Source code for sympy.polys.rootoftools

"""Implementation of RootOf class and related tools. """

from __future__ import print_function, division

from sympy.core import (S, Expr, Integer, Float, I, Add, Lambda, symbols,
        sympify, Rational, Dummy)
from sympy.core.cache import cacheit
from sympy.core.function import AppliedUndef
from sympy.functions.elementary.miscellaneous import root as _root

from sympy.polys.polytools import Poly, PurePoly, factor
from sympy.polys.rationaltools import together
from sympy.polys.polyfuncs import symmetrize, viete

from sympy.polys.rootisolation import (
    dup_isolate_complex_roots_sqf,
    dup_isolate_real_roots_sqf)

from sympy.polys.polyroots import (
    roots_linear, roots_quadratic, roots_binomial,
    preprocess_roots, roots)

from sympy.polys.polyerrors import (
    MultivariatePolynomialError,
    GeneratorsNeeded,
    PolynomialError,
    DomainError)

from sympy.polys.domains import QQ

from mpmath import mpf, mpc, findroot, workprec
from mpmath.libmp.libmpf import prec_to_dps

from sympy.utilities import lambdify, public, sift

from sympy.core.compatibility import range

from math import log as mathlog

__all__ = ['CRootOf']

def _ispow2(i):
    v = mathlog(i, 2)
    return v == int(v)

