def test_basic_math(types):
for x in types["all"]:
for y in types["all"]:
### Arithmetic
x + y
x - y
x * y
x / y
np.sum(x) # Sum of all entries of array-like object x
### Exponentials & Powers
x**y
np.power(x, y)
np.exp(x)
np.log(x)
np.log10(x)
np.sqrt(x) # Note: do x ** 0.5 rather than np.sqrt(x).
### Trig
np.sin(x)
np.cos(x)
np.tan(x)
np.arcsin(x)
np.arccos(x)
np.arctan(x)
np.arctan2(y, x)
np.sinh(x)
np.cosh(x)
np.tanh(x)
np.arcsinh(x)
np.arccosh(x)
np.arctanh(x - 0.5) # `- 0.5` to give valid argument
def sigmoid(
x,
sigmoid_type: str = "tanh",
normalization_range: Tuple[Union[float, int], Union[float, int]] = (0, 1)
):
"""
A sigmoid function. From Wikipedia (https://en.wikipedia.org/wiki/Sigmoid_function):
A sigmoid function is a mathematical function having a characteristic "S"-shaped curve
or sigmoid curve.
Args:
x: The input
sigmoid_type: Type of sigmoid function to use [str]. Can be one of:
* "tanh" or "logistic" (same thing)
* "arctan"
* "polynomial"
normalization_type: Range in which to normalize the sigmoid, shorthanded here in the
documentation as "N". This parameter is given as a two-element tuple (min, max).
After normalization:
>>> sigmoid(-Inf) == normalization_range[0]
>>> sigmoid(Inf) == normalization_range[1]
* In the special case of N = (0, 1):
>>> sigmoid(-Inf) == 0
>>> sigmoid(Inf) == 1
>>> sigmoid(0) == 0.5
>>> d(sigmoid)/dx at x=0 == 0.5
* In the special case of N = (-1, 1):
>>> sigmoid(-Inf) == -1
>>> sigmoid(Inf) == 1
>>> sigmoid(0) == 0
>>> d(sigmoid)/dx at x=0 == 1
Returns: The value of the sigmoid.
"""
### Sigmoid equations given here under the (-1, 1) normalization:
if sigmoid_type == ("tanh" or "logistic"):
# Note: tanh(x) is simply a scaled and shifted version of a logistic curve; after
# normalization these functions are identical.
s = _np.tanh(x)
elif sigmoid_type == "arctan":
s = 2 / _np.pi * _np.arctan(_np.pi / 2 * x)
elif sigmoid_type == "polynomial":
s = x / (1 + x ** 2) ** 0.5
else:
raise ValueError("Bad value of parameter 'type'!")
### Normalize
min = normalization_range[0]
max = normalization_range[1]
s_normalized = s * (max - min) / 2 + (max + min) / 2
return s_normalized
) # From AVF Eq. 4.53
c_f = 2 / Re_theta * np.where(
H < 6.2,
-0.066 + 0.066 * np.abs(6.2 - H) ** 1.5 / (H - 1),
-0.066 + 0.066 * (H - 6.2) ** 2 / (H - 4) ** 2
) # From AVF Eq. 4.54
c_D = H_star / 2 / Re_theta * np.where(
H < 4,
0.207 + 0.00205 * np.abs(4 - H) ** 5.5,
0.207 - 0.100 * (H - 4) ** 2 / H ** 2
) # From AVF Eq. 4.55
Re_theta_o = 10 ** (
2.492 / (H - 1) ** 0.43 +
0.7 * (
np.tanh(
14 / (H - 1) - 9.24
) + 1
)
) # From AVF Eq. 6.38
d_theta_dx = np.diff(theta) / np.diff(x)
d_ue_dx = np.diff(ue) / np.diff(x)
d_H_star_dx = np.diff(H_star) / np.diff(x)
def int(x):
return (x[1:] + x[:-1]) / 2
def logint(x):
# return int(x)
H_0,
n_vars=N,
)
Re_theta = ue * theta / nu
H_star = np.where(H < 4, 1.515 + 0.076 * (H - 4)**2 / H,
1.515 + 0.040 * (H - 4)**2 / H) # From AVF Eq. 4.53
c_f = 2 / Re_theta * np.where(H < 6.2, -0.066 + 0.066 * np.abs(6.2 - H)**1.5 /
(H - 1), -0.066 + 0.066 * (H - 6.2)**2 /
(H - 4)**2) # From AVF Eq. 4.54
c_D = H_star / 2 / Re_theta * np.where(
H < 4, 0.207 + 0.00205 * np.abs(4 - H)**5.5, 0.207 - 0.100 *
(H - 4)**2 / H**2) # From AVF Eq. 4.55
Re_theta_o = 10**(2.492 / (H - 1)**0.43 + 0.7 *
(np.tanh(14 / (H - 1) - 9.24) + 1)) # From AVF Eq. 6.38
d_theta_dx = np.diff(theta) / np.diff(x)
d_ue_dx = np.diff(ue) / np.diff(x)
d_H_star_dx = np.diff(H_star) / np.diff(x)
def int(x):
return (x[1:] + x[:-1]) / 2
def logint(x):
# return int(x)
logx = np.log(x)
return np.exp((logx[1:] + logx[:-1]) / 2)