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Editing: numeric.cpython-311.pyc
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Parameters ---------- a : array_like The shape and data-type of `a` define these same attributes of the returned array. dtype : data-type, optional Overrides the data type of the result. .. versionadded:: 1.6.0 order : {'C', 'F', 'A', or 'K'}, optional Overrides the memory layout of the result. 'C' means C-order, 'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous, 'C' otherwise. 'K' means match the layout of `a` as closely as possible. .. versionadded:: 1.6.0 subok : bool, optional. If True, then the newly created array will use the sub-class type of `a`, otherwise it will be a base-class array. Defaults to True. shape : int or sequence of ints, optional. Overrides the shape of the result. If order='K' and the number of dimensions is unchanged, will try to keep order, otherwise, order='C' is implied. .. versionadded:: 1.17.0 Returns ------- out : ndarray Array of zeros with the same shape and type as `a`. See Also -------- empty_like : Return an empty array with shape and type of input. ones_like : Return an array of ones with shape and type of input. full_like : Return a new array with shape of input filled with value. zeros : Return a new array setting values to zero. Examples -------- >>> x = np.arange(6) >>> x = x.reshape((2, 3)) >>> x array([[0, 1, 2], [3, 4, 5]]) >>> np.zeros_like(x) array([[0, 0, 0], [0, 0, 0]]) >>> y = np.arange(3, dtype=float) >>> y array([0., 1., 2.]) >>> np.zeros_like(y) array([0., 0., 0.]) �r r| r} r~ r �r �unsafe��casting)r r2 r r r )r{ r r| r} r~ �res�zs r rU rU C sN � �| �Q�e�5��U� K� K� K�C� �a�s�y�!�!�!�A���c�1�h�/�/�/�/� �Jr� �C)�likec � � |�t || ||�� � S t | ||� � }t j |dd�� � |S )a� Return a new array of given shape and type, filled with ones. Parameters ---------- shape : int or sequence of ints Shape of the new array, e.g., ``(2, 3)`` or ``2``. dtype : data-type, optional The desired data-type for the array, e.g., `numpy.int8`. Default is `numpy.float64`. order : {'C', 'F'}, optional, default: C Whether to store multi-dimensional data in row-major (C-style) or column-major (Fortran-style) order in memory. ${ARRAY_FUNCTION_LIKE} .. versionadded:: 1.20.0 Returns ------- out : ndarray Array of ones with the given shape, dtype, and order. See Also -------- ones_like : Return an array of ones with shape and type of input. empty : Return a new uninitialized array. zeros : Return a new array setting values to zero. full : Return a new array of given shape filled with value. Examples -------- >>> np.ones(5) array([1., 1., 1., 1., 1.]) >>> np.ones((5,), dtype=int) array([1, 1, 1, 1, 1]) >>> np.ones((2, 1)) array([[1.], [1.]]) >>> s = (2,2) >>> np.ones(s) array([[1., 1.], [1., 1.]]) N�r r| r r� r� )�_ones_with_liker r r )r~ r r| r� r{ s r rh rh � sP � �h �� �t�U�%�u�E�E�E�E� �e�U�E�"�"�A���a��H�-�-�-�-� �Hr� c � � | fS rx ry rz s r �_ones_like_dispatcherr� � r� r� c �\ � t | ||||�� � }t j |dd�� � |S )a Return an array of ones with the same shape and type as a given array. Parameters ---------- a : array_like The shape and data-type of `a` define these same attributes of the returned array. dtype : data-type, optional Overrides the data type of the result. .. versionadded:: 1.6.0 order : {'C', 'F', 'A', or 'K'}, optional Overrides the memory layout of the result. 'C' means C-order, 'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous, 'C' otherwise. 'K' means match the layout of `a` as closely as possible. .. versionadded:: 1.6.0 subok : bool, optional. If True, then the newly created array will use the sub-class type of `a`, otherwise it will be a base-class array. Defaults to True. shape : int or sequence of ints, optional. Overrides the shape of the result. If order='K' and the number of dimensions is unchanged, will try to keep order, otherwise, order='C' is implied. .. versionadded:: 1.17.0 Returns ------- out : ndarray Array of ones with the same shape and type as `a`. See Also -------- empty_like : Return an empty array with shape and type of input. zeros_like : Return an array of zeros with shape and type of input. full_like : Return a new array with shape of input filled with value. ones : Return a new array setting values to one. Examples -------- >>> x = np.arange(6) >>> x = x.reshape((2, 3)) >>> x array([[0, 1, 2], [3, 4, 5]]) >>> np.ones_like(x) array([[1, 1, 1], [1, 1, 1]]) >>> y = np.arange(3, dtype=float) >>> y array([0., 1., 2.]) >>> np.ones_like(y) array([1., 1., 1.]) r� r r� r� �r r r )r{ r r| r} r~ r� s r rV rV � s: � �| �Q�e�5��U� K� K� K�C���c�1�h�/�/�/�/� �Jr� c � � |fS rx ry )r~ � fill_valuer r| r� s r �_full_dispatcherr� s � � �'�Mr� c � � |�t || |||�� � S |�t |� � }|j }t | ||� � }t j ||d�� � |S )a� Return a new array of given shape and type, filled with `fill_value`. Parameters ---------- shape : int or sequence of ints Shape of the new array, e.g., ``(2, 3)`` or ``2``. fill_value : scalar or array_like Fill value. dtype : data-type, optional The desired data-type for the array The default, None, means ``np.array(fill_value).dtype``. order : {'C', 'F'}, optional Whether to store multidimensional data in C- or Fortran-contiguous (row- or column-wise) order in memory. ${ARRAY_FUNCTION_LIKE} .. versionadded:: 1.20.0 Returns ------- out : ndarray Array of `fill_value` with the given shape, dtype, and order. See Also -------- full_like : Return a new array with shape of input filled with value. empty : Return a new uninitialized array. ones : Return a new array setting values to one. zeros : Return a new array setting values to zero. Examples -------- >>> np.full((2, 2), np.inf) array([[inf, inf], [inf, inf]]) >>> np.full((2, 2), 10) array([[10, 10], [10, 10]]) >>> np.full((2, 2), [1, 2]) array([[1, 2], [1, 2]]) Nr� r� r� )�_full_with_liker r r r r )r~ r� r r| r� r{ s r ru ru s{ � �` �� ��e�Z�u�E�C� C� C� C� �}� �Z�(�(� �� �� �e�U�E�"�"�A���a��X�6�6�6�6� �Hr� c � � | fS rx ry )r{ r� r r| r} r~ s r �_full_like_dispatcherr� Q r� r� c �\ � t | ||||�� � }t j ||d�� � |S )af Return a full array with the same shape and type as a given array. Parameters ---------- a : array_like The shape and data-type of `a` define these same attributes of the returned array. fill_value : array_like Fill value. dtype : data-type, optional Overrides the data type of the result. order : {'C', 'F', 'A', or 'K'}, optional Overrides the memory layout of the result. 'C' means C-order, 'F' means F-order, 'A' means 'F' if `a` is Fortran contiguous, 'C' otherwise. 'K' means match the layout of `a` as closely as possible. subok : bool, optional. If True, then the newly created array will use the sub-class type of `a`, otherwise it will be a base-class array. Defaults to True. shape : int or sequence of ints, optional. Overrides the shape of the result. If order='K' and the number of dimensions is unchanged, will try to keep order, otherwise, order='C' is implied. .. versionadded:: 1.17.0 Returns ------- out : ndarray Array of `fill_value` with the same shape and type as `a`. See Also -------- empty_like : Return an empty array with shape and type of input. ones_like : Return an array of ones with shape and type of input. zeros_like : Return an array of zeros with shape and type of input. full : Return a new array of given shape filled with value. Examples -------- >>> x = np.arange(6, dtype=int) >>> np.full_like(x, 1) array([1, 1, 1, 1, 1, 1]) >>> np.full_like(x, 0.1) array([0, 0, 0, 0, 0, 0]) >>> np.full_like(x, 0.1, dtype=np.double) array([0.1, 0.1, 0.1, 0.1, 0.1, 0.1]) >>> np.full_like(x, np.nan, dtype=np.double) array([nan, nan, nan, nan, nan, nan]) >>> y = np.arange(6, dtype=np.double) >>> np.full_like(y, 0.1) array([0.1, 0.1, 0.1, 0.1, 0.1, 0.1]) >>> y = np.zeros([2, 2, 3], dtype=int) >>> np.full_like(y, [0, 0, 255]) array([[[ 0, 0, 255], [ 0, 0, 255]], [[ 0, 0, 255], [ 0, 0, 255]]]) r� r� r� r� )r{ r� r r| r} r~ r� s r rv rv U s: � �B �Q�e�5��U� K� K� K�C���c�:�x�8�8�8�8� �Jr� )�keepdimsc � � | fS rx ry )r{ �axisr� s r �_count_nonzero_dispatcherr� � r� r� Fc �Z � |�|st j | � � S t | � � } t j | j t j � � r | | j � � � k }n!| � t j d�� � }|� |t j |�� � S )a\ Counts the number of non-zero values in the array ``a``. The word "non-zero" is in reference to the Python 2.x built-in method ``__nonzero__()`` (renamed ``__bool__()`` in Python 3.x) of Python objects that tests an object's "truthfulness". For example, any number is considered truthful if it is nonzero, whereas any string is considered truthful if it is not the empty string. Thus, this function (recursively) counts how many elements in ``a`` (and in sub-arrays thereof) have their ``__nonzero__()`` or ``__bool__()`` method evaluated to ``True``. Parameters ---------- a : array_like The array for which to count non-zeros. axis : int or tuple, optional Axis or tuple of axes along which to count non-zeros. Default is None, meaning that non-zeros will be counted along a flattened version of ``a``. .. versionadded:: 1.12.0 keepdims : bool, optional If this is set to True, the axes that are counted are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array. .. versionadded:: 1.19.0 Returns ------- count : int or array of int Number of non-zero values in the array along a given axis. Otherwise, the total number of non-zero values in the array is returned. See Also -------- nonzero : Return the coordinates of all the non-zero values. Examples -------- >>> np.count_nonzero(np.eye(4)) 4 >>> a = np.array([[0, 1, 7, 0], ... [3, 0, 2, 19]]) >>> np.count_nonzero(a) 5 >>> np.count_nonzero(a, axis=0) array([1, 1, 2, 1]) >>> np.count_nonzero(a, axis=1) array([2, 3]) >>> np.count_nonzero(a, axis=1, keepdims=True) array([[2], [3]]) NF)�copy)r� r r� ) r rR