Пример #1
0
def reduce_inequalities(inequalities, symbols=[]):
    """Reduce a system of inequalities with rational coefficients.

    Examples
    ========

    >>> from sympy import sympify as S, Symbol
    >>> from sympy.abc import x, y
    >>> from sympy.solvers.inequalities import reduce_inequalities

    >>> reduce_inequalities(S(0) <= x + 3, [])
    And(-3 <= x, x < oo)

    >>> reduce_inequalities(S(0) <= x + y*2 - 1, [x])
    -2*y + 1 <= x
    """
    if not iterable(inequalities):
        inequalities = [inequalities]

    # prefilter
    keep = []
    for i in inequalities:
        if isinstance(i, Relational):
            i = i.func(i.lhs.as_expr() - i.rhs.as_expr(), 0)
        elif i not in (True, False):
            i = Eq(i, 0)
        if i == True:
            continue
        elif i == False:
            return S.false
        if i.lhs.is_number:
            raise NotImplementedError(
                "could not determine truth value of %s" % i)
        keep.append(i)
    inequalities = keep
    del keep

    gens = reduce(set.union, [i.free_symbols for i in inequalities], set())

    if not iterable(symbols):
        symbols = [symbols]
    symbols = set(symbols) or gens

    # make vanilla symbol real
    recast = dict([(i, Dummy(i.name, real=True))
        for i in gens if i.is_real is None])
    inequalities = [i.xreplace(recast) for i in inequalities]
    symbols = set([i.xreplace(recast) for i in symbols])

    # solve system
    rv = _reduce_inequalities(inequalities, symbols)

    # restore original symbols and return
    return rv.xreplace(dict([(v, k) for k, v in recast.items()]))
Пример #2
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def eval_sum_symbolic(f, limits):
    from sympy.functions import harmonic, bernoulli

    f_orig = f
    (i, a, b) = limits
    if not f.has(i):
        return f*(b - a + 1)

    # Linearity
    if f.is_Mul:
        L, R = f.as_two_terms()

        if not L.has(i):
            sR = eval_sum_symbolic(R, (i, a, b))
            if sR:
                return L*sR

        if not R.has(i):
            sL = eval_sum_symbolic(L, (i, a, b))
            if sL:
                return R*sL

        try:
            f = apart(f, i)  # see if it becomes an Add
        except PolynomialError:
            pass

    if f.is_Add:
        L, R = f.as_two_terms()
        lrsum = telescopic(L, R, (i, a, b))

        if lrsum:
            return lrsum

        lsum = eval_sum_symbolic(L, (i, a, b))
        rsum = eval_sum_symbolic(R, (i, a, b))

        if None not in (lsum, rsum):
            r = lsum + rsum
            if not r is S.NaN:
                return r

    # Polynomial terms with Faulhaber's formula
    n = Wild('n')
    result = f.match(i**n)

    if result is not None:
        n = result[n]

        if n.is_Integer:
            if n >= 0:
                if (b is S.Infinity and not a is S.NegativeInfinity) or \
                   (a is S.NegativeInfinity and not b is S.Infinity):
                    return S.Infinity
                return ((bernoulli(n + 1, b + 1) - bernoulli(n + 1, a))/(n + 1)).expand()
            elif a.is_Integer and a >= 1:
                if n == -1:
                    return harmonic(b) - harmonic(a - 1)
                else:
                    return harmonic(b, abs(n)) - harmonic(a - 1, abs(n))

    if not (a.has(S.Infinity, S.NegativeInfinity) or
            b.has(S.Infinity, S.NegativeInfinity)):
        # Geometric terms
        c1 = Wild('c1', exclude=[i])
        c2 = Wild('c2', exclude=[i])
        c3 = Wild('c3', exclude=[i])

        e = f.match(c1**(c2*i + c3))

        if e is not None:
            p = (c1**c3).subs(e)
            q = (c1**c2).subs(e)

            r = p*(q**a - q**(b + 1))/(1 - q)
            l = p*(b - a + 1)

            return Piecewise((l, Eq(q, S.One)), (r, True))

        r = gosper_sum(f, (i, a, b))

        if not r in (None, S.NaN):
            return r

    return eval_sum_hyper(f_orig, (i, a, b))
Пример #3
0
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 do not 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 no general solution that is always applicable for all
    coefficients is known to this reviewer, certain conditions are tested
    to determine the simplest 4 expressions that can be 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
        >>> 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. https://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
    7. eqworld.ipmnet.ru/en/solutions/ae/ae0108.pdf
    """
    _, 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 = _mexpand(c + a * (a2 / 8 - b / 2))
        aon4 = a / 4
        g = _mexpand(d - aon4 * (a * (3 * a2 / 64 - b / 4) + c))

        if f.is_zero:
            y1, y2 = [sqrt(tmp) for tmp in roots([1, e, g], multiple=True)]
            return [tmp - aon4 for tmp in [-y1, -y2, y1, y2]]
        if g.is_zero:
            y = [S.Zero] + roots([1, 0, e, f], multiple=True)
            return [tmp - aon4 for tmp in y]
        else:
            # Descartes-Euler method, see [7]
            sols = _roots_quartic_euler(e, f, g, aon4)
            if sols:
                return sols
            # Ferrari method, see [1, 2]
            p = -e**2 / 12 - g
            q = -e**3 / 108 + e * g / 3 - f**2 / 8
            TH = Rational(1, 3)

            def _ans(y):
                w = sqrt(e + 2 * y)
                arg1 = 3 * e + 2 * y
                arg2 = 2 * f / w
                ans = []
                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

            # whether a Piecewise is returned or not
            # depends on knowing p, so try to put
            # in a simple form
            p = _mexpand(p)

            # p == 0 case
            y1 = e * Rational(-5, 6) - q**TH
            if p.is_zero:
                return _ans(y1)

            # if 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)
            y2 = e * Rational(-5, 6) + u - p / u / 3
            if fuzzy_not(p.is_zero):
                return _ans(y2)

            # sort it out once they know the values of the coefficients
            return [
                Piecewise((a1, Eq(p, 0)), (a2, True))
                for a1, a2 in zip(_ans(y1), _ans(y2))
            ]
Пример #4
0
def test_Equality_rewrite_as_Add():
    eq = Eq(x + y, y - x)
    assert eq.rewrite(Add) == 2 * x
    assert eq.rewrite(Add, evaluate=None).args == (x, x, y, -y)
    assert eq.rewrite(Add, evaluate=False).args == (x, y, x, -y)
Пример #5
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def test_issue_10633():
    assert Eq(True, False) == False
    assert Eq(False, True) == False
    assert Eq(True, True) == True
    assert Eq(False, False) == True
Пример #6
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def test_issue_10304():
    d = cos(1)**2 + sin(1)**2 - 1
    assert d.is_comparable is False  # if this fails, find a new d
    e = 1 + d * I
    assert simplify(Eq(e, 0)) is S.false
Пример #7
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def test_new_relational():
    x = Symbol('x')

    assert Eq(x) == Relational(x, 0)  # None ==> Equality
    assert Eq(x) == Relational(x, 0, '==')
    assert Eq(x) == Relational(x, 0, 'eq')
    assert Eq(x) == Equality(x, 0)
    assert Eq(x, -1) == Relational(x, -1)  # None ==> Equality
    assert Eq(x, -1) == Relational(x, -1, '==')
    assert Eq(x, -1) == Relational(x, -1, 'eq')
    assert Eq(x, -1) == Equality(x, -1)
    assert Eq(x) != Relational(x, 1)  # None ==> Equality
    assert Eq(x) != Relational(x, 1, '==')
    assert Eq(x) != Relational(x, 1, 'eq')
    assert Eq(x) != Equality(x, 1)
    assert Eq(x, -1) != Relational(x, 1)  # None ==> Equality
    assert Eq(x, -1) != Relational(x, 1, '==')
    assert Eq(x, -1) != Relational(x, 1, 'eq')
    assert Eq(x, -1) != Equality(x, 1)

    assert Ne(x, 0) == Relational(x, 0, '!=')
    assert Ne(x, 0) == Relational(x, 0, '<>')
    assert Ne(x, 0) == Relational(x, 0, 'ne')
    assert Ne(x, 0) == Unequality(x, 0)
    assert Ne(x, 0) != Relational(x, 1, '!=')
    assert Ne(x, 0) != Relational(x, 1, '<>')
    assert Ne(x, 0) != Relational(x, 1, 'ne')
    assert Ne(x, 0) != Unequality(x, 1)

