def test_roots_quadratic(): assert roots_quadratic(Poly(2*x**2, x)) == [0, 0] assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [-Rational(3, 2), 0] assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2] assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2] f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c) assert roots_quadratic(Poly(f, x)) == \ [-e*(a + c)/(a - c) - sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2), -e*(a + c)/(a - c) + sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)**2)] # check for simplification f = Poly(y*x**2 - 2*x - 2*y, x) assert roots_quadratic(f) == \ [-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y] f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x) assert roots_quadratic(f) == \ [y**2/2 - sqrt(y**4)/2 + 1, y**2/2 + sqrt(y**4)/2 + 1] f = Poly(sqrt(2)*x**2 - 1, x) r = roots_quadratic(f) assert r == _nsort(r) # issue 8255 f = Poly(-24*x**2 - 180*x + 264) assert [w.n(2) for w in f.all_roots(radicals=True)] == \ [w.n(2) for w in f.all_roots(radicals=False)] for _a, _b, _c in cartes((-2, 2), (-2, 2), (0, -1)): f = Poly(_a*x**2 + _b*x + _c) roots = roots_quadratic(f) assert roots == _nsort(roots)
def test_roots_quadratic(): assert roots_quadratic(Poly(2*x**2, x)) == [0, 0] assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [Rational(-3, 2), 0] assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2] assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2] _check(Poly(2*x**2 + 4*x + 3, x).all_roots()) f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c) assert roots_quadratic(Poly(f, x)) == \ [-e*(a + c)/(a - c) - sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2))/(a - c), -e*(a + c)/(a - c) + sqrt((a*b + c*d - a*d - b*c + 4*a*c*e**2))/(a - c)] # check for simplification f = Poly(y*x**2 - 2*x - 2*y, x) assert roots_quadratic(f) == \ [-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y] f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x) assert roots_quadratic(f) == \ [1,y**2 + 1] f = Poly(sqrt(2)*x**2 - 1, x) r = roots_quadratic(f) assert r == _nsort(r) # issue 8255 f = Poly(-24*x**2 - 180*x + 264) assert [w.n(2) for w in f.all_roots(radicals=True)] == \ [w.n(2) for w in f.all_roots(radicals=False)] for _a, _b, _c in cartes((-2, 2), (-2, 2), (0, -1)): f = Poly(_a*x**2 + _b*x + _c) roots = roots_quadratic(f) assert roots == _nsort(roots)
def _solve_as_poly(f, symbol, solveset_solver, invert_func): """ Solve the equation using polynomial techniques if it already is a polynomial equation or, with a change of variables, can be made so. """ result = None if f.is_polynomial(symbol): solns = roots(f, symbol, cubics=True, quartics=True, quintics=True, domain='EX') num_roots = sum(solns.values()) if degree(f, symbol) <= num_roots: result = FiniteSet(*solns.keys()) else: poly = Poly(f, symbol) solns = poly.all_roots() if poly.degree() <= len(solns): result = FiniteSet(*solns) else: result = ConditionSet(symbol, Eq(f, 0), S.Complexes) else: poly = Poly(f) if poly is None: result = ConditionSet(symbol, Eq(f, 0), S.Complexes) gens = [g for g in poly.gens if g.has(symbol)] if len(gens) == 1: poly = Poly(poly, gens[0]) gen = poly.gen deg = poly.degree() poly = Poly(poly.as_expr(), poly.gen, composite=True) poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True, quintics=True).keys()) if len(poly_solns) < deg: result = ConditionSet(symbol, Eq(f, 0), S.Complexes) if gen != symbol: y = Dummy('y') lhs, rhs_s = invert_func(gen, y, symbol) if lhs is symbol: result = Union(*[rhs_s.subs(y, s) for s in poly_solns]) else: result = ConditionSet(symbol, Eq(f, 0), S.Complexes) else: result = ConditionSet(symbol, Eq(f, 0), S.Complexes) if result is not None: if isinstance(result, FiniteSet): # this is to simplify solutions like -sqrt(-I) to sqrt(2)/2 # - sqrt(2)*I/2. We are not expanding for solution with free # variables because that makes the solution more complicated. For # example expand_complex(a) returns re(a) + I*im(a) if all([s.free_symbols == set() and not isinstance(s, RootOf) for s in result]): s = Dummy('s') result = imageset(Lambda(s, expand_complex(s)), result) return result else: return ConditionSet(symbol, Eq(f, 0), S.Complexes)
def _solve_as_poly(f, symbol, solveset_solver, invert_func): """ Solve the equation using polynomial techniques if it already is a polynomial equation or, with a change of variables, can be made so. """ result = None if f.is_polynomial(symbol): solns = roots(f, symbol, cubics=True, quartics=True, quintics=True, domain='EX') num_roots = sum(solns.values()) if degree(f, symbol) <= num_roots: result = FiniteSet(*solns.keys()) else: poly = Poly(f, symbol) solns = poly.all_roots() if poly.degree() <= len(solns): result = FiniteSet(*solns) else: result = ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes) else: poly = Poly(f) if poly is None: result = ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes) gens = [g for g in poly.gens if g.has(symbol)] if len(gens) == 1: poly = Poly(poly, gens[0]) gen = poly.gen deg = poly.degree() poly = Poly(poly.as_expr(), poly.gen, composite=True) poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True, quintics=True).keys()) if len(poly_solns) < deg: result = ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes) if gen != symbol: y = Dummy('y') lhs, rhs_s = invert_func(gen, y, symbol) if lhs is symbol: result = Union(*[rhs_s.subs(y, s) for s in poly_solns]) else: result = ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes) else: result = ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes) if result is not None: if isinstance(result, FiniteSet): # this is to simplify solutions like -sqrt(-I) to sqrt(2)/2 # - sqrt(2)*I/2. We are not expanding for solution with free # variables because that makes the solution more complicated. For # example expand_complex(a) returns re(a) + I*im(a) if all([s.free_symbols == set() and not isinstance(s, RootOf) for s in result]): s = Dummy('s') result = imageset(Lambda(s, expand_complex(s)), result) return result else: return ConditionSet(Lambda(symbol, Eq(f, 0)), S.Complexes)