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Created starter files for the project.

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Batuhan Berk Başoğlu 2020-10-02 21:26:03 -04:00
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"""
Wrapper functions to more user-friendly calling of certain math functions
whose output data-type is different than the input data-type in certain
domains of the input.
For example, for functions like `log` with branch cuts, the versions in this
module provide the mathematically valid answers in the complex plane::
>>> import math
>>> from numpy.lib import scimath
>>> scimath.log(-math.exp(1)) == (1+1j*math.pi)
True
Similarly, `sqrt`, other base logarithms, `power` and trig functions are
correctly handled. See their respective docstrings for specific examples.
"""
import numpy.core.numeric as nx
import numpy.core.numerictypes as nt
from numpy.core.numeric import asarray, any
from numpy.core.overrides import array_function_dispatch
from numpy.lib.type_check import isreal
__all__ = [
'sqrt', 'log', 'log2', 'logn', 'log10', 'power', 'arccos', 'arcsin',
'arctanh'
]
_ln2 = nx.log(2.0)
def _tocomplex(arr):
"""Convert its input `arr` to a complex array.
The input is returned as a complex array of the smallest type that will fit
the original data: types like single, byte, short, etc. become csingle,
while others become cdouble.
A copy of the input is always made.
Parameters
----------
arr : array
Returns
-------
array
An array with the same input data as the input but in complex form.
Examples
--------
First, consider an input of type short:
>>> a = np.array([1,2,3],np.short)
>>> ac = np.lib.scimath._tocomplex(a); ac
array([1.+0.j, 2.+0.j, 3.+0.j], dtype=complex64)
>>> ac.dtype
dtype('complex64')
If the input is of type double, the output is correspondingly of the
complex double type as well:
>>> b = np.array([1,2,3],np.double)
>>> bc = np.lib.scimath._tocomplex(b); bc
array([1.+0.j, 2.+0.j, 3.+0.j])
>>> bc.dtype
dtype('complex128')
Note that even if the input was complex to begin with, a copy is still
made, since the astype() method always copies:
>>> c = np.array([1,2,3],np.csingle)
>>> cc = np.lib.scimath._tocomplex(c); cc
array([1.+0.j, 2.+0.j, 3.+0.j], dtype=complex64)
>>> c *= 2; c
array([2.+0.j, 4.+0.j, 6.+0.j], dtype=complex64)
>>> cc
array([1.+0.j, 2.+0.j, 3.+0.j], dtype=complex64)
"""
if issubclass(arr.dtype.type, (nt.single, nt.byte, nt.short, nt.ubyte,
nt.ushort, nt.csingle)):
return arr.astype(nt.csingle)
else:
return arr.astype(nt.cdouble)
def _fix_real_lt_zero(x):
"""Convert `x` to complex if it has real, negative components.
Otherwise, output is just the array version of the input (via asarray).
Parameters
----------
x : array_like
Returns
-------
array
Examples
--------
>>> np.lib.scimath._fix_real_lt_zero([1,2])
array([1, 2])
>>> np.lib.scimath._fix_real_lt_zero([-1,2])
array([-1.+0.j, 2.+0.j])
"""
x = asarray(x)
if any(isreal(x) & (x < 0)):
x = _tocomplex(x)
return x
def _fix_int_lt_zero(x):
"""Convert `x` to double if it has real, negative components.
Otherwise, output is just the array version of the input (via asarray).
Parameters
----------
x : array_like
Returns
-------
array
Examples
--------
>>> np.lib.scimath._fix_int_lt_zero([1,2])
array([1, 2])
>>> np.lib.scimath._fix_int_lt_zero([-1,2])
array([-1., 2.])
"""
x = asarray(x)
if any(isreal(x) & (x < 0)):
x = x * 1.0
return x
def _fix_real_abs_gt_1(x):
"""Convert `x` to complex if it has real components x_i with abs(x_i)>1.
Otherwise, output is just the array version of the input (via asarray).
Parameters
----------
x : array_like
Returns
-------
array
Examples
--------
>>> np.lib.scimath._fix_real_abs_gt_1([0,1])
array([0, 1])
>>> np.lib.scimath._fix_real_abs_gt_1([0,2])
array([0.+0.j, 2.+0.j])
"""
x = asarray(x)
if any(isreal(x) & (abs(x) > 1)):
x = _tocomplex(x)
return x
def _unary_dispatcher(x):
return (x,)
@array_function_dispatch(_unary_dispatcher)
def sqrt(x):
"""
Compute the square root of x.
For negative input elements, a complex value is returned
(unlike `numpy.sqrt` which returns NaN).
Parameters
----------
x : array_like
The input value(s).
Returns
-------
out : ndarray or scalar
The square root of `x`. If `x` was a scalar, so is `out`,
otherwise an array is returned.
See Also
--------
numpy.sqrt
Examples
--------
For real, non-negative inputs this works just like `numpy.sqrt`:
>>> np.lib.scimath.sqrt(1)
1.0
>>> np.lib.scimath.sqrt([1, 4])
array([1., 2.])