    assert Ge(x, 0) == Relational(x, 0, '>=')
    assert Ge(x, 0) == Relational(x, 0, 'ge')
    assert Ge(x, 0) == GreaterThan(x, 0)
    assert Ge(x, 1) != Relational(x, 0, '>=')
    assert Ge(x, 1) != Relational(x, 0, 'ge')
    assert Ge(x, 1) != GreaterThan(x, 0)
    assert (x >= 1) == Relational(x, 1, '>=')
    assert (x >= 1) == Relational(x, 1, 'ge')
    assert (x >= 1) == GreaterThan(x, 1)
    assert (x >= 0) != Relational(x, 1, '>=')
    assert (x >= 0) != Relational(x, 1, 'ge')
    assert (x >= 0) != GreaterThan(x, 1)

    assert Le(x, 0) == Relational(x, 0, '<=')
    assert Le(x, 0) == Relational(x, 0, 'le')
    assert Le(x, 0) == LessThan(x, 0)
    assert Le(x, 1) != Relational(x, 0, '<=')
    assert Le(x, 1) != Relational(x, 0, 'le')
    assert Le(x, 1) != LessThan(x, 0)
    assert (x <= 1) == Relational(x, 1, '<=')
    assert (x <= 1) == Relational(x, 1, 'le')
    assert (x <= 1) == LessThan(x, 1)
    assert (x <= 0) != Relational(x, 1, '<=')
    assert (x <= 0) != Relational(x, 1, 'le')
    assert (x <= 0) != LessThan(x, 1)

    assert Gt(x, 0) == Relational(x, 0, '>')
    assert Gt(x, 0) == Relational(x, 0, 'gt')
    assert Gt(x, 0) == StrictGreaterThan(x, 0)
    assert Gt(x, 1) != Relational(x, 0, '>')
    assert Gt(x, 1) != Relational(x, 0, 'gt')
    assert Gt(x, 1) != StrictGreaterThan(x, 0)
    assert (x > 1) == Relational(x, 1, '>')
    assert (x > 1) == Relational(x, 1, 'gt')
    assert (x > 1) == StrictGreaterThan(x, 1)
    assert (x > 0) != Relational(x, 1, '>')
    assert (x > 0) != Relational(x, 1, 'gt')
    assert (x > 0) != StrictGreaterThan(x, 1)

    assert Lt(x, 0) == Relational(x, 0, '<')
    assert Lt(x, 0) == Relational(x, 0, 'lt')
    assert Lt(x, 0) == StrictLessThan(x, 0)
    assert Lt(x, 1) != Relational(x, 0, '<')
    assert Lt(x, 1) != Relational(x, 0, 'lt')
    assert Lt(x, 1) != StrictLessThan(x, 0)
    assert (x < 1) == Relational(x, 1, '<')
    assert (x < 1) == Relational(x, 1, 'lt')
    assert (x < 1) == StrictLessThan(x, 1)
    assert (x < 0) != Relational(x, 1, '<')
    assert (x < 0) != Relational(x, 1, 'lt')
    assert (x < 0) != StrictLessThan(x, 1)

    # finally, some fuzz testing
    from random import randint
    from sympy.core.compatibility import unichr
    for i in range(100):
        while 1:
            strtype, length = (unichr, 65535) if randint(0, 1) else (chr, 255)
            relation_type = strtype(randint(0, length))
            if randint(0, 1):
                relation_type += strtype(randint(0, length))
            if relation_type not in ('==', 'eq', '!=', '<>', 'ne', '>=', 'ge',
                                     '<=', 'le', '>', 'gt', '<', 'lt', ':=',
                                     '+=', '-=', '*=', '/=', '%='):
                break

        raises(ValueError, lambda: Relational(x, 1, relation_type))
    assert all(Relational(x, 0, op).rel_op == '==' for op in ('eq', '=='))
    assert all(
        Relational(x, 0, op).rel_op == '!=' for op in ('ne', '<>', '!='))
    assert all(Relational(x, 0, op).rel_op == '>' for op in ('gt', '>'))
    assert all(Relational(x, 0, op).rel_op == '<' for op in ('lt', '<'))
    assert all(Relational(x, 0, op).rel_op == '>=' for op in ('ge', '>='))
    assert all(Relational(x, 0, op).rel_op == '<=' for op in ('le', '<='))
Пример #8
0
def _preprocess_eqs(eqs):
    processed_eqs = []
    for eq in eqs:
        processed_eqs.append(eq if isinstance(eq, Equality) else Eq(eq, 0))

    return processed_eqs
Пример #9
0
def test_BooleanFunction_diff():
    assert And(x, y).diff(x) == Piecewise((0, Eq(y, False)), (1, True))
Пример #10
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def test_simplification():
    """
    Test working of simplification methods.
    """
    set1 = [[0, 0, 1], [0, 1, 1], [1, 0, 0], [1, 1, 0]]
    set2 = [[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]]
    assert SOPform([x, y, z], set1) == Or(And(Not(x), z), And(Not(z), x))
    assert Not(SOPform([x, y, z], set2)) == \
        Not(Or(And(Not(x), Not(z)), And(x, z)))
    assert POSform([x, y, z], set1 + set2) is true
    assert SOPform([x, y, z], set1 + set2) is true
    assert SOPform([Dummy(), Dummy(), Dummy()], set1 + set2) is true

    minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
                [1, 1, 1, 1]]
    dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]
    assert (SOPform([w, x, y, z], minterms,
                    dontcares) == Or(And(y, z), And(Not(w), Not(x))))
    assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)

    minterms = [1, 3, 7, 11, 15]
    dontcares = [0, 2, 5]
    assert (SOPform([w, x, y, z], minterms,
                    dontcares) == Or(And(y, z), And(Not(w), Not(x))))
    assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)

    minterms = [1, [0, 0, 1, 1], 7, [1, 0, 1, 1], [1, 1, 1, 1]]
    dontcares = [0, [0, 0, 1, 0], 5]
    assert (SOPform([w, x, y, z], minterms,
                    dontcares) == Or(And(y, z), And(Not(w), Not(x))))
    assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)

    minterms = [1, {y: 1, z: 1}]
    dontcares = [0, [0, 0, 1, 0], 5]
    assert (SOPform([w, x, y, z], minterms,
                    dontcares) == Or(And(y, z), And(Not(w), Not(x))))
    assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)

    minterms = [{y: 1, z: 1}, 1]
    dontcares = [[0, 0, 0, 0]]

    minterms = [[0, 0, 0]]
    raises(ValueError, lambda: SOPform([w, x, y, z], minterms))
    raises(ValueError, lambda: POSform([w, x, y, z], minterms))

    raises(TypeError, lambda: POSform([w, x, y, z], ["abcdefg"]))

    # test simplification
    ans = And(A, Or(B, C))
    assert simplify_logic(A & (B | C)) == ans
    assert simplify_logic((A & B) | (A & C)) == ans
    assert simplify_logic(Implies(A, B)) == Or(Not(A), B)
    assert simplify_logic(Equivalent(A, B)) == \
        Or(And(A, B), And(Not(A), Not(B)))
    assert simplify_logic(And(Equality(A, 2), C)) == And(Equality(A, 2), C)
    assert simplify_logic(And(Equality(A, 2), A)) is S.false
    assert simplify_logic(And(Equality(A, 2), A)) == And(Equality(A, 2), A)
    assert simplify_logic(And(Equality(A, B), C)) == And(Equality(A, B), C)
    assert simplify_logic(Or(And(Equality(A, 3), B), And(Equality(A, 3), C))) \
        == And(Equality(A, 3), Or(B, C))
    b = (~x & ~y & ~z) | (~x & ~y & z)
    e = And(A, b)
    assert simplify_logic(e) == A & ~x & ~y
    raises(ValueError, lambda: simplify_logic(A & (B | C), form='blabla'))

    # Check that expressions with nine variables or more are not simplified
    # (without the force-flag)
    a, b, c, d, e, f, g, h, j = symbols('a b c d e f g h j')
    expr = a & b & c & d & e & f & g & h & j | \
        a & b & c & d & e & f & g & h & ~j
    # This expression can be simplified to get rid of the j variables
    assert simplify_logic(expr) == expr

    # check input
    ans = SOPform([x, y], [[1, 0]])
    assert SOPform([x, y], [[1, 0]]) == ans
    assert POSform([x, y], [[1, 0]]) == ans

    raises(ValueError, lambda: SOPform([x], [[1]], [[1]]))
    assert SOPform([x], [[1]], [[0]]) is true
    assert SOPform([x], [[0]], [[1]]) is true
    assert SOPform([x], [], []) is false

    raises(ValueError, lambda: POSform([x], [[1]], [[1]]))
    assert POSform([x], [[1]], [[0]]) is true
    assert POSform([x], [[0]], [[1]]) is true
    assert POSform([x], [], []) is false