r �np� issubdtyper � character�type�astyperG �sum�intp)r{ r� r� �a_bools r rR rR � s� � �x �|�H�|��'��*�*�*��1� � �A� �}�Q�W�b�l�+�+� 0��a�g�l�l�n�n�$������"�(���/�/�� �:�:�4�r�w��:� B� B�Br� c � � | j j S )a Check if the array is Fortran contiguous but *not* C contiguous. This function is obsolete and, because of changes due to relaxed stride checking, its return value for the same array may differ for versions of NumPy >= 1.10.0 and previous versions. If you only want to check if an array is Fortran contiguous use ``a.flags.f_contiguous`` instead. Parameters ---------- a : ndarray Input array. Returns ------- isfortran : bool Returns True if the array is Fortran contiguous but *not* C contiguous. Examples -------- np.array allows to specify whether the array is written in C-contiguous order (last index varies the fastest), or FORTRAN-contiguous order in memory (first index varies the fastest). >>> a = np.array([[1, 2, 3], [4, 5, 6]], order='C') >>> a array([[1, 2, 3], [4, 5, 6]]) >>> np.isfortran(a) False >>> b = np.array([[1, 2, 3], [4, 5, 6]], order='F') >>> b array([[1, 2, 3], [4, 5, 6]]) >>> np.isfortran(b) True The transpose of a C-ordered array is a FORTRAN-ordered array. >>> a = np.array([[1, 2, 3], [4, 5, 6]], order='C') >>> a array([[1, 2, 3], [4, 5, 6]]) >>> np.isfortran(a) False >>> b = a.T >>> b array([[1, 4], [2, 5], [3, 6]]) >>> np.isfortran(b) True C-ordered arrays evaluate as False even if they are also FORTRAN-ordered. >>> np.isfortran(np.array([1, 2], order='F')) False )�flags�fnc�r{ s r rT rT � s � �B �7�;�r� c � � | fS rx ry r� s r �_argwhere_dispatcherr� - r� r� c �� � t j | � � dk r/t j | � � } t | � � dd�dd�f S t t | � � � � S )a� Find the indices of array elements that are non-zero, grouped by element. Parameters ---------- a : array_like Input data. Returns ------- index_array : (N, a.ndim) ndarray Indices of elements that are non-zero. Indices are grouped by element. This array will have shape ``(N, a.ndim)`` where ``N`` is the number of non-zero items. See Also -------- where, nonzero Notes ----- ``np.argwhere(a)`` is almost the same as ``np.transpose(np.nonzero(a))``, but produces a result of the correct shape for a 0D array. The output of ``argwhere`` is not suitable for indexing arrays. For this purpose use ``nonzero(a)`` instead. Examples -------- >>> x = np.arange(6).reshape(2,3) >>> x array([[0, 1, 2], [3, 4, 5]]) >>> np.argwhere(x>1) array([[0, 2], [1, 0], [1, 1], [1, 2]]) r N)r� �ndimr9 � atleast_1drS � transpose�nonzeror� s r rS rS 1 sW � �V �w�q�z�z�Q��� � !�!� $� $����{�{�1�1�1�R�a�R�4� � � �W�Q�Z�Z� � � r� c � � | fS rx ry r� s r �_flatnonzero_dispatcherr� c r� r� c �Z � t j t j | � � � � d S )aH Return indices that are non-zero in the flattened version of a. This is equivalent to ``np.nonzero(np.ravel(a))[0]``. Parameters ---------- a : array_like Input data. Returns ------- res : ndarray Output array, containing the indices of the elements of ``a.ravel()`` that are non-zero. See Also -------- nonzero : Return the indices of the non-zero elements of the input array. ravel : Return a 1-D array containing the elements of the input array. Examples -------- >>> x = np.arange(-2, 3) >>> x array([-2, -1, 0, 1, 2]) >>> np.flatnonzero(x) array([0, 1, 3, 4]) Use the indices of the non-zero elements as an index array to extract these elements: >>> x.ravel()[np.flatnonzero(x)] array([-2, -1, 1, 2]) r )r� r� �ravelr� s r rk rk g s" � �L �:�b�h�q�k�k� "� "�1� %�%r� c � � | |fS rx ry �r{ �v�modes r �_correlate_dispatcherr� � � � � �q�6�Mr� �validc �. � t j | ||� � S )a� Cross-correlation of two 1-dimensional sequences. This function computes the correlation as generally defined in signal processing texts: .. math:: c_k = \sum_n a_{n+k} \cdot \overline{v}_n with a and v sequences being zero-padded where necessary and :math:`\overline x` denoting complex conjugation. Parameters ---------- a, v : array_like Input sequences. mode : {'valid', 'same', 'full'}, optional Refer to the `convolve` docstring. Note that the default is 'valid', unlike `convolve`, which uses 'full'. old_behavior : bool `old_behavior` was removed in NumPy 1.10. If you need the old behavior, use `multiarray.correlate`. Returns ------- out : ndarray Discrete cross-correlation of `a` and `v`. See Also -------- convolve : Discrete, linear convolution of two one-dimensional sequences. multiarray.correlate : Old, no conjugate, version of correlate. scipy.signal.correlate : uses FFT which has superior performance on large arrays. Notes ----- The definition of correlation above is not unique and sometimes correlation may be defined differently. Another common definition is: .. math:: c'_k = \sum_n a_{n} \cdot \overline{v_{n+k}} which is related to :math:`c_k` by :math:`c'_k = c_{-k}`. `numpy.correlate` may perform slowly in large arrays (i.e. n = 1e5) because it does not use the FFT to compute the