_reals_cache = {}
_complexes_cache = {}


[docs]@public def rootof(f, x, index=None, radicals=True, expand=True): """An indexed root of a univariate polynomial. Returns either a ``ComplexRootOf`` object or an explicit expression involving radicals. Parameters ---------- f : Expr Univariate polynomial. x : Symbol, optional Generator for ``f``. index : int or Integer radicals : bool Return a radical expression if possible. expand : bool Expand ``f``. """ return CRootOf(f, x, index=index, radicals=radicals, expand=expand)
[docs]@public class RootOf(Expr): """Represents a root of a univariate polynomial. Base class for roots of different kinds of polynomials. Only complex roots are currently supported. """ __slots__ = ['poly'] def __new__(cls, f, x, index=None, radicals=True, expand=True): """Construct a new ``CRootOf`` object for ``k``-th root of ``f``.""" return rootof(f, x, index=index, radicals=radicals, expand=expand)
[docs]@public class ComplexRootOf(RootOf): """Represents an indexed complex root of a polynomial. Roots of a univariate polynomial separated into disjoint real or complex intervals and indexed in a fixed order. Currently only rational coefficients are allowed. Can be imported as ``CRootOf``. """ __slots__ = ['index'] is_complex = True is_number = True def __new__(cls, f, x, index=None, radicals=False, expand=True): """ Construct an indexed complex root of a polynomial. See ``rootof`` for the parameters. The default value of ``radicals`` is ``False`` to satisfy ``eval(srepr(expr) == expr``. """ x = sympify(x) if index is None and x.is_Integer: x, index = None, x else: index = sympify(index) if index is not None and index.is_Integer: index = int(index) else: raise ValueError("expected an integer root index, got %s" % index) poly = PurePoly(f, x, greedy=False, expand=expand) if not poly.is_univariate: raise PolynomialError("only univariate polynomials are allowed") degree = poly.degree() if degree <= 0: raise PolynomialError("can't construct CRootOf object for %s" % f) if index < -degree or index >= degree: raise IndexError("root index out of [%d, %d] range, got %d" % (-degree, degree - 1, index)) elif index < 0: index += degree dom = poly.get_domain() if not dom.is_Exact: poly = poly.to_exact() roots = cls._roots_trivial(poly, radicals) if roots is not None: return roots[index] coeff, poly = preprocess_roots(poly) dom = poly.get_domain() if not dom.is_ZZ: raise NotImplementedError("CRootOf is not supported over %s" % dom) root = cls._indexed_root(poly, index) return coeff * cls._postprocess_root(root, radicals) @classmethod def _new(cls, poly, index): """Construct new ``CRootOf`` object from raw data. """ obj = Expr.__new__(cls) obj.poly = PurePoly(poly) obj.index = index try: _reals_cache[obj.poly] = _reals_cache[poly] _complexes_cache[obj.poly] = _complexes_cache[poly] except KeyError: pass return obj def _hashable_content(self): return (self.poly, self.index) @property def expr(self): return self.poly.as_expr() @property def args(self): return (self.expr, Integer(self.index)) @property def free_symbols(self): # CRootOf currently only works with univariate expressions and although # the poly attribute is often a PurePoly, sometimes it is a Poly. In # either case no free symbols should be reported. return set() def _eval_is_real(self): """Return ``True`` if the root is real. """ return self.index < len(_reals_cache[self.poly]) @classmethod def real_roots(cls, poly, radicals=True): """Get real roots of a polynomial. """ return cls._get_roots("_real_roots", poly, radicals) @classmethod def all_roots(cls, poly, radicals=True): """Get real and complex roots of a polynomial. """ return cls._get_roots("_all_roots", poly, radicals) @classmethod def _get_reals_sqf(cls, factor): """Get real root isolating intervals for a square-free factor.""" if factor in _reals_cache: real_part = _reals_cache[factor] else: _reals_cache[factor] = real_part = \ dup_isolate_real_roots_sqf( factor.rep.rep, factor.rep.dom, blackbox=True) return real_part @classmethod def _get_complexes_sqf(cls, factor): """Get complex root isolating intervals for a square-free factor.""" if factor in _complexes_cache: complex_part = _complexes_cache[factor] else: _complexes_cache[factor] = complex_part = \ dup_isolate_complex_roots_sqf( factor.rep.rep, factor.rep.dom, blackbox=True) return complex_part @classmethod def _get_reals(cls, factors): """Compute real root isolating intervals for a list of factors. """ reals = [] for factor, k in factors: real_part = cls._get_reals_sqf(factor) reals.extend([(root, factor, k) for root in real_part]) return