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(L$ BL$ ((T$蛧L$ ((\$ dT$\$ Y\$ ? NULL expect '->' expect '(' expect dimension name expect ',' or ')' expect ',' %s at position %d in "%s" %s encountered in %s NN write divide by zero overflow underflow invalid value UFUNC_PYVALS %s must be a length 3 list. invalid error mask (%d) NO wrong shape for output function not supported OO|iO&O& O(O)i cannot %s on a scalar axis not in array add multiply reduce accumulate index out-of-bounds (0, %d) reduceat __array_wrap__ invalid number of arguments OOi __array_priority__ __radd__ subtract __rsub__ __rmul__ __rdiv__ true_divide __rtruediv__ floor_divide __rfloordiv__ remainder __rmod__ power __rpow__ left_shift __rlshift__ right_shift __rrshift__ bitwise_and __rand__ bitwise_xor __rxor__ bitwise_or __ror__ __array__ invalid return array shape return array is not writeable __array_prepare__ O(OOi) invalid keyword extobj sig (OO) unknown user-defined type test Oii function must be callable __name__ umath numpy.core.multiarray _ARRAY_API _UFUNC_API 0.4.0 __version__ DO NOT USE, ONLY FOR TESTING _arg absolute arccos arccosh arcsin arcsinh arctan arctan2 arctanh ceil conjugate copysign deg2rad degrees expm1 fabs floor fmax fmin fmod greater greater_equal hypot invert isfinite isinf isnan less less_equal log log10 log1p log2 logaddexp logaddexp2 logical_and logical_not logical_or logical_xor maximum minimum modf negative nextafter not_equal ones_like rad2deg radians reciprocal rint signbit spacing sqrt square trunc frexp Compute y = x1 * 2**x2. ldexp pi ERR_IGNORE ERR_WARN ERR_CALL ERR_RAISE ERR_PRINT ERR_LOG ERR_DEFAULT ERR_DEFAULT2 SHIFT_DIVIDEBYZERO SHIFT_OVERFLOW SHIFT_UNDERFLOW SHIFT_INVALID FPE_DIVIDEBYZERO FPE_OVERFLOW FPE_UNDERFLOW FPE_INVALID FLOATING_POINT_SUPPORT UFUNC_PYVALS_NAME UFUNC_BUFSIZE_DEFAULT PINF NINF PZERO NZERO NAN conj cannot load umath module. numpy.ufunc indices axis dtype frompyfunc seterrobj geterrobj outer __doc__ nin nout nargs ntypes identity signature a type-tuple must be specified of length 1 or %d for %s a type-string for %s, requires %d typecode(s) before and %d after the -> sign user-defined type used in ufunc with no registered loops dimension of output variable exceeds limit ',' must not be followed by ')' incomplete signature: not all arguments found python callback specified for %s (in %s) but no function found. Warning: %s encountered in %s
log specified for %s (in %s) but no object with write method found. buffer size (%d) is not in range (%ld - %ld) or not a multiple of 16 python object must be callable or have a callable write method construct_reduce not allowed on ufunc with signature zero-size array to ufunc.%s without identity UFUNC_MAXIDENTITY (%d) is too small (needs to be at least %d) Reduction not defined on ufunc with signature %s only supported for binary functions %s only supported for functions returning a single value cannot perform %s with flexible type return arrays must be of ArrayType __array_prepare__ must return an ndarray or subclass thereof never reached; copy should have been made Object type not allowed in ufunc with signature '%s' is an invalid keyword to %s illegal loop method for ufunc with signature method outer is not allowed in ufunc with non-trivial signature outer product only supported for binary functions exactly two arguments expected dynamic ufunc based on a python function module compiled against ABI version %x but this version of numpy is %x module compiled against API version %x but this version of numpy is %x FATAL: module compiled as unknown endian FATAL: module compiled as little endian, but detected different endianness at runtime umath failed: Could not import array core. Calculate the absolute value element-wise.
Parameters
----------
x : array_like
Input array.
Returns
-------
res : ndarray
An ndarray containing the absolute value of
each element in `x`. For complex input, ``a + ib``, the
absolute value is :math:`\sqrt{ a^2 + b^2 }`.
Examples
--------
>>> x = np.array([-1.2, 1.2])
>>> np.absolute(x)
array([ 1.2, 1.2])
>>> np.absolute(1.2 + 1j)
1.5620499351813308
Plot the function over ``[-10, 10]``:
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-10, 10, 101)
>>> plt.plot(x, np.absolute(x))
>>> plt.show()
Plot the function over the complex plane:
>>> xx = x + 1j * x[:, np.newaxis]
>>> plt.imshow(np.abs(xx), extent=[-10, 10, -10, 10])
>>> plt.show() Add arguments element-wise.
Parameters
----------
x1, x2 : array_like
The arrays to be added.
Returns
-------
y : {ndarray, scalar}
The sum of `x1` and `x2`, element-wise. Returns scalar if
both `x1` and `x2` are scalars.
Notes
-----
Equivalent to `x1` + `x2` in terms of array broadcasting.
Examples
--------
>>> np.add(1.0, 4.0)
5.0
>>> x1 = np.arange(9.0).reshape((3, 3))
>>> x2 = np.arange(3.0)
>>> np.add(x1, x2)
array([[ 0., 2., 4.],
[ 3., 5., 7.],
[ 6., 8., 10.]]) Trigonometric inverse cosine, element-wise.
The inverse of `cos` so that, if ``y = cos(x)``, then ``x = arccos(y)``.
Parameters
----------
x : array_like
`x`-coordinate on the unit circle.
For real arguments, the domain is [-1, 1].
out : ndarray, optional
Array of the same shape as `a`, to store results in. See
`doc.ufuncs` (Section "Output arguments") for more details.
Returns
-------
angle : ndarray
The angle of the ray intersecting the unit circle at the given
`x`-coordinate in radians [0, pi]. If `x` is a scalar then a
scalar is returned, otherwise an array of the same shape as `x`
is returned.
See Also
--------
cos, arctan, arcsin, emath.arccos
Notes
-----
`arccos` is a multivalued function: for each `x` there are infinitely
many numbers `z` such that `cos(z) = x`. The convention is to return the
angle `z` whose real part lies in `[0, pi]`.
For real-valued input data types, `arccos` always returns real output.
For each value that cannot be expressed as a real number or infinity, it
yields ``nan`` and sets the `invalid` floating point error flag.
For complex-valued input, `arccos` is a complex analytical function that
has branch cuts `[-inf, -1]` and `[1, inf]` and is continuous from above
on the former and from below on the latter.
The inverse `cos` is also known as `acos` or cos^-1.
References
----------
.. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
10th printing, 1964, pp. 79. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Inverse trigonometric function",
http://en.wikipedia.org/wiki/Inverse_trigonometric_function
Examples
--------
We expect the arccos of 1 to be 0, and of -1 to be pi:
>>> np.arccos([1, -1])
array([ 0. , 3.14159265])
Plot arccos:
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-1, 1, num=100)
>>> plt.plot(x, np.arccos(x))
>>> plt.axis('tight')
>>> plt.show() Inverse hyperbolic cosine, elementwise.
Parameters
----------
x : array_like
Input array.
out : ndarray, optional
Array of the same shape as `x`, to store results in.
See `doc.ufuncs` (Section "Output arguments") for details.
Returns
-------
y : ndarray
Array of the same shape as `x`.
See Also
--------
cosh, arcsinh, sinh, arctanh, tanh
Notes
-----
`arccosh` is a multivalued function: for each `x` there are infinitely
many numbers `z` such that `cosh(z) = x`. The convention is to return the
`z` whose imaginary part lies in `[-pi, pi]` and the real part in
``[0, inf]``.
For real-valued input data types, `arccosh` always returns real output.
For each value that cannot be expressed as a real number or infinity, it
yields ``nan`` and sets the `invalid` floating point error flag.
For complex-valued input, `arccosh` is a complex analytical function that
has a branch cut `[-inf, 1]` and is continuous from above on it.
References
----------
.. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
10th printing, 1964, pp. 86. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Inverse hyperbolic function",
http://en.wikipedia.org/wiki/Arccosh
Examples
--------
>>> np.arccosh([np.e, 10.0])
array([ 1.65745445, 2.99322285])
>>> np.arccosh(1)
0.0 Inverse sine elementwise.
Parameters
----------
x : array_like
`y`-coordinate on the unit circle.
out : ndarray, optional
Array of the same shape as `x`, to store results in. See
`doc.ufuncs` (Section "Output arguments") for more details.
Returns
-------
angle : ndarray
The angle of the ray intersecting the unit circle at the given
`y`-coordinate in radians ``[-pi, pi]``. If `x` is a scalar then
a scalar is returned, otherwise an array is returned.
See Also
--------
sin, cos, arccos, tan, arctan, arctan2, emath.arcsin
Notes
-----
`arcsin` is a multivalued function: for each `x` there are infinitely
many numbers `z` such that `sin(z) = x`. The convention is to return the
angle `z` whose real part lies in `[-pi/2, pi/2]`.
For real-valued input data types, `arcsin` always returns real output.
For each value that cannot be expressed as a real number or infinity, it
yields ``nan`` and sets the `invalid` floating point error flag.
For complex-valued input, `arcsin` is a complex analytical function that
has branch cuts `[-inf, -1]` and `[1, inf]` and is continuous from above
on the former and from below on the latter.
The inverse sine is also known as `asin` or ``sin^-1``.
