def test_single_variable_trig_sin():
"""analytic hessian is -25 sin (5 * x + 3)"""
x = ad.Variable()
f = ad.Sin(5 * x + 3)
assert (equals(f.hessian({x: 1}), -25 * np.sin(5 * 1 + 3)))
assert (equals(f.hessian({x: 2}), -25 * np.sin(5 * 2 + 3)))
assert (equals(f.hessian({x: 3}), -25 * np.sin(5 * 3 + 3)))
def test_single_variable_trig_hyperbolic_2():
x = ad.Variable()
# x^2 Cosh[Sin[x] + Tanh[Exp[3 * x] + Log[x]]]
g = ad.Sin(x) + ad.Tanh(ad.Exp(3 * x) + ad.Log(x))
f = x * x * ad.Cosh(g)
assert (equals(f.hessian({x: 1}), 11.464317742))
assert (equals(f.hessian({x: 2}), -13.704377252))
def test_sine():
x = ad.Variable("x")
y = ad.Sin(x)
yd1 = y.d_expr()
assert np.isclose(yd1.eval({x: pi / 2}), 0.0)
assert np.isclose(yd1.eval({x: 0.0}), 1.0)
assert np.isclose(yd1.d({x: 0.0}), 0.0)
def test_logexp_multivar_hessian_2():
x, y, z = ad.Variable(), ad.Variable(), ad.Variable()
f = ad.Sinh(ad.Exp(x - 3.0) * y) + ad.Log(y + x**2) * z * ad.Sin(x)
h = f.hessian({x: 1, y: 3, z: 5})
assert (equals(h[x][x], -1.5705707143))
assert (equals(h[y][y], -0.2553174360))
assert (equals(h[z][z], 0))
assert (equals(h[x][y], 0.31902899408))
assert (equals(h[y][z], 0.2103677462))
assert (equals(h[x][z], 1.1697535323))
def test_trig_multivar_hessian_1():
x, y, z = ad.Variable(), ad.Variable(), ad.Variable()
f = ad.Sin(x * y) + z * ad.Cos(z * ad.Tan(1 / z))
h = f.hessian({x: 1, y: 3, z: 5})
assert (equals(h[x][x], -1.2700800725))
assert (equals(h[y][y], -0.14112000805))
assert (equals(h[z][z], -0.0050593837))
assert (equals(h[x][y], -1.41335252))
assert (equals(h[y][z], 0))
assert (equals(h[x][z], 0))
def test_exp_exceptions():
x = ad.Variable('x')
y = ad.Exp(x)
z = ad.Cos(x)
a = ad.Log(x)
b = ad.Sin(x)
f1 = y*2 + y
f2 = z*2 + z
f3 = a*2 + a
f4 = b*2 + b
assert np.isclose(f1.d_n(1, 1), 8.154845485377136)
assert np.isclose(f2.d_n(1, 1), -2.5244129544236893)
assert np.isclose(f3.d_n(1, 1), 3.0)
assert np.isclose(f4.d_n(1, 1), 1.6209069176044193)
def test_variable_inheritance_three():
x = ad.Variable()
y = ad.Variable()
z = ad.Variable()
f = ad.Cos(x) * y
g = ad.Sin(f) + z * z * ad.Log(z) + 1
assert (x in f.dep_vars)
assert (y in f.dep_vars)
assert (len(f.dep_vars) == 2)
assert (y not in x.dep_vars)
assert (x not in y.dep_vars)
assert (len(g.dep_vars) == 3)
assert (x in g.dep_vars)
assert (y in g.dep_vars)
assert (z in g.dep_vars)
assert (z not in f.dep_vars)
def test_inverse_trig_exceptions():
x = ad.Variable('x')
y = ad.Variable('y')
f5 = ad.Arccos(x)
f6 = ad.Arcsin(x)
f7 = ad.Arctan(x)
f8 = ad.Log(x)
f9 = ad.Sin(x)
assert f5._d_expr(y).eval({x:0}) == 0
assert np.isclose(f5._d_expr(x).eval({x:0.5}), f5.d({x: 0.5}))
assert f6._d_expr(y).eval({x:0}) == 0
assert np.isclose(f6._d_expr(x).eval({x:0}), f6.d({x: 0}))
assert f7._d_expr(y).eval({x:0}) == 0
assert np.isclose(f7._d_expr(x).eval({x:0}), f7.d({x: 0}))
assert f8._d_expr(y).eval({x:1}) == 0
assert np.isclose(f8._d_expr(x).eval({x:2}), f8.d({x: 2}))
assert f9._d_expr(y).eval({x:1}) == 0
assert np.isclose(f9._d_expr(x).eval({x:2}), f9.d({x: 2}))
def test_unop():
x = ad.Variable("x")
y = -x
assert np.isclose(y.d_n(n=0, val=2.0), -2.0)
assert np.isclose(y.d_n(n=1, val=2.0), -1.0)
assert np.isclose(y.d_n(n=2, val=2.0), 0.0)
y = ad.Sin(2 * x)
assert np.isclose(y.d_n(n=0, val=0.0), 0.0)
assert np.isclose(y.d_n(n=1, val=0.0), 2.0)
assert np.isclose(y.d_n(n=3, val=0.0), -8.0)
y = ad.Exp(3 * x)
assert np.isclose(y.d_n(n=0, val=0.0), 1.0)
assert np.isclose(y.d_n(n=1, val=0.0), 3.0)
assert np.isclose(y.d_n(n=3, val=0.0), 27.0)
y = ad.Log(2 * x)
assert np.isclose(y.d_n(n=0, val=0.5), 0.0)
assert np.isclose(y.d_n(n=1, val=0.5), 2.0)
assert np.isclose(y.d_n(n=3, val=0.5), 2.0 / (0.5**3))
def test_complex():
x = ad.Variable("x")
y = -12 * ad.Cos(x**2) + 8 * (x**3) * ad.Sin(x**2)
yd5 = y.d_expr(5)
assert np.isclose(y.d_n(5, 2.0), yd5.eval({x: 2.0}))
def test_sine_expression():
a = ad.Variable('a')
b = ad.Sin(a)
assert np.isclose(b.eval({a: pi/2}), 1)
assert np.isclose(b.d({a: pi/2}), 0)
def test_addition():
x = ad.Variable("x")
y = x + ad.Sin(x)
yd1 = y.d_expr()
assert np.isclose(yd1.eval({x: 0.0}), 2.0)