reals @classmethod def _get_complexes(cls, factors): """Compute complex root isolating intervals for a list of factors. """ complexes = [] for factor, k in factors: complex_part = cls._get_complexes_sqf(factor) complexes.extend([(root, factor, k) for root in complex_part]) return complexes @classmethod def _reals_sorted(cls, reals): """Make real isolating intervals disjoint and sort roots. """ cache = {} for i, (u, f, k) in enumerate(reals): for j, (v, g, m) in enumerate(reals[i + 1:]): u, v = u.refine_disjoint(v) reals[i + j + 1] = (v, g, m) reals[i] = (u, f, k) reals = sorted(reals, key=lambda r: r[0].a) for root, factor, _ in reals: if factor in cache: cache[factor].append(root) else: cache[factor] = [root] for factor, roots in cache.items(): _reals_cache[factor] = roots return reals @classmethod def _separate_imaginary_from_complex(cls, complexes): def is_imag(c): ''' return True if all roots are imaginary (ax**2 + b) return False if no roots are imaginary return None if 2 roots are imaginary (ax**N''' u, f, k = c deg = f.degree() if f.length() == 2: if deg == 2: return True # both imag elif _ispow2(deg): if f.LC()*f.TC() < 0: return None # 2 are imag return False # none are imag # separate according to the function sifted = sift(complexes, lambda c: c[1]) del complexes imag = [] complexes = [] for f in sifted: isift = sift(sifted[f], lambda c: is_imag(c)) imag.extend(isift.pop(True, [])) complexes.extend(isift.pop(False, [])) mixed = isift.pop(None, []) assert not isift if not mixed: continue while True: # the non-imaginary ones will be on one side or the other # of the y-axis i = 0 while i < len(mixed): u, f, k = mixed[i] if u.ax*u.bx > 0: complexes.append(mixed.pop(i)) else: i += 1 if len(mixed) == 2: imag.extend(mixed) break # refine for i, (u, f, k) in enumerate(mixed): u = u._inner_refine() mixed[i] = u, f, k return imag, complexes @classmethod def _refine_complexes(cls, complexes): """return complexes such that no bounding rectangles of non-conjugate roots would intersect if slid horizontally or vertically/ """ while complexes: # break when all are distinct # get the intervals pairwise-disjoint. # If rectangles were drawn around the coordinates of the bounding # rectangles, no rectangles would intersect after this procedure. for i, (u, f, k) in enumerate(complexes): for j, (v, g, m) in enumerate(complexes[i + 1:]): u, v = u.refine_disjoint(v) complexes[i + j + 1] = (v, g, m) complexes[i] = (u, f, k) # Although there are no intersecting rectangles, a given rectangle # might intersect another when slid horizontally. We have to refine # intervals until this is not true so we can sort the roots # unambiguously. Since complex roots come in conjugate pairs, we # will always have 2 rectangles above each other but we should not # have more than that. N = len(complexes)//2 - 1 # check x (real) parts: there must be N + 1 disjoint x ranges, i.e. # the first one must be different from N others uu = set([(u.ax, u.bx) for u, _, _ in complexes]) u = uu.pop() if sum([u[1] <= v[0] or v[1] <= u[0] for v in uu]) < N: # refine for i, (u, f, k) in enumerate(complexes): u = u._inner_refine() complexes[i] = u, f, k else: # intervals with identical x-values have disjoint y-values or # else they would not be disjoint so there is no need for # further checks break return complexes @classmethod def _complexes_sorted(cls, complexes): """Make complex isolating intervals disjoint and sort roots. """ if not complexes: return [] cache = {} # imaginary roots can cause a problem in terms of sorting since # their x-intervals will never refine as distinct from others # so we handle them separately imag, complexes = cls._separate_imaginary_from_complex(complexes) complexes = cls._refine_complexes(complexes) # sort imaginary roots def key(c): '''return, for ax**n+b, +/-root(abs(b/a), b) according to the apparent sign of the imaginary interval, e.g. if the interval were (0, 3) the positive root would be returned. ''' u, f, k = c r = _root(abs(f.TC()/f.LC()), f.degree()) if u.ay < 0 or u.by < 0: return -r return r imag = sorted(imag, key=lambda c: key(c)) # sort complexes and combine with imag if complexes: # key is (x1, y1) e.g. (1, 2)x(3, 4) -> (1,3) complexes = sorted(complexes, key=lambda c: c[0].a) # find insertion point for imaginary for i, c in enumerate(reversed(complexes)): if c[0].bx <= 0: break i = len(complexes) - i - 1 if i: i += 1 complexes = complexes[:i] + imag + complexes[i:] else: complexes = imag # update cache for root, factor, _ in complexes: if factor in cache: cache[factor].append(root) else: cache[factor] = [root] for