References
----------
.. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
10th printing, 1964, pp. 79. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Inverse trigonometric function",
http://en.wikipedia.org/wiki/Inverse_trigonometric_function
Examples
--------
>>> np.arcsin(1) # pi/2
1.5707963267948966
>>> np.arcsin(-1) # -pi/2
-1.5707963267948966
>>> np.arcsin(0)
0.0 Inverse hyperbolic sine elementwise.
Parameters
----------
x : array_like
Input array.
out : ndarray, optional
Array into which the output is placed. Its type is preserved and it
must be of the right shape to hold the output. See `doc.ufuncs`.
Returns
-------
out : ndarray
Array of of the same shape as `x`.
Notes
-----
`arcsinh` is a multivalued function: for each `x` there are infinitely
many numbers `z` such that `sinh(z) = x`. The convention is to return the
`z` whose imaginary part lies in `[-pi/2, pi/2]`.
For real-valued input data types, `arcsinh` always returns real output.
For each value that cannot be expressed as a real number or infinity, it
returns ``nan`` and sets the `invalid` floating point error flag.
For complex-valued input, `arccos` is a complex analytical function that
has branch cuts `[1j, infj]` and `[-1j, -infj]` and is continuous from
the right on the former and from the left on the latter.
The inverse hyperbolic sine is also known as `asinh` or ``sinh^-1``.
References
----------
.. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
10th printing, 1964, pp. 86. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Inverse hyperbolic function",
http://en.wikipedia.org/wiki/Arcsinh
Examples
--------
>>> np.arcsinh(np.array([np.e, 10.0]))
array([ 1.72538256, 2.99822295]) Trigonometric inverse tangent, element-wise.
The inverse of tan, so that if ``y = tan(x)`` then
``x = arctan(y)``.
Parameters
----------
x : array_like
Input values. `arctan` is applied to each element of `x`.
Returns
-------
out : ndarray
Out has the same shape as `x`. Its real part is
in ``[-pi/2, pi/2]``. It is a scalar if `x` is a scalar.
See Also
--------
arctan2 : Calculate the arctan of y/x.
Notes
-----
`arctan` is a multivalued function: for each `x` there are infinitely
many numbers `z` such that `tan(z) = x`. The convention is to return the
angle `z` whose real part lies in `[-pi/2, pi/2]`.
For real-valued input data types, `arctan` always returns real output.
For each value that cannot be expressed as a real number or infinity, it
yields ``nan`` and sets the `invalid` floating point error flag.
For complex-valued input, `arctan` is a complex analytical function that
has branch cuts `[1j, infj]` and `[-1j, -infj]` and is continuous from the
left on the former and from the right on the latter.
The inverse tangent is also known as `atan` or ``tan^-1``.
References
----------
.. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
10th printing, 1964, pp. 79. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Inverse trigonometric function",
http://en.wikipedia.org/wiki/Arctan
Examples
--------
We expect the arctan of 0 to be 0, and of 1 to be :math:`\pi/4`:
>>> np.arctan([0, 1])
array([ 0. , 0.78539816])
>>> np.pi/4
0.78539816339744828
Plot arctan:
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-10, 10)
>>> plt.plot(x, np.arctan(x))
>>> plt.axis('tight')
>>> plt.show() Elementwise arc tangent of ``x1/x2`` choosing the quadrant correctly.
The quadrant (ie. branch) is chosen so that ``arctan2(x1, x2)``
is the signed angle in radians between the line segments
``(0,0) - (1,0)`` and ``(0,0) - (x2,x1)``. This function is defined
also for `x2` = 0.
`arctan2` is not defined for complex-valued arguments.
Parameters
----------
x1 : array_like, real-valued
y-coordinates.
x2 : array_like, real-valued
x-coordinates. `x2` must be broadcastable to match the shape of `x1`,
or vice versa.
Returns
-------
angle : ndarray
Array of angles in radians, in the range ``[-pi, pi]``.
See Also
--------
arctan, tan
Notes
-----
`arctan2` is identical to the `atan2` function of the underlying
C library. The following special values are defined in the C standard [2]:
====== ====== ================
`x1` `x2` `arctan2(x1,x2)`
====== ====== ================
+/- 0 +0 +/- 0
+/- 0 -0 +/- pi
> 0 +/-inf +0 / +pi
< 0 +/-inf -0 / -pi
+/-inf +inf +/- (pi/4)
+/-inf -inf +/- (3*pi/4)
====== ====== ================
Note that +0 and -0 are distinct floating point numbers.
References
----------
.. [1] Wikipedia, "atan2",
http://en.wikipedia.org/wiki/Atan2
.. [2] ISO/IEC standard 9899:1999, "Programming language C", 1999.
Examples
--------
Consider four points in different quadrants:
>>> x = np.array([-1, +1, +1, -1])
>>> y = np.array([-1, -1, +1, +1])
>>> np.arctan2(y, x) * 180 / np.pi
array([-135., -45., 45., 135.])
Note the order of the parameters. `arctan2` is defined also when `x2` = 0
and at several other special points, obtaining values in
the range ``[-pi, pi]``:
>>> np.arctan2([1., -1.], [0., 0.])
array([ 1.57079633, -1.57079633])
>>> np.arctan2([0., 0., np.inf], [+0., -0., np.inf])
array([ 0. , 3.14159265, 0.78539816]) Inverse hyperbolic tangent elementwise.
Parameters
----------
x : array_like
Input array.
Returns
-------
out : ndarray
Array of the same shape as `x`.
See Also
--------
emath.arctanh
Notes
-----
`arctanh` is a multivalued function: for each `x` there are infinitely
many numbers `z` such that `tanh(z) = x`. The convention is to return the
`z` whose imaginary part lies in `[-pi/2, pi/2]`.
For real-valued input data types, `arctanh` always returns real output.
For each value that cannot be expressed as a real number or infinity, it
yields ``nan`` and sets the `invalid` floating point error flag.
For complex-valued input, `arctanh` is a complex analytical function that
has branch cuts `[-1, -inf]` and `[1, inf]` and is continuous from
above on the former and from below on the latter.
The inverse hyperbolic tangent is also known as `atanh` or ``tanh^-1``.
References
----------
.. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
10th printing, 1964, pp. 86. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Inverse hyperbolic function",
http://en.wikipedia.org/wiki/Arctanh
Examples
--------
>>> np.arctanh([0, -0.5])
array([ 0. , -0.54930614]) Compute the bit-wise AND of two arrays element-wise.
Computes the bit-wise AND of the underlying binary representation of
the integers in the input arrays. This ufunc implements the C/Python
operator ``&``.
Parameters
----------
x1, x2 : array_like
Only integer types are handled (including booleans).
Returns
-------
out : array_like
Result.
See Also
--------
logical_and
bitwise_or
bitwise_xor
binary_repr :
Return the binary representation of the input number as a string.
Examples
--------
The number 13 is represented by ``00001101``. Likewise, 17 is
represented by ``00010001``. The bit-wise AND of 13 and 17 is
therefore ``000000001``, or 1:
>>> np.bitwise_and(13, 17)
1
>>> np.bitwise_and(14, 13)
12
>>> np.binary_repr(12)
'1100'
>>> np.bitwise_and([14,3], 13)
array([12, 1])
>>> np.bitwise_and([11,7], [4,25])
array([0, 1])
>>> np.bitwise_and(np.array([2,5,255]), np.array([3,14,16]))
array([ 2, 4, 16])
>>> np.bitwise_and([True, True], [False, True])
array([False, True], dtype=bool) Compute the bit-wise OR of two arrays element-wise.
Computes the bit-wise OR of the underlying binary representation of
the integers in the input arrays. This ufunc implements the C/Python
operator ``|``.
Parameters
----------
x1, x2 : array_like
Only integer types are handled (including booleans).
out : ndarray, optional
Array into which the output is placed. Its type is preserved and it
must be of the right shape to hold the output. See doc.ufuncs.
Returns
-------
out : array_like
Result.
See Also
--------
logical_or
bitwise_and
bitwise_xor
binary_repr :
Return the binary representation of the input number as a string.
Examples
--------
The number 13 has the binaray representation ``00001101``. Likewise,
16 is represented by ``00010000``. The bit-wise OR of 13 and 16 is
then ``000111011``, or 29:
>>> np.bitwise_or(13, 16)
29
>>> np.binary_repr(29)
'11101'
>>> np.bitwise_or(32, 2)
34
>>> np.bitwise_or([33, 4], 1)
array([33, 5])
>>> np.bitwise_or([33, 4], [1, 2])
array([33, 6])
>>> np.bitwise_or(np.array([2, 5, 255]), np.array([4, 4, 4]))
array([ 6, 5, 255])
>>> np.array([2, 5, 255]) | np.array([4, 4, 4])
array([ 6, 5, 255])
>>> np.bitwise_or(np.array([2, 5, 255, 2147483647L], dtype=np.int32),
... np.array([4, 4, 4, 2147483647L], dtype=np.int32))
array([ 6, 5, 255, 2147483647])
>>> np.bitwise_or([True, True], [False, True])
array([ True, True], dtype=bool) Compute the bit-wise XOR of two arrays element-wise.
Computes the bit-wise XOR of the underlying binary representation of
the integers in the input arrays. This ufunc implements the C/Python
operator ``^``.
Parameters
----------
x1, x2 : array_like
Only integer types are handled (including booleans).
Returns
-------
out : array_like
Result.