    # check working of simplify
    assert simplify((A & B) | (A & C)) == And(A, Or(B, C))
    assert simplify(And(x, Not(x))) == False
    assert simplify(Or(x, Not(x))) == True
    assert simplify(And(Eq(x, 0), Eq(x, y))) == And(Eq(x, 0), Eq(y, 0))
    assert And(Eq(x - 1, 0), Eq(x, y)).simplify() == And(Eq(x, 1), Eq(y, 1))
    assert And(Ne(x - 1, 0), Ne(x, y)).simplify() == And(Ne(x, 1), Ne(x, y))
    assert And(Eq(x - 1, 0), Ne(x, y)).simplify() == And(Eq(x, 1), Ne(y, 1))
    assert And(Eq(x - 1, 0), Eq(x, z + y),
               Eq(y + x, 0)).simplify() == And(Eq(x, 1), Eq(y, -1), Eq(z, 2))
    assert And(Eq(x - 1, 0), Eq(x + 2, 3)).simplify() == Eq(x, 1)
    assert And(Ne(x - 1, 0), Ne(x + 2, 3)).simplify() == Ne(x, 1)
    assert And(Eq(x - 1, 0), Eq(x + 2, 2)).simplify() == False
    assert And(Ne(x - 1, 0), Ne(x + 2,
                                2)).simplify() == And(Ne(x, 1), Ne(x, 0))
Пример #11
0
def test_relational_simplification():
    w, x, y, z = symbols('w x y z', real=True)
    d, e = symbols('d e', real=False)
    # Test all combinations or sign and order
    assert Or(x >= y, x < y).simplify() == S.true
    assert Or(x >= y, y > x).simplify() == S.true
    assert Or(x >= y, -x > -y).simplify() == S.true
    assert Or(x >= y, -y < -x).simplify() == S.true
    assert Or(-x <= -y, x < y).simplify() == S.true
    assert Or(-x <= -y, -x > -y).simplify() == S.true
    assert Or(-x <= -y, y > x).simplify() == S.true
    assert Or(-x <= -y, -y < -x).simplify() == S.true
    assert Or(y <= x, x < y).simplify() == S.true
    assert Or(y <= x, y > x).simplify() == S.true
    assert Or(y <= x, -x > -y).simplify() == S.true
    assert Or(y <= x, -y < -x).simplify() == S.true
    assert Or(-y >= -x, x < y).simplify() == S.true
    assert Or(-y >= -x, y > x).simplify() == S.true
    assert Or(-y >= -x, -x > -y).simplify() == S.true
    assert Or(-y >= -x, -y < -x).simplify() == S.true

    assert Or(x < y, x >= y).simplify() == S.true
    assert Or(y > x, x >= y).simplify() == S.true
    assert Or(-x > -y, x >= y).simplify() == S.true
    assert Or(-y < -x, x >= y).simplify() == S.true
    assert Or(x < y, -x <= -y).simplify() == S.true
    assert Or(-x > -y, -x <= -y).simplify() == S.true
    assert Or(y > x, -x <= -y).simplify() == S.true
    assert Or(-y < -x, -x <= -y).simplify() == S.true
    assert Or(x < y, y <= x).simplify() == S.true
    assert Or(y > x, y <= x).simplify() == S.true
    assert Or(-x > -y, y <= x).simplify() == S.true
    assert Or(-y < -x, y <= x).simplify() == S.true
    assert Or(x < y, -y >= -x).simplify() == S.true
    assert Or(y > x, -y >= -x).simplify() == S.true
    assert Or(-x > -y, -y >= -x).simplify() == S.true
    assert Or(-y < -x, -y >= -x).simplify() == S.true

    # Some other tests
    assert Or(x >= y, w < z, x <= y).simplify() == S.true
    assert And(x >= y, x < y).simplify() == S.false
    assert Or(x >= y, Eq(y, x)).simplify() == (x >= y)
    assert And(x >= y, Eq(y, x)).simplify() == Eq(x, y)
    assert Or(Eq(x, y), x >= y, w < y, z < y).simplify() == \
        Or(x >= y, y > Min(w, z))
    assert And(Eq(x, y), x >= y, w < y, y >= z, z < y).simplify() == \
        And(Eq(x, y), y > Max(w, z))
    assert Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
        (Eq(x, y) | (x >= 1) | (y > Min(2, z)))
    assert And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
        (Eq(x, y) & (x >= 1) & (y >= 5) & (y > z))
    assert (Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)).simplify() == \
        (Eq(x, y) & Eq(d, e) & (d >= e))
    assert And(Eq(x, y), Eq(x, -y)).simplify() == And(Eq(x, 0), Eq(y, 0))
    assert Xor(x >= y, x <= y).simplify() == Ne(x, y)
Пример #12
0
def test_Piecewise():
    assert refine(Piecewise((1, x < 0), (3, True)), Q.is_true(x < 0)) == 1
    assert refine(Piecewise((1, x < 0), (3, True)), ~Q.is_true(x < 0)) == 3
    assert refine(Piecewise((1, x < 0), (3, True)), Q.is_true(y < 0)) == \
        Piecewise((1, x < 0), (3, True))
    assert refine(Piecewise((1, x > 0), (3, True)), Q.is_true(x > 0)) == 1
    assert refine(Piecewise((1, x > 0), (3, True)), ~Q.is_true(x > 0)) == 3
    assert refine(Piecewise((1, x > 0), (3, True)), Q.is_true(y > 0)) == \
        Piecewise((1, x > 0), (3, True))
    assert refine(Piecewise((1, x <= 0), (3, True)), Q.is_true(x <= 0)) == 1
    assert refine(Piecewise((1, x <= 0), (3, True)), ~Q.is_true(x <= 0)) == 3
    assert refine(Piecewise((1, x <= 0), (3, True)), Q.is_true(y <= 0)) == \
        Piecewise((1, x <= 0), (3, True))
    assert refine(Piecewise((1, x >= 0), (3, True)), Q.is_true(x >= 0)) == 1
    assert refine(Piecewise((1, x >= 0), (3, True)), ~Q.is_true(x >= 0)) == 3
    assert refine(Piecewise((1, x >= 0), (3, True)), Q.is_true(y >= 0)) == \
        Piecewise((1, x >= 0), (3, True))
    assert refine(Piecewise((1, Eq(x, 0)), (3, True)), Q.is_true(Eq(x, 0)))\
        == 1
    assert refine(Piecewise((1, Eq(x, 0)), (3, True)), Q.is_true(Eq(0, x)))\
        == 1
    assert refine(Piecewise((1, Eq(x, 0)), (3, True)), ~Q.is_true(Eq(x, 0)))\
        == 3
    assert refine(Piecewise((1, Eq(x, 0)), (3, True)), ~Q.is_true(Eq(0, x)))\
        == 3
    assert refine(Piecewise((1, Eq(x, 0)), (3, True)), Q.is_true(Eq(y, 0)))\
        == Piecewise((1, Eq(x, 0)), (3, True))
    assert refine(Piecewise((1, Ne(x, 0)), (3, True)), Q.is_true(Ne(x, 0)))\
        == 1
    assert refine(Piecewise((1, Ne(x, 0)), (3, True)), ~Q.is_true(Ne(x, 0)))\
        == 3
    assert refine(Piecewise((1, Ne(x, 0)), (3, True)), Q.is_true(Ne(y, 0)))\
        == Piecewise((1, Ne(x, 0)), (3, True))
Пример #13
0
def heurisch_wrapper(f,
                     x,
                     rewrite=False,
                     hints=None,
                     mappings=None,
                     retries=3,
                     degree_offset=0,
                     unnecessary_permutations=None):
    """
    A wrapper around the heurisch integration algorithm.

    This method takes the result from heurisch and checks for poles in the
    denominator. For each of these poles, the integral is reevaluated, and
    the final integration result is given in terms of a Piecewise.