convolution; in that case, `scipy.signal.correlate` might be preferable. Examples -------- >>> np.correlate([1, 2, 3], [0, 1, 0.5]) array([3.5]) >>> np.correlate([1, 2, 3], [0, 1, 0.5], "same") array([2. , 3.5, 3. ]) >>> np.correlate([1, 2, 3], [0, 1, 0.5], "full") array([0.5, 2. , 3.5, 3. , 0. ]) Using complex sequences: >>> np.correlate([1+1j, 2, 3-1j], [0, 1, 0.5j], 'full') array([ 0.5-0.5j, 1.0+0.j , 1.5-1.5j, 3.0-1.j , 0.0+0.j ]) Note that you get the time reversed, complex conjugated result (:math:`\overline{c_{-k}}`) when the two input sequences a and v change places: >>> np.correlate([0, 1, 0.5j], [1+1j, 2, 3-1j], 'full') array([ 0.0+0.j , 3.0+1.j , 1.5+1.5j, 1.0+0.j , 0.5+0.5j]) )r � correlate2r� s r rW rW � s � �N � ��A�t� ,� ,�,r� c � � | |fS rx ry r� s r �_convolve_dispatcherr� � r� r� ru c �X � t | dd�� � t |dd�� � }} t |� � t | � � k r|| }} t | � � dk rt d� � �t |� � dk rt d� � �t j | |ddd� |� � S ) a Returns the discrete, linear convolution of two one-dimensional sequences. The convolution operator is often seen in signal processing, where it models the effect of a linear time-invariant system on a signal [1]_. In probability theory, the sum of two independent random variables is distributed according to the convolution of their individual distributions. If `v` is longer than `a`, the arrays are swapped before computation. Parameters ---------- a : (N,) array_like First one-dimensional input array. v : (M,) array_like Second one-dimensional input array. mode : {'full', 'valid', 'same'}, optional 'full': By default, mode is 'full'. This returns the convolution at each point of overlap, with an output shape of (N+M-1,). At the end-points of the convolution, the signals do not overlap completely, and boundary effects may be seen. 'same': Mode 'same' returns output of length ``max(M, N)``. Boundary effects are still visible. 'valid': Mode 'valid' returns output of length ``max(M, N) - min(M, N) + 1``. The convolution product is only given for points where the signals overlap completely. Values outside the signal boundary have no effect. Returns ------- out : ndarray Discrete, linear convolution of `a` and `v`. See Also -------- scipy.signal.fftconvolve : Convolve two arrays using the Fast Fourier Transform. scipy.linalg.toeplitz : Used to construct the convolution operator. polymul : Polynomial multiplication. Same output as convolve, but also accepts poly1d objects as input. Notes ----- The discrete convolution operation is defined as .. math:: (a * v)_n = \sum_{m = -\infty}^{\infty} a_m v_{n - m} It can be shown that a convolution :math:`x(t) * y(t)` in time/space is equivalent to the multiplication :math:`X(f) Y(f)` in the Fourier domain, after appropriate padding (padding is necessary to prevent circular convolution). Since multiplication is more efficient (faster) than convolution, the function `scipy.signal.fftconvolve` exploits the FFT to calculate the convolution of large data-sets. References ---------- .. [1] Wikipedia, "Convolution", https://en.wikipedia.org/wiki/Convolution Examples -------- Note how the convolution operator flips the second array before "sliding" the two across one another: >>> np.convolve([1, 2, 3], [0, 1, 0.5]) array([0. , 1. , 2.5, 4. , 1.5]) Only return the middle values of the convolution. Contains boundary effects, where zeros are taken into account: >>> np.convolve([1,2,3],[0,1,0.5], 'same') array([1. , 2.5, 4. ]) The two arrays are of the same length, so there is only one position where they completely overlap: >>> np.convolve([1,2,3],[0,1,0.5], 'valid') array([2.5]) Fr )r� �ndminr za cannot be emptyzv cannot be emptyN���)r �len� ValueErrorr rW r� s r rX rX � s� � �r ���a� (� (� (�%���Q�*G�*G�*G�q�A� �A����Q������!�1�� �1�v�v��{�{��,�-�-�-� �1�v�v��{�{��,�-�-�-� � ��1�T�T�r�T�7�D� 1� 1�1r� c � � | ||fS rx ry �r{ �b�outs r �_outer_dispatcherr� E s � � �q�#�;�r� c �� � t | � � } t |� � }t | � � � dd�t f |� � � t dd�f |� � S )a' Compute the outer product of two vectors. Given two vectors `a` and `b` of length ``M`` and ``N``, repsectively, the outer product [1]_ is:: [[a_0*b_0 a_0*b_1 ... a_0*b_{N-1} ] [a_1*b_0 . [ ... . [a_{M-1}*b_0 a_{M-1}*b_{N-1} ]] Parameters ---------- a : (M,) array_like First input vector. Input is flattened if not already 1-dimensional. b : (N,) array_like Second input vector. Input is flattened if not already 1-dimensional. out : (M, N) ndarray, optional A location where the result is stored .. versionadded:: 1.9.0 Returns ------- out : (M, N) ndarray ``out[i, j] = a[i] * b[j]`` See also -------- inner einsum : ``einsum('i,j->ij', a.ravel(), b.ravel())`` is the equivalent. ufunc.outer : A generalization to dimensions other than 1D and other operations. ``np.multiply.outer(a.ravel(), b.ravel())`` is the equivalent. tensordot : ``np.tensordot(a.ravel(), b.ravel(), axes=((), ()))`` is the equivalent. References ---------- .