See Also
--------
logical_xor
bitwise_and
bitwise_or
binary_repr :
Return the binary representation of the input number as a string.
Examples
--------
The number 13 is represented by ``00001101``. Likewise, 17 is
represented by ``00010001``. The bit-wise XOR of 13 and 17 is
therefore ``00011100``, or 28:
>>> np.bitwise_xor(13, 17)
28
>>> np.binary_repr(28)
'11100'
>>> np.bitwise_xor(31, 5)
26
>>> np.bitwise_xor([31,3], 5)
array([26, 6])
>>> np.bitwise_xor([31,3], [5,6])
array([26, 5])
>>> np.bitwise_xor([True, True], [False, True])
array([ True, False], dtype=bool) Return the ceiling of the input, element-wise.
The ceil of the scalar `x` is the smallest integer `i`, such that
`i >= x`. It is often denoted as :math:`\lceil x \rceil`.
Parameters
----------
x : array_like
Input data.
Returns
-------
y : {ndarray, scalar}
The ceiling of each element in `x`, with `float` dtype.
See Also
--------
floor, trunc, rint
Examples
--------
>>> a = np.array([-1.7, -1.5, -0.2, 0.2, 1.5, 1.7, 2.0])
>>> np.ceil(a)
array([-1., -1., -0., 1., 2., 2., 2.]) Return the complex conjugate, element-wise.
The complex conjugate of a complex number is obtained by changing the
sign of its imaginary part.
Parameters
----------
x : array_like
Input value.
Returns
-------
y : ndarray
The complex conjugate of `x`, with same dtype as `y`.
Examples
--------
>>> np.conjugate(1+2j)
(1-2j)
>>> x = np.eye(2) + 1j * np.eye(2)
>>> np.conjugate(x)
array([[ 1.-1.j, 0.-0.j],
[ 0.-0.j, 1.-1.j]]) Change the sign of x1 to that of x2, element-wise.
If both arguments are arrays or sequences, they have to be of the same
length. If `x2` is a scalar, its sign will be copied to all elements of
`x1`.
Parameters
----------
x1: array_like
Values to change the sign of.
x2: array_like
The sign of `x2` is copied to `x1`.
out : ndarray, optional
Array into which the output is placed. Its type is preserved and it
must be of the right shape to hold the output. See doc.ufuncs.
Returns
-------
out : array_like
The values of `x1` with the sign of `x2`.
Examples
--------
>>> np.copysign(1.3, -1)
-1.3
>>> 1/np.copysign(0, 1)
inf
>>> 1/np.copysign(0, -1)
-inf
>>> np.copysign([-1, 0, 1], -1.1)
array([-1., -0., -1.])
>>> np.copysign([-1, 0, 1], np.arange(3)-1)
array([-1., 0., 1.]) Cosine elementwise.
Parameters
----------
x : array_like
Input array in radians.
out : ndarray, optional
Output array of same shape as `x`.
Returns
-------
y : ndarray
The corresponding cosine values.
Raises
------
ValueError: invalid return array shape
if `out` is provided and `out.shape` != `x.shape` (See Examples)
Notes
-----
If `out` is provided, the function writes the result into it,
and returns a reference to `out`. (See Examples)
References
----------
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions.
New York, NY: Dover, 1972.
Examples
--------
>>> np.cos(np.array([0, np.pi/2, np.pi]))
array([ 1.00000000e+00, 6.12303177e-17, -1.00000000e+00])
>>>
>>> # Example of providing the optional output parameter
>>> out2 = np.cos([0.1], out1)
>>> out2 is out1
True
>>>
>>> # Example of ValueError due to provision of shape mis-matched `out`
>>> np.cos(np.zeros((3,3)),np.zeros((2,2)))
Traceback (most recent call last):
File "", line 1, in
ValueError: invalid return array shape Hyperbolic cosine, element-wise.
Equivalent to ``1/2 * (np.exp(x) + np.exp(-x))`` and ``np.cos(1j*x)``.
Parameters
----------
x : array_like
Input array.
Returns
-------
out : ndarray
Output array of same shape as `x`.
Examples
--------
>>> np.cosh(0)
1.0
The hyperbolic cosine describes the shape of a hanging cable:
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-4, 4, 1000)
>>> plt.plot(x, np.cosh(x))
>>> plt.show() Convert angles from degrees to radians.
Parameters
----------
x : array_like
Angles in degrees.
Returns
-------
y : ndarray
The corresponding angle in radians.
See Also
--------
rad2deg : Convert angles from radians to degrees.
unwrap : Remove large jumps in angle by wrapping.
Notes
-----
.. versionadded:: 1.3.0
``deg2rad(x)`` is ``x * pi / 180``.
Examples
--------
>>> np.deg2rad(180)
3.1415926535897931 Convert angles from radians to degrees.
Parameters
----------
x : array_like
Input array in radians.
out : ndarray, optional
Output array of same shape as x.
Returns
-------
y : ndarray of floats
The corresponding degree values.
See Also
--------
rad2deg : equivalent function
Examples
--------
Convert a radian array to degrees
>>> rad = np.arange(12.)*np.pi/6
>>> np.degrees(rad)
array([ 0., 30., 60., 90., 120., 150., 180., 210., 240.,
270., 300., 330.])
>>> out = np.zeros((rad.shape))
>>> r = degrees(rad, out)
>>> np.all(r == out)
True Divide arguments element-wise.
Parameters
----------
x1 : array_like
Dividend array.
x2 : array_like
Divisor array.
out : ndarray, optional
Array into which the output is placed. Its type is preserved and it
must be of the right shape to hold the output. See doc.ufuncs.
Returns
-------
y : {ndarray, scalar}
The quotient `x1/x2`, element-wise. Returns a scalar if
both `x1` and `x2` are scalars.
See Also
--------
seterr : Set whether to raise or warn on overflow, underflow and division
by zero.
Notes
-----
Equivalent to `x1` / `x2` in terms of array-broadcasting.
Behavior on division by zero can be changed using `seterr`.
When both `x1` and `x2` are of an integer type, `divide` will return
integers and throw away the fractional part. Moreover, division by zero
always yields zero in integer arithmetic.
Examples
--------
>>> np.divide(2.0, 4.0)
0.5
>>> x1 = np.arange(9.0).reshape((3, 3))
>>> x2 = np.arange(3.0)
>>> np.divide(x1, x2)
array([[ NaN, 1. , 1. ],
[ Inf, 4. , 2.5],
[ Inf, 7. , 4. ]])
Note the behavior with integer types:
>>> np.divide(2, 4)
0
>>> np.divide(2, 4.)
0.5
Division by zero always yields zero in integer arithmetic, and does not
raise an exception or a warning:
>>> np.divide(np.array([0, 1], dtype=int), np.array([0, 0], dtype=int))
array([0, 0])
Division by zero can, however, be caught using `seterr`:
>>> old_err_state = np.seterr(divide='raise')
>>> np.divide(1, 0)
Traceback (most recent call last):
File "", line 1, in
FloatingPointError: divide by zero encountered in divide
>>> ignored_states = np.seterr(**old_err_state)
>>> np.divide(1, 0)
0 Return (x1 == x2) element-wise.
Parameters
----------
x1, x2 : array_like
Input arrays of the same shape.
Returns
-------
out : {ndarray, bool}
Output array of bools, or a single bool if x1 and x2 are scalars.
See Also
--------
not_equal, greater_equal, less_equal, greater, less
Examples
--------
>>> np.equal([0, 1, 3], np.arange(3))
array([ True, True, False], dtype=bool)
What is compared are values, not types. So an int (1) and an array of
length one can evaluate as True:
>>> np.equal(1, np.ones(1))
array([ True], dtype=bool) Calculate the exponential of all elements in the input array.
Parameters
----------
x : array_like
Input values.
Returns
-------
out : ndarray
Output array, element-wise exponential of `x`.
See Also
--------
expm1 : Calculate ``exp(x) - 1`` for all elements in the array.
exp2 : Calculate ``2**x`` for all elements in the array.
Notes
-----
The irrational number ``e`` is also known as Euler's number. It is
approximately 2.718281, and is the base of the natural logarithm,
``ln`` (this means that, if :math:`x = \ln y = \log_e y`,
then :math:`e^x = y`. For real input, ``exp(x)`` is always positive.
For complex arguments, ``x = a + ib``, we can write
:math:`e^x = e^a e^{ib}`. The first term, :math:`e^a`, is already
known (it is the real argument, described above). The second term,
:math:`e^{ib}`, is :math:`\cos b + i \sin b`, a function with magnitude
1 and a periodic phase.
References
----------
.. [1] Wikipedia, "Exponential function",
http://en.wikipedia.org/wiki/Exponential_function
.. [2] M. Abramovitz and I. A. Stegun, "Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Tables," Dover, 1964, p. 69,
http://www.math.sfu.ca/~cbm/aands/page_69.htm
Examples
--------
Plot the magnitude and phase of ``exp(x)`` in the complex plane:
>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-2*np.pi, 2*np.pi, 100)
>>> xx = x + 1j * x[:, np.newaxis] # a + ib over complex plane
>>> out = np.exp(xx)
>>> plt.subplot(121)
>>> plt.imshow(np.abs(out),
... extent=[-2*np.pi, 2*np.pi, -2*np.pi, 2*np.pi])
>>> plt.title('Magnitude of exp(x)')
>>> plt.subplot(122)
>>> plt.imshow(np.angle(out),
... extent=[-2*np.pi, 2*np.pi, -2*np.pi, 2*np.pi])
>>> plt.title('Phase (angle) of exp(x)')
>>> plt.show() Calculate `2**p` for all `p` in the input array.