But it automatically handles negative inputs:
>>> np.lib.scimath.sqrt(-1)
1j
>>> np.lib.scimath.sqrt([-1,4])
array([0.+1.j, 2.+0.j])
"""
x = _fix_real_lt_zero(x)
return nx.sqrt(x)
@array_function_dispatch(_unary_dispatcher)
def log(x):
"""
Compute the natural logarithm of `x`.
Return the "principal value" (for a description of this, see `numpy.log`)
of :math:`log_e(x)`. For real `x > 0`, this is a real number (``log(0)``
returns ``-inf`` and ``log(np.inf)`` returns ``inf``). Otherwise, the
complex principle value is returned.
Parameters
----------
x : array_like
The value(s) whose log is (are) required.
Returns
-------
out : ndarray or scalar
The log of the `x` value(s). If `x` was a scalar, so is `out`,
otherwise an array is returned.
See Also
--------
numpy.log
Notes
-----
For a log() that returns ``NAN`` when real `x < 0`, use `numpy.log`
(note, however, that otherwise `numpy.log` and this `log` are identical,
i.e., both return ``-inf`` for `x = 0`, ``inf`` for `x = inf`, and,
notably, the complex principle value if ``x.imag != 0``).
Examples
--------
>>> np.emath.log(np.exp(1))
1.0
Negative arguments are handled "correctly" (recall that
``exp(log(x)) == x`` does *not* hold for real ``x < 0``):
>>> np.emath.log(-np.exp(1)) == (1 + np.pi * 1j)
True
"""
x = _fix_real_lt_zero(x)
return nx.log(x)
@array_function_dispatch(_unary_dispatcher)
def log10(x):
"""
Compute the logarithm base 10 of `x`.
Return the "principal value" (for a description of this, see
`numpy.log10`) of :math:`log_{10}(x)`. For real `x > 0`, this
is a real number (``log10(0)`` returns ``-inf`` and ``log10(np.inf)``
returns ``inf``). Otherwise, the complex principle value is returned.
Parameters
----------
x : array_like or scalar
The value(s) whose log base 10 is (are) required.
Returns
-------
out : ndarray or scalar
The log base 10 of the `x` value(s). If `x` was a scalar, so is `out`,
otherwise an array object is returned.
See Also
--------
numpy.log10
Notes
-----
For a log10() that returns ``NAN`` when real `x < 0`, use `numpy.log10`
(note, however, that otherwise `numpy.log10` and this `log10` are
identical, i.e., both return ``-inf`` for `x = 0`, ``inf`` for `x = inf`,
and, notably, the complex principle value if ``x.imag != 0``).
Examples
--------
(We set the printing precision so the example can be auto-tested)
>>> np.set_printoptions(precision=4)
>>> np.emath.log10(10**1)
1.0
>>> np.emath.log10([-10**1, -10**2, 10**2])
array([1.+1.3644j, 2.+1.3644j, 2.+0.j ])
"""
x = _fix_real_lt_zero(x)
return nx.log10(x)
def _logn_dispatcher(n, x):
return (n, x,)
@array_function_dispatch(_logn_dispatcher)
def logn(n, x):
"""
Take log base n of x.
If `x` contains negative inputs, the answer is computed and returned in the
complex domain.
Parameters
----------
n : array_like
The integer base(s) in which the log is taken.
x : array_like
The value(s) whose log base `n` is (are) required.
Returns
-------
out : ndarray or scalar
The log base `n` of the `x` value(s). If `x` was a scalar, so is
`out`, otherwise an array is returned.
Examples
--------
>>> np.set_printoptions(precision=4)
>>> np.lib.scimath.logn(2, [4, 8])
array([2., 3.])
>>> np.lib.scimath.logn(2, [-4, -8, 8])
array([2.+4.5324j, 3.+4.5324j, 3.+0.j ])
"""
x = _fix_real_lt_zero(x)
n = _fix_real_lt_zero(n)
return nx.log(x)/nx.log(n)
@array_function_dispatch(_unary_dispatcher)
def log2(x):
"""
Compute the logarithm base 2 of `x`.
Return the "principal value" (for a description of this, see
`numpy.log2`) of :math:`log_2(x)`. For real `x > 0`, this is
a real number (``log2(0)`` returns ``-inf`` and ``log2(np.inf)`` returns
``inf``). Otherwise, the complex principle value is returned.
Parameters
----------
x : array_like
The value(s) whose log base 2 is (are) required.
Returns
-------
out : ndarray or scalar
The log base 2 of the `x` value(s). If `x` was a scalar, so is `out`,
otherwise an array is returned.
See Also
--------
numpy.log2
Notes
-----
For a log2() that returns ``NAN`` when real `x < 0`, use `numpy.log2`
(note, however, that otherwise `numpy.log2` and this `log2` are
identical, i.e., both return ``-inf`` for `x = 0`, ``inf`` for `x = inf`,
and, notably, the complex principle value if ``x.imag != 0``).
