Source code for sympy.polys.polyroots

"""Algorithms for computing symbolic roots of polynomials. """

from sympy.core.symbol import Dummy
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core import S, I, Basic
from sympy.core.sympify import sympify
from sympy.core.numbers import Rational, igcd

from sympy.ntheory import divisors, isprime, nextprime
from sympy.functions import exp, sqrt, re, im

from sympy.polys.polytools import Poly, cancel, factor, gcd_list
from sympy.polys.specialpolys import cyclotomic_poly
from sympy.polys.polyerrors import PolynomialError, GeneratorsNeeded, DomainError

from sympy.simplify import simplify
from sympy.utilities import default_sort_key

from sympy.core.compatibility import reduce

import math

def roots_linear(f):
    """Returns a list of roots of a linear polynomial."""
    r = -f.nth(0)/f.nth(1)
    dom = f.get_domain()

    if not dom.is_Numerical:
        if dom.is_Composite:
            r = factor(r)
        else:
            r = simplify(r)

    return [r]

def roots_quadratic(f):
    """Returns a list of roots of a quadratic polynomial."""
    a, b, c = f.all_coeffs()
    dom = f.get_domain()

    def _simplify(expr):
        if dom.is_Composite:
            return factor(expr)
        else:
            return simplify(expr)

    if c is S.Zero:
        r0, r1 = S.Zero, -b/a

        if not dom.is_Numerical:
            r1 = _simplify(r1)
    elif b is S.Zero:
        r = -c/a

        if not dom.is_Numerical:
            R = sqrt(_simplify(r))
        else:
            R = sqrt(r)

        r0 =  R
        r1 = -R
    else:
        d = b**2 - 4*a*c

        if dom.is_Numerical:
            D = sqrt(d)

            r0 = (-b + D) / (2*a)
            r1 = (-b - D) / (2*a)
        else:
            D = sqrt(_simplify(d))
            A = 2*a

            E = _simplify(-b/A)
            F = D/A

            r0 = E + F
            r1 = E - F

    return sorted([r0, r1], key=default_sort_key)

def roots_cubic(f):
    """Returns a list of roots of a cubic polynomial."""
    _, a, b, c = f.monic().all_coeffs()

    if c is S.Zero:
        x1, x2 = roots([1,a,b], multiple = True)
        return [x1, S.Zero, x2]

    p = b - a**2/3
    q = c - a*b/3 + 2*a**3/27

    pon3 = p/3
    aon3 = a/3

    if p is S.Zero:
        if q is S.Zero:
            return [-aon3]*3
        else:
            u1 = q**Rational(1, 3)
    elif q is S.Zero:
        y1, y2 = roots([1, 0, p], multiple=True)
        return [tmp - aon3 for tmp in [y1, S.Zero, y2]]
    else:
        u1 = (q/2 + sqrt(q**2/4 + pon3**3))**Rational(1, 3)

    coeff = S.ImaginaryUnit*sqrt(3)/2

    u2 = u1*(-S.Half + coeff)
    u3 = u1*(-S.Half - coeff)

    soln = [
        -u1 + pon3/u1 - aon3,
        -u2 + pon3/u2 - aon3,
        -u3 + pon3/u3 - aon3
    ]

    return soln

def roots_quartic(f):
    r"""
    Returns a list of roots of a quartic polynomial.

    There are many references for solving quartic expressions available [1-5].
    This reviewer has found that many of them require one to select from among
    2 or more possible sets of solutions and that some solutions work when one
    is searching for real roots but don't work when searching for complex roots
    (though this is not always stated clearly). The following routine has been
    tested and found to be correct for 0, 2 or 4 complex roots.

    The quasisymmetric case solution [6] looks for quartics that have the form
    `x**4 + A*x**3 + B*x**2 + C*x + D = 0` where `(C/A)**2 = D`.

    Although there is a general solution, simpler results can be obtained for
    certain values of the coefficients. In all cases, 4 roots are returned:

      1) `f = c + a*(a**2/8 - b/2) == 0`
      2) `g = d - a*(a*(3*a**2/256 - b/16) + c/4) = 0`
      3) if `f != 0` and `g != 0` and `p = -d + a*c/4 - b**2/12` then
        a) `p == 0`
        b) `p != 0`

    **Examples**

        >>> from sympy import Poly, symbols, I
        >>> from sympy.polys.polyroots import roots_quartic

        >>> r = roots_quartic(Poly('x**4-6*x**3+17*x**2-26*x+20'))

        >>> # 4 complex roots: 1+-I*sqrt(3), 2+-I
        >>> sorted(str(tmp.evalf(n=2)) for tmp in r)
        ['1.0 + 1.7*I', '1.0 - 1.7*I', '2.0 + 1.0*I', '2.0 - 1.0*I']

    **References**

    1. http://mathforum.org/dr.math/faq/faq.cubic.equations.html
    2. http://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method
    3. http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html
    4. http://staff.bath.ac.uk/masjhd/JHD-CA.pdf
    5. http://www.albmath.org/files/Math_5713.pdf
    6. http://www.statemaster.com/encyclopedia/Quartic-equation