Parameters
----------
x : array_like
Input values.
out : ndarray, optional
Array to insert results into.
Returns
-------
out : ndarray
Element-wise 2 to the power `x`.
See Also
--------
exp : calculate x**p.
Notes
-----
.. versionadded:: 1.3.0
Examples
--------
>>> np.exp2([2, 3])
array([ 4., 8.]) Calculate ``exp(x) - 1`` for all elements in the array.
Parameters
----------
x : array_like
Input values.
Returns
-------
out : ndarray
Element-wise exponential minus one: ``out = exp(x) - 1``.
See Also
--------
log1p : ``log(1 + x)``, the inverse of expm1.
Notes
-----
This function provides greater precision than the formula ``exp(x) - 1``
for small values of ``x``.
Examples
--------
The true value of ``exp(1e-10) - 1`` is ``1.00000000005e-10`` to
about 32 significant digits. This example shows the superiority of
expm1 in this case.
>>> np.expm1(1e-10)
1.00000000005e-10
>>> np.exp(1e-10) - 1
1.000000082740371e-10 Compute the absolute values elementwise.
This function returns the absolute values (positive magnitude) of the data
in `x`. Complex values are not handled, use `absolute` to find the
absolute values of complex data.
Parameters
----------
x : array_like
The array of numbers for which the absolute values are required. If
`x` is a scalar, the result `y` will also be a scalar.
out : ndarray, optional
Array into which the output is placed. Its type is preserved and it
must be of the right shape to hold the output. See doc.ufuncs.
Returns
-------
y : {ndarray, scalar}
The absolute values of `x`, the returned values are always floats.
See Also
--------
absolute : Absolute values including `complex` types.
Examples
--------
>>> np.fabs(-1)
1.0
>>> np.fabs([-1.2, 1.2])
array([ 1.2, 1.2]) Return the floor of the input, element-wise.
The floor of the scalar `x` is the largest integer `i`, such that
`i <= x`. It is often denoted as :math:`\lfloor x \rfloor`.
Parameters
----------
x : array_like
Input data.
Returns
-------
y : {ndarray, scalar}
The floor of each element in `x`.
See Also
--------
ceil, trunc, rint
Notes
-----
Some spreadsheet programs calculate the "floor-towards-zero", in other
words ``floor(-2.5) == -2``. NumPy, however, uses the a definition of
`floor` such that `floor(-2.5) == -3`.
Examples
--------
>>> a = np.array([-1.7, -1.5, -0.2, 0.2, 1.5, 1.7, 2.0])
>>> np.floor(a)
array([-2., -2., -1., 0., 1., 1., 2.]) Return the largest integer smaller or equal to the division of the inputs.
Parameters
----------
x1 : array_like
Numerator.
x2 : array_like
Denominator.
Returns
-------
y : ndarray
y = floor(`x1`/`x2`)
See Also
--------
divide : Standard division.
floor : Round a number to the nearest integer toward minus infinity.
ceil : Round a number to the nearest integer toward infinity.
Examples
--------
>>> np.floor_divide(7,3)
2
>>> np.floor_divide([1., 2., 3., 4.], 2.5)
array([ 0., 0., 1., 1.]) Element-wise maximum of array elements.
Compare two arrays and returns a new array containing the element-wise
maxima. If one of the elements being compared is a nan, then the non-nan
element is returned. If both elements are nans then the first is returned.
The latter distinction is important for complex nans, which are defined as
at least one of the real or imaginary parts being a nan. The net effect is
that nans are ignored when possible.
Parameters
----------
x1, x2 : array_like
The arrays holding the elements to be compared. They must have
the same shape.
Returns
-------
y : {ndarray, scalar}
The minimum of `x1` and `x2`, element-wise. Returns scalar if
both `x1` and `x2` are scalars.
See Also
--------
fmin :
element-wise minimum that ignores nans unless both inputs are nans.
maximum :
element-wise maximum that propagates nans.
minimum :
element-wise minimum that propagates nans.
Notes
-----
.. versionadded:: 1.3.0
The fmax is equivalent to ``np.where(x1 >= x2, x1, x2)`` when neither
x1 nor x2 are nans, but it is faster and does proper broadcasting.
Examples
--------
>>> np.fmax([2, 3, 4], [1, 5, 2])
array([ 2., 5., 4.])
>>> np.fmax(np.eye(2), [0.5, 2])
array([[ 1. , 2. ],
[ 0.5, 2. ]])
>>> np.fmax([np.nan, 0, np.nan],[0, np.nan, np.nan])
array([ 0., 0., NaN]) fmin(x1, x2[, out])
Element-wise minimum of array elements.
Compare two arrays and returns a new array containing the element-wise
minima. If one of the elements being compared is a nan, then the non-nan
element is returned. If both elements are nans then the first is returned.
The latter distinction is important for complex nans, which are defined as
at least one of the real or imaginary parts being a nan. The net effect is
that nans are ignored when possible.
Parameters
----------
x1, x2 : array_like
The arrays holding the elements to be compared. They must have
the same shape.
Returns
-------
y : {ndarray, scalar}
The minimum of `x1` and `x2`, element-wise. Returns scalar if
both `x1` and `x2` are scalars.
See Also
--------
fmax :
element-wise maximum that ignores nans unless both inputs are nans.
maximum :
element-wise maximum that propagates nans.
minimum :
element-wise minimum that propagates nans.
Notes
-----
.. versionadded:: 1.3.0
The fmin is equivalent to ``np.where(x1 <= x2, x1, x2)`` when neither
x1 nor x2 are nans, but it is faster and does proper broadcasting.
Examples
--------
>>> np.fmin([2, 3, 4], [1, 5, 2])
array([2, 5, 4])
>>> np.fmin(np.eye(2), [0.5, 2])
array([[ 1. , 2. ],
[ 0.5, 2. ]])
>>> np.fmin([np.nan, 0, np.nan],[0, np.nan, np.nan])
array([ 0., 0., NaN]) Return the element-wise remainder of division.
This is the NumPy implementation of the Python modulo operator `%`.
Parameters
----------
x1 : array_like
Dividend.
x2 : array_like
Divisor.
Returns
-------
y : array_like
The remainder of the division of `x1` by `x2`.
See Also
--------
remainder : Modulo operation where the quotient is `floor(x1/x2)`.
divide
Notes
-----
The result of the modulo operation for negative dividend and divisors is
bound by conventions. In `fmod`, the sign of the remainder is the sign of
the dividend. In `remainder`, the sign of the divisor does not affect the
sign of the result.
Examples
--------
>>> np.fmod([-3, -2, -1, 1, 2, 3], 2)
array([-1, 0, -1, 1, 0, 1])
>>> np.remainder([-3, -2, -1, 1, 2, 3], 2)
array([1, 0, 1, 1, 0, 1])
>>> np.fmod([5, 3], [2, 2.])
array([ 1., 1.])
>>> a = np.arange(-3, 3).reshape(3, 2)
>>> a
array([[-3, -2],
[-1, 0],
[ 1, 2]])
>>> np.fmod(a, [2,2])
array([[-1, 0],
[-1, 0],
[ 1, 0]]) Return (x1 > x2) element-wise.
Parameters
----------
x1, x2 : array_like
Input arrays.
Returns
-------
Out : {ndarray, bool}
Output array of bools, or a single bool if `x1` and `x2` are scalars.
See Also
--------
greater_equal, less, less_equal, equal, not_equal
Examples
--------
>>> np.greater([4,2],[2,2])
array([ True, False], dtype=bool)
If the inputs are ndarrays, then np.greater is equivalent to '>'.
>>> a = np.array([4,2])
>>> b = np.array([2,2])
>>> a > b
array([ True, False], dtype=bool) Return (x1 >= x2) element-wise.
Parameters
----------
x1, x2 : array_like
Input arrays.
Returns
-------
out : {ndarray, bool}
Output array of bools, or a single bool if x1 and x2 are scalars.
See Also
--------
greater, less, less_equal, equal, not_equal
Examples
--------
>>> np.greater_equal([4, 2, 1], [2, 2, 2])
array([ True, True, False], dtype=bool) Given the "legs" of a right triangle, return its hypotenuse.
Equivalent to ``sqrt(x1**2 + x2**2)``, element-wise. If `x1` or
`x2` is scalar_like (i.e., unambiguously cast-able to a scalar type),
it is broadcast for use with each element of the other argument.
(See Examples)
Parameters
----------
x1, x2 : array_like
Leg of the triangle(s).
out : ndarray, optional
Array into which the output is placed. Its type is preserved and it
must be of the right shape to hold the output. See doc.ufuncs.
Returns
-------
z : ndarray
The hypotenuse of the triangle(s).
Examples
--------
>>> np.hypot(3*np.ones((3, 3)), 4*np.ones((3, 3)))
array([[ 5., 5., 5.],
[ 5., 5., 5.],
[ 5., 5., 5.]])
Example showing broadcast of scalar_like argument:
>>> np.hypot(3*np.ones((3, 3)), [4])
array([[ 5., 5., 5.],
[ 5., 5., 5.],
[ 5., 5., 5.]]) Compute bit-wise inversion, or bit-wise NOT, element-wise.
Computes the bit-wise NOT of the underlying binary representation of
the integers in the input arrays. This ufunc implements the C/Python
operator ``~``.