    Examples
    ========

    >>> from sympy.core import symbols
    >>> from sympy.functions import cos
    >>> from sympy.integrals.heurisch import heurisch, heurisch_wrapper
    >>> n, x = symbols('n x')
    >>> heurisch(cos(n*x), x)
    sin(n*x)/n
    >>> heurisch_wrapper(cos(n*x), x)
    Piecewise((x, n == 0), (sin(n*x)/n, True))

    See Also
    ========

    heurisch
    """
    f = sympify(f)
    if x not in f.free_symbols:
        return f * x

    res = heurisch(f, x, rewrite, hints, mappings, retries, degree_offset,
                   unnecessary_permutations)
    if not isinstance(res, Basic):
        return res
    # We consider each denominator in the expression, and try to find
    # cases where one or more symbolic denominator might be zero. The
    # conditions for these cases are stored in the list slns.
    slns = []
    for d in denoms(res):
        try:
            slns += solve(d, dict=True, exclude=(x, ))
        except NotImplementedError:
            pass
    if not slns:
        return res
    slns = list(uniq(slns))
    # Remove the solutions corresponding to poles in the original expression.
    slns0 = []
    for d in denoms(f):
        try:
            slns0 += solve(d, dict=True, exclude=(x, ))
        except NotImplementedError:
            pass
    slns = [s for s in slns if s not in slns0]
    if not slns:
        return res
    if len(slns) > 1:
        eqs = []
        for sub_dict in slns:
            eqs.extend([Eq(key, value) for key, value in sub_dict.iteritems()])
        slns = solve(eqs, dict=True, exclude=(x, )) + slns
    # For each case listed in the list slns, we reevaluate the integral.
    pairs = []
    for sub_dict in slns:
        expr = heurisch(f.subs(sub_dict), x, rewrite, hints, mappings, retries,
                        degree_offset, unnecessary_permutations)
        cond = And(*[Eq(key, value) for key, value in sub_dict.iteritems()])
        pairs.append((expr, cond))
    pairs.append((heurisch(f, x, rewrite, hints, mappings, retries,
                           degree_offset, unnecessary_permutations), True))
    return Piecewise(*pairs)
Пример #14
0
def eval_sum_symbolic(f, limits):
    from sympy.functions import harmonic, bernoulli

    f_orig = f
    (i, a, b) = limits
    if not f.has(i):
        return f * (b - a + 1)

    # Linearity
    if f.is_Mul:
        # Try factor out everything not including i
        without_i, with_i = f.as_independent(i)
        if without_i != 1:
            s = eval_sum_symbolic(with_i, (i, a, b))
            if s:
                r = without_i * s
                if r is not S.NaN:
                    return r
        else:
            # Try term by term
            L, R = f.as_two_terms()

            if not L.has(i):
                sR = eval_sum_symbolic(R, (i, a, b))
                if sR:
                    return L * sR

            if not R.has(i):
                sL = eval_sum_symbolic(L, (i, a, b))
                if sL:
                    return sL * R
        try:
            f = apart(f, i)  # see if it becomes an Add
        except PolynomialError:
            pass

    if f.is_Add:
        L, R = f.as_two_terms()
        lrsum = telescopic(L, R, (i, a, b))

        if lrsum:
            return lrsum

        # Try factor out everything not including i
        without_i, with_i = f.as_independent(i)
        if without_i != 0:
            s = eval_sum_symbolic(with_i, (i, a, b))
            if s:
                r = without_i * (b - a + 1) + s
                if r is not S.NaN:
                    return r
        else:
            # Try term by term
            lsum = eval_sum_symbolic(L, (i, a, b))
            rsum = eval_sum_symbolic(R, (i, a, b))

            if None not in (lsum, rsum):
                r = lsum + rsum
                if r is not S.NaN:
                    return r

    # Polynomial terms with Faulhaber's formula
    n = Wild('n')
    result = f.match(i**n)

    if result is not None:
        n = result[n]

        if n.is_Integer:
            if n >= 0:
                if (b is S.Infinity and not a is S.NegativeInfinity) or \
                   (a is S.NegativeInfinity and not b is S.Infinity):
                    return S.Infinity
                return ((bernoulli(n + 1, b + 1) - bernoulli(n + 1, a)) /
                        (n + 1)).expand()
            elif a.is_Integer and a >= 1:
                if n == -1:
                    return harmonic(b) - harmonic(a - 1)
                else:
                    return harmonic(b, abs(n)) - harmonic(a - 1, abs(n))

    if not (a.has(S.Infinity, S.NegativeInfinity)
            or b.has(S.Infinity, S.NegativeInfinity)):
        # Geometric terms
        c1 = Wild('c1', exclude=[i])
        c2 = Wild('c2', exclude=[i])
        c3 = Wild('c3', exclude=[i])
        wexp = Wild('wexp')

        # Here we first attempt powsimp on f for easier matching with the
        # exponential pattern, and attempt expansion on the exponent for easier
        # matching with the linear pattern.
        e = f.powsimp().match(c1**wexp)
        if e is not None:
            e_exp = e.pop(wexp).expand().match(c2 * i + c3)
            if e_exp is not None:
                e.update(e_exp)

                p = (c1**c3).subs(e)
                q = (c1**c2).subs(e)
                r = p * (q**a - q**(b + 1)) / (1 - q)
                l = p * (b - a + 1)
                return Piecewise((l, Eq(q, S.One)), (r, True))

        r = gosper_sum(f, (i, a, b))

        if isinstance(r, (Mul, Add)):
            from sympy import ordered, Tuple
            non_limit = r.free_symbols - Tuple(*limits[1:]).free_symbols
            den = denom(together(r))
            den_sym = non_limit & den.free_symbols
            args = []
            for v in ordered(den_sym):
                try:
                    s = solve(den, v)
                    m = Eq(v, s[0]) if s else S.false
                    if m != False:
                        args.append((Sum(f_orig.subs(*m.args),
                                         limits).doit(), m))
                    break
                except NotImplementedError:
                    continue

            args.append((r, True))
            return Piecewise(*args)

        if not r in (None, S.NaN):
            return r

    h = eval_sum_hyper(f_orig, (i, a, b))
    if h is not None:
        return h

    factored = f_orig.factor()
    if factored != f_orig:
        return eval_sum_symbolic(factored, (i, a, b))
Пример #15
0
 def recurrence(self):
     """Equation defining recurrence."""
     return Eq(self.y(self.n), self._recurrence)
Пример #16
0
def _eval_is_eq(lhs, rhs):  # noqa: F811
    return And(Eq(lhs.left, rhs.left), Eq(lhs.right, rhs.right),
               lhs.left_open == rhs.left_open,
               lhs.right_open == rhs.right_open)
Пример #17
0
 def check(pdf, set):
     x = Dummy('x')
     val = Sum(pdf(x), (x, set._inf, set._sup)).doit()
     _value_check(Eq(val, 1) != S.false, "The pdf is incorrect on the given set.")
Пример #18
0
def canonical_odes(eqs, funcs, t):
    r"""
    Function that solves for highest order derivatives in a system

    Explanation
    ===========

    This function inputs a system of ODEs and based on the system,
    the dependent variables and their highest order, returns the system
    in the following form:

    .. math::
        X'(t) = A(t) X(t) + b(t)

    Here, $X(t)$ is the vector of dependent variables of lower order, $A(t)$ is
    the coefficient matrix, $b(t)$ is the non-homogeneous term and $X'(t)$ is the
    vector of dependent variables in their respective highest order. We use the term
    canonical form to imply the system of ODEs which is of the above form.

    If the system passed has a non-linear term with multiple solutions, then a list of
    systems is returned in its canonical form.

    Parameters
    ==========

    eqs : List
        List of the ODEs
    funcs : List
        List of dependent variables
    t : Symbol
        Independent variable

    Examples
    ========

    >>> from sympy import symbols, Function, Eq, Derivative
    >>> from sympy.solvers.ode.systems import canonical_odes
    >>> f, g = symbols("f g", cls=Function)
    >>> x, y = symbols("x y")
    >>> funcs = [f(x), g(x)]
    >>> eqs = [Eq(f(x).diff(x) - 7*f(x), 12*g(x)), Eq(g(x).diff(x) + g(x), 20*f(x))]

    >>> canonical_eqs = canonical_odes(eqs, funcs, x)
    >>> canonical_eqs
    [Eq(Derivative(f(x), x), 7*f(x) + 12*g(x)), Eq(Derivative(g(x), x), 20*f(x) - g(x))]

    >>> system = [Eq(Derivative(f(x), x)**2 - 2*Derivative(f(x), x) + 1, 4), Eq(-y*f(x) + Derivative(g(x), x), 0)]

    >>> canonical_system = canonical_odes(system, funcs, x)
    >>> canonical_system
    [[Eq(Derivative(f(x), x), -1), Eq(Derivative(g(x), x), y*f(x))], [Eq(Derivative(f(x), x), 3), Eq(Derivative(g(x), x), y*f(x))]]

