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