For signed integer inputs, the two's complement is returned.
In a two's-complement system negative numbers are represented by the two's
complement of the absolute value. This is the most common method of
representing signed integers on computers [1]_. A N-bit two's-complement
system can represent every integer in the range
:math:`-2^{N-1}` to :math:`+2^{N-1}-1`.
Parameters
----------
x1 : array_like
Only integer types are handled (including booleans).
Returns
-------
out : array_like
Result.
See Also
--------
bitwise_and, bitwise_or, bitwise_xor
logical_not
binary_repr :
Return the binary representation of the input number as a string.
Notes
-----
`bitwise_not` is an alias for `invert`:
>>> np.bitwise_not is np.invert
True
References
----------
.. [1] Wikipedia, "Two's complement",
http://en.wikipedia.org/wiki/Two's_complement
Examples
--------
We've seen that 13 is represented by ``00001101``.
The invert or bit-wise NOT of 13 is then:
>>> np.invert(np.array([13], dtype=uint8))
array([242], dtype=uint8)
>>> np.binary_repr(x, width=8)
'00001101'
>>> np.binary_repr(242, width=8)
'11110010'
The result depends on the bit-width:
>>> np.invert(np.array([13], dtype=uint16))
array([65522], dtype=uint16)
>>> np.binary_repr(x, width=16)
'0000000000001101'
>>> np.binary_repr(65522, width=16)
'1111111111110010'
When using signed integer types the result is the two's complement of
the result for the unsigned type:
>>> np.invert(np.array([13], dtype=int8))
array([-14], dtype=int8)
>>> np.binary_repr(-14, width=8)
'11110010'
Booleans are accepted as well:
>>> np.invert(array([True, False]))
array([False, True], dtype=bool) Test element-wise for finite-ness (not infinity or not Not a Number).
The result is returned as a boolean array.
Parameters
----------
x : array_like
Input values.
out : ndarray, optional
Array into which the output is placed. Its type is preserved and it
must be of the right shape to hold the output. See `doc.ufuncs`.
Returns
-------
y : ndarray, bool
For scalar input, the result is a new boolean with value True
if the input is finite; otherwise the value is False (input is
either positive infinity, negative infinity or Not a Number).
For array input, the result is a boolean array with the same
dimensions as the input and the values are True if the corresponding
element of the input is finite; otherwise the values are False (element
is either positive infinity, negative infinity or Not a Number).
See Also
--------
isinf, isneginf, isposinf, isnan
Notes
-----
Not a Number, positive infinity and negative infinity are considered
to be non-finite.
Numpy uses the IEEE Standard for Binary Floating-Point for Arithmetic
(IEEE 754). This means that Not a Number is not equivalent to infinity.
Also that positive infinity is not equivalent to negative infinity. But
infinity is equivalent to positive infinity.
Errors result if the second argument is also supplied when `x` is a scalar
input, or if first and second arguments have different shapes.
Examples
--------
>>> np.isfinite(1)
True
>>> np.isfinite(0)
True
>>> np.isfinite(np.nan)
False
>>> np.isfinite(np.inf)
False
>>> np.isfinite(np.NINF)
False
>>> np.isfinite([np.log(-1.),1.,np.log(0)])
array([False, True, False], dtype=bool)
>>> x = np.array([-np.inf, 0., np.inf])
>>> y = np.array([2, 2, 2])
>>> np.isfinite(x, y)
array([0, 1, 0])
>>> y
array([0, 1, 0]) Test element-wise for positive or negative infinity, return result as bool
array.
Parameters
----------
x : array_like
Input values
y : array_like, optional
An array with the same shape as `x` to store the result.
Returns
-------
y : {ndarray, bool}
For scalar input, the result is a new boolean with value True
if the input is positive or negative infinity; otherwise the value
is False.
For array input, the result is a boolean array with the same
dimensions as the input and the values are True if the corresponding
element of the input is positive or negative infinity; otherwise the
values are False. If a second argument is supplied the result is
stored there. If the type of that array is a numeric type the result
is represented as zeros and ones, if the type is boolean then as
False and True. The return value `y` is then a reference to that array.
See Also
--------
isneginf, isposinf, isnan, isfinite
Notes
-----
Numpy uses the IEEE Standard for Binary Floating-Point for Arithmetic
(IEEE 754).
Errors result if second argument is also supplied with scalar input or
if first and second arguments have different shapes.
Examples
--------
>>> np.isinf(np.inf)
True
>>> np.isinf(np.nan)
False
>>> np.isinf(np.NINF)
True
>>> np.isinf([np.inf, -np.inf, 1.0, np.nan])
array([ True, True, False, False], dtype=bool)
>>> x = np.array([-np.inf, 0., np.inf])
>>> y = np.array([2, 2, 2])
>>> np.isinf(x, y)
array([1, 0, 1])
>>> y
array([1, 0, 1]) Test element-wise for Not a Number (NaN), return result as a bool array.
Parameters
----------
x : array_like
Input array.
Returns
-------
y : {ndarray, bool}
For scalar input, the result is a new boolean with value True
if the input is NaN; otherwise the value is False.
For array input, the result is a boolean array with the same
dimensions as the input and the values are True if the corresponding
element of the input is NaN; otherwise the values are False.
See Also
--------
isinf, isneginf, isposinf, isfinite
Notes
-----
Numpy uses the IEEE Standard for Binary Floating-Point for Arithmetic
(IEEE 754). This means that Not a Number is not equivalent to infinity.
Examples
--------
>>> np.isnan(np.nan)
True
>>> np.isnan(np.inf)
False
>>> np.isnan([np.log(-1.),1.,np.log(0)])
array([ True, False, False], dtype=bool) Shift the bits of an integer to the left.
Bits are shifted to the left by appending `x2` 0s at the right of `x1`.
Since the internal representation of numbers is in binary format, this
operation is equivalent to multiplying `x1` by ``2**x2``.
Parameters
----------
x1 : array_like of integer type
Input values.
x2 : array_like of integer type
Number of zeros to append to `x1`.
Returns
-------
out : array of integer type
Return `x1` with bits shifted `x2` times to the left.
See Also
--------
right_shift : Shift the bits of an integer to the right.
binary_repr : Return the binary representation of the input number
as a string.
Examples
--------
>>> np.binary_repr(5)
'101'
>>> np.left_shift(5, 2)
20
>>> np.binary_repr(20)
'10100'
>>> np.left_shift(5, [1,2,3])
array([10, 20, 40]) Return (x1 < x2) element-wise.
Parameters
----------
x1, x2 : array_like
Input arrays.
Returns
-------
Out : ndarray of bools
Output array of bools, or a single bool if `x1` and `x2` are scalars.
See Also
--------
less_equal, greater, greater_equal, equal, not_equal
Examples
--------
>>> np.less([1, 2], [2, 2])
array([ True, False], dtype=bool) Return (x1 <= x2) element-wise.
Parameters
----------
x1, x2 : array_like
Input arrays.
Returns
-------
Out : {ndarray, bool}
Output array of bools, or a single bool if `x1` and `x2` are scalars.
See Also
--------
less, greater_equal, greater, equal, not_equal
Examples
--------
>>> np.less_equal([1, 2, 3], [2, 2, 2])
array([ True, True, False], dtype=bool) Natural logarithm, element-wise.
The natural logarithm `log` is the inverse of the exponential function,
so that `log(exp(x)) = x`. The natural logarithm is logarithm in base `e`.
Parameters
----------
x : array_like
Input value.
Returns
-------
y : ndarray
The natural logarithm of `x`, element-wise.
See Also
--------
log10, log2, log1p, emath.log
Notes
-----
Logarithm is a multivalued function: for each `x` there is an infinite
number of `z` such that `exp(z) = x`. The convention is to return the `z`
whose imaginary part lies in `[-pi, pi]`.
For real-valued input data types, `log` always returns real output. For
each value that cannot be expressed as a real number or infinity, it
yields ``nan`` and sets the `invalid` floating point error flag.
For complex-valued input, `log` is a complex analytical function that
has a branch cut `[-inf, 0]` and is continuous from above on it. `log`
handles the floating-point negative zero as an infinitesimal negative
number, conforming to the C99 standard.
References
----------
.. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
10th printing, 1964, pp. 67. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Logarithm". http://en.wikipedia.org/wiki/Logarithm
Examples
--------
>>> np.log([1, np.e, np.e**2, 0])
array([ 0., 1., 2., -Inf]) Return the base 10 logarithm of the input array, element-wise.
Parameters
----------
x : array_like
Input values.
Returns
-------
y : ndarray
The logarithm to the base 10 of `x`, element-wise. NaNs are
returned where x is negative.
See Also
--------
emath.log10
Notes
-----
Logarithm is a multivalued function: for each `x` there is an infinite
number of `z` such that `10**z = x`. The convention is to return the `z`
whose imaginary part lies in `[-pi, pi]`.
For real-valued input data types, `log10` always returns real output. For
each value that cannot be expressed as a real number or infinity, it
yields ``nan`` and sets the `invalid` floating point error flag.
For complex-valued input, `log10` is a complex analytical function that
has a branch cut `[-inf, 0]` and is continuous from above on it. `log10`
handles the floating-point negative zero as an infinitesimal negative
number, conforming to the C99 standard.
References
----------
.. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
10th printing, 1964, pp. 67. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Logarithm". http://en.wikipedia.org/wiki/Logarithm
Examples
--------
>>> np.log10([1e-15, -3.])
array([-15., NaN]) Return the natural logarithm of one plus the input array, element-wise.