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*, 3rd ed., Baltimore, MD, Johns Hopkins University Press, 1996, pg. 8. Examples -------- Make a (*very* coarse) grid for computing a Mandelbrot set: >>> rl = np.outer(np.ones((5,)), np.linspace(-2, 2, 5)) >>> rl array([[-2., -1., 0., 1., 2.], [-2., -1., 0., 1., 2.], [-2., -1., 0., 1., 2.], [-2., -1., 0., 1., 2.], [-2., -1., 0., 1., 2.]]) >>> im = np.outer(1j*np.linspace(2, -2, 5), np.ones((5,))) >>> im array([[0.+2.j, 0.+2.j, 0.+2.j, 0.+2.j, 0.+2.j], [0.+1.j, 0.+1.j, 0.+1.j, 0.+1.j, 0.+1.j], [0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j, 0.+0.j], [0.-1.j, 0.-1.j, 0.-1.j, 0.-1.j, 0.-1.j], [0.-2.j, 0.-2.j, 0.-2.j, 0.-2.j, 0.-2.j]]) >>> grid = rl + im >>> grid array([[-2.+2.j, -1.+2.j, 0.+2.j, 1.+2.j, 2.+2.j], [-2.+1.j, -1.+1.j, 0.+1.j, 1.+1.j, 2.+1.j], [-2.+0.j, -1.+0.j, 0.+0.j, 1.+0.j, 2.+0.j], [-2.-1.j, -1.-1.j, 0.-1.j, 1.-1.j, 2.-1.j], [-2.-2.j, -1.-2.j, 0.-2.j, 1.-2.j, 2.-2.j]]) An example using a "vector" of letters: >>> x = np.array(['a', 'b', 'c'], dtype=object) >>> np.outer(x, [1, 2, 3]) array([['a', 'aa', 'aaa'], ['b', 'bb', 'bbb'], ['c', 'cc', 'ccc']], dtype=object) N)r r< r� rP r� s r rY rY I sW � �d �� � �A��� � �A� �A�G�G�I�I�a�a�a��j�)�1�7�7�9�9�W�a�a�a�Z�+@�#� F� F�Fr� c � � | |fS rx ry )r{ r� �axess r �_tensordot_dispatcherr� � r� r� c �� ����� t |� � |\ ��nK# t $ r> t t | d� � � � �t t d|� � � � �Y nw xY w t �� � }t �� � �n# t $ r �g�d}Y nw xY w t �� � }t �� � �n# t $ r �g�d}Y nw xY wt | � � t |� � }} | j �| j }|j �|j }d}||k rd}nlt |� � D ]\}��| ��| k rd} n9�| dk r�|xx |z cc< �| dk r�|xx |z cc< �]|st d� � ��fd�t |� � D � � } | �z } d} �D ] } | �| z } �t t j �fd�| D � � � � � � | f} �fd�| D � � }�fd �t |� � D � � } �| z }d} �D ] } | �| z } �| t t j �fd �| D � � � � � � f}�fd �| D � � }| � | � � � | � � }|� |� � � |� � }t ||� � }|� ||z � � S ) a� Compute tensor dot product along specified axes. Given two tensors, `a` and `b`, and an array_like object containing two array_like objects, ``(a_axes, b_axes)``, sum the products of `a`'s and `b`'s elements (components) over the axes specified by ``a_axes`` and ``b_axes``. The third argument can be a single non-negative integer_like scalar, ``N``; if it is such, then the last ``N`` dimensions of `a` and the first ``N`` dimensions of `b` are summed over. Parameters ---------- a, b : array_like Tensors to "dot". axes : int or (2,) array_like * integer_like If an int N, sum over the last N axes of `a` and the first N axes of `b` in order. The sizes of the corresponding axes must match. * (2,) array_like Or, a list of axes to be summed over, first sequence applying to `a`, second to `b`. Both elements array_like must be of the same length. Returns ------- output : ndarray The tensor dot product of the input. See Also -------- dot, einsum Notes ----- Three common use cases are: * ``axes = 0`` : tensor product :math:`a\otimes b` * ``axes = 1`` : tensor dot product :math:`a\cdot b` * ``axes = 2`` : (default) tensor double contraction :math:`a:b` When `axes` is integer_like, the sequence for evaluation will be: first the -Nth axis in `a` and 0th axis in `b`, and the -1th axis in `a` and Nth axis in `b` last. When there is more than one axis to sum over - and they are not the last (first) axes of `a` (`b`) - the argument `axes` should consist of two sequences of the same length, with the first axis to sum over given first in both sequences, the second axis second, and so forth. The shape of the result consists of the non-contracted axes of the first tensor, followed by the non-contracted axes of the second. Examples -------- A "traditional" example: >>> a = np.arange(60.).reshape(3,4,5) >>> b = np.arange(24.).reshape(4,3,2) >>> c = np.tensordot(a,b, axes=([1,0],[0,1])) >>> c.shape (5, 2) >>> c array([[4400., 4730.], [4532., 4874.], [4664., 5018.], [4796., 5162.], [4928., 5306.]]) >>> # A slower but equivalent way of computing the same... >>> d = np.zeros((5,2)) >>> for i in range(5): ... for j in range(2): ... for k in range(3): ... for n in range(4): ... d[i,j] += a[k,n,i] * b[n,k,j] >>> c == d array([[ True, True], [ True, True], [ True, True], [ True, True], [ True, True]]) An extended example taking advantage of the overloading of + and \*: >>> a = np.array(range(1, 9)) >>> a.shape = (2, 2, 2) >>> A = np.array(('a', 'b', 'c', 'd'), dtype=object) >>> A.shape = (2, 2) >>> a; A array([[[1, 2], [3, 4]], [[5, 6], [7, 8]]]) array([['a', 'b'], ['c', 'd']], dtype=object) >>> np.tensordot(a, A) # third argument default is 2 for double-contraction array(['abbcccdddd', 'aaaaabbbbbbcccccccdddddddd'], dtype=object) >>> np.tensordot(a, A, 1) array([[['acc', 'bdd'], ['aaacccc', 'bbbdddd']], [['aaaaacccccc', 'bbbbbdddddd'], ['aaaaaaacccccccc', 'bbbbbbbdddddddd']]], dtype=object) >>> np.tensordot(a, A, 0) # tensor product (result too long to incl.) array([[[[['a', 'b'], ['c', 'd']], ... >>> np.tensordot(a, A, (0, 1)) array([[['abbbbb', 'cddddd'], ['aabbbbbb', 'ccdddddd']], [['aaabbbbbbb', 'cccddddddd'], ['aaaabbbbbbbb', 'ccccdddddddd']]], dtype=object) >>> np.tensordot(a, A, (2, 1)) array([[['abb', 'cdd'], ['aaabbbb', 'cccdddd']], [['aaaaabbbbbb', 'cccccdddddd'], ['aaaaaaabbbbbbbb', 'cccccccdddddddd']]], dtype=object) >>> np.tensordot(a, A, ((0, 1), (0, 1))) array(['abbbcccccddddddd', 'aabbbbccccccdddddddd'], dtype=object) >>> np.tensordot(a, A, ((2, 1), (1, 0))) array(['acccbbdddd', 'aaaaacccccccbbbbbbdddddddd'], dtype=object) r r TFzshape-mismatch for sumc � �� g | ]}|�v�|�� S ry ry )�.0�k�axes_as �r � <listcomp>z tensordot.