def log_normcdf(x):
"""Elementwise log of the cumulative distribution function of a standard normal random variable.
Implementation is a quadratic approximation with modest accuracy over [-4, 4].
"""
A = scipy.sparse.diags(
np.sqrt(
[
0.02301291,
0.08070214,
0.16411522,
0.09003495,
0.08200854,
0.01371543,
0.04641081,
]
)
)
b = np.array([[3.0, 2.0, 1.0, 0.0, -1.0, -2.5, -3.5]]).reshape(-1, 1)
x = Expression.cast_to_const(x)
flat_x = reshape(x, (1, x.size))
y = A @ (b @ np.ones(flat_x.shape) - np.ones(b.shape) @ flat_x)
out = -sum_(maximum(y, 0) ** 2, axis=0)
return reshape(out, x.shape)
def log_normcdf(x): # noqa: E501
"""Elementwise log of the cumulative distribution function of a standard normal random variable.
The implementation is a quadratic approximation with modest accuracy over [-4, 4].
For details on the nature of the approximation, refer to
`CVXPY GitHub PR #1224 <https://github.com/cvxpy/cvxpy/pull/1224#issue-793221374>`_.
.. note::
SciPy's analog of ``log_normcdf`` is called `log_ndtr <https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.log_ndtr.html>`_.
We opted not to use that name because its meaning would not be obvious to the casual user.
"""
A = scipy.sparse.diags(
np.sqrt(
[
0.02301291,
0.08070214,
0.16411522,
0.09003495,
0.08200854,
0.01371543,
0.04641081,
]
)
)
b = np.array([[3.0, 2.0, 1.0, 0.0, -1.0, -2.5, -3.5]]).reshape(-1, 1)
x = Expression.cast_to_const(x)
flat_x = reshape(x, (1, x.size))
y = A @ (b @ np.ones(flat_x.shape) - np.ones(b.shape) @ flat_x)
out = -sum_(maximum(y, 0) ** 2, axis=0)
return reshape(out, x.shape)
def loggamma(x):
"""Elementwise log of the gamma function.
Implementation has modest accuracy over the full range, approaching perfect
accuracy as x goes to infinity.
"""
return maximum(
2.18382 - 3.62887 * x,
1.79241 - 2.4902 * x,
1.21628 - 1.37035 * x,
0.261474 - 0.28904 * x,
0.577216 - 0.577216 * x,
-0.175517 + 0.03649 * x,
-1.27572 + 0.621514 * x,
-0.845568 + 0.422784 * x,
-0.577216 * x - log(x),
0.918939 - x - entr(x) - 0.5 * log(x),
)
def loggamma(x):
"""Elementwise log of the gamma function.
Implementation has modest accuracy over the full range, approaching perfect
accuracy as x goes to infinity. For details on the nature of the approximation,
refer to `CVXPY GitHub Issue #228 <https://github.com/cvxpy/cvxpy/issues/228#issuecomment-544281906>`_.
"""
return maximum(
2.18382 - 3.62887*x,
1.79241 - 2.4902*x,
1.21628 - 1.37035*x,
0.261474 - 0.28904*x,
0.577216 - 0.577216*x,
-0.175517 + 0.03649*x,
-1.27572 + 0.621514*x,
-0.845568 + 0.422784*x,
-0.577216*x - log(x),
0.918939 - x - entr(x) - 0.5*log(x),
)
def minimum(*args):
return -maximum(*[-Elementwise.cast_to_const(arg) for arg in args])
def pos(x):
""" Alias for maximum{x,0}.
"""
return maximum(x, 0)
def minimum_canon(expr, args):
del expr
tmp = maximum(*[-arg for arg in args])
canon, constr = maximum_canon(tmp, tmp.args)
return -canon, constr
def minimum(*args):
"""Elementwise minimum of a sequence of Expressions."""
return -maximum(*[-Elementwise.cast_to_const(arg) for arg in args])