Calculates ``log(1 + x)``.
Parameters
----------
x : array_like
Input values.
Returns
-------
y : ndarray
Natural logarithm of `1 + x`, element-wise.
See Also
--------
expm1 : ``exp(x) - 1``, the inverse of `log1p`.
Notes
-----
For real-valued input, `log1p` is accurate also for `x` so small
that `1 + x == 1` in floating-point accuracy.
Logarithm is a multivalued function: for each `x` there is an infinite
number of `z` such that `exp(z) = 1 + x`. The convention is to return
the `z` whose imaginary part lies in `[-pi, pi]`.
For real-valued input data types, `log1p` always returns real output. For
each value that cannot be expressed as a real number or infinity, it
yields ``nan`` and sets the `invalid` floating point error flag.
For complex-valued input, `log1p` is a complex analytical function that
has a branch cut `[-inf, -1]` and is continuous from above on it. `log1p`
handles the floating-point negative zero as an infinitesimal negative
number, conforming to the C99 standard.
References
----------
.. [1] M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions",
10th printing, 1964, pp. 67. http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Logarithm". http://en.wikipedia.org/wiki/Logarithm
Examples
--------
>>> np.log1p(1e-99)
1e-99
>>> np.log(1 + 1e-99)
0.0 Base-2 logarithm of `x`.
Parameters
----------
x : array_like
Input values.
Returns
-------
y : ndarray
Base-2 logarithm of `x`.
See Also
--------
log, log10, log1p, emath.log2
Notes
-----
.. versionadded:: 1.3.0
Logarithm is a multivalued function: for each `x` there is an infinite
number of `z` such that `2**z = x`. The convention is to return the `z`
whose imaginary part lies in `[-pi, pi]`.
For real-valued input data types, `log2` always returns real output. For
each value that cannot be expressed as a real number or infinity, it
yields ``nan`` and sets the `invalid` floating point error flag.
For complex-valued input, `log2` is a complex analytical function that
has a branch cut `[-inf, 0]` and is continuous from above on it. `log2`
handles the floating-point negative zero as an infinitesimal negative
number, conforming to the C99 standard. Logarithm of the sum of exponentiations of the inputs.
Calculates ``log(exp(x1) + exp(x2))``. This function is useful in
statistics where the calculated probabilities of events may be so small
as to exceed the range of normal floating point numbers. In such cases
the logarithm of the calculated probability is stored. This function
allows adding probabilities stored in such a fashion.
Parameters
----------
x1, x2 : array_like
Input values.
Returns
-------
result : ndarray
Logarithm of ``exp(x1) + exp(x2)``.
See Also
--------
logaddexp2: Logarithm of the sum of exponentiations of inputs in base-2.
Notes
-----
.. versionadded:: 1.3.0
Examples
--------
>>> prob1 = np.log(1e-50)
>>> prob2 = np.log(2.5e-50)
>>> prob12 = np.logaddexp(prob1, prob2)
>>> prob12
-113.87649168120691
>>> np.exp(prob12)
3.5000000000000057e-50 Logarithm of the sum of exponentiations of the inputs in base-2.
Calculates ``log2(2**x1 + 2**x2)``. This function is useful in machine
learning when the calculated probabilities of events may be so small
as to exceed the range of normal floating point numbers. In such cases
the base-2 logarithm of the calculated probability can be used instead.
This function allows adding probabilities stored in such a fashion.
Parameters
----------
x1, x2 : array_like
Input values.
out : ndarray, optional
Array to store results in.
Returns
-------
result : ndarray
Base-2 logarithm of ``2**x1 + 2**x2``.
See Also
--------
logaddexp: Logarithm of the sum of exponentiations of the inputs.
Notes
-----
.. versionadded:: 1.3.0
Examples
--------
>>> prob1 = np.log2(1e-50)
>>> prob2 = np.log2(2.5e-50)
>>> prob12 = np.logaddexp2(prob1, prob2)
>>> prob1, prob2, prob12
(-166.09640474436813, -164.77447664948076, -164.28904982231052)
>>> 2**prob12
3.4999999999999914e-50 Compute the truth value of x1 AND x2 elementwise.
Parameters
----------
x1, x2 : array_like
Input arrays. `x1` and `x2` must be of the same shape.
Returns
-------
y : {ndarray, bool}
Boolean result with the same shape as `x1` and `x2` of the logical
AND operation on corresponding elements of `x1` and `x2`.
See Also
--------
logical_or, logical_not, logical_xor
bitwise_and
Examples
--------
>>> np.logical_and(True, False)
False
>>> np.logical_and([True, False], [False, False])
array([False, False], dtype=bool)
>>> x = np.arange(5)
>>> np.logical_and(x>1, x<4)
array([False, False, True, True, False], dtype=bool) Compute the truth value of NOT x elementwise.
Parameters
----------
x : array_like
Logical NOT is applied to the elements of `x`.
Returns
-------
y : {ndarray, bool}
Boolean result with the same shape as `x` of the NOT operation
on elements of `x`.
See Also
--------
logical_and, logical_or, logical_xor
Examples
--------
>>> np.logical_not(3)
False
>>> np.logical_not([True, False, 0, 1])
array([False, True, True, False], dtype=bool)
>>> x = np.arange(5)
>>> np.logical_not(x<3)
array([False, False, False, True, True], dtype=bool) Compute the truth value of x1 OR x2 elementwise.
Parameters
----------
x1, x2 : array_like
Logical OR is applied to the elements of `x1` and `x2`.
They have to be of the same shape.
Returns
-------
y : {ndarray, bool}
Boolean result with the same shape as `x1` and `x2` of the logical
OR operation on elements of `x1` and `x2`.
See Also
--------
logical_and, logical_not, logical_xor
bitwise_or
Examples
--------
>>> np.logical_or(True, False)
True
>>> np.logical_or([True, False], [False, False])
array([ True, False], dtype=bool)
>>> x = np.arange(5)
>>> np.logical_or(x < 1, x > 3)
array([ True, False, False, False, True], dtype=bool) Compute the truth value of x1 XOR x2 elementwise.
Parameters
----------
x1, x2 : array_like
Logical XOR is applied to the elements of `x1` and `x2`.
They have to be of the same shape.
Returns
-------
y : {ndarray, bool}
Boolean result with the same shape as `x1` and `x2` of the logical
XOR operation on elements of `x1` and `x2`.
See Also
--------
logical_and, logical_or, logical_not
bitwise_xor
Examples
--------
>>> np.logical_xor(True, False)
True
>>> np.logical_xor([True, True, False, False], [True, False, True, False])
array([False, True, True, False], dtype=bool)
>>> x = np.arange(5)
>>> np.logical_xor(x < 1, x > 3)
array([ True, False, False, False, True], dtype=bool) Element-wise maximum of array elements.
Compare two arrays and returns a new array containing
the element-wise maxima. If one of the elements being
compared is a nan, then that element is returned. If
both elements are nans then the first is returned. The
latter distinction is important for complex nans,
which are defined as at least one of the real or
imaginary parts being a nan. The net effect is that
nans are propagated.
Parameters
----------
x1, x2 : array_like
The arrays holding the elements to be compared. They must have
the same shape, or shapes that can be broadcast to a single shape.
Returns
-------
y : {ndarray, scalar}
The maximum of `x1` and `x2`, element-wise. Returns scalar if
both `x1` and `x2` are scalars.
See Also
--------
minimum :
element-wise minimum
fmax :
element-wise maximum that ignores nans unless both inputs are nans.
fmin :
element-wise minimum that ignores nans unless both inputs are nans.
Notes
-----
Equivalent to ``np.where(x1 > x2, x1, x2)`` but faster and does proper
broadcasting.
Examples
--------
>>> np.maximum([2, 3, 4], [1, 5, 2])
array([2, 5, 4])
>>> np.maximum(np.eye(2), [0.5, 2])
array([[ 1. , 2. ],
[ 0.5, 2. ]])
>>> np.maximum([np.nan, 0, np.nan], [0, np.nan, np.nan])
array([ NaN, NaN, NaN])
>>> np.maximum(np.Inf, 1)
inf Element-wise minimum of array elements.
Compare two arrays and returns a new array containing the element-wise
minima. If one of the elements being compared is a nan, then that element
is returned. If both elements are nans then the first is returned. The
latter distinction is important for complex nans, which are defined as at
least one of the real or imaginary parts being a nan. The net effect is
that nans are propagated.
Parameters
----------
x1, x2 : array_like
The arrays holding the elements to be compared. They must have
the same shape, or shapes that can be broadcast to a single shape.
Returns
-------
y : {ndarray, scalar}
The minimum of `x1` and `x2`, element-wise. Returns scalar if
both `x1` and `x2` are scalars.
See Also
--------
maximum :
element-wise minimum that propagates nans.
fmax :
element-wise maximum that ignores nans unless both inputs are nans.
fmin :
element-wise minimum that ignores nans unless both inputs are nans.
Notes
-----
The minimum is equivalent to ``np.where(x1 <= x2, x1, x2)`` when neither
x1 nor x2 are nans, but it is faster and does proper broadcasting.
Examples
--------
>>> np.minimum([2, 3, 4], [1, 5, 2])
array([1, 3, 2])
>>> np.minimum(np.eye(2), [0.5, 2]) # broadcasting
array([[ 0.5, 0. ],
[ 0. , 1. ]])
>>> np.minimum([np.nan, 0, np.nan],[0, np.nan, np.nan])
array([ NaN, NaN, NaN]) Return the fractional and integral parts of an array, element-wise.