<locals>.<listcomp>O � �� � 6� 6� 6�1�a�v�o�o�Q�o�o�or� c � �� g | ] }�| �� S ry ry )r� �ax�as_s �r r� z tensordot.<locals>.<listcomp>T s �� �&?�&?�&?�2�s�2�w�&?�&?�&?r� c � �� g | ] }�| �� S ry ry )r� r� r� s �r r� z tensordot.<locals>.<listcomp>U s �� � (� (� (�$�C��I� (� (� (r� c � �� g | ]}|�v�|�� S ry ry )r� r� �axes_bs �r r� z tensordot.<locals>.<listcomp>W r� r� c � �� g | ] }�| �� S ry ry )r� r� �bss �r r� z tensordot.<locals>.<listcomp>\ s �� �*B�*B�*B�b�2�b�6�*B�*B�*Br� c � �� g | ] }�| �� S ry ry )r� r� r� s �r r� z tensordot.<locals>.<listcomp>] s �� � '� '� '��B�t�H� '� '� 'r� )�iter� Exception�list�ranger� � TypeErrorr r~ r� r� �intr< �reducer� �reshaper )r{ r� r� �na�nb�nda�ndb�equalr� �notin� newaxes_a�N2r� � newshape_a�olda� newaxes_b� newshape_b�oldb�at�btr� r� r� r� r� s @@@@r r^ r^ � sn ����� �B � �T� � � � ������ � &� &� &��e�T�E�1�o�o�&�&���e�A�t�n�n�%�%����&���� � ��[�[���f� � ���� � � � ���� ��������� ��[�[���f� � ���� � � � ���� �������� �1�:�:�w�q�z�z�q�A� �'�C� �&�C� ��B� �&�C� �E� �R�x�x�����r��� !� !�A��6�!�9�~��F�1�I� �.�.� �����a�y�1�}�}��q� � � �S� � � � ��a�y�1�}�}��q� � � �S� � � � �� � 3��1�2�2�2� 7� 6� 6� 6��c� � � 6� 6� 6�E����I� �B�� � �� �c�$�i�����h�o�&?�&?�&?�&?��&?�&?�&?�@�@�A�A�2�F�J� (� (� (� (�%� (� (� (�D� 6� 6� 6� 6��c� � � 6� 6� 6�E����I� �B�� � �� �b��h�����c�(�/�*B�*B�*B�*B�E�*B�*B�*B�C�C�D�D�E�J� '� '� '� '�� '� '� '�D� ���Y� � � '� '� � 3� 3�B� ���Y� � � '� '� � 3� 3�B� �b�"�+�+�C� �;�;�t�d�{� #� #�#s4 � �AA#�"A#�' B �B�B� B; �;C � C c � � | fS rx ry )r{ �shiftr� s r �_roll_dispatcherr� e r� r� c �0 � t | � � } |�;t | � � � |d� � � | j � � S t || j d�� � }t ||� � }|j dk rt d� � �d� t | j � � D � � }|D ]\ }}||xx |z cc< �t d� � t d� � ffg| j z }|� � � D ]^\ }}|| j | pdz }|rEt d| � � t |d� � ft | d� � t d|� � ff||< �_t | � � } t j |� D ]} t | � \ } } | | | | < �| S )a) Roll array elements along a given axis. Elements that roll beyond the last position are re-introduced at the first. Parameters ---------- a : array_like Input array. shift : int or tuple of ints The number of places by which elements are shifted. If a tuple, then `axis` must be a tuple of the same size, and each of the given axes is shifted by the corresponding number. If an int while `axis` is a tuple of ints, then the same value is used for all given axes. axis : int or tuple of ints, optional Axis or axes along which elements are shifted. By default, the array is flattened before shifting, after which the original shape is restored. Returns ------- res : ndarray Output array, with the same shape as `a`. See Also -------- rollaxis : Roll the specified axis backwards, until it lies in a given position. Notes ----- .. versionadded:: 1.12.0 Supports rolling over multiple dimensions simultaneously. Examples -------- >>> x = np.arange(10) >>> np.roll(x, 2) array([8, 9, 0, 1, 2, 3, 4, 5, 6, 7]) >>> np.roll(x, -2) array([2, 3, 4, 5, 6, 7, 8, 9, 0, 1]) >>> x2 = np.reshape(x, (2, 5)) >>> x2 array([[0, 1, 2, 3, 4], [5, 6, 7, 8, 9]]) >>> np.roll(x2, 1) array([[9, 0, 1, 2, 3], [4, 5, 6, 7, 8]]) >>> np.roll(x2, -1) array([[1, 2, 3, 4, 5], [6, 7, 8, 9, 0]]) >>> np.roll(x2, 1, axis=0) array([[5, 6, 7, 8, 9], [0, 1, 2, 3, 4]]) >>> np.roll(x2, -1, axis=0) array([[5, 6, 7, 8, 9], [0, 1, 2, 3, 4]]) >>> np.roll(x2, 1, axis=1) array([[4, 0, 1, 2, 3], [9, 5, 6, 7, 8]]) >>> np.roll(x2, -1, axis=1) array([[1, 2, 3, 4, 0], [6, 7, 8, 9, 5]]) >>> np.roll(x2, (1, 1), axis=(1, 0)) array([[9, 5, 6, 7, 8], [4, 0, 1, 2, 3]]) >>> np.roll(x2, (2, 1), axis=(1, 0)) array([[8, 9, 5, 6, 7], [3, 4, 0, 1, 2]]) Nr T)�allow_duplicater z4'shift' and 'axis' should be scalars or 1D sequencesc � � i | ]}|d ��S �r ry )r� r� s r � <dictcomp>zroll.<locals>.<dictcomp>� s � �0�0�0�B�"�a�0�0�0r� )r rZ r� r� r~ �normalize_axis_tupler� r r� r� �slice�itemsr � itertools�product�zip) r{ r� r� � broadcasted�shifts�shr� �rolls�offset�resultrb � arr_index� res_indexs r rZ rZ i s� � �Z �1� � �A� �|��A�G�G�I�I�u�a�(�(�0�0���9�9�9� $�D�!�&�$�G�G�G�� ��t�,�,� � � �a� � � �F�H� H� H�0�0�%���-�-�0�0�0��!� � �F�B�� �2�J�J�J�"� �J�J�J�J���;�;��d� � �,�.�/�!�&�8�� �,�,�.�.� J� J�J�B�� �a�g�b�k�&�Q� &�F�� J� #�D�6�'� 2� 2�E�&�$�4G�4G� H� #�V�G�T� 2� 2�E�$��4G�4G� H� J��b� �� �A� � �� �(�%�0� -� -�G�#&��=� �I�y� !