default a dictionary is returned giving a compact result in
    case of multiple roots.  However to get a list containing all
    those roots set the ``multiple`` flag to True; the list will
    have identical roots appearing next to each other in the result.
    (For a given Poly, the all_roots method will give the roots in
    sorted numerical order.)

    Examples
    ========

    >>> from sympy import Poly, roots
    >>> from sympy.abc import x, y

    >>> roots(x**2 - 1, x)
    {-1: 1, 1: 1}

    >>> p = Poly(x**2-1, x)
    >>> roots(p)
    {-1: 1, 1: 1}

    >>> p = Poly(x**2-y, x, y)

    >>> roots(Poly(p, x))
    {-sqrt(y): 1, sqrt(y): 1}

    >>> roots(x**2 - y, x)
    {-sqrt(y): 1, sqrt(y): 1}

    >>> roots([1, 0, -1])
    {-1: 1, 1: 1}


    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method

    """
    from sympy.polys.polytools import to_rational_coeffs
    flags = dict(flags)

    auto = flags.pop('auto', True)
    cubics = flags.pop('cubics', True)
    trig = flags.pop('trig', False)
    quartics = flags.pop('quartics', True)
    quintics = flags.pop('quintics', False)
    multiple = flags.pop('multiple', False)
    filter = flags.pop('filter', None)
    predicate = flags.pop('predicate', None)

    if isinstance(f, list):
        if gens:
            raise ValueError('redundant generators given')

        x = Dummy('x')

        poly, i = {}, len(f) - 1

        for coeff in f:
            poly[i], i = sympify(coeff), i - 1

        f = Poly(poly, x, field=True)
    else:
        try:
            f = Poly(f, *gens, **flags)
            if f.length == 2 and f.degree() != 1:
                # check for foo**n factors in the constant
                n = f.degree()
                npow_bases = []
                others = []
                expr = f.as_expr()
                con = expr.as_independent(*gens)[0]
                for p in Mul.make_args(con):
                    if p.is_Pow and not p.exp % n:
                        npow_bases.append(p.base**(p.exp/n))
                    else:
                        others.append(p)
                    if npow_bases:
#.........这里部分代码省略.........

# 需要导入模块: from sympy.polys.polytools import Poly [as 别名]
# 或者: from sympy.polys.polytools.Poly import length [as 别名]

#.........这里部分代码省略.........

            h = compose_(factors[i:]).as_expr()
            add_comment("Therefore")
            for root in previous:
                add_eq(h, root)

            for root in previous:
                if i < len(factors) - 1:
                    add_comment("Solve the equation")
                    add_eq(h, root)
                    add_comment("Use the substitution")
                    t = Dummy("t" + str(i+1))
                    add_eq(t, compose_(factors[(i+1):]).as_expr())
                    g = Poly(factor.as_expr().subs(f.gen, t), t) - Poly(root, t)
                else:
                    g = factor - Poly(root, f.gen)

                for root in _try_heuristics(g):
                    roots.append(root)
            i += 1

        return (roots, True)

    def _try_heuristics(f):
        """Find roots using formulas and some tricks. """
        
        if f.is_ground:
            add_comment("The polynomial is constant. Therefore there is no root.")
            return []
        if f.is_monomial:
            add_comment("The root of the equation is zero")
            rr = [S(0)]*f.degree()
            return rr
        if f.length() == 2:
            if f.degree() == 1:
                rr = list(map(cancel, roots_linear(f)))
                return rr
            else:
                return roots_binomial(f)

        result = []

        for i in [-1, 1]:
            if not f.eval(i):
                f = f.quo(Poly(f.gen - i, f.gen))
                result.append(i)
                break

        n = f.degree()
        if n == 1:
            result += list(map(cancel, roots_linear(f)))
        elif n == 2:
            result += list(map(cancel, roots_quadratic(f)))
        elif f.is_cyclotomic:
            result += roots_cyclotomic(f)
        elif n == 3 and cubics:
            result += roots_cubic(f)
        elif n == 4 and quartics:
            result += roots_quartic(f)
        elif n == 5 and quintics:
            result += roots_quintic(f)

        return result

    tmp_ = f
    (k,), f = f.terms_gcd()