def test_image_interval(): from sympy.core.numbers import Rational x = Symbol('x', real=True) a = Symbol('a', real=True) assert imageset(x, 2*x, Interval(-2, 1)) == Interval(-4, 2) assert imageset(x, 2*x, Interval(-2, 1, True, False)) == \ Interval(-4, 2, True, False) assert imageset(x, x**2, Interval(-2, 1, True, False)) == \ Interval(0, 4, False, True) assert imageset(x, x**2, Interval(-2, 1)) == Interval(0, 4) assert imageset(x, x**2, Interval(-2, 1, True, False)) == \ Interval(0, 4, False, True) assert imageset(x, x**2, Interval(-2, 1, True, True)) == \ Interval(0, 4, False, True) assert imageset(x, (x - 2)**2, Interval(1, 3)) == Interval(0, 1) assert imageset(x, 3*x**4 - 26*x**3 + 78*x**2 - 90*x, Interval(0, 4)) == \ Interval(-35, 0) # Multiple Maxima assert imageset(x, x + 1/x, Interval(-oo, oo)) == Interval(-oo, -2) \ + Interval(2, oo) # Single Infinite discontinuity assert imageset(x, 1/x + 1/(x-1)**2, Interval(0, 2, True, False)) == \ Interval(Rational(3, 2), oo, False) # Multiple Infinite discontinuities # Test for Python lambda assert imageset(lambda x: 2*x, Interval(-2, 1)) == Interval(-4, 2) assert imageset(Lambda(x, a*x), Interval(0, 1)) == \ ImageSet(Lambda(x, a*x), Interval(0, 1)) assert imageset(Lambda(x, sin(cos(x))), Interval(0, 1)) == \ ImageSet(Lambda(x, sin(cos(x))), Interval(0, 1))
def test_imageset(): ints = S.Integers assert imageset(x, x - 1, S.Naturals) is S.Naturals0 assert imageset(x, x + 1, S.Naturals0) is S.Naturals assert imageset(x, abs(x), S.Naturals0) is S.Naturals0 assert imageset(x, abs(x), S.Naturals) is S.Naturals assert imageset(x, abs(x), S.Integers) is S.Naturals0 # issue 16878a r = symbols('r', real=True) assert (1, r) not in imageset(x, (x, x), S.Reals) assert (r, r) in imageset(x, (x, x), S.Reals) assert 1 + I in imageset(x, x + I, S.Reals) assert {1} not in imageset(x, (x,), S.Reals) assert (1, 1) not in imageset(x, (x,) , S.Reals) raises(TypeError, lambda: imageset(x, ints)) raises(ValueError, lambda: imageset(x, y, z, ints)) raises(ValueError, lambda: imageset(Lambda(x, cos(x)), y)) raises(ValueError, lambda: imageset(Lambda(x, x), ints, ints)) assert imageset(cos, ints) == ImageSet(Lambda(x, cos(x)), ints) def f(x): return cos(x) assert imageset(f, ints) == imageset(x, cos(x), ints) f = lambda x: cos(x) assert imageset(f, ints) == ImageSet(Lambda(x, cos(x)), ints) assert imageset(x, 1, ints) == FiniteSet(1) assert imageset(x, y, ints) == {y} assert imageset((x, y), (1, z), ints*S.Reals) == {(1, z)} clash = Symbol('x', integer=true) assert (str(imageset(lambda x: x + clash, Interval(-2, 1)).lamda.expr) in ('_x + x', 'x + _x')) x1, x2 = symbols("x1, x2") assert imageset(lambda x,y: Add(x,y), Interval(1,2), Interval(2, 3)) == \ ImageSet(Lambda((x1, x2), x1+x2), Interval(1,2), Interval(2,3))
def test_imageset(): ints = S.Integers raises(TypeError, lambda: imageset(x, ints)) raises(ValueError, lambda: imageset(x, y, z, ints)) raises(ValueError, lambda: imageset(Lambda(x, cos(x)), y)) assert imageset(cos, ints) == ImageSet(Lambda(x, cos(x)), ints) def f(x): return cos(x) raises(TypeError, lambda: imageset(f, ints)) f = lambda x: cos(x) assert imageset(f, ints) == ImageSet(Lambda(x, cos(x)), ints) assert imageset(x, 1, ints) == FiniteSet(1) assert imageset(x, y, ints) == FiniteSet(y) assert (str(imageset(lambda y: x + y, Interval(-2, 1)).lamda.expr) in ('_x + x', 'x + _x'))
def test_imageset(): ints = S.Integers raises(TypeError, lambda: imageset(x, ints)) raises(ValueError, lambda: imageset(x, y, z, ints)) raises(ValueError, lambda: imageset(Lambda(x, cos(x)), y)) assert imageset(cos, ints) == ImageSet(Lambda(x, cos(x)), ints) def f(x): return cos(x) assert imageset(f, ints) == imageset(x, cos(x), ints) f = lambda x: cos(x) assert imageset(f, ints) == ImageSet(Lambda(x, cos(x)), ints) assert imageset(x, 1, ints) == FiniteSet(1) assert imageset(x, y, ints) == FiniteSet(y) clash = Symbol('x', integer=true) assert (str(imageset(lambda x: x + clash, Interval(-2, 1)).lamda.expr) in ('_x + x', 'x + _x')) x1, x2 = symbols("x1, x2") assert imageset(lambda x,y: Add(x,y), Interval(1,2), Interval(2, 3)) == \ ImageSet(Lambda((x1, x2), x1+x2), Interval(1,2), Interval(2,3))
def test_minima(): from sympy import Integers x, y, n = symbols('x y n') assert minima(sin(x), x) == ImageSet(Lambda(n, 2*pi*n + 3*pi/2), Integers) assert minima(sin(x), x, Interval(0, 2*pi)) == {3*pi/2} assert minima(sin(x)-x, x) == EmptySet() assert minima(x - x**3, x, Interval(-1, 2)) == {-sqrt(3)/3} assert minima(sin(x)**2, x, Interval(-pi/2, pi/2)) == {0} assert minima(sin(x)*cos(x), x, Interval(0, 2*pi)) == {3*pi/4, 7*pi/4} #assert minima((x+3)*(x-2), x) == {-1/2} assert minima((x+3)/(x-2), x) == EmptySet() assert minima(x**5-3*x**3+2*x**2+6, x) == {0, 1} assert minima(exp(x), x) == EmptySet() assert minima(log(x) - x, x) == EmptySet() assert minima(y, x) == EmptySet() raises(ValueError, lambda : maxima(sin(x), x, S.EmptySet)) raises(ValueError, lambda : maxima(log(cos(x)), x, S.EmptySet)) raises(ValueError, lambda : maxima(1/(x**2 + y**2 + 1), x, S.EmptySet)) raises(ValueError, lambda : maxima(sin(x), sin(x))) raises(ValueError, lambda : maxima(sin(x), x*y, S.EmptySet)) raises(ValueError, lambda : maxima(sin(x), S(1)))