    Returns
    =======

    List

    """
    from sympy.solvers.solvers import solve

    order = _get_func_order(eqs, funcs)

    canon_eqs = solve(eqs,
                      *[func.diff(t, order[func]) for func in funcs],
                      dict=True)

    systems = []
    for eq in canon_eqs:
        system = [
            Eq(func.diff(t, order[func]), eq[func.diff(t, order[func])])
            for func in funcs
        ]
        systems.append(system)

    if len(canon_eqs) == 1:
        systems = systems[0]

    return systems
Пример #19
0
def test_reduce_inequalities_boolean():
    assert reduce_inequalities(
        [Eq(x**2, 0), True]) == Eq(x, 0)
    assert reduce_inequalities([Eq(x**2, 0), False]) == False
    assert reduce_inequalities(x**2 >= 0) is S.true  # issue 10196
Пример #20
0
def test_equals():
    w, x, y, z = symbols('w:z')
    f = Function('f')
    assert Eq(x, 1).equals(Eq(x * (y + 1) - x * y - x + 1, x))
    assert Eq(x, y).equals(x < y, True) == False
    assert Eq(x, f(1)).equals(Eq(x, f(2)), True) == f(1) - f(2)
    assert Eq(f(1), y).equals(Eq(f(2), y), True) == f(1) - f(2)
    assert Eq(x, f(1)).equals(Eq(f(2), x), True) == f(1) - f(2)
    assert Eq(f(1), x).equals(Eq(x, f(2)), True) == f(1) - f(2)
    assert Eq(w, x).equals(Eq(y, z), True) == False
    assert Eq(f(1), f(2)).equals(Eq(f(3), f(4)), True) == f(1) - f(3)
    assert (x < y).equals(y > x, True) == True
    assert (x < y).equals(y >= x, True) == False
    assert (x < y).equals(z < y, True) == False
    assert (x < y).equals(x < z, True) == False
    assert (x < f(1)).equals(x < f(2), True) == f(1) - f(2)
    assert (f(1) < x).equals(f(2) < x, True) == f(1) - f(2)
Пример #21
0
def test__solve_inequalities():
    assert reduce_inequalities(x + y < 1, symbols=[x]) == (x < 1 - y)
    assert reduce_inequalities(x + y >= 1, symbols=[x]) == (x < oo) & (x >= -y + 1)
    assert reduce_inequalities(Eq(0, x - y), symbols=[x]) == Eq(x, y)
    assert reduce_inequalities(Ne(0, x - y), symbols=[x]) == Ne(x, y)
Пример #22
0
def test_issue_10401():
    x = symbols('x')
    fin = symbols('inf', finite=True)
    inf = symbols('inf', infinite=True)
    inf2 = symbols('inf2', infinite=True)
    zero = symbols('z', zero=True)
    nonzero = symbols('nz', zero=False, finite=True)

    assert Eq(1 / (1 / x + 1), 1).func is Eq
    assert Eq(1 / (1 / x + 1), 1).subs(x, S.ComplexInfinity) is S.true
    assert Eq(1 / (1 / fin + 1), 1) is S.false

    T, F = S.true, S.false
    assert Eq(fin, inf) is F
    assert Eq(inf, inf2) is T and inf != inf2
    assert Eq(inf / inf2, 0) is F
    assert Eq(inf / fin, 0) is F
    assert Eq(fin / inf, 0) is T
    assert Eq(zero / nonzero, 0) is T and ((zero / nonzero) != 0)
    assert Eq(inf, -inf) is F

    assert Eq(fin / (fin + 1), 1) is S.false

    o = symbols('o', odd=True)
    assert Eq(o, 2 * o) is S.false

    p = symbols('p', positive=True)
    assert Eq(p / (p - 1), 1) is F
Пример #23
0
def test_reduce_poly_inequalities_real_interval():
    assert reduce_rational_inequalities(
        [[Eq(x**2, 0)]], x, relational=False) == FiniteSet(0)
    assert reduce_rational_inequalities(
        [[Le(x**2, 0)]], x, relational=False) == FiniteSet(0)
    assert reduce_rational_inequalities(
        [[Lt(x**2, 0)]], x, relational=False) == S.EmptySet
    assert reduce_rational_inequalities(
        [[Ge(x**2, 0)]], x, relational=False) == \
        S.Reals if x.is_real else Interval(-oo, oo)
    assert reduce_rational_inequalities(
        [[Gt(x**2, 0)]], x, relational=False) == \
        FiniteSet(0).complement(S.Reals)
    assert reduce_rational_inequalities(
        [[Ne(x**2, 0)]], x, relational=False) == \
        FiniteSet(0).complement(S.Reals)

    assert reduce_rational_inequalities(
        [[Eq(x**2, 1)]], x, relational=False) == FiniteSet(-1, 1)
    assert reduce_rational_inequalities(
        [[Le(x**2, 1)]], x, relational=False) == Interval(-1, 1)
    assert reduce_rational_inequalities(
        [[Lt(x**2, 1)]], x, relational=False) == Interval(-1, 1, True, True)
    assert reduce_rational_inequalities(
        [[Ge(x**2, 1)]], x, relational=False) == \
        Union(Interval(-oo, -1), Interval(1, oo))
    assert reduce_rational_inequalities(
        [[Gt(x**2, 1)]], x, relational=False) == \
        Interval(-1, 1).complement(S.Reals)
    assert reduce_rational_inequalities(
        [[Ne(x**2, 1)]], x, relational=False) == \
        FiniteSet(-1, 1).complement(S.Reals)
    assert reduce_rational_inequalities([[Eq(
        x**2, 1.0)]], x, relational=False) == FiniteSet(-1.0, 1.0).evalf()
    assert reduce_rational_inequalities(
        [[Le(x**2, 1.0)]], x, relational=False) == Interval(-1.0, 1.0)
    assert reduce_rational_inequalities([[Lt(
        x**2, 1.0)]], x, relational=False) == Interval(-1.0, 1.0, True, True)
    assert reduce_rational_inequalities(
        [[Ge(x**2, 1.0)]], x, relational=False) == \
        Union(Interval(-inf, -1.0), Interval(1.0, inf))
    assert reduce_rational_inequalities(
        [[Gt(x**2, 1.0)]], x, relational=False) == \
        Union(Interval(-inf, -1.0, right_open=True),
        Interval(1.0, inf, left_open=True))
    assert reduce_rational_inequalities([[Ne(
        x**2, 1.0)]], x, relational=False) == \
        FiniteSet(-1.0, 1.0).complement(S.Reals)

    s = sqrt(2)

    assert reduce_rational_inequalities([[Lt(
        x**2 - 1, 0), Gt(x**2 - 1, 0)]], x, relational=False) == S.EmptySet
    assert reduce_rational_inequalities([[Le(x**2 - 1, 0), Ge(
        x**2 - 1, 0)]], x, relational=False) == FiniteSet(-1, 1)
    assert reduce_rational_inequalities(
        [[Le(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x, relational=False
        ) == Union(Interval(-s, -1, False, False), Interval(1, s, False, False))
    assert reduce_rational_inequalities(
        [[Le(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x, relational=False
        ) == Union(Interval(-s, -1, False, True), Interval(1, s, True, False))
    assert reduce_rational_inequalities(
        [[Lt(x**2 - 2, 0), Ge(x**2 - 1, 0)]], x, relational=False
        ) == Union(Interval(-s, -1, True, False), Interval(1, s, False, True))
    assert reduce_rational_inequalities(
        [[Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]], x, relational=False
        ) == Union(Interval(-s, -1, True, True), Interval(1, s, True, True))
    assert reduce_rational_inequalities(
        [[Lt(x**2 - 2, 0), Ne(x**2 - 1, 0)]], x, relational=False
        ) == Union(Interval(-s, -1, True, True), Interval(-1, 1, True, True),
        Interval(1, s, True, True))

    assert reduce_rational_inequalities([[Lt(x**2, -1.)]], x) is S.false
Пример #24
0
def test_issue_10927():
    x = symbols('x')
    assert str(Eq(x, oo)) == 'Eq(x, oo)'
    assert str(Eq(x, -oo)) == 'Eq(x, -oo)'
Пример #25
0
def test_nonpolymonial_relations():
    assert Eq(cos(x), 0).simplify() == Eq(cos(x), 0)
Пример #26
0
def _linear_neq_order1_type3(match_):
    r"""
    System of n first-order nonconstant-coefficient linear homogeneous differential equations

    .. math::
        X' = A(t) X

    where $X$ is the vector of $n$ dependent variables, $t$ is the dependent variable, $X'$
    is the first order differential of $X$ with respect to $t$ and $A(t)$ is a $n \times n$
    coefficient matrix.

    Let us define $B$ as antiderivative of coefficient matrix $A$:

    .. math::
        B(t) = \int A(t) dt

    If the system of ODEs defined above is such that its antiderivative $B(t)$ commutes with
    $A(t)$ itself, then, the solution of the above system is given as:

    .. math::
        X = \exp(B(t)) C

    where $C$ is the vector of constants.