    """
    _, a, b, c, d = f.monic().all_coeffs()

    if not d:
        return [S.Zero] + roots([1, a, b, c], multiple=True)
    elif (c/a)**2 == d:
        x, m = f.gen, c/a

        g = Poly(x**2 + a*x + b - 2*m, x)

        z1, z2 = roots_quadratic(g)

        h1 = Poly(x**2 - z1*x + m, x)
        h2 = Poly(x**2 - z2*x + m, x)

        r1 = roots_quadratic(h1)
        r2 = roots_quadratic(h2)

        return r1 + r2
    else:
        a2 = a**2
        e = b - 3*a2/8
        f = c + a*(a2/8 - b/2)
        g = d - a*(a*(3*a2/256 - b/16) + c/4)
        aon4 = a/4
        ans = []

        if f is S.Zero:
            y1, y2 = [tmp**S.Half for tmp in
                      roots([1, e, g], multiple = True)]
            return [tmp - aon4 for tmp in [-y1, -y2, y1, y2]]
        if g is S.Zero:
            y = [S.Zero] + roots([1, 0, e, f], multiple = True)
            return [tmp - aon4 for tmp in y]
        else:
            p = -e**2/12 - g
            q = -e**3/108 + e*g/3 - f**2/8
            TH = Rational(1, 3)
            if p is S.Zero:
                y = -5*e/6 - q**TH
            else:
                # with p !=0 then u below is not 0
                root = sqrt(q**2/4 + p**3/27)
                r = -q/2 + root # or -q/2 - root
                u = r**TH # primary root of solve(x**3-r, x)
                y = -5*e/6 + u - p/u/3
            w = sqrt(e + 2*y)
            arg1 = 3*e + 2*y
            arg2 = 2*f/w
            for s in [-1, 1]:
                root = sqrt(-(arg1 + s*arg2))
                for t in [-1, 1]:
                    ans.append((s*w - t*root)/2 - aon4)

    return ans

def roots_binomial(f):
    """Returns a list of roots of a binomial polynomial."""
    n = f.degree()

    a, b = f.nth(n), f.nth(0)
    alpha = (-cancel(b/a))**Rational(1, n)

    if alpha.is_number:
        alpha = alpha.expand(complex=True)

    roots, I = [], S.ImaginaryUnit

    for k in xrange(n):
        zeta = exp(2*k*S.Pi*I/n).expand(complex=True)
        roots.append((alpha*zeta).expand(power_base=False))

    return sorted(roots, key=default_sort_key)

def _inv_totient_estimate(m):
    """
    Find ``(L, U)`` such that ``L <= phi^-1(m) <= U``.

    **Examples**

    >>> from sympy.polys.polyroots import _inv_totient_estimate

    >>> _inv_totient_estimate(192)
    (192, 840)
    >>> _inv_totient_estimate(400)
    (400, 1750)

    """
    primes = [ d + 1 for d in divisors(m) if isprime(d + 1) ]

    a, b = 1, 1

    for p in primes:
        a *= p
        b *= p - 1

    L = m
    U = int(math.ceil(m*(float(a)/b)))

    P = p = 2
    primes = []

    while P <= U:
        p = nextprime(p)
        primes.append(p)
        P *= p

    P //= p
    b = 1

    for p in primes[:-1]:
        b *= p - 1

    U = int(math.ceil(m*(float(P)/b)))

    return L, U

def roots_cyclotomic(f, factor=False):
    """Compute roots of cyclotomic polynomials. """
    L, U = _inv_totient_estimate(f.degree())

    for n in xrange(L, U+1):
        g = cyclotomic_poly(n, f.gen, polys=True)

        if f == g:
            break
    else: # pragma: no cover
        raise RuntimeError("failed to find index of a cyclotomic polynomial")

    roots = []

    if not factor:
        for k in xrange(1, n+1):
            if igcd(k, n) == 1:
                roots.append(exp(2*k*S.Pi*I/n).expand(complex=True))
    else:
        g = Poly(f, extension=(-1)**Rational(1, n))

        for h, _ in g.factor_list()[1]:
            roots.append(-h.TC())

    return sorted(roots, key=default_sort_key)

def roots_rational(f):
    """Returns a list of rational roots of a polynomial."""
    domain = f.get_domain()

    if domain.is_QQ:
        _, f = f.clear_denoms()
    elif domain.is_ZZ:
        f = f.set_domain('QQ')
    else:
        return []

