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Returns ------- absolute : 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(start=-10, stop=10, num=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], cmap='gray') >>> plt.show()Add arguments element-wise. Parameters ---------- x1, x2 : array_like The arrays to be added. If ``x1.shape != x2.shape``, they must be broadcastable to a common shape (which may be the shape of one or the other). out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- add : ndarray or scalar The sum 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.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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 analytic 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 ---------- M. Abramowitz and I.A. Stegun, "Handbook of Mathematical Functions", 10th printing, 1964, pp. 79. http://www.math.sfu.ca/~cbm/aands/ 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, element-wise. Parameters ---------- x : array_like Input array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- arccosh : 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) 0.0Inverse sine, element-wise. Parameters ---------- x : array_like `y`-coordinate on the unit circle. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- angle : ndarray The inverse sine of each element in `x`, in radians and in the closed interval ``[-pi/2, pi/2]``. If `x` is a scalar, a scalar is returned, otherwise an array. 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 :math:`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 analytic function that has, by convention, the 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 ---------- Abramowitz, M. and Stegun, I. A., *Handbook of Mathematical Functions*, 10th printing, New York: Dover, 1964, pp. 79ff. http://www.math.sfu.ca/~cbm/aands/ Examples -------- >>> np.arcsin(1) # pi/2 1.5707963267948966 >>> np.arcsin(-1) # -pi/2 -1.5707963267948966 >>> np.arcsin(0) 0.0Inverse hyperbolic sine element-wise. Parameters ---------- x : array_like Input array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- out : ndarray Out has the same shape as `x`. Its real part is in ``[-pi/2, pi/2]`` (``arctan(+/-inf)`` returns ``+/-pi/2``). It is a scalar if `x` is a scalar. See Also -------- arctan2 : The "four quadrant" arctan of the angle formed by (`x`, `y`) and the positive `x`-axis. angle : Argument of complex values. Notes ----- `arctan` is a multi-valued 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 analytic function that has [`1j, infj`] and [`-1j, -infj`] as branch cuts, 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 ---------- Abramowitz, M. and Stegun, I. A., *Handbook of Mathematical Functions*, 10th printing, New York: Dover, 1964, pp. 79. http://www.math.sfu.ca/~cbm/aands/ Examples -------- We expect the arctan of 0 to be 0, and of 1 to be 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()Element-wise arc tangent of ``x1/x2`` choosing the quadrant correctly. The quadrant (i.e., branch) is chosen so that ``arctan2(x1, x2)`` is the signed angle in radians between the ray ending at the origin and passing through the point (1,0), and the ray ending at the origin and passing through the point (`x2`, `x1`). (Note the role reversal: the "`y`-coordinate" is the first function parameter, the "`x`-coordinate" is the second.) By IEEE convention, this function is defined for `x2` = +/-0 and for either or both of `x1` and `x2` = +/-inf (see Notes for specific values). This function is not defined for complex-valued arguments; for the so-called argument of complex values, use `angle`. 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- angle : ndarray Array of angles in radians, in the range ``[-pi, pi]``. See Also -------- arctan, tan, angle Notes ----- *arctan2* is identical to the `atan2` function of the underlying C library. The following special values are defined in the C standard: [1]_ ====== ====== ================ `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, as are +inf and -inf. References ---------- .. [1] ISO/IEC standard 9899:1999, "Programming language C." 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 element-wise. Parameters ---------- x : array_like Input array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 and boolean types are handled. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 and boolean types are handled. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 and boolean types are handled. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 cube-root of an array, element-wise. .. versionadded:: 1.10.0 Parameters ---------- x : array_like The values whose cube-roots are required. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray An array of the same shape as `x`, containing the cube cube-root of each element in `x`. If `out` was provided, `y` is a reference to it. Examples -------- >>> np.cbrt([1,8,27]) array([ 1., 2., 3.])