�)� �F�9� � �� r� c � � | fS rx ry )r{ r� �starts r �_rollaxis_dispatcherr � r� r� c � � | j }t ||� � }|dk r||z }d}d|cxk r |dz k s n t |d| d|dz |fz � � �||k r|dz }||k r| d S t t d|� � � � }|� |� � |� ||� � | � |� � S )a Roll the specified axis backwards, until it lies in a given position. This function continues to be supported for backward compatibility, but you should prefer `moveaxis`. The `moveaxis` function was added in NumPy 1.11. Parameters ---------- a : ndarray Input array. axis : int The axis to be rolled. The positions of the other axes do not change relative to one another. start : int, optional When ``start <= axis``, the axis is rolled back until it lies in this position. When ``start > axis``, the axis is rolled until it lies before this position. The default, 0, results in a "complete" roll. The following table describes how negative values of ``start`` are interpreted: .. table:: :align: left +-------------------+----------------------+ | ``start`` | Normalized ``start`` | +===================+======================+ | ``-(arr.ndim+1)`` | raise ``AxisError`` | +-------------------+----------------------+ | ``-arr.ndim`` | 0 | +-------------------+----------------------+ | |vdots| | |vdots| | +-------------------+----------------------+ | ``-1`` | ``arr.ndim-1`` | +-------------------+----------------------+ | ``0`` | ``0`` | +-------------------+----------------------+ | |vdots| | |vdots| | +-------------------+----------------------+ | ``arr.ndim`` | ``arr.ndim`` | +-------------------+----------------------+ | ``arr.ndim + 1`` | raise ``AxisError`` | +-------------------+----------------------+ .. |vdots| unicode:: U+22EE .. Vertical Ellipsis Returns ------- res : ndarray For NumPy >= 1.10.0 a view of `a` is always returned. For earlier NumPy versions a view of `a` is returned only if the order of the axes is changed, otherwise the input array is returned. See Also -------- moveaxis : Move array axes to new positions. roll : Roll the elements of an array by a number of positions along a given axis. Examples -------- >>> a = np.ones((3,4,5,6)) >>> np.rollaxis(a, 3, 1).shape (3, 6, 4, 5) >>> np.rollaxis(a, 2).shape (5, 3, 4, 6) >>> np.rollaxis(a, 1, 4).shape (3, 5, 6, 4) r z5'%s' arg requires %d <= %s < %d, but %d was passed inr r .)r� r3 rK r� r� �remove�insertr� )r{ r� r �n�msgr� s r r[ r[ � s� � �P ��A� ��a� (� (�D� �q�y�y� �� �� A�C� �� � � � ��Q�� � � � ���w���G�Q��U�E� B�B�C�C�C� �e�|�|� �� �� �u�}�}���v� � ��a�� � � � �D��K�K������K�K��t� � � � �;�;�t� � � r� c � ��� t | � � t t fvr' t j | � � g} n# t $ r Y nw xY wt ��fd�| D � � � � } |s`t t | � � � � t | � � k r3�r"t d� �� � � � �t d� � �| S )a� Normalizes an axis argument into a tuple of non-negative integer axes. This handles shorthands such as ``1`` and converts them to ``(1,)``, as well as performing the handling of negative indices covered by `normalize_axis_index`. By default, this forbids axes from being specified multiple times. Used internally by multi-axis-checking logic. .. versionadded:: 1.13.0 Parameters ---------- axis : int, iterable of int The un-normalized index or indices of the axis. ndim : int The number of dimensions of the array that `axis` should be normalized against. argname : str, optional A prefix to put before the error message, typically the name of the argument. allow_duplicate : bool, optional If False, the default, disallow an axis from being specified twice. Returns ------- normalized_axes : tuple of int The normalized axis index, such that `0 <= normalized_axis < ndim` Raises ------ AxisError If any axis provided is out of range ValueError If an axis is repeated See also -------- normalize_axis_index : normalizing a single scalar axis c �2 �� g | ]}t |��� � ��S ry )r3 )r� r� �argnamer� s ��r r� z(normalize_axis_tuple.<locals>.<listcomp>d s& �� �I�I�I�b�&�r�4��9�9�I�I�Ir� z repeated axis in `{}` argumentz repeated axis) r� �tupler� �operator�indexr� r� �setr� �format)r� r� r r� s `` r r r 2 s� ��� �X �D�z�z�%�� �&�&� � �N�4�(�(�)�D�D��� � � � �D� ���� �I�I�I�I�I�D�I�I�I� J� J�D� � .�s�3�t�9�9�~�~��T��� :� :� � .� � =� D� D�W� M� M�N�N� N� �_�-�-� -� �Ks �7 � A�Ac � � | fS rx ry )r{ �source� destinations r �_moveaxis_dispatcherr# m r� r� c �� �� | j }n&# t $ r t | � � } | j }Y nw xY wt �| j d� � �t || j d� � }t �� � t |� � k rt d� � ��fd�t | j � � D � � }t t |�� � � � D ]\ }}|� ||� � � ||� � }|S )aw Move axes of an array to new positions. Other axes remain in their original order. .. versionadded:: 1.11.0 Parameters ---------- a : np.ndarray The array whose axes should be reordered. source : int or sequence of int Original positions of the axes to move. These must be unique. destination : int or sequence of int Destination positions for each of the original axes. These must also be unique. Returns ------- result : np.ndarray Array with moved axes. This array is a view of the input array. See Also -------- transpose : Permute the dimensions of an array. swapaxes : Interchange two axes of an array. Examples -------- >>> x = np.zeros((3, 4, 5)) >>> np.moveaxis(x, 0, -1).shape (4, 5, 3) >>> np.moveaxis(x, -1, 0).shape (5, 3, 4) These all achieve the same result: >>> np.transpose(x).shape (5, 4, 3) >>> np.swapaxes(x, 0, -1).shape (5, 4, 3) >>> np.moveaxis(x, [0, 1], [-1, -2]).shape (5, 4, 3) >>> np.moveaxis(x, [0, 1, 2], [-1, -2, -3]).shape (5, 4, 3) r! r" zJ`source` and `destination` arguments must have the same number of elementsc � �� g | ]}|�v�|�� S ry ry )r� r r! s �r r� z moveaxis.