    LC_divs = divisors(int(f.LC()))
    EC_divs = divisors(int(f.EC()))

    if not f.eval(S.Zero):
        zeros = [S.Zero]
    else:
        zeros = []

    for p in LC_divs:
        for q in EC_divs:
            zero = Rational(p, q)

            if not f.eval(zero):
                zeros.append(zero)

            if not f.eval(-zero):
                zeros.append(-zero)

    return sorted(zeros, key=default_sort_key)

def _integer_basis(poly):
    """Compute coefficient basis for a polynomial over integers. """
    monoms, coeffs = zip(*poly.terms())

    monoms, = zip(*monoms)
    coeffs = map(abs, coeffs)

    if coeffs[0] < coeffs[-1]:
        coeffs = list(reversed(coeffs))
    else:
        return None

    monoms = monoms[:-1]
    coeffs = coeffs[:-1]

    divs = reversed(divisors(gcd_list(coeffs))[1:])

    try:
        div = divs.next()
    except StopIteration:
        return None

    while True:
        for monom, coeff in zip(monoms, coeffs):
            if coeff % div**monom != 0:
                try:
                    div = divs.next()
                except StopIteration:
                    return None
                else:
                    break
        else:
            return div

def preprocess_roots(poly):
    """Try to get rid of symbolic coefficients from ``poly``. """
    coeff = S.One

    try:
        _, poly = poly.clear_denoms(convert=True)
    except DomainError:
        return coeff, poly

    poly = poly.primitive()[1]
    poly = poly.retract()

    if poly.get_domain().is_Poly and all(c.is_monomial for c in poly.rep.coeffs()):
        poly = poly.inject()

        strips = zip(*poly.monoms())
        gens = list(poly.gens[1:])

        base, strips = strips[0], strips[1:]

        for gen, strip in zip(list(gens), strips):
            reverse = False

            if strip[0] < strip[-1]:
                strip = reversed(strip)
                reverse = True

            ratio = None

            for a, b in zip(base, strip):
                if not a and not b:
                    continue
                elif not a or not b:
                    break
                elif b % a != 0:
                    break
                else:
                    _ratio = b // a

                    if ratio is None:
                        ratio = _ratio
                    elif ratio != _ratio:
                        break
            else:
                if reverse:
                    ratio = -ratio

                poly = poly.eval(gen, 1)
                coeff *= gen**(-ratio)
                gens.remove(gen)

        if gens:
            poly = poly.eject(*gens)

    if poly.is_univariate and poly.get_domain().is_ZZ:
        basis = _integer_basis(poly)

        if basis is not None:
            n = poly.degree()

            def func(k, coeff):
                return coeff//basis**(n-k[0])

            poly = poly.termwise(func)
            coeff *= basis

    return coeff, poly

[docs]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. 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 tuple containing all those roots set the ``multiple`` flag to True. **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)) {-y**(1/2): 1, y**(1/2): 1} >>> roots(x**2 - y, x) {-y**(1/2): 1, y**(1/2): 1} >>> roots([1, 0, -1]) {-1: 1, 1: 1} """ flags = dict(flags) auto = flags.pop('auto', True) cubics = flags.pop('cubics', True) quartics = flags.pop('quartics', True) 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) except GeneratorsNeeded: if multiple: return [] else: return {} if f.is_multivariate: 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) return result (k,), f = f.terms_gcd() if not k: zeros = {} else: zeros = {S(0) : k} coeff, f = preprocess_roots(f) if auto and f.get_domain().has_Ring: f = f.to_field() result = {} if not f.is_ground: if not f.get_domain().is_Exact: for r in f.nroots(): _update_dict(result, r, 1) elif f.degree() == 1: result[roots_linear(f)[0]] = 1 elif f.degree() == 2: for r in roots_quadratic(f): _update_dict(result, r, 1) elif f.length() == 2: for r in roots_binomial(f): _update_dict(result, r, 1) else: _, factors = Poly(f.as_expr()).factor_list() if len(factors) == 1 and factors[0][1] == 1: for root in _try_decompose(f): _update_dict(result, root, 1) else: for factor, k in factors: for r in _try_heuristics(Poly(factor, f.gen, field=True)): _update_dict(result, r, k) if coeff is not S.One: _result, result, = result, {} for root, k in _result.iteritems(): result[coeff*root] = k result.update(zeros) if filter not in [None, 'C']: handlers = { 'Z' : lambda r: r.is_Integer, 'Q' : lambda r: r.is_Rational, 'R' : lambda r: r.is_real, 'I' : lambda r: r.is_imaginary, } try: query = handlers[filter] except KeyError: raise ValueError("Invalid filter: %s" % filter) for zero in dict(result).iterkeys(): if not query(zero): del result[zero] if predicate is not None: for zero in dict(result).iterkeys(): if not predicate(zero): del result[zero] if not multiple: return result else: zeros = [] for zero, k in result.iteritems(): zeros.extend([zero]*k) return sorted(zeros, key=default_sort_key)
def root_factors(f, *gens, **args): """ Returns all factors of a univariate polynomial. **Examples** >>> from sympy.abc import x, y >>> from sympy.polys.polyroots import root_factors >>> root_factors(x**2-y, x) [x - y**(1/2), x + y**(1/2)] """ args = dict(args) filter = args.pop('filter', None) F = Poly(f, *gens, **args) if not F.is_Poly: return [f] if F.is_multivariate: raise ValueError('multivariate polynomials not supported') x = F.gens[0] zeros = roots(F, filter=filter) if not zeros: factors = [F] else: factors, N = [], 0 for r, n in zeros.iteritems(): factors, N = factors + [Poly(x-r, x)]*n, N + n if N < F.degree(): G = reduce(lambda p,q: p*q, factors) factors.append(F.quo(G)) if not isinstance(f, Poly): return [ f.as_expr() for f in factors ] else: return factors