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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 element-wise. Parameters ---------- x : array_like Input array in radians. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray The corresponding cosine values. 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 shapeHyperbolic cosine, element-wise. Equivalent to ``1/2 * (np.exp(x) + np.exp(-x))`` and ``np.cos(1j*x)``. Parameters ---------- x : array_like Input array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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) 3.1415926535897931Convert angles from radians to degrees. Parameters ---------- x : array_like Input array in radians. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray of floats The corresponding degree values; if `out` was supplied this is a reference to it. 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) TrueDivide arguments element-wise. Parameters ---------- x1 : array_like Dividend array. x2 : array_like Divisor array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or 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``. In Python 2, when both ``x1`` and ``x2`` are of an integer type, ``divide`` will behave like ``floor_divide``. In Python 3, it behaves like ``true_divide``. 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 (Python 2 only): >>> np.divide(2, 4) 0 >>> np.divide(2, 4.) 0.5 Division by zero always yields zero in integer arithmetic (again, Python 2 only), 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) 0Return element-wise quotient and remainder simultaneously. .. versionadded:: 1.13.0 ``np.divmod(x, y)`` is equivalent to ``(x // y, x % y)``, but faster because it avoids redundant work. It is used to implement the Python built-in function ``divmod`` on NumPy arrays. Parameters ---------- x1 : array_like Dividend array. x2 : array_like Divisor array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- out1 : ndarray Element-wise quotient resulting from floor division. out2 : ndarray Element-wise remainder from floor division. See Also -------- floor_divide : Equivalent to Python's ``//`` operator. remainder : Equivalent to Python's ``%`` operator. modf : Equivalent to ``divmod(x, 1)`` for positive ``x`` with the return values switched. Examples -------- >>> np.divmod(np.arange(5), 3) (array([0, 0, 0, 1, 1]), array([0, 1, 2, 0, 1]))Return (x1 == x2) element-wise. Parameters ---------- x1, x2 : array_like Input arrays of the same shape. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- out : ndarray or 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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], cmap='gray') >>> 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], cmap='hsv') >>> 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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- out : ndarray Element-wise 2 to the power `x`. See Also -------- power 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 ``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-10Compute the absolute values element-wise. 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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or 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])First array elements raised to powers from second array, element-wise. Raise each base in `x1` to the positionally-corresponding power in `x2`. `x1` and `x2` must be broadcastable to the same shape. This differs from the power function in that integers, float16, and float32 are promoted to floats with a minimum precision of float64 so that the result is always inexact. The intent is that the function will return a usable result for negative powers and seldom overflow for positive powers. .. versionadded:: 1.12.0 Parameters ---------- x1 : array_like The bases. x2 : array_like The exponents. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray The bases in `x1` raised to the exponents in `x2`. See Also -------- power : power function that preserves type Examples -------- Cube each element in a list. >>> x1 = range(6) >>> x1 [0, 1, 2, 3, 4, 5] >>> np.float_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.float_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.float_power(x1, x2) array([[ 0., 1., 8., 27., 16., 5.], [ 0., 1., 8., 27., 16., 5.]])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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or 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 instead uses the definition of `floor` where `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. It is equivalent to the Python ``//`` operator and pairs with the Python ``%`` (`remainder`), function so that ``b = a % b + b * (a // b)`` up to roundoff. Parameters ---------- x1 : array_like Numerator. x2 : array_like Denominator. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray y = floor(`x1`/`x2`) See Also -------- remainder : Remainder complementary to floor_divide. divmod : Simultaneous floor division and remainder. 