    """
    # Some parts of code is repeated, this needs to be taken care of
    # The constant vector obtained here can be done so in the match
    # function itself.
    eq = match_['eq']
    func = match_['func']
    fc = match_['func_coeff']
    n = len(eq)
    t = list(list(eq[0].atoms(Derivative))[0].atoms(Symbol))[0]
    constants = numbered_symbols(prefix='C', cls=Symbol, start=1)

    # This needs to be modified in future so that fc is only of type Matrix
    M = -fc if type(fc) is Matrix else Matrix(n, n,
                                              lambda i, j: -fc[i, func[j], 0])

    Cvect = Matrix(list(next(constants) for _ in range(n)))

    # The code in if block will be removed when it is made sure
    # that the code works without the statements in if block.
    if "commutative_antiderivative" not in match_:
        B, is_commuting = _is_commutative_anti_derivative(M, t)

        # This course is subject to change
        if not is_commuting:
            return None

    else:
        B = match_['commutative_antiderivative']

    sol_vector = B.exp() * Cvect

    # The expand_mul is added to handle the solutions so that
    # the exponential terms are collected properly.
    sol_vector = [
        collect(expand_mul(s), ordered(s.atoms(exp)), exact=True)
        for s in sol_vector
    ]

    sol_dict = [Eq(func[i], sol_vector[i]) for i in range(n)]
    return sol_dict
Пример #27
0
def test_simplify_relational():
    assert simplify(x * (y + 1) - x * y - x + 1 < x) == (x > 1)
    assert simplify(x * (y + 1) - x * y - x - 1 < x) == (x > -1)
    assert simplify(x < x * (y + 1) - x * y - x + 1) == (x < 1)
    r = S.One < x
    # canonical operations are not the same as simplification,
    # so if there is no simplification, canonicalization will
    # be done unless the measure forbids it
    assert simplify(r) == r.canonical
    assert simplify(r, ratio=0) != r.canonical
    # this is not a random test; in _eval_simplify
    # this will simplify to S.false and that is the
    # reason for the 'if r.is_Relational' in Relational's
    # _eval_simplify routine
    assert simplify(-(2**(pi * Rational(3, 2)) + 6**pi)**(1 / pi) + 2 *
                    (2**(pi / 2) + 3**pi)**(1 / pi) < 0) is S.false
    # canonical at least
    assert Eq(y, x).simplify() == Eq(x, y)
    assert Eq(x - 1, 0).simplify() == Eq(x, 1)
    assert Eq(x - 1, x).simplify() == S.false
    assert Eq(2 * x - 1, x).simplify() == Eq(x, 1)
    assert Eq(2 * x, 4).simplify() == Eq(x, 2)
    z = cos(1)**2 + sin(1)**2 - 1  # z.is_zero is None
    assert Eq(z * x, 0).simplify() == S.true

    assert Ne(y, x).simplify() == Ne(x, y)
    assert Ne(x - 1, 0).simplify() == Ne(x, 1)
    assert Ne(x - 1, x).simplify() == S.true
    assert Ne(2 * x - 1, x).simplify() == Ne(x, 1)
    assert Ne(2 * x, 4).simplify() == Ne(x, 2)
    assert Ne(z * x, 0).simplify() == S.false

    # No real-valued assumptions
    assert Ge(y, x).simplify() == Le(x, y)
    assert Ge(x - 1, 0).simplify() == Ge(x, 1)
    assert Ge(x - 1, x).simplify() == S.false
    assert Ge(2 * x - 1, x).simplify() == Ge(x, 1)
    assert Ge(2 * x, 4).simplify() == Ge(x, 2)
    assert Ge(z * x, 0).simplify() == S.true
    assert Ge(x, -2).simplify() == Ge(x, -2)
    assert Ge(-x, -2).simplify() == Le(x, 2)
    assert Ge(x, 2).simplify() == Ge(x, 2)
    assert Ge(-x, 2).simplify() == Le(x, -2)

    assert Le(y, x).simplify() == Ge(x, y)
    assert Le(x - 1, 0).simplify() == Le(x, 1)
    assert Le(x - 1, x).simplify() == S.true
    assert Le(2 * x - 1, x).simplify() == Le(x, 1)
    assert Le(2 * x, 4).simplify() == Le(x, 2)
    assert Le(z * x, 0).simplify() == S.true
    assert Le(x, -2).simplify() == Le(x, -2)
    assert Le(-x, -2).simplify() == Ge(x, 2)
    assert Le(x, 2).simplify() == Le(x, 2)
    assert Le(-x, 2).simplify() == Ge(x, -2)

    assert Gt(y, x).simplify() == Lt(x, y)
    assert Gt(x - 1, 0).simplify() == Gt(x, 1)
    assert Gt(x - 1, x).simplify() == S.false
    assert Gt(2 * x - 1, x).simplify() == Gt(x, 1)
    assert Gt(2 * x, 4).simplify() == Gt(x, 2)
    assert Gt(z * x, 0).simplify() == S.false
    assert Gt(x, -2).simplify() == Gt(x, -2)
    assert Gt(-x, -2).simplify() == Lt(x, 2)
    assert Gt(x, 2).simplify() == Gt(x, 2)
    assert Gt(-x, 2).simplify() == Lt(x, -2)

    assert Lt(y, x).simplify() == Gt(x, y)
    assert Lt(x - 1, 0).simplify() == Lt(x, 1)
    assert Lt(x - 1, x).simplify() == S.true
    assert Lt(2 * x - 1, x).simplify() == Lt(x, 1)
    assert Lt(2 * x, 4).simplify() == Lt(x, 2)
    assert Lt(z * x, 0).simplify() == S.false
    assert Lt(x, -2).simplify() == Lt(x, -2)
    assert Lt(-x, -2).simplify() == Gt(x, 2)
    assert Lt(x, 2).simplify() == Lt(x, 2)
    assert Lt(-x, 2).simplify() == Gt(x, -2)
Пример #28
0
 def as_relational(self, symbol):
     """Rewrite a FiniteSet in terms of equalities and logic operators.
     """
     from sympy.core.relational import Eq
     from sympy.logic.boolalg import Or
     return Or(*[Eq(symbol, elem) for elem in self])
Пример #29
0
def test_issue_10401():
    x = symbols('x')
    fin = symbols('inf', finite=True)
    inf = symbols('inf', infinite=True)
    inf2 = symbols('inf2', infinite=True)
    infx = symbols('infx', infinite=True, extended_real=True)
    infx2 = symbols('infx2', infinite=True, extended_real=True)
    infnx = symbols('inf~x', infinite=True, extended_real=False)
    infnx2 = symbols('inf~x2', infinite=True, extended_real=False)
    infp = symbols('infp', infinite=True, extended_positive=True)
    infp1 = symbols('infp1', infinite=True, extended_positive=True)
    infn = symbols('infn', infinite=True, extended_negative=True)
    zero = symbols('z', zero=True)
    nonzero = symbols('nz', zero=False, finite=True)

    assert Eq(1 / (1 / x + 1), 1).func is Eq
    assert Eq(1 / (1 / x + 1), 1).subs(x, S.ComplexInfinity) is S.true
    assert Eq(1 / (1 / fin + 1), 1) is S.false

    T, F = S.true, S.false
    assert Eq(fin, inf) is F
    assert Eq(inf, inf2) not in (T, F) and inf != inf2
    assert Eq(1 + inf, 2 + inf2) not in (T, F) and inf != inf2
    assert Eq(infp, infp1) is T
    assert Eq(infp, infn) is F
    assert Eq(1 + I * oo, I * oo) is F
    assert Eq(I * oo, 1 + I * oo) is F
    assert Eq(1 + I * oo, 2 + I * oo) is F
    assert Eq(1 + I * oo, 2 + I * infx) is F
    assert Eq(1 + I * oo, 2 + infx) is F
    # FIXME: The test below fails because (-infx).is_extended_positive is True
    # (should be None)
    #assert Eq(1 + I*infx, 1 + I*infx2) not in (T, F) and infx != infx2
    #
    assert Eq(zoo, sqrt(2) + I * oo) is F
    assert Eq(zoo, oo) is F
    r = Symbol('r', real=True)
    i = Symbol('i', imaginary=True)
    assert Eq(i * I, r) not in (T, F)
    assert Eq(infx, infnx) is F
    assert Eq(infnx, infnx2) not in (T, F) and infnx != infnx2
    assert Eq(zoo, oo) is F
    assert Eq(inf / inf2, 0) is F
    assert Eq(inf / fin, 0) is F
    assert Eq(fin / inf, 0) is T
    assert Eq(zero / nonzero, 0) is T and ((zero / nonzero) != 0)
    # The commented out test below is incorrect because:
    assert zoo == -zoo
    assert Eq(zoo, -zoo) is T
    assert Eq(oo, -oo) is F
    assert Eq(inf, -inf) not in (T, F)

    assert Eq(fin / (fin + 1), 1) is S.false

    o = symbols('o', odd=True)
    assert Eq(o, 2 * o) is S.false

    p = symbols('p', positive=True)
    assert Eq(p / (p - 1), 1) is F
Пример #30
0
def test_Equality_rewrite_as_Add():
    eq = Eq(x + y, y - x)
    assert eq.rewrite(Add) == 2*x
    assert eq.rewrite(Add, evaluate=None).args == (x, x, y, -y)
    assert eq.rewrite(Add, evaluate=False).args == (x, y, x, -y)
Пример #31
0
class EllipticCurve:
    """
    Create the following Elliptic Curve over domain.

