def f(value):
"""
f takes in a dictionary of independent variables
"""
X = ad.Variable(value['x'], name='x')
Y = ad.Variable(value['y'], name='y')
Z1 = ad.exp(-X**2 - Y**2)
Z2 = ad.exp(-(X - 1)**2 - (Y - 1)**2)
return (Z1 - Z2) * 2
def test_hyberbolic(self):
x1 = Variable(1, name='x1')
x2 = Variable(1, name='x2')
x3 = Variable(1, name='x3')
x4 = AutoDiff.sinh(x1) + AutoDiff.cosh(x2) + AutoDiff.tanh(x3)
assert x4.val == 3.4798759844148099
assert x4.der == {'x1': 1.5430806348152437,
'x2': 1.1752011936438014,
'x3': 0.41997434161402614}
def test_trigonometric(self):
x1 = Variable(np.pi/4, name='x1')
x2 = Variable(np.pi/4, name='x2')
x3 = Variable(np.pi/4, name='x3')
x4 = AutoDiff.sin(x1) + AutoDiff.cos(x2) + AutoDiff.tan(x3)
assert x4.val == 2**0.5 + 1
assert x4.der == {'x1': 0.70710678118654757,
'x2': -0.70710678118654746,
'x3': 1.9999999999999998}
def test_inverse_trigonometric(self):
x1 = Variable(0.1, name='x1')
x2 = Variable(0.2, name='x2')
x3 = Variable(0.3, name='x3')
x4 = AutoDiff.arcsin(x1) + AutoDiff.arccos(x2) + AutoDiff.arctan(x3)
assert x4.val == 1.7610626216439926
assert x4.der == {'x1': 1.005037815259212,
'x2': -1.0206207261596576,
'x3': 0.9174311926605504}
def test_gmres_autodiff():
b = [1, 2, 3]
x1 = ad.Variable(1, name='x1')
x2 = ad.Variable(1, name='x2')
x3 = ad.Variable(1, name='x3')
f1 = 2*x1+3*x2+2*x3
f2 = 3*x1+2*x2+1*x3
f3 = 3*x1+3*x2+3*x3
F = [f1, f2, f3]
x = gmres_autodiff(F, b)
np.testing.assert_allclose(x, [1, -1, 1])
def test_hyperbolics(self):
x1 = Variable(1, name='x1')
x2 = Variable(1, name='x2')
x = AutoDiff.sinh(x1)
y = AutoDiff.cosh(x2)
z = AutoDiff.tanh(x2)
z1 = x*y+z
assert z1.der2['x1x1'][0] == 1.8134302039235093
assert z1.der2['x1x2'][0] == 1.8134302039235093
assert z1.der2['x2x1'][0] == 1.8134302039235093
assert z1.der2['x2x2'][0] == 1.1737301954742847
def test_exp_log(self):
x1 = Variable(1, name='x1')
x2 = Variable(1, name='x2')
x = AutoDiff.exp(x1)
y = AutoDiff.log(x2)
z1 = x*y
assert z1.der2['x1x1'][0] == 0.
assert z1.der2['x1x2'][0] == 2.7182818284590451
assert z1.der2['x2x1'][0] == 2.7182818284590451
assert z1.der2['x2x2'][0] == -2.7182818284590451
def test_inverse_trigonometric(self):
x1 = Variable(0.5, name='x1')
x2 = Variable(0.6, name='x2')
x = AutoDiff.arcsin(x1)
y = AutoDiff.arccos(x2)
z = AutoDiff.arctan(x2)
z1 = x*y+z
assert z1.der2['x1x1'][0] == 0.71383219164197809
assert z1.der2['x1x2'][0] == -1.4433756729740643
assert z1.der2['x2x1'][0] == -1.4433756729740643
assert z1.der2['x2x2'][0] == -1.262381242489897
def test_trigonometric(self):
x1 = Variable(1, name='x1')
x2 = Variable(1, name='x2')
x = AutoDiff.sin(x1)
y = AutoDiff.cos(x2)
z = AutoDiff.tan(x2)
z1 = x*y+z
assert z1.der2['x1x1'][0] == -0.45464871341284091
assert z1.der2['x1x2'][0] == -0.45464871341284091
assert z1.der2['x2x1'][0] == -0.45464871341284091
assert z1.der2['x2x2'][0] == 10.215210231562478
def test_pow(self):
x1 = Variable(1, name='x1')
x2 = Variable(1, name='x2')
x = AutoDiff.sin(x1)
y = AutoDiff.sin(x2)