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or scalar The maximum of `x1` and `x2`, element-wise. Returns scalar if both `x1` and `x2` are scalars. See Also -------- fmin : Element-wise minimum of two arrays, ignores NaNs. maximum : Element-wise maximum of two arrays, propagates NaNs. amax : The maximum value of an array along a given axis, propagates NaNs. nanmax : The maximum value of an array along a given axis, ignores NaNs. minimum, amin, nanmin 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])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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or scalar The minimum of `x1` and `x2`, element-wise. Returns scalar if both `x1` and `x2` are scalars. See Also -------- fmax : Element-wise maximum of two arrays, ignores NaNs. minimum : Element-wise minimum of two arrays, propagates NaNs. amin : The minimum value of an array along a given axis, propagates NaNs. nanmin : The minimum value of an array along a given axis, ignores NaNs. maximum, amax, nanmax 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([1, 3, 2]) >>> np.fmin(np.eye(2), [0.5, 2]) array([[ 0.5, 0. ], [ 0. , 1. ]]) >>> 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 C library function fmod, the remainder has the same sign as the dividend `x1`. It is equivalent to the Matlab(TM) ``rem`` function and should not be confused with the Python modulus operator ``x1 % x2``. Parameters ---------- x1 : array_like Dividend. x2 : array_like Divisor. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : array_like The remainder of the division of `x1` by `x2`. See Also -------- remainder : Equivalent to the Python ``%`` operator. divide Notes ----- The result of the modulo operation for negative dividend and divisors is bound by conventions. For `fmod`, the sign of result is the sign of the dividend, while for `remainder` the sign of the result is the sign of the divisor. The `fmod` function is equivalent to the Matlab(TM) ``rem`` function. 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]])Decompose the elements of x into mantissa and twos exponent. Returns (`mantissa`, `exponent`), where `x = mantissa * 2**exponent``. The mantissa is lies in the open interval(-1, 1), while the twos exponent is a signed integer. Parameters ---------- x : array_like Array of numbers to be decomposed. out1 : ndarray, optional Output array for the mantissa. Must have the same shape as `x`. out2 : ndarray, optional Output array for the exponent. Must have the same shape as `x`. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- (mantissa, exponent) : tuple of ndarrays, (float, int) `mantissa` is a float array with values between -1 and 1. `exponent` is an int array which represents the exponent of 2. See Also -------- ldexp : Compute ``y = x1 * 2**x2``, the inverse of `frexp`. Notes ----- Complex dtypes are not supported, they will raise a TypeError. Examples -------- >>> x = np.arange(9) >>> y1, y2 = np.frexp(x) >>> y1 array([ 0. , 0.5 , 0.5 , 0.75 , 0.5 , 0.625, 0.75 , 0.875, 0.5 ]) >>> y2 array([0, 1, 2, 2, 3, 3, 3, 3, 4]) >>> y1 * 2**y2 array([ 0., 1., 2., 3., 4., 5., 6., 7., 8.])Return the truth value of (x1 > x2) element-wise. Parameters ---------- x1, x2 : array_like Input arrays. If ``x1.shape != x2.shape``, they must be broadcastable to a common shape (which may be the shape of one or the other). out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- out : bool or ndarray of bool 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 the truth value of (x1 >= x2) element-wise. Parameters ---------- x1, x2 : array_like Input arrays. If ``x1.shape != x2.shape``, they must be broadcastable to a common shape (which may be the shape of one or the other). out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- out : bool or ndarray of bool 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)Compute the Heaviside step function. The Heaviside step function is defined as:: 0 if x1 < 0 heaviside(x1, x2) = x2 if x1 == 0 1 if x1 > 0 where `x2` is often taken to be 0.5, but 0 and 1 are also sometimes used. Parameters ---------- x1 : array_like Input values. x2 : array_like The value of the function when x1 is 0. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- out : ndarray The output array, element-wise Heaviside step function of `x1`. Notes ----- .. versionadded:: 1.13.0 References ---------- .. Wikipedia, "Heaviside step function", https://en.wikipedia.org/wiki/Heaviside_step_function Examples -------- >>> np.heaviside([-1.5, 0, 2.0], 0.5) array([ 0. , 0.5, 1. ]) >>> np.heaviside([-1.5, 0, 2.0], 1) array([ 0., 1., 1.])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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 ---------- x : array_like Only integer and boolean types are handled. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 finiteness (not infinity or not Not a Number). The result is returned as a boolean array. Parameters ---------- x : array_like Input values. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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. Returns a boolean array of the same shape as `x`, True where ``x == +/-inf``, otherwise False. Parameters ---------- x : array_like Input values out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : bool (scalar) or boolean ndarray 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 shape as the input and the values are True where the corresponding element of the input is positive or negative infinity; elsewhere the values are False. If a second argument was 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, respectively. 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 the second argument is supplied when the first argument is a scalar, or if the 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 NaN and return result as a boolean array. Parameters ---------- x : array_like Input array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or 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 of 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, isnat 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)Test element-wise for NaT (not a time) and return result as a boolean array. Parameters ---------- x : array_like Input array with datetime or timedelta data type. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or bool For scalar input, the result is a new boolean with value True if the input is NaT; otherwise the value is False. For array input, the result is a boolean array of the same dimensions as the input and the values are True if the corresponding element of the input is NaT; otherwise the values are False. See Also -------- isnan, isinf, isneginf, isposinf, isfinite Examples -------- >>> np.isnat(np.datetime64("NaT")) True >>> np.isnat(np.datetime64("2016-01-01")) False >>> np.isnat(np.array(["NaT", "2016-01-01"], dtype="datetime64[ns]")) array([ True, False], dtype=bool)Returns x1 * 2**x2, element-wise. The mantissas `x1` and twos exponents `x2` are used to construct floating point numbers ``x1 * 2**x2``. Parameters ---------- x1 : array_like Array of multipliers. x2 : array_like, int Array of twos exponents. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or scalar The result of ``x1 * 2**x2``. See Also -------- frexp : Return (y1, y2) from ``x = y1 * 2**y2``, inverse to `ldexp`. Notes ----- Complex dtypes are not supported, they will raise a TypeError. `ldexp` is useful as the inverse of `frexp`, if used by itself it is more clear to simply use the expression ``x1 * 2**x2``. Examples -------- >>> np.ldexp(5, np.arange(4)) array([ 5., 10., 20., 40.], dtype=float32) >>> x = np.arange(6) >>> np.ldexp(*np.frexp(x)) array([ 0., 1., 2., 3., 4., 5.])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`. Has to be non-negative. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 the truth value of (x1 < x2) element-wise. Parameters ---------- x1, x2 : array_like Input arrays. If ``x1.shape != x2.shape``, they must be broadcastable to a common shape (which may be the shape of one or the other). out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- out : bool or ndarray of bool Array of bools, or a single bool if `x1` and `x2` are scalars. See Also -------- greater, less_equal, greater_equal, equal, not_equal Examples -------- >>> np.less([1, 2], [2, 2]) array([ True, False], dtype=bool)Return the truth value of (x1 =< x2) element-wise. Parameters ---------- x1, x2 : array_like Input arrays. If ``x1.shape != x2.shape``, they must be broadcastable to a common shape (which may be the shape of one or the other). out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- out : bool or ndarray of bool Array of bools, or a single bool if `x1` and `x2` are scalars. See Also -------- greater, less, greater_equal, equal, not_equal Examples -------- >>> np.less_equal([4, 2, 1], [2, 2, 2]) array([False, True, True], 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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) 0.0Base-2 logarithm of `x`. Parameters ---------- x : array_like Input values. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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. Examples -------- >>> x = np.array([0, 1, 2, 2**4]) >>> np.log2(x) array([-Inf, 0., 1., 4.]) >>> xi = np.array([0+1.j, 1, 2+0.j, 4.j]) >>> np.log2(xi) array([ 0.+2.26618007j, 0.+0.j , 1.+0.j , 2.+2.26618007j])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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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-50Logarithm 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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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-50Compute the truth value of x1 AND x2 element-wise. Parameters ---------- x1, x2 : array_like Input arrays. `x1` and `x2` must be of the same shape. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or 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 element-wise. Parameters ---------- x : array_like Logical NOT is applied to the elements of `x`. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : bool or ndarray of 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 element-wise. Parameters ---------- x1, x2 : array_like Logical OR is applied to the elements of `x1` and `x2`. They have to be of the same shape. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or 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, element-wise. Parameters ---------- x1, x2 : array_like Logical XOR is applied to the elements of `x1` and `x2`. They must be broadcastable to the same shape. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : bool or ndarray of bool Boolean result of the logical XOR operation applied to the elements of `x1` and `x2`; the shape is determined by whether or not broadcasting of one or both arrays was required. 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) Simple example showing support of broadcasting >>> np.logical_xor(0, np.eye(2)) array([[ True, 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or scalar The maximum of `x1` and `x2`, element-wise. Returns scalar if both `x1` and `x2` are scalars. See Also -------- minimum : Element-wise minimum of two arrays, propagates NaNs. fmax : Element-wise maximum of two arrays, ignores NaNs. amax : The maximum value of an array along a given axis, propagates NaNs. nanmax : The maximum value of an array along a given axis, ignores NaNs. fmin, amin, nanmin Notes ----- The maximum 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.maximum([2, 3, 4], [1, 5, 2]) array([2, 5, 4]) >>> np.maximum(np.eye(2), [0.5, 2]) # broadcasting 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) infElement-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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or scalar The minimum of `x1` and `x2`, element-wise. Returns scalar if both `x1` and `x2` are scalars. See Also -------- maximum : Element-wise maximum of two arrays, propagates NaNs. fmin : Element-wise minimum of two arrays, ignores NaNs. amin : The minimum value of an array along a given axis, propagates NaNs. nanmin : The minimum value of an array along a given axis, ignores NaNs. fmax, amax, nanmax 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]) >>> np.minimum(-np.Inf, 1) -infReturn 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y1 : ndarray Fractional part of `x`. y2 : ndarray Integral part of `x`. Notes ----- For integer input the return values are floats. See Also -------- divmod : ``divmod(x, 1)`` is equivalent to ``modf`` with the return values switched, except it always has a positive remainder. 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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.]])Numerical negative, element-wise. Parameters ---------- x : array_like or scalar Input array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or scalar Returned array or scalar: `y = -x`. Examples -------- >>> np.negative([1.,-1.]) array([-1., 1.])Return the next floating-point value after x1 towards 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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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)Numerical positive, element-wise. .. versionadded:: 1.13.0 Parameters ---------- x : array_like or scalar Input array. Returns ------- y : ndarray or scalar Returned array or scalar: `y = +x`. Notes ----- Equivalent to `x.copy()`, but only defined for types that support arithmetic.First array elements raised to powers from second array, element-wise. Raise each base in `x1` to the positionally-corresponding power in `x2`. `x1` and `x2` must be broadcastable to the same shape. Note that an integer type raised to a negative integer power will raise a ValueError. Parameters ---------- x1 : array_like The bases. x2 : array_like The exponents. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray The bases in `x1` raised to the exponents in `x2`. See Also -------- float_power : power function that promotes integers to float 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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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) 90.0Convert angles from degrees to radians. Parameters ---------- x : array_like Input array in degrees. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 TrueReturn the reciprocal of the argument, element-wise. Calculates ``1/x``. Parameters ---------- x : array_like Input array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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 the remainder complementary to the `floor_divide` function. It is equivalent to the Python modulus operator``x1 % x2`` and has the same sign as the divisor `x2`. It should not be confused with the Matlab(TM) ``rem`` function. Parameters ---------- x1 : array_like Dividend array. x2 : array_like Divisor array. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray The element-wise remainder of the quotient ``floor_divide(x1, x2)``. Returns a scalar if both `x1` and `x2` are scalars. See Also -------- floor_divide : Equivalent of Python ``//`` operator. divmod : Simultaneous floor division and remainder. fmod : Equivalent of the Matlab(TM) ``rem`` function. 