The fractional and integral parts are negative if the given number is
negative.
Parameters
----------
x : array_like
Input array.
Returns
-------
y1 : ndarray
Fractional part of `x`.
y2 : ndarray
Integral part of `x`.
Notes
-----
For integer input the return values are floats.
Examples
--------
>>> np.modf([0, 3.5])
(array([ 0. , 0.5]), array([ 0., 3.]))
>>> np.modf(-0.5)
(-0.5, -0) Multiply arguments element-wise.
Parameters
----------
x1, x2 : array_like
Input arrays to be multiplied.
Returns
-------
y : ndarray
The product of `x1` and `x2`, element-wise. Returns a scalar if
both `x1` and `x2` are scalars.
Notes
-----
Equivalent to `x1` * `x2` in terms of array broadcasting.
Examples
--------
>>> np.multiply(2.0, 4.0)
8.0
>>> x1 = np.arange(9.0).reshape((3, 3))
>>> x2 = np.arange(3.0)
>>> np.multiply(x1, x2)
array([[ 0., 1., 4.],
[ 0., 4., 10.],
[ 0., 7., 16.]]) Returns an array with the negative of each element of the original array.
Parameters
----------
x : {array_like, scalar}
Input array.
Returns
-------
y : {ndarray, scalar}
Returned array or scalar `y=-x`.
Examples
--------
>>> np.negative([1.,-1.])
array([-1., 1.]) Return the next representable floating-point value after x1 in the direction
of x2 element-wise.
Parameters
----------
x1 : array_like
Values to find the next representable value of.
x2 : array_like
The direction where to look for the next representable value of `x1`.
out : ndarray, optional
Array into which the output is placed. Its type is preserved and it
must be of the right shape to hold the output. See `doc.ufuncs`.
Returns
-------
out : array_like
The next representable values of `x1` in the direction of `x2`.
Examples
--------
>>> eps = np.finfo(np.float64).eps
>>> np.nextafter(1, 2) == eps + 1
True
>>> np.nextafter([1, 2], [2, 1]) == [eps + 1, 2 - eps]
array([ True, True], dtype=bool) Return (x1 != x2) element-wise.
Parameters
----------
x1, x2 : array_like
Input arrays.
out : ndarray, optional
A placeholder the same shape as `x1` to store the result.
See `doc.ufuncs` (Section "Output arguments") for more details.
Returns
-------
not_equal : ndarray bool, scalar bool
For each element in `x1, x2`, return True if `x1` is not equal
to `x2` and False otherwise.
See Also
--------
equal, greater, greater_equal, less, less_equal
Examples
--------
>>> np.not_equal([1.,2.], [1., 3.])
array([False, True], dtype=bool)
>>> np.not_equal([1, 2], [[1, 3],[1, 4]])
array([[False, True],
[False, True]], dtype=bool) Returns an array of ones with the same shape and type as a given array.
Equivalent to ``a.copy().fill(1)``.
Please refer to the documentation for `zeros_like`.
See Also
--------
zeros_like
Examples
--------
>>> a = np.array([[1, 2, 3], [4, 5, 6]])
>>> np.ones_like(a)
array([[1, 1, 1],
[1, 1, 1]]) Returns element-wise base array raised to power from second array.
Raise each base in `x1` to the power of the exponents in `x2`. This
requires that `x1` and `x2` must be broadcastable to the same shape.
Parameters
----------
x1 : array_like
The bases.
x2 : array_like
The exponents.
Returns
-------
y : ndarray
The bases in `x1` raised to the exponents in `x2`.
Examples
--------
Cube each element in a list.
>>> x1 = range(6)
>>> x1
[0, 1, 2, 3, 4, 5]
>>> np.power(x1, 3)
array([ 0, 1, 8, 27, 64, 125])
Raise the bases to different exponents.
>>> x2 = [1.0, 2.0, 3.0, 3.0, 2.0, 1.0]
>>> np.power(x1, x2)
array([ 0., 1., 8., 27., 16., 5.])
The effect of broadcasting.
>>> x2 = np.array([[1, 2, 3, 3, 2, 1], [1, 2, 3, 3, 2, 1]])
>>> x2
array([[1, 2, 3, 3, 2, 1],
[1, 2, 3, 3, 2, 1]])
>>> np.power(x1, x2)
array([[ 0, 1, 8, 27, 16, 5],
[ 0, 1, 8, 27, 16, 5]]) Convert angles from radians to degrees.
Parameters
----------
x : array_like
Angle in radians.
out : ndarray, optional
Array into which the output is placed. Its type is preserved and it
must be of the right shape to hold the output. See doc.ufuncs.
Returns
-------
y : ndarray
The corresponding angle in degrees.
See Also
--------
deg2rad : Convert angles from degrees to radians.
unwrap : Remove large jumps in angle by wrapping.
Notes
-----
.. versionadded:: 1.3.0
rad2deg(x) is ``180 * x / pi``.
Examples
--------
>>> np.rad2deg(np.pi/2)
90.0 Convert angles from degrees to radians.
Parameters
----------
x : array_like
Input array in degrees.
out : ndarray, optional
Output array of same shape as x.
Returns
-------
y : ndarray
The corresponding radian values.
See Also
--------
deg2rad : equivalent function
Examples
--------
Convert a degree array to radians
>>> deg = np.arange(12.) * 30.
>>> np.radians(deg)
array([ 0. , 0.52359878, 1.04719755, 1.57079633, 2.0943951 ,
2.61799388, 3.14159265, 3.66519143, 4.1887902 , 4.71238898,
5.23598776, 5.75958653])
>>> out = np.zeros((deg.shape))
>>> ret = np.radians(deg, out)
>>> ret is out
True Return the reciprocal of the argument, element-wise.
Calculates ``1/x``.
Parameters
----------
x : array_like
Input array.
Returns
-------
y : ndarray
Return array.
Notes
-----
.. note::
This function is not designed to work with integers.
For integer arguments with absolute value larger than 1 the result is
always zero because of the way Python handles integer division.
For integer zero the result is an overflow.
Examples
--------
>>> np.reciprocal(2.)
0.5
>>> np.reciprocal([1, 2., 3.33])
array([ 1. , 0.5 , 0.3003003]) Return element-wise remainder of division.
Computes ``x1 - floor(x1 / x2) * x2``.
Parameters
----------
x1 : array_like
Dividend array.
x2 : array_like
Divisor array.
out : ndarray, optional
Array into which the output is placed. Its type is preserved and it
must be of the right shape to hold the output. See doc.ufuncs.
Returns
-------
y : ndarray
The remainder of the quotient ``x1/x2``, element-wise. Returns a scalar
if both `x1` and `x2` are scalars.
See Also
--------
divide, floor
Notes
-----
Returns 0 when `x2` is 0 and both `x1` and `x2` are (arrays of) integers.
Examples
--------
>>> np.remainder([4, 7], [2, 3])
array([0, 1])
>>> np.remainder(np.arange(7), 5)
array([0, 1, 2, 3, 4, 0, 1]) Shift the bits of an integer to the right.
Bits are shifted to the right by removing `x2` bits at the right of `x1`.
Since the internal representation of numbers is in binary format, this
operation is equivalent to dividing `x1` by ``2**x2``.
Parameters
----------
x1 : array_like, int
Input values.
x2 : array_like, int
Number of bits to remove at the right of `x1`.
Returns
-------
out : ndarray, int
Return `x1` with bits shifted `x2` times to the right.
See Also
--------
left_shift : Shift the bits of an integer to the left.
binary_repr : Return the binary representation of the input number
as a string.
Examples
--------
>>> np.binary_repr(10)
'1010'
>>> np.right_shift(10, 1)
5
>>> np.binary_repr(5)
'101'
>>> np.right_shift(10, [1,2,3])
array([5, 2, 1]) Round elements of the array to the nearest integer.
Parameters
----------
x : array_like
Input array.
Returns
-------
out : {ndarray, scalar}
Output array is same shape and type as `x`.
See Also
--------
ceil, floor, trunc
Examples
--------
>>> a = np.array([-1.7, -1.5, -0.2, 0.2, 1.5, 1.7, 2.0])
>>> np.rint(a)
array([-2., -2., -0., 0., 2., 2., 2.]) Returns an element-wise indication of the sign of a number.
The `sign` function returns ``-1 if x < 0, 0 if x==0, 1 if x > 0``.
Parameters
----------
x : array_like
Input values.
Returns
-------
y : ndarray
The sign of `x`.
Examples
--------
>>> np.sign([-5., 4.5])
array([-1., 1.])
>>> np.sign(0)
0 Returns element-wise True where signbit is set (less than zero).
Parameters
----------
x: array_like
The input value(s).
out : ndarray, optional
Array into which the output is placed. Its type is preserved
and it must be of the right shape to hold the output. See doc.ufuncs.
Returns
-------
out : array_like, bool
Output array.
Examples
--------
>>> np.signbit(-1.2)
True
>>> np.signbit(np.array([1, -2.3, 2.1]))
array([False, True, False], dtype=bool) Trigonometric sine, element-wise.
Parameters
----------
x : array_like
Angle, in radians (:math:`2 \pi` rad equals 360 degrees).
Returns
-------
y : array_like
The sine of each element of x.