    `y^{2} + a_{1} x y + a_{3} y = x^{3} + a_{2} x^{2} + a_{4} x + a_{6}`

    The default domain is ``QQ``. If no coefficient ``a1``, ``a2``, ``a3``,
    it create curve as following form.

    `y^{2} = x^{3} + a_{4} x + a_{6}`

    Examples
    ========

    References
    ==========

    [1] J. Silverman "A Friendly Introduction to Number Theory" Third Edition
    [2] http://mathworld.wolfram.com/EllipticDiscriminant.html
    [3] G. Hardy, E. Wright "An Introduction to the Theory of Numbers" Sixth Edition

    """
    def __init__(self, a4, a6, a1=0, a2=0, a3=0, modulus=0):
        if modulus == 0:
            domain = QQ
        else:
            domain = FF(modulus)
        a1, a2, a3, a4, a6 = map(domain.convert, (a1, a2, a3, a4, a6))
        self._domain = domain
        self.modulus = modulus
        # Calculate discriminant
        b2 = a1**2 + 4 * a2
        b4 = 2 * a4 + a1 * a3
        b6 = a3**2 + 4 * a6
        b8 = a1**2 * a6 + 4 * a2 * a6 - a1 * a3 * a4 + a2 * a3**2 - a4**2
        self._b2, self._b4, self._b6, self._b8 = b2, b4, b6, b8
        self._discrim = -b2**2 * b8 - 8 * b4**3 - 27 * b6**2 + 9 * b2 * b4 * b6
        self._a1 = a1
        self._a2 = a2
        self._a3 = a3
        self._a4 = a4
        self._a6 = a6
        x, y, z = symbols('x y z')
        self.x, self.y, self.z = x, y, z
        self._eq = Eq(y**2 * z + a1 * x * y * z + a3 * y * z**2,
                      x**3 + a2 * x**2 * z + a4 * x * z**2 + a6 * z**3)
        if isinstance(self._domain, FiniteField):
            self._rank = 0
        elif isinstance(self._domain, RationalField):
            self._rank = None

    def __call__(self, x, y, z=1):
        return EllipticCurvePoint(x, y, z, self)

    def __contains__(self, point):
        if is_sequence(point):
            if len(point) == 2:
                z1 = 1
            else:
                z1 = point[2]
            x1, y1 = point[:2]
        elif isinstance(point, EllipticCurvePoint):
            x1, y1, z1 = point.x, point.y, point.z
        else:
            raise ValueError('Invalid point.')
        if self.characteristic == 0 and z1 == 0:
            return True
        return self._eq.subs({self.x: x1, self.y: y1, self.z: z1})

    def __repr__(self):
        return 'E({}): {}'.format(self._domain, self._eq)

    def minimal(self):
        """
        Return minimal Weierstrass equation.

        Examples
        ========

        >>> from sympy.ntheory.elliptic_curve import EllipticCurve

        >>> e1 = EllipticCurve(-10, -20, 0, -1, 1)
        >>> e1.minimal()
        E(QQ): Eq(y**2*z, x**3 - 13392*x*z**2 - 1080432*z**3)

        """
        char = self.characteristic
        if char == 2:
            return self
        if char == 3:
            return EllipticCurve(self._b4 / 2,
                                 self._b6 / 4,
                                 a2=self._b2 / 4,
                                 modulus=self.modulus)
        c4 = self._b2**2 - 24 * self._b4
        c6 = -self._b2**3 + 36 * self._b2 * self._b4 - 216 * self._b6
        return EllipticCurve(-27 * c4, -54 * c6, modulus=self.modulus)

    def points(self):
        """
        Return points of curve over Finite Field.

        Examples
        ========

        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
        >>> e2 = EllipticCurve(1, 1, 1, 1, 1, modulus=5)
        >>> e2.points()
        {(0, 2), (1, 4), (2, 0), (2, 2), (3, 0), (3, 1), (4, 0)}

        """

        char = self.characteristic
        all_pt = set()
        if char >= 1:
            for i in range(char):
                congruence_eq = ((self._eq.lhs - self._eq.rhs).subs({
                    self.x: i,
                    self.z: 1
                }))
                sol = polynomial_congruence(congruence_eq, char)
                for num in sol:
                    all_pt.add((i, num))
            return all_pt
        else:
            raise ValueError("Infinitely many points")

    def points_x(self, x):
        "Returns points on with curve where xcoordinate = x"
        pt = []
        if self._domain == QQ:
            for y in solve(self._eq.subs(self.x, x)):
                pt.append((x, y))
        congruence_eq = ((self._eq.lhs - self._eq.rhs).subs({
            self.x: x,
            self.z: 1
        }))
        for y in polynomial_congruence(congruence_eq, self.characteristic):
            pt.append((x, y))
        return pt

    def torsion_points(self):
        """
        Return torsion points of curve over Rational number.

        Return point objects those are finite order.
        According to Nagell-Lutz theorem, torsion point p(x, y)
        x and y are integers, either y = 0 or y**2 is divisor
        of discriminent. According to Mazur's theorem, there are
        at most 15 points in torsion collection.

        Examples
        ========

        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
        >>> e2 = EllipticCurve(-43, 166)
        >>> sorted(e2.torsion_points())
        [(-5, -16), (-5, 16), O, (3, -8), (3, 8), (11, -32), (11, 32)]

        """
        if self.characteristic > 0:
            raise ValueError("No torsion point for Finite Field.")
        l = [EllipticCurvePoint.point_at_infinity(self)]
        for xx in solve(self._eq.subs({self.y: 0, self.z: 1})):
            if xx.is_rational:
                l.append(self(xx, 0))
        for i in divisors(self.discriminant, generator=True):
            j = int(i**.5)
            if j**2 == i:
                for xx in solve(self._eq.subs({self.y: j, self.z: 1})):
                    if not xx.is_rational:
                        continue
                    p = self(xx, j)
                    if p.order() != oo:
                        l.extend([p, -p])
        return l

    @property
    def characteristic(self):
        """
        Return domain characteristic.

        Examples
        ========

        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
        >>> e2 = EllipticCurve(-43, 166)
        >>> e2.characteristic
        0

        """
        return self._domain.characteristic()

    @property
    def discriminant(self):
        """
        Return curve discriminant.

        Examples
        ========

        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
        >>> e2 = EllipticCurve(0, 17)
        >>> e2.discriminant
        -124848

        """
        return int(self._discrim)

    @property
    def is_singular(self):
        """
        Return True if curve discriminant is equal to zero.
        """
        return self.discriminant == 0

    @property
    def j_invariant(self):
        """
        Return curve j-invariant.

        Examples
        ========

        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
        >>> e1 = EllipticCurve(-2, 0, 0, 1, 1)
        >>> e1.j_invariant
        1404928/389

        """
        c4 = self._b2**2 - 24 * self._b4
        return self._domain.to_sympy(c4**3 / self._discrim)

    @property
    def order(self):
        """
        Number of points in Finite field.

        Examples
        ========

        >>> from sympy.polys.domains import FF
        >>> from sympy.ntheory.elliptic_curve import EllipticCurve
        >>> e2 = EllipticCurve(1, 0, modulus=19)
        >>> e2.order
        19

        """
        if self.characteristic == 0:
            raise NotImplementedError("Still not implemented")
        return len(list(self.points()))

    @property
    def rank(self):
        """
        Number of independent points of infinite order.

        For Finite field, it must be 0.
        """
        if self._rank is not None:
            return self._rank
        raise NotImplementedError("Still not implemented")
Пример #32
0
def checkpdesol(pde, sol, func=None, solve_for_func=True):
    """
    Checks if the given solution satisfies the partial differential
    equation.

    pde is the partial differential equation which can be given in the
    form of an equation or an expression. sol is the solution for which
    the pde is to be checked. This can also be given in an equation or
    an expression form. If the function is not provided, the helper
    function _preprocess from deutils is used to identify the function.