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 `x2`. Because 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`. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- out : ndarray or 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``. nan is returned for nan inputs. For complex inputs, the `sign` function returns ``sign(x.real) + 0j if x.real != 0 else sign(x.imag) + 0j``. complex(nan, 0) is returned for complex nan inputs. Parameters ---------- x : array_like Input values. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray The sign of `x`. Notes ----- There is more than one definition of sign in common use for complex numbers. The definition used here is equivalent to :math:`x/\sqrt{x*x}` which is different from a common alternative, :math:`x/|x|`. Examples -------- >>> np.sign([-5., 4.5]) array([-1., 1.]) >>> np.sign(0) 0 >>> np.sign(5-2j) (1+0j)Returns element-wise True where signbit is set (less than zero). Parameters ---------- x : array_like The input value(s). out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- result : ndarray of bool Output array, or reference to `out` if that was supplied. 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). out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray The corresponding hyperbolic sine values. 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 shapeReturn the distance between x and the nearest adjacent number. Parameters ---------- x : array_like Values to find the spacing of. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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) == np.finfo(np.float64).eps TrueReturn the positive square-root of an array, element-wise. Parameters ---------- x : array_like The values whose square-roots are required. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray An array of the same shape as `x`, containing the positive square-root of each element in `x`. If any element in `x` is complex, a complex array is returned (and the square-roots of negative reals are calculated). If all of the elements in `x` are real, so is `y`, with negative elements returning ``nan``. If `out` was provided, `y` is a reference to it. See Also -------- lib.scimath.sqrt A version which returns complex numbers when given negative reals. Notes ----- *sqrt* has--consistent with common convention--as its branch cut the real "interval" [`-inf`, 0), and is continuous from above on it. A branch cut is a curve in the complex plane across which a given complex function fails to be continuous. 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray The corresponding tangent values. 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 shapeCompute 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, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray The corresponding hyperbolic tangent values. 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 shapeReturns 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. 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. out : ndarray, None, or tuple of ndarray and None, optional A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or `None`, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs. where : array_like, optional Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. **kwargs For other keyword-only arguments, see the :ref:`ufunc docs `. Returns ------- y : ndarray or 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) array([-1., -1., -0., 0., 1., 1., 2.])ý.@5?q(7[??.eB5<K;?[>r1??cܥL@9RFߑ?0C+eG?&{?9B.?-DT! @iW @ox?output parameter for reduction operation %s has the wrong number of dimensions (must match the operand's when keepdims=True)output parameter for reduction operation %s has a reduction dimension not equal to one (required when keepdims=True)output parameter for reduction operation %s does not have enough dimensionsoutput parameter for reduction operation %s has too many dimensionsreduction operation '%s' is not reorderable, so only one axis may be specifiedzero-size array to reduction operation %s which has no identityReduce operations in NumPy do not yet support a where maskreduction operation %s did not supply an inner loop function(O)Integers to negative integer powers are not allowed.In the future, 'NAT == x' and 'x == NAT' will always be False.In the future, 'NAT > x' and 'x > NAT' will always be False.In the future, 'NAT >= x' and 'x >= NAT' will always be False.In the future, 'NAT < x' and 'x < NAT' will always be False.In the future, 'NAT <= x' and 'x <= NAT' will always be False.In the future, NAT != NAT will be True rather than False.unorderable types for comparison??AAAA????C?    ????AAAAAAAA?Warning: %s encountered in %s python callback specified for %s (in %s) but no function found.