z1 = x**y
z2 = 2**x
z3 = x**2
assert z1.der2['x1x1'][0] == -0.77527835938046719
assert z1.der2['x1x2'][0] == 0.25644885963975589
assert z1.der2['x2x1'][0] == 0.25644885963975589
assert z1.der2['x2x2'][0] == 0.13312782635352052
assert z2.der2['x1x1'][0] == -0.79381233880392665
assert z3.der2['x1x1'][0] == -0.83229367309428459
def test_add_sub(self):
x1 = Variable(1, name='x1')
x2 = Variable(1, name='x2')
x = AutoDiff.sin(x1)
y = AutoDiff.sin(x2)
z1 = x+y
z2 = x-y
z3 = +x
z4 = -x
assert z1.der2 == {'x1x1': np.array([-0.8414709848078965]),
'x2x2': np.array([-0.8414709848078965]),
'x1x2':np.array([0.0]),'x2x1':np.array([0.0])}
assert z2.der2 == {'x1x1': np.array([-0.8414709848078965]),
'x2x2': np.array([0.8414709848078965]),
'x1x2': np.array([0.0]),'x2x1':np.array([0.0])}
assert z3.der2 == {'x1x1': np.array([-0.8414709848078965])}
assert z4.der2 == {'x1x1': np.array([0.8414709848078965])}
def test_mul_div(self):
x1 = Variable(1, name='x1')
x2 = Variable(1, name='x2')
x = AutoDiff.sin(x1)
y = AutoDiff.sin(x2)
z1 = x*y
z2 = x/y
z3 = 1/x
z4 = x/1
assert z1.der2['x1x1'][0] == -0.70807341827357118
assert z1.der2['x1x2'][0] == 0.29192658172642888
assert z1.der2['x2x1'][0] == 0.29192658172642888
assert z1.der2['x2x2'][0] == -0.70807341827357118
assert z2.der2['x1x1'][0] == -1.0
assert z2.der2['x1x2'][0] == -0.41228292743739203
assert z2.der2['x2x1'][0] == -0.41228292743739203
assert z2.der2['x2x2'][0] == 1.8245658548747841
assert z3.der2['x1x1'][0] == 2.1683051321030673
assert z4.der2['x1x1'][0] == -0.8414709848078965
def test_sqrt(self):
x1 = Variable(1, name='x1')
z1 = AutoDiff.sqrt(x1)
assert z1.der2['x1x1'][0] == -0.25
def test_exp(self):
x1 = Variable(1, name='x1')
x2 = Variable(1, name='x2')
x3 = AutoDiff.exp(x1 + x2)
assert x3.val == np.exp(1+1)
assert x3.der == {'x1': np.exp(1+1), 'x2': np.exp(1+1)}
def test_sigmoid(self):
x1 = Variable(0, name='x1')
x2 = AutoDiff.sigmoid(x1)
assert x2.val == 0.5
assert x2.der == {'x1': 0.25}
def f1(values): x1 = ad.Variable(values['x1'], name='x1') f = 2 * (x1 ** 2) + 5 return f
import AutoDiff.AutoDiff as ad from AutoDiff.GMRes import gmres_autodiff import numpy as np # define variables and vector of vector functions x1 = ad.Variable(1, name='x1') x2 = ad.Variable(1, name='x2') x3 = ad.Variable(1, name='x3') f1 = 2 * x1 + 3 * x2 + 2 * x3 f2 = 3 * x1 + 2 * x2 + 1 * x3 f3 = 3 * x1 + 3 * x2 + 3 * x3 F = [f1, f2, f3] b = [1, 2, 3] # GMRES to get x x = gmres_autodiff(F, b) print(x)
def f2(values):
x1 = ad.Variable(values['x1'], name='x1')
x2 = ad.Variable(values['x2'], name='x2')
return 2 * (x1**2) + ad.sin(x2)
def test_log(self):
x1 = Variable(1, name='x1')
x2 = Variable(1, name='x2')
x3 = AutoDiff.log(x1 + x2)
assert x3.val == np.log(1+1)
assert x3.der == {'x1': 1/2, 'x2': 1/2}
def test_sqrt(self):
x1 = Variable(1, name='x1')
x2 = AutoDiff.sqrt(x1)
assert x2.val == 1
assert x2.der == {'x1': 0.5}
def test_sigmoid(self):
x1 = Variable(0, name='x1')
z1 = AutoDiff.sigmoid(x1)
assert z1.der2['x1x1'][0] == 0