See Also
--------
arcsin, sinh, cos
Notes
-----
The sine is one of the fundamental functions of trigonometry
(the mathematical study of triangles). Consider a circle of radius
1 centered on the origin. A ray comes in from the :math:`+x` axis,
makes an angle at the origin (measured counter-clockwise from that
axis), and departs from the origin. The :math:`y` coordinate of
the outgoing ray's intersection with the unit circle is the sine
of that angle. It ranges from -1 for :math:`x=3\pi / 2` to
+1 for :math:`\pi / 2.` The function has zeroes where the angle is
a multiple of :math:`\pi`. Sines of angles between :math:`\pi` and
:math:`2\pi` are negative. The numerous properties of the sine and
related functions are included in any standard trigonometry text.
Examples
--------
Print sine of one angle:
>>> np.sin(np.pi/2.)
1.0
Print sines of an array of angles given in degrees:
>>> np.sin(np.array((0., 30., 45., 60., 90.)) * np.pi / 180. )
array([ 0. , 0.5 , 0.70710678, 0.8660254 , 1. ])
Plot the sine function:
>>> import matplotlib.pylab as plt
>>> x = np.linspace(-np.pi, np.pi, 201)
>>> plt.plot(x, np.sin(x))
>>> plt.xlabel('Angle [rad]')
>>> plt.ylabel('sin(x)')
>>> plt.axis('tight')
>>> plt.show() Hyperbolic sine, element-wise.
Equivalent to ``1/2 * (np.exp(x) - np.exp(-x))`` or
``-1j * np.sin(1j*x)``.
Parameters
----------
x : array_like
Input array.
out : ndarray, optional
Output array of same shape as `x`.
Returns
-------
y : ndarray
The corresponding hyperbolic sine values.
Raises
------
ValueError: invalid return array shape
if `out` is provided and `out.shape` != `x.shape` (See Examples)
Notes
-----
If `out` is provided, the function writes the result into it,
and returns a reference to `out`. (See Examples)
References
----------
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions.
New York, NY: Dover, 1972, pg. 83.
Examples
--------
>>> np.sinh(0)
0.0
>>> np.sinh(np.pi*1j/2)
1j
>>> np.sinh(np.pi*1j) # (exact value is 0)
1.2246063538223773e-016j
>>> # Discrepancy due to vagaries of floating point arithmetic.
>>> # Example of providing the optional output parameter
>>> out2 = np.sinh([0.1], out1)
>>> out2 is out1
True
>>> # Example of ValueError due to provision of shape mis-matched `out`
>>> np.sinh(np.zeros((3,3)),np.zeros((2,2)))
Traceback (most recent call last):
File "", line 1, in
ValueError: invalid return array shape Return the distance between x and the nearest adjacent number.
Parameters
----------
x1: array_like
Values to find the spacing of.
Returns
-------
out : array_like
The spacing of values of `x1`.
Notes
-----
It can be considered as a generalization of EPS:
``spacing(np.float64(1)) == np.finfo(np.float64).eps``, and there
should not be any representable number between ``x + spacing(x)`` and
x for any finite x.
Spacing of +- inf and nan is nan.
Examples
--------
>>> np.spacing(1, 2) == np.finfo(np.float64).eps
True Return the positive square-root of an array, element-wise.
Parameters
----------
x : array_like
The square root of each element in this array is calculated.
Returns
-------
y : ndarray
An array of the same shape as `x`, containing the square-root of
each element in `x`. If any element in `x`
is complex, a complex array is returned. If all of the elements
of `x` are real, negative elements return numpy.nan elements.
See Also
--------
numpy.lib.scimath.sqrt
A version which returns complex numbers when given negative reals.
Notes
-----
`sqrt` has a branch cut ``[-inf, 0)`` and is continuous from above on it.
Examples
--------
>>> np.sqrt([1,4,9])
array([ 1., 2., 3.])
>>> np.sqrt([4, -1, -3+4J])
array([ 2.+0.j, 0.+1.j, 1.+2.j])
>>> np.sqrt([4, -1, numpy.inf])
array([ 2., NaN, Inf]) Return the element-wise square of the input.
Parameters
----------
x : array_like
Input data.
Returns
-------
out : ndarray
Element-wise `x*x`, of the same shape and dtype as `x`.
Returns scalar if `x` is a scalar.
See Also
--------
numpy.linalg.matrix_power
sqrt
power
Examples
--------
>>> np.square([-1j, 1])
array([-1.-0.j, 1.+0.j]) Subtract arguments, element-wise.
Parameters
----------
x1, x2 : array_like
The arrays to be subtracted from each other.
Returns
-------
y : ndarray
The difference of `x1` and `x2`, element-wise. Returns a scalar if
both `x1` and `x2` are scalars.
Notes
-----
Equivalent to ``x1 - x2`` in terms of array broadcasting.
Examples
--------
>>> np.subtract(1.0, 4.0)
-3.0
>>> x1 = np.arange(9.0).reshape((3, 3))
>>> x2 = np.arange(3.0)
>>> np.subtract(x1, x2)
array([[ 0., 0., 0.],
[ 3., 3., 3.],
[ 6., 6., 6.]]) Compute tangent element-wise.
Equivalent to ``np.sin(x)/np.cos(x)`` element-wise.
Parameters
----------
x : array_like
Input array.
out : ndarray, optional
Output array of same shape as `x`.
Returns
-------
y : ndarray
The corresponding tangent values.
Raises
------
ValueError: invalid return array shape
if `out` is provided and `out.shape` != `x.shape` (See Examples)
Notes
-----
If `out` is provided, the function writes the result into it,
and returns a reference to `out`. (See Examples)
References
----------
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions.
New York, NY: Dover, 1972.
Examples
--------
>>> from math import pi
>>> np.tan(np.array([-pi,pi/2,pi]))
array([ 1.22460635e-16, 1.63317787e+16, -1.22460635e-16])
>>>
>>> # Example of providing the optional output parameter illustrating
>>> # that what is returned is a reference to said parameter
>>> out2 = np.cos([0.1], out1)
>>> out2 is out1
True
>>>
>>> # Example of ValueError due to provision of shape mis-matched `out`
>>> np.cos(np.zeros((3,3)),np.zeros((2,2)))
Traceback (most recent call last):
File "", line 1, in
ValueError: invalid return array shape Compute hyperbolic tangent element-wise.
Equivalent to ``np.sinh(x)/np.cosh(x)`` or
``-1j * np.tan(1j*x)``.
Parameters
----------
x : array_like
Input array.
out : ndarray, optional
Output array of same shape as `x`.
Returns
-------
y : ndarray
The corresponding hyperbolic tangent values.
Raises
------
ValueError: invalid return array shape
if `out` is provided and `out.shape` != `x.shape` (See Examples)
Notes
-----
If `out` is provided, the function writes the result into it,
and returns a reference to `out`. (See Examples)
References
----------
.. [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions.
New York, NY: Dover, 1972, pg. 83.
http://www.math.sfu.ca/~cbm/aands/
.. [2] Wikipedia, "Hyperbolic function",
http://en.wikipedia.org/wiki/Hyperbolic_function
Examples
--------
>>> np.tanh((0, np.pi*1j, np.pi*1j/2))
array([ 0. +0.00000000e+00j, 0. -1.22460635e-16j, 0. +1.63317787e+16j])
>>> # Example of providing the optional output parameter illustrating
>>> # that what is returned is a reference to said parameter
>>> out2 = np.tanh([0.1], out1)
>>> out2 is out1
True
>>> # Example of ValueError due to provision of shape mis-matched `out`
>>> np.tanh(np.zeros((3,3)),np.zeros((2,2)))
Traceback (most recent call last):
File "", line 1, in
ValueError: invalid return array shape Returns a true division of the inputs, element-wise.
Instead of the Python traditional 'floor division', this returns a true
division. True division adjusts the output type to present the best
answer, regardless of input types.
Parameters
----------
x1 : array_like
Dividend array.
x2 : array_like
Divisor array.
Returns
-------
out : ndarray
Result is scalar if both inputs are scalar, ndarray otherwise.
Notes
-----
The floor division operator ``//`` was added in Python 2.2 making ``//``
and ``/`` equivalent operators. The default floor division operation of
``/`` can be replaced by true division with
``from __future__ import division``.
In Python 3.0, ``//`` is the floor division operator and ``/`` the
true division operator. The ``true_divide(x1, x2)`` function is
equivalent to true division in Python.
Examples
--------
>>> x = np.arange(5)
>>> np.true_divide(x, 4)
array([ 0. , 0.25, 0.5 , 0.75, 1. ])
>>> x/4
array([0, 0, 0, 0, 1])
>>> x//4
array([0, 0, 0, 0, 1])
>>> from __future__ import division
>>> x/4
array([ 0. , 0.25, 0.5 , 0.75, 1. ])
>>> x//4
array([0, 0, 0, 0, 1]) Return the truncated value of the input, element-wise.
The truncated value of the scalar `x` is the nearest integer `i` which
is closer to zero than `x` is. In short, the fractional part of the
signed number `x` is discarded.
Parameters
----------
x : array_like
Input data.
Returns
-------
y : {ndarray, scalar}
The truncated value of each element in `x`.
See Also
--------
ceil, floor, rint
Notes
-----
.. versionadded:: 1.3.0
Examples
--------
>>> a = np.array([-1.7, -1.5, -0.2, 0.2, 1.5, 1.7, 2.0])
>>> np.trunc(a)
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