factor, roots in cache.items(): _complexes_cache[factor] = roots return complexes @classmethod def _reals_index(cls, reals, index): """ Map initial real root index to an index in a factor where the root belongs. """ i = 0 for j, (_, factor, k) in enumerate(reals): if index < i + k: poly, index = factor, 0 for _, factor, _ in reals[:j]: if factor == poly: index += 1 return poly, index else: i += k @classmethod def _complexes_index(cls, complexes, index): """ Map initial complex root index to an index in a factor where the root belongs. """ index, i = index, 0 for j, (_, factor, k) in enumerate(complexes): if index < i + k: poly, index = factor, 0 for _, factor, _ in complexes[:j]: if factor == poly: index += 1 index += len(_reals_cache[poly]) return poly, index else: i += k @classmethod def _count_roots(cls, roots): """Count the number of real or complex roots with multiplicities.""" return sum([k for _, _, k in roots]) @classmethod def _indexed_root(cls, poly, index): """Get a root of a composite polynomial by index. """ (_, factors) = poly.factor_list() reals = cls._get_reals(factors) reals_count = cls._count_roots(reals) if index < reals_count: reals = cls._reals_sorted(reals) return cls._reals_index(reals, index) else: complexes = cls._get_complexes(factors) complexes = cls._complexes_sorted(complexes) return cls._complexes_index(complexes, index - reals_count) @classmethod def _real_roots(cls, poly): """Get real roots of a composite polynomial. """ (_, factors) = poly.factor_list() reals = cls._get_reals(factors) reals = cls._reals_sorted(reals) reals_count = cls._count_roots(reals) roots = [] for index in range(0, reals_count): roots.append(cls._reals_index(reals, index)) return roots @classmethod def _all_roots(cls, poly): """Get real and complex roots of a composite polynomial. """ (_, factors) = poly.factor_list() reals = cls._get_reals(factors) reals = cls._reals_sorted(reals) reals_count = cls._count_roots(reals) roots = [] for index in range(0, reals_count): roots.append(cls._reals_index(reals, index)) complexes = cls._get_complexes(factors) complexes = cls._complexes_sorted(complexes) complexes_count = cls._count_roots(complexes) for index in range(0, complexes_count): roots.append(cls._complexes_index(complexes, index)) return roots @classmethod @cacheit def _roots_trivial(cls, poly, radicals): """Compute roots in linear, quadratic and binomial cases. """ if poly.degree() == 1: return roots_linear(poly) if not radicals: return None if poly.degree() == 2: return roots_quadratic(poly) elif poly.length() == 2 and poly.TC(): return roots_binomial(poly) else: return None @classmethod def _preprocess_roots(cls, poly): """Take heroic measures to make ``poly`` compatible with ``CRootOf``.""" dom = poly.get_domain() if not dom.is_Exact: poly = poly.to_exact() coeff, poly = preprocess_roots(poly) dom = poly.get_domain() if not dom.is_ZZ: raise NotImplementedError( "sorted roots not supported over %s" % dom) return coeff, poly @classmethod def _postprocess_root(cls, root, radicals): """Return the root if it is trivial or a ``CRootOf`` object. """ poly, index = root roots = cls._roots_trivial(poly, radicals) if roots is not None: return roots[index] else: return cls._new(poly, index) @classmethod def _get_roots(cls, method, poly, radicals): """Return postprocessed roots of specified kind. """ if not poly.is_univariate: raise PolynomialError("only univariate polynomials are allowed") coeff, poly = cls._preprocess_roots(poly) roots = [] for root in getattr(cls, method)(poly): roots.append(coeff*cls._postprocess_root(root, radicals)) return roots def _get_interval(self): """Internal function for retrieving isolation interval from cache. """ if self.is_real: return _reals_cache[self.poly][self.index] else: reals_count = len(_reals_cache[self.poly]) return _complexes_cache[self.poly][self.index - reals_count] def _set_interval(self, interval): """Internal function for updating isolation interval in cache. """ if self.is_real: _reals_cache[self.poly][self.index] = interval else: reals_count = len(_reals_cache[self.poly]) _complexes_cache[self.poly][self.index - reals_count] = interval def _eval_subs(self, old, new): # don't allow subs to change anything return self def _eval_evalf(self, prec): """Evaluate this complex root to the given precision. """ with workprec(prec): g = self.poly.gen if not g.is_Symbol: d = Dummy('x') func = lambdify(d, self.expr.subs(g, d)) else: func = lambdify(g, self.expr) interval = self._get_interval() if not self.is_real: # For complex intervals, we need to keep refining until the # imaginary interval is disjunct with other roots, that is, # until both ends get refined. ay = interval.ay by = interval.by while interval.ay == ay or interval.by == by: interval = interval.refine() while True: if self.is_real: a = mpf(str(interval.a)) b = mpf(str(interval.b)) if a == b: root = a break x0 = mpf(str(interval.center)) else: ax = mpf(str(interval.ax)) bx = mpf(str(interval.bx)) ay = mpf(str(interval.ay)) by = mpf(str(interval.by)) if ax == bx and ay == by: # the sign of the imaginary part will be assigned # according to the desired index using the fact that # roots are sorted with negative imag parts coming # before positive (and all imag roots coming after real # roots) deg = self.poly.degree() i = self.index # a positive attribute after creation if (deg - i) % 2: if ay < 0: ay = -ay else: if ay > 0: ay = -ay root = mpc(ax, ay) break x0 = mpc(*map(str, interval.center)) try: root = findroot(func, x0) # If the (real or complex) root is not in the 'interval', # then keep refining the interval. This happens if findroot # accidentally finds a different root outside of this # interval because our initial estimate 'x0' was not close # enough. It is also possible that the secant method will # get trapped by a max/min in the interval; the root # verification by findroot will raise a ValueError in this # case and the interval will then be tightened -- and # eventually the root will be found. # # It is also possible that findroot will not have any # successful iterations to process (in which case it # will fail to initialize a variable that is tested # after the iterations and raise an UnboundLocalError). if self.is_real: if (a <= root <= b): break elif (ax <= root.real <= bx and ay <= root.imag <= by): break except (UnboundLocalError, ValueError): pass interval = interval.refine() return (Float._new(root.real._mpf_, prec) + I*Float._new(root.imag._mpf_, prec)) def eval_rational(self, tol): """ Return a Rational approximation to ``self`` with the tolerance ``tol``. This method uses bisection, which is very robust and it will always converge. The returned Rational instance will be at most 'tol' from the exact root. The following example first obtains Rational approximation to 1e-7 accuracy for all roots of the 4-th order Legendre polynomial, and then evaluates it to 5 decimal digits (so all digits will be correct including rounding): >>> from sympy import S, legendre_poly, Symbol >>> x = Symbol("x") >>> p = legendre_poly(4, x, polys=True) >>> roots = [r.eval_rational(S(1)/10**7) for r in p.real_roots()] >>> roots = [str(r.n(5)) for r in roots] >>> roots ['-0.86114', '-0.33998', '0.33998', '0.86114'] """ if not self.is_real: raise NotImplementedError( "eval_rational() only works for real polynomials so far") func = lambdify(self.poly.gen, self.expr) interval = self._get_interval() a = Rational(str(interval.a)) b = Rational(str(interval.b)) return bisect(func, a, b, tol) def _eval_Eq(self, other): # CRootOf represents a Root, so if other is that root, it should set # the expression to zero *and* it should be in the interval of the # CRootOf instance. It must also be a number that agrees with the # is_real value of the CRootOf instance. if type(self) == type(other): return sympify(self == other) if not (other.is_number and not other.has(AppliedUndef)): return S.false if not other.is_finite: return S.false z = self.expr.subs(self.expr.free_symbols.pop(), other).is_zero if z is False: # all roots will make z True but we don't know # whether this is the right root if z is True return S.false o = other.is_real, other.is_imaginary s = self.is_real, self.is_imaginary if o != s and None not in o and None not in s: return S.false i = self._get_interval() was = i.a, i.b need = [True]*2 # make sure it would be distinct from others while any(need): i = i.refine() a, b = i.a, i.b if need[0] and a != was[0]: need[0] = False if need[1] and b != was[1]: need[1] = False re, im = other.as_real_imag() if not im: if self.is_real: a, b = [Rational(str(i)) for i in (a, b)] return sympify(a < other and other < b) return S.false if self.is_real: return S.false z = r1, r2, i1, i2 = [Rational(str(j)) for j in ( i.ax, i.bx, i.ay, i.by)] return sympify(( r1 < re and re < r2) and ( i1 < im and im < i2))
CRootOf = ComplexRootOf