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 16python object must be callable or have a callable write methodOnly unary and binary ufuncs supported at this timeOnly single output ufuncs supported at this timesecond operand needed for ufuncreturn arrays must be of ArrayType'out' must be a tuple of arrays'out' must be an array or a tuple of a single arrayelementwise comparison failed; returning scalar instead, but in the future will perform elementwise comparisonunorderable dtypes; returning scalar but in the future this will be an errorcannot specify both 'sig' and 'dtype'cannot specify 'out' as both a positional and keyword argumentThe 'out' tuple must have exactly one entry per ufunc outputpassing a single array to the 'out' keyword argument of a ufunc with more than one output will result in an error in the futurecannot specify both 'sig' and 'signature''%s' is an invalid keyword to ufunc '%s'__array_prepare__ must return an ndarray or subclass thereof which is otherwise identical to its input__array_prepare__ must return an ndarray or subclass thereof_ufunc_doc_signature_formatterReduction not defined on ufunc with signature%s only supported for binary functions%s only supported for functions returning a single valueThe 'out' tuple must have exactly one entrykeepdims argument has no effect on accumulate, and will be removed in futurecannot perform %s with flexible typeufunc %s has an invalid identity for reductiontype resolution returned NotImplemented to reduce ufunc %scould not find a type resolution appropriate for reduce ufunc %saccumulate does not allow multiple axescould not find a matching type for %s.accumulate, requested type has type code '%c'provided out is the wrong size for the reductionreduceat does not allow multiple axesindex %d out-of-bounds in %s.%s [0, %d)could not find a matching type for %s.%s, requested type has type code '%c'too many dimensions for generalized ufunc %s%s: %s operand %d does not have enough dimensions (has %d, gufunc core with signature %s requires %d)%s: %s operand %d has a mismatch in its core dimension %d, with gufunc signature %s (size %zd is different from %zd)%s: Output operand %d has core dimension %d unspecified, with gufunc signature %sToo many operands when including where= parameterThe __array_prepare__ functions modified the data pointer addresses in an invalid fashionXX can't happen, please report a bug XXmethod outer is not allowed in ufunc with non-trivial signatureouter product only supported for binary functionsexactly two arguments expectedError object must be a list of length 3',' must not be followed by ')'incomplete signature: not all arguments foundunknown user defined struct dtypeuserloop for user dtype not found%s encountered in %sNNwriteNO%s must be a length 3 list.invalid error mask (%d)OO|O:atfirst operand must be arrayoutput arrayinvalid number of argumentsOOiinvalid keyword argumentcastingdtype(N)extobjordersigsignature'subok' must be a booleanwhereO(OOi)numpy.core._internal %sfunction not supportedOO|OO&O&:reduceatO|OO&O&O:accumulateO|OO&O&i:reduceO(O)itoo many values for 'axis'AxisErroriiOcannot %s on a scalarduplicate value in 'axis'divide by zerooverflowunderflowinvalid valueOutputInputufunc %s __call__outer(OO)O:seterrobjtestexpect dimension nameexpect '('expect '->'expect ','expect ',' or ')'%s at position %d in "%s"unknown user-defined typeaxiskeepdimsindicesnumpy.ufunc__doc__ninnoutnargsntypesidentity5|\lMXX(numpy.coreComplexWarningubyte_scalarsushort_scalarsuint_scalarsulong_scalarsulonglong_scalarshalf_scalarscfloat_scalarscdouble_scalarsclongdouble_scalarsCasting complex values to real discards the imaginary part,WW WWV'no''safe''same_kind''unsafe''equiv' to with casting rule %s(unknown) and numpy: classic int divisionCannot cast ufunc %s input from Cannot cast ufunc %s output from ufunc 'isnat' is only defined for datetime and timedelta.ufunc '%s' did not contain a loop with signature matching types the ufunc default masked inner loop selector doesn't yet support wrapping the new inner loop selector, it still only wraps the legacy inner loop selectoronly boolean masks are supported in ufunc inner loops presentlyufunc '%s' output (typecode '%c') could not be coerced to provided output parameter (typecode '%c') according to the casting rule '%s'ufunc '%s' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule '%s'a type-tuple must be specified of length 1 or %d for ufunc '%s'the type-tuple provided to the ufunc must specify at least one none-None dtypea type-string for %s, requires 1 typecode, or %d typecode(s) before and %d after the -> signfound a user loop for ufunc '%s' matching the type-tuple, but the inputs and/or outputs could not be cast according to the casting ruleNo loop matching the specified signature and casting was found for ufunc %sufunc %s is configured to use binary comparison type resolution but has the wrong number of inputs or outputsrequire data type in the type tupleufunc %s is configured to use unary operation type resolution but has the wrong number of inputs or outputsThe numpy boolean negative, the `-` operator, is not supported, use the `~` operator or the logical_not function instead.ufunc %s is configured to use binary operation type resolution but has