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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. 

 

""" 

from __future__ import division, absolute_import, print_function 

 

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) 

(0.0+1.0j) 

>>> 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) 

 

>>> 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)