    If a sequence of solutions is passed, the same sort of container will be
    used to return the result for each solution.

    The following methods are currently being implemented to check if the
    solution satisfies the PDE:

        1. Directly substitute the solution in the PDE and check. If the
           solution hasn't been solved for f, then it will solve for f
           provided solve_for_func hasn't been set to False.

    If the solution satisfies the PDE, then a tuple (True, 0) is returned.
    Otherwise a tuple (False, expr) where expr is the value obtained
    after substituting the solution in the PDE. However if a known solution
    returns False, it may be due to the inability of doit() to simplify it to zero.

    Examples
    ========
    >>> from sympy import Function, symbols, diff
    >>> from sympy.solvers.pde import checkpdesol, pdsolve
    >>> x, y = symbols('x y')
    >>> f = Function('f')
    >>> eq = 2*f(x,y) + 3*f(x,y).diff(x) + 4*f(x,y).diff(y)
    >>> sol = pdsolve(eq)
    >>> assert checkpdesol(eq, sol)[0]
    >>> eq = x*f(x,y) + f(x,y).diff(x)
    >>> checkpdesol(eq, sol)
    (False, (x*F(4*x - 3*y) - 6*F(4*x - 3*y)/25 + 4*Subs(Derivative(F(_xi_1), _xi_1), (_xi_1,), (4*x - 3*y,)))*exp(-6*x/25 - 8*y/25))
    """

    # Converting the pde into an equation
    if not isinstance(pde, Equality):
        pde = Eq(pde, 0)

    # If no function is given, try finding the function present.
    if func is None:
        try:
            _, func = _preprocess(pde.lhs)
        except ValueError:
            funcs = [s.atoms(AppliedUndef) for s in (
                sol if is_sequence(sol, set) else [sol])]
            funcs = reduce(set.union, funcs, set())
            if len(funcs) != 1:
                raise ValueError(
                    'must pass func arg to checkpdesol for this case.')
            func = funcs.pop()

    # If the given solution is in the form of a list or a set
    # then return a list or set of tuples.
    if is_sequence(sol, set):
        return type(sol)(map(lambda i: checkpdesol(pde, i,
            solve_for_func=solve_for_func), sol))

    # Convert solution into an equation
    if not isinstance(sol, Equality):
        sol = Eq(func, sol)

    # Try solving for the function
    if solve_for_func and not (sol.lhs == func and not sol.rhs.has(func)) and not \
            (sol.rhs == func and not sol.lhs.has(func)):
        try:
            solved = solve(sol, func)
            if not solved:
                raise NotImplementedError
        except NotImplementedError:
            pass
        else:
            if len(solved) == 1:
                result = checkpdesol(pde, Eq(func, solved[0]),
                    order=order, solve_for_func=False)
            else:
                result = checkpdesol(pde, [Eq(func, t) for t in solved],
                order=order, solve_for_func=False)

    # The first method includes direct substitution of the solution in
    # the PDE and simplifying.
    pde = pde.lhs - pde.rhs
    if sol.lhs == func:
        s = pde.subs(func, sol.rhs).doit()
    elif sol.rhs == func:
        s = pde.subs(func, sol.lhs).doit()
    if s:
        ss = simplify(s)
        if ss:
            return False, ss
        else:
            return True, 0
    else:
        return True, 0
Пример #33
0
def ufuncify(args, expr, language=None, backend='numpy', tempdir=None,
             flags=None, verbose=False, helpers=None, **kwargs):
    """Generates a binary function that supports broadcasting on numpy arrays.

    Parameters
    ----------
    args : iterable
        Either a Symbol or an iterable of symbols. Specifies the argument
        sequence for the function.
    expr
        A SymPy expression that defines the element wise operation.
    language : string, optional
        If supplied, (options: 'C' or 'F95'), specifies the language of the
        generated code. If ``None`` [default], the language is inferred based
        upon the specified backend.
    backend : string, optional
        Backend used to wrap the generated code. Either 'numpy' [default],
        'cython', or 'f2py'.
    tempdir : string, optional
        Path to directory for temporary files. If this argument is supplied,
        the generated code and the wrapper input files are left intact in
        the specified path.
    flags : iterable, optional
        Additional option flags that will be passed to the backend.
    verbose : bool, optional
        If True, autowrap will not mute the command line backends. This can
        be helpful for debugging.
    helpers : iterable, optional
        Used to define auxiliary expressions needed for the main expr. If
        the main expression needs to call a specialized function it should
        be put in the ``helpers`` iterable. Autowrap will then make sure
        that the compiled main expression can link to the helper routine.
        Items should be tuples with (<funtion_name>, <sympy_expression>,
        <arguments>). It is mandatory to supply an argument sequence to
        helper routines.
    kwargs : dict
        These kwargs will be passed to autowrap if the `f2py` or `cython`
        backend is used and ignored if the `numpy` backend is used.

    Note
    ----
    The default backend ('numpy') will create actual instances of
    ``numpy.ufunc``. These support ndimensional broadcasting, and implicit type
    conversion. Use of the other backends will result in a "ufunc-like"
    function, which requires equal length 1-dimensional arrays for all
    arguments, and will not perform any type conversions.

    References
    ----------
    [1] http://docs.scipy.org/doc/numpy/reference/ufuncs.html

    Examples
    ========

    >>> from sympy.utilities.autowrap import ufuncify
    >>> from sympy.abc import x, y
    >>> import numpy as np
    >>> f = ufuncify((x, y), y + x**2)
    >>> type(f)
    <class 'numpy.ufunc'>
    >>> f([1, 2, 3], 2)
    array([  3.,   6.,  11.])
    >>> f(np.arange(5), 3)
    array([  3.,   4.,   7.,  12.,  19.])

    For the 'f2py' and 'cython' backends, inputs are required to be equal length
    1-dimensional arrays. The 'f2py' backend will perform type conversion, but
    the Cython backend will error if the inputs are not of the expected type.

    >>> f_fortran = ufuncify((x, y), y + x**2, backend='f2py')
    >>> f_fortran(1, 2)
    array([ 3.])
    >>> f_fortran(np.array([1, 2, 3]), np.array([1.0, 2.0, 3.0]))
    array([  2.,   6.,  12.])
    >>> f_cython = ufuncify((x, y), y + x**2, backend='Cython')
    >>> f_cython(1, 2)  # doctest: +ELLIPSIS
    Traceback (most recent call last):
      ...
    TypeError: Argument '_x' has incorrect type (expected numpy.ndarray, got int)
    >>> f_cython(np.array([1.0]), np.array([2.0]))
    array([ 3.])
    """

    if isinstance(args, Symbol):
        args = (args,)
    else:
        args = tuple(args)

    if language:
        _validate_backend_language(backend, language)
    else:
        language = _infer_language(backend)

    helpers = helpers if helpers else ()
    flags = flags if flags else ()

    if backend.upper() == 'NUMPY':
        # maxargs is set by numpy compile-time constant NPY_MAXARGS
        # If a future version of numpy modifies or removes this restriction
        # this variable should be changed or removed
        maxargs = 32
        helps = []
        for name, expr, args in helpers:
            helps.append(make_routine(name, expr, args))
        code_wrapper = UfuncifyCodeWrapper(C99CodeGen("ufuncify"), tempdir,
                                           flags, verbose)
        if not isinstance(expr, (list, tuple)):
            expr = [expr]
        if len(expr) == 0:
            raise ValueError('Expression iterable has zero length')
        if (len(expr) + len(args)) > maxargs:
            msg = ('Cannot create ufunc with more than {0} total arguments: '
                   'got {1} in, {2} out')
            raise ValueError(msg.format(maxargs, len(args), len(expr)))
        routines = [make_routine('autofunc{}'.format(idx), exprx, args) for
                    idx, exprx in enumerate(expr)]
        return code_wrapper.wrap_code(routines, helpers=helps)
    else:
        # Dummies are used for all added expressions to prevent name clashes
        # within the original expression.
        y = IndexedBase(Dummy('y'))
        m = Dummy('m', integer=True)
        i = Idx(Dummy('i', integer=True), m)
        f_dummy = Dummy('f')
        f = implemented_function('%s_%d' % (f_dummy.name, f_dummy.dummy_index), Lambda(args, expr))
        # For each of the args create an indexed version.
        indexed_args = [IndexedBase(Dummy(str(a))) for a in args]
        # Order the arguments (out, args, dim)
        args = [y] + indexed_args + [m]
        args_with_indices = [a[i] for a in indexed_args]
        return autowrap(Eq(y[i], f(*args_with_indices)), language, backend,
                        tempdir, args, flags, verbose, helpers, **kwargs)