the wrong number of inputs or outputsufunc %s cannot use operands with types numpy boolean subtract, the `-` operator, is deprecated, use the bitwise_xor, the `^` operator, or the logical_xor function instead.MMMMMoMMMMMVVVVVVW,W5W#Warray_ufunc_errmsg_formatterThe 'out' tuple must have exactly %d entries: one per ufunc outputpassing a single argument to the 'out' keyword argument of a ufunc with more than one output will result in an error in the future'out' must be a tuple of argumentsufunc() missing %ld of %ldrequired positional argument(s)ufunc() takes from %ld to %ldarguments but %ld were givenargument given by name ('out') and position (%ld)ufunc.reduce() takes from 1 to 5 positional arguments but %ld were givenargument given by name ('%s') and position (%ld)ufunc.accumulate() takes from 1 to 4 positional arguments but %ld were givenufunc.reduceat() takes from 2 to 4 positional arguments but %ld were givenufunc.outer() missing %ld of %ldrequired positional argument(s)ufunc.outer() takes %ld arguments but%ld were givenufunc.at() takes from 2 to 3 positional arguments but %ld were givenInternal Numpy error: unknown ufunc method '%s' in call to PyUFunc_CheckOverrideInternal Numpy error: call to PyUFunc_HasOverride with non-tupleInternal Numpy error: too many arguments in call to PyUFunc_HasOverrideoperand '%.200s' does not support ufuncs (__array_ufunc__=None)ndarrayyrxrxrxyxxx_z4x4x4x4x4x4x4x_z4x4xIz4x:z_zIzIz4x4x4x4x4x$z:z4x4x4xz4x$zzIz$z4x4x4x4x4x4x4xy4x4xy4xyyyy4x4x4x4x4xyyy4x4xy4x4xyyy4x4x4x4x4xyyy4x4xy4x4xyy4x4x4x4x4x4x4xly4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x4x]y4x4x4x4x4x4x4x4xy4x4x4x4x4x4x4x4x4x4x4x4x4x4x4xy;:?G : ]H:r}`u}Py`:`yt:py:y:y:y:y:@z:Pz;`z;z(;z<;zP; {d;@{x;p{;{;{;{;0|;`|<<<P<d<@x<<< <@<P =` =p4=h==0=@=P=`=Ђ>,> H>`p>>>P>> ? ?4?ЄH?@|?p?????Ѕ@@0,@@@T@Ph@@Ї@@@ADAXAlAAAЈAAAAA,B`BBBB0BPC CЋTc0>hc`>|c>cHcQcpSd@VdY0d[Dd`gXdqld0ddPdd@e`epeeeeef`ftf f@ff`f0 f f g`gpg hp&hh0,h1i`<> p?T@@PA,At BBCLpCCPD$DlEPFGDGHI IhJJJ J K4PKH`K\KL@M8MpNPOO0PQ4SHS\ TpT@UU VV`W(WTPXhX|YpYYZZ0[[<\\p]^^ _` p` `4`bTchd|ee0ffgg$PhHht@iii`jjkk0ll\mmn0oopp@q,q@qTrps`tuupvvPwx@xd@yyyPzz {{ | P|0 | 0} 0~ ~  0 `D @   d Ѕ  PX pl   0 p  Љ( < P d x Ќ pd`P4X0TPh|М`Ш|``D`@p00pH\@8@ !P#8$x**-p`000001$P18p1L1`1t11202223(@3H3p334P4444805XP5l66778$8H@9l9:P:`:p:::8;L@;l;;;< <`< <4  @> >!>0!>P!?p!@?!P?!?!?!?! @ "`@,"p@@"@T"@h"@"A"@A"PA"A"A#A$#A8#BL#0B`#`Bt#C# D#D$E,$FP$PGt$H$H$I%J$%`KH%PLl%M%M%N%pO& P@&Qd&R&R&@S&T'T8'U\'0W'PW'W'X (YD(PZh([([(\(`])0^<)_`)p`)`)Pa)b*c`*cx*Pd*d*e*@f+gX+gp+h+i+i+j,0k$,lt,l,pm,n,n,`o-pl-p-q-0r-r-s.Pt8.u.u.v.0w.w /x0/ z/@z/{/{/P|0}(0}L0@00 00p 1p111P11Ѕ2<22 22p20343`33@3Ѝ34P,4P4x44P445В<5Pd555P55646ЖH6 66`7D7x707@7080080\8888 989d9999 :0@: l:0:0:0: ; H;t;`;@;0;0$<0P<@|<0< <=p,=PX=@=P=@=P>@4>0`>>>`>P?`DP@EPBDEPDpEEEIE MEQ4FR`FUFVFXFZ G\LG^xG`GbGdHf8H0idH kH@mHoHpqIPsHI0utIwI yI{J|,J~XJJJJpKP p> > >J? ? ?JL? 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D?A$?8?L?%`?t?'???(?oEAG X AAA 4@"0P@IFKH DP  AABC 8lC LhC$`CAtC80CdEJ@VHCPCXB`U@[ AA (D!NR 0A(A BBBD 0A(A BDBGhPXL@BBB E(A0A8G 8D0A(B BBBG HXeBBE E(D0D8G`f 8A0A(B BBBC HeBBE E(D0D8Gpb 8A0A(B BBBG L[BBE E(D0D8Gv 8A0A(B BBBC D@ hFBB B(A0A8DPK8A0A(B BBBD nFBB B(A0A8DPQ8A0A(B BBBD DnFBB B(A0A8DPQ8A0A(B BBBD!lfFBB B(A0A8DPI8A0A(B BBBD`!rFBB B(A0A8DPU8A0A(B BBBD!FBB B(A0A8D`c8A0A(B BBBD!FBB B(A0A8D`8A0A(B BBBD8"\FBB B(A0A8D`8A0A(B BBBD"FBB B(A0A8D`f8A0A(B BBBD"FBB B(A0A8D`v8A0A(B BBBD#TfFBB B(A0A8DPI8A0A(B BBBDX#|FBB B(A0A8D`f8A0A(B BBBP#hFBB B(A0A8DPjXH`OXAPI8A0A(B BBBT#FBB B(A0A8D`zhHpDxCZ`I8A0A(B BBBHL$FBB B(A0A8D`v 8A0A(B BBBA H$lFBB B(A0A8D 8A0A(B BBBA H$FBB B(A0A8D 8A0A(B BBBA L0%0FBB B(A0A8D 8A0A(B BBBA H%dFBB B(A0A8Dpx 8A0A(B BBBA H%FBB B(A0A8D 8A0A(B BBBA H&LFBB B(A0A8D 8A0A(B BBBA Hd&FBB B(A0A8G 8A0A(B BBBA D&TFBB B(A0A8DP~8A0A(B BBBD&FBB B(A0A8DP8A0A(B BBBD@'FBB B(A0A8D`8A0A(B BBBD'FBB B(A0A8Dp8A0A(B BBBd'FBB B(A0A8G0 8A0A(B BBBA e 8A0A(B BBBB 8(|QEKT(OEIp(QEK(8QEK(|QEK(QEK0(EX C U K  A  A )()<)P)0d)0x)L)X FBB B(A0A8G 8A0A(B BBBF )()T*`*,*@* T*p Dh*\ \FBB B(A0A8^ 0A(B BBBK *t ** *l6 L*AEb I  H x H  A I A B A W A d A dL<+XHEb I  H x H  A I A B A W A d A d+N +W +` +i +r +{ T, E` K O A ? 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A l A d4y 4 4D 5НD 5 D ,5HD X@5'E` K M C F J  G & A  A } A } A c A 5Xl 5OEI55n5P6606TD6@gC A E  F@ ` A  A  A  A W A T6gC A E  G@ e A { A  A  A W A T61gC A E  F@  A  A  A  A W A TL7gC A E  F@ ` A  A  A  A W A T7`gC A E  F@ ` A  A  A  A W A T7 gC A E  F@ ` A  A  A  A W A T80a EC Ef M x8|a EC Ef M P8#EC e C  I  A  A  A  A Y A P8T)EC e C  I  A  A  A  A Y A PD9.(EC e C  I  A  A  A  A Y A P94(EC e C  I  A  A  A  A Y A P9:(EC e C  I  A  A  A  A Y A P@:t@(EC e C  I  A  A  A  A Y A :PFEC El G P:L#EC d D  G  A  A  A  A X A  ;RWEQ(;SEM F pH;SEM F p4h;SFBA A(w  ABBJ D;dTFBB B(A0A8DPr8A0A(B BBB;T5;T<TU,\$<pW FBB B(A0A8 0A(B BBBJ s 0A(B BBBA <0_<`*<b(<f< h<oZ<Hwo=$=8=\PL=8Ec H w I w I @ H B A B A 8 A k A eP=tEc H w I w I @ H B A B A 8 A k A e=z >z >z 0>z D>`z X>z Tl>8 Eb I  G  G  I f A o A  A  A d A > >lQEK><? 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