[docs]@public class RootSum(Expr): """Represents a sum of all roots of a univariate polynomial. """ __slots__ = ['poly', 'fun', 'auto'] def __new__(cls, expr, func=None, x=None, auto=True, quadratic=False): """Construct a new ``RootSum`` instance of roots of a polynomial.""" coeff, poly = cls._transform(expr, x) if not poly.is_univariate: raise MultivariatePolynomialError( "only univariate polynomials are allowed") if func is None: func = Lambda(poly.gen, poly.gen) else: try: is_func = func.is_Function except AttributeError: is_func = False if is_func and 1 in func.nargs: if not isinstance(func, Lambda): func = Lambda(poly.gen, func(poly.gen)) else: raise ValueError( "expected a univariate function, got %s" % func) var, expr = func.variables[0], func.expr if coeff is not S.One: expr = expr.subs(var, coeff*var) deg = poly.degree() if not expr.has(var): return deg*expr if expr.is_Add: add_const, expr = expr.as_independent(var) else: add_const = S.Zero if expr.is_Mul: mul_const, expr = expr.as_independent(var) else: mul_const = S.One func = Lambda(var, expr) rational = cls._is_func_rational(poly, func) (_, factors), terms = poly.factor_list(), [] for poly, k in factors: if poly.is_linear: term = func(roots_linear(poly)[0]) elif quadratic and poly.is_quadratic: term = sum(map(func, roots_quadratic(poly))) else: if not rational or not auto: term = cls._new(poly, func, auto) else: term = cls._rational_case(poly, func) terms.append(k*term) return mul_const*Add(*terms) + deg*add_const @classmethod def _new(cls, poly, func, auto=True): """Construct new raw ``RootSum`` instance. """ obj = Expr.__new__(cls) obj.poly = poly obj.fun = func obj.auto = auto return obj @classmethod def new(cls, poly, func, auto=True): """Construct new ``RootSum`` instance. """ if not func.expr.has(*func.variables): return func.expr rational = cls._is_func_rational(poly, func) if not rational or not auto: return cls._new(poly, func, auto) else: return cls._rational_case(poly, func) @classmethod def _transform(cls, expr, x): """Transform an expression to a polynomial. """ poly = PurePoly(expr, x, greedy=False) return preprocess_roots(poly) @classmethod def _is_func_rational(cls, poly, func): """Check if a lambda is areational function. """ var, expr = func.variables[0], func.expr return expr.is_rational_function(var) @classmethod def _rational_case(cls, poly, func): """Handle the rational function case. """ roots = symbols('r:%d' % poly.degree()) var, expr = func.variables[0], func.expr f = sum(expr.subs(var, r) for r in roots) p, q = together(f).as_numer_denom() domain = QQ[roots] p = p.expand() q = q.expand() try: p = Poly(p, domain=domain, expand=False) except GeneratorsNeeded: p, p_coeff = None, (p,) else: p_monom, p_coeff = zip(*p.terms()) try: q = Poly(q, domain=domain, expand=False) except GeneratorsNeeded: q, q_coeff = None, (q,) else: q_monom, q_coeff = zip(*q.terms()) coeffs, mapping = symmetrize(p_coeff + q_coeff, formal=True) formulas, values = viete(poly, roots), [] for (sym, _), (_, val) in zip(mapping, formulas): values.append((sym, val)) for i, (coeff, _) in enumerate(coeffs): coeffs[i] = coeff.subs(values) n = len(p_coeff) p_coeff = coeffs[:n] q_coeff = coeffs[n:] if p is not None: p = Poly(dict(zip(p_monom, p_coeff)), *p.gens).as_expr() else: (p,) = p_coeff if q is not None: q = Poly(dict(zip(q_monom, q_coeff)), *q.gens).as_expr() else: (q,) = q_coeff return factor(p/q) def _hashable_content(self): return (self.poly, self.fun) @property def expr(self): return self.poly.as_expr() @property def args(self): return (self.expr, self.fun, self.poly.gen) @property def free_symbols(self): return self.poly.free_symbols | self.fun.free_symbols @property def is_commutative(self): return True def doit(self, **hints): if not hints.get('roots', True): return self _roots = roots(self.poly, multiple=True) if len(_roots) < self.poly.degree(): return self else: return Add(*[self.fun(r) for r in _roots]) def _eval_evalf(self, prec): try: _roots = self.poly.nroots(n=prec_to_dps(prec)) except (DomainError, PolynomialError): return self else: return Add(*[self.fun(r) for r in _roots]) def _eval_derivative(self, x): var, expr = self.fun.args func = Lambda(var, expr.diff(x)) return self.new(self.poly, func, self.auto)
def bisect(f, a, b, tol): """ Implements bisection. This function is used in RootOf.eval_rational() and it needs to be robust. Examples ======== >>> from sympy import S >>> from sympy.polys.rootoftools import bisect >>> bisect(lambda x: x**2-1, -10, 0, S(1)/10**2) -1025/1024 >>> bisect(lambda x: x**2-1, -10, 0, S(1)/10**4) -131075/131072 """ a = sympify(a) b = sympify(b) fa = f(a) fb = f(b) if fa * fb >= 0: raise ValueError("bisect: f(a) and f(b) must have opposite signs") while (b - a > tol): c = (a + b)/2 fc = f(c) if (fc == 0): return c # We need to make sure f(c) is not zero below if (fa * fc < 0): b = c fb = fc else: a = c fa = fc return (a + b)/2