Examples
--------
We set the printing precision so the example can be auto-tested:
>>> np.set_printoptions(precision=4)
>>> np.emath.log2(8)
3.0
>>> np.emath.log2([-4, -8, 8])
array([2.+4.5324j, 3.+4.5324j, 3.+0.j ])
"""
x = _fix_real_lt_zero(x)
return nx.log2(x)
def _power_dispatcher(x, p):
return (x, p)
@array_function_dispatch(_power_dispatcher)
def power(x, p):
"""
Return x to the power p, (x**p).
If `x` contains negative values, the output is converted to the
complex domain.
Parameters
----------
x : array_like
The input value(s).
p : array_like of ints
The power(s) to which `x` is raised. If `x` contains multiple values,
`p` has to either be a scalar, or contain the same number of values
as `x`. In the latter case, the result is
``x[0]**p[0], x[1]**p[1], ...``.
Returns
-------
out : ndarray or scalar
The result of ``x**p``. If `x` and `p` are scalars, so is `out`,
otherwise an array is returned.
See Also
--------
numpy.power
Examples
--------
>>> np.set_printoptions(precision=4)
>>> np.lib.scimath.power([2, 4], 2)
array([ 4, 16])
>>> np.lib.scimath.power([2, 4], -2)
array([0.25 , 0.0625])
>>> np.lib.scimath.power([-2, 4], 2)
array([ 4.-0.j, 16.+0.j])
"""
x = _fix_real_lt_zero(x)
p = _fix_int_lt_zero(p)
return nx.power(x, p)
@array_function_dispatch(_unary_dispatcher)
def arccos(x):
"""
Compute the inverse cosine of x.
Return the "principal value" (for a description of this, see
`numpy.arccos`) of the inverse cosine of `x`. For real `x` such that
`abs(x) <= 1`, this is a real number in the closed interval
:math:`[0, \\pi]`. Otherwise, the complex principle value is returned.
Parameters
----------
x : array_like or scalar
The value(s) whose arccos is (are) required.
Returns
-------
out : ndarray or scalar
The inverse cosine(s) of the `x` value(s). If `x` was a scalar, so
is `out`, otherwise an array object is returned.
See Also
--------
numpy.arccos
Notes
-----
For an arccos() that returns ``NAN`` when real `x` is not in the
interval ``[-1,1]``, use `numpy.arccos`.
Examples
--------
>>> np.set_printoptions(precision=4)
>>> np.emath.arccos(1) # a scalar is returned
0.0
>>> np.emath.arccos([1,2])
array([0.-0.j , 0.-1.317j])
"""
x = _fix_real_abs_gt_1(x)
return nx.arccos(x)
@array_function_dispatch(_unary_dispatcher)
def arcsin(x):
"""
Compute the inverse sine of x.
Return the "principal value" (for a description of this, see
`numpy.arcsin`) of the inverse sine of `x`. For real `x` such that
`abs(x) <= 1`, this is a real number in the closed interval
:math:`[-\\pi/2, \\pi/2]`. Otherwise, the complex principle value is
returned.
Parameters
----------
x : array_like or scalar
The value(s) whose arcsin is (are) required.
Returns
-------
out : ndarray or scalar
The inverse sine(s) of the `x` value(s). If `x` was a scalar, so
is `out`, otherwise an array object is returned.
See Also
--------
numpy.arcsin
Notes
-----
For an arcsin() that returns ``NAN`` when real `x` is not in the
interval ``[-1,1]``, use `numpy.arcsin`.
Examples
--------
>>> np.set_printoptions(precision=4)
>>> np.emath.arcsin(0)
0.0
>>> np.emath.arcsin([0,1])
array([0. , 1.5708])
"""
x = _fix_real_abs_gt_1(x)
return nx.arcsin(x)
@array_function_dispatch(_unary_dispatcher)
def arctanh(x):
"""
Compute the inverse hyperbolic tangent of `x`.
Return the "principal value" (for a description of this, see
`numpy.arctanh`) of `arctanh(x)`. For real `x` such that
`abs(x) < 1`, this is a real number. If `abs(x) > 1`, or if `x` is
complex, the result is complex. Finally, `x = 1` returns``inf`` and
`x=-1` returns ``-inf``.
Parameters
----------
x : array_like
The value(s) whose arctanh is (are) required.
Returns
-------
out : ndarray or scalar
The inverse hyperbolic tangent(s) of the `x` value(s). If `x` was
a scalar so is `out`, otherwise an array is returned.
See Also
--------
numpy.arctanh
Notes
-----
For an arctanh() that returns ``NAN`` when real `x` is not in the
interval ``(-1,1)``, use `numpy.arctanh` (this latter, however, does
return +/-inf for `x = +/-1`).
Examples
--------
>>> np.set_printoptions(precision=4)
>>> from numpy.testing import suppress_warnings
>>> with suppress_warnings() as sup:
... sup.filter(RuntimeWarning)
... np.emath.arctanh(np.eye(2))
array([[inf, 0.],
[ 0., inf]])
>>> np.emath.arctanh([1j])
array([0.+0.7854j])
"""
x = _fix_real_abs_gt_1(x)
return nx.arctanh(x)