<locals>.<listcomp>� s �� � 9� 9� 9�1��&� � �Q� � � r� ) r� �AttributeErrorr r r� r� r� r� �sortedr r )r{ r! r" r� r| �dest�srcr s ` r r\ r\ q s �� �b ��K� � �� � � � � �A�J�J���K� � � � ���� "�&�!�&�(� ;� ;�F�&�{�A�F�M�J�J�K� �6�{�{�c�+�&�&�&�&�� 7� 8� 8� 8� :� 9� 9� 9��a�f� � � 9� 9� 9�E��C� �V� 4� 4�5�5� � � ��c� � � �T�3� � � � � �Y�u� � �F� �Ms � � .�.c � � | |fS rx ry )r{ r� �axisa�axisb�axiscr� s r �_cross_dispatcherr. � r� r� r� c �@ � |� |fdz \ }}}t | � � } t |� � }t || j d�� � }t ||j d�� � }t | |d� � } t ||d� � }d}| j d dvs|j d dvrt |� � �t | d |d � � j }| j d dk s|j d dk r$|d z }t |t |� � d �� � }t | j |j � � }t ||� � } | � |� � } |� |� � }| d } | d } | j d dk r| d } |d } |d }|j d dk r|d }| j dk r)| j d dk r| d }| d }| d }| j d dk r�|j d dk r t | || �� � | | | z z } | S |j d dk sJ �t | ||�� � t | ||�� � t ||�� � t | ||�� � || | z z }�n| j d dk sJ �|j d dk r|t | ||�� � t | |z � � }||z }t | | |�� � t | ||�� � ||z }t | ||�� � t | | |�� � ||z }nb|j d dk sJ �t | ||�� � t ||�� � t | | |�� � t | ||�� � || | z z }t | d|� � S )a Return the cross product of two (arrays of) vectors. The cross product of `a` and `b` in :math:`R^3` is a vector perpendicular to both `a` and `b`. If `a` and `b` are arrays of vectors, the vectors are defined by the last axis of `a` and `b` by default, and these axes can have dimensions 2 or 3. Where the dimension of either `a` or `b` is 2, the third component of the input vector is assumed to be zero and the cross product calculated accordingly. In cases where both input vectors have dimension 2, the z-component of the cross product is returned. Parameters ---------- a : array_like Components of the first vector(s). b : array_like Components of the second vector(s). axisa : int, optional Axis of `a` that defines the vector(s). By default, the last axis. axisb : int, optional Axis of `b` that defines the vector(s). By default, the last axis. axisc : int, optional Axis of `c` containing the cross product vector(s). Ignored if both input vectors have dimension 2, as the return is scalar. By default, the last axis. axis : int, optional If defined, the axis of `a`, `b` and `c` that defines the vector(s) and cross product(s). Overrides `axisa`, `axisb` and `axisc`. Returns ------- c : ndarray Vector cross product(s). Raises ------ ValueError When the dimension of the vector(s) in `a` and/or `b` does not equal 2 or 3. See Also -------- inner : Inner product outer : Outer product. ix_ : Construct index arrays. Notes ----- .. versionadded:: 1.9.0 Supports full broadcasting of the inputs. Examples -------- Vector cross-product. >>> x = [1, 2, 3] >>> y = [4, 5, 6] >>> np.cross(x, y) array([-3, 6, -3]) One vector with dimension 2. >>> x = [1, 2] >>> y = [4, 5, 6] >>> np.cross(x, y) array([12, -6, -3]) Equivalently: >>> x = [1, 2, 0] >>> y = [4, 5, 6] >>> np.cross(x, y) array([12, -6, -3]) Both vectors with dimension 2. >>> x = [1,2] >>> y = [4,5] >>> np.cross(x, y) array(-3) Multiple vector cross-products. Note that the direction of the cross product vector is defined by the *right-hand rule*. >>> x = np.array([[1,2,3], [4,5,6]]) >>> y = np.array([[4,5,6], [1,2,3]]) >>> np.cross(x, y) array([[-3, 6, -3], [ 3, -6, 3]]) The orientation of `c` can be changed using the `axisc` keyword. >>> np.cross(x, y, axisc=0) array([[-3, 3], [ 6, -6], [-3, 3]]) Change the vector definition of `x` and `y` using `axisa` and `axisb`. >>> x = np.array([[1,2,3], [4,5,6], [7, 8, 9]]) >>> y = np.array([[7, 8, 9], [4,5,6], [1,2,3]]) >>> np.cross(x, y) array([[ -6, 12, -6], [ 0, 0, 0], [ 6, -12, 6]]) >>> np.cross(x, y, axisa=0, axisb=0) array([[-24, 48, -24], [-30, 60, -30], [-36, 72, -36]]) N� r+ )� msg_prefixr, r� zDincompatible dimensions for cross product (dimension must be 2 or 3))rH r0 ).r )r0 r- ).r ).rH r rH )r� )r r3 r� r\ r~ r� r r� r+ r r r� r< �negativer )r{ r� r+ r, r- r� r r~ r �cp�a0�a1�a2�b0�b1�b2�cp0�cp1�cp2�tmps r r] r] � s� � �d �� #�g��k���u�e��� � �A��� � �A� ����7� C� C� C�E� ����7� C� C� C�E� ��E�2� � �A���E�2� � �A� (�C��w�r�{�&� � �A�G�B�K�v�$=�$=���o�o� � �a��i��6�� +� +� 1�E��w�r�{�a���1�7�2�;�!�+�+� �� ��$�U�C��J�J�7�K�K�K�� �!�'�1�7� +� +�E� �u�e� � �B� ������A� ������A� �6��B� �6��B��w�r�{�a��� �v�Y�� �6��B� �6��B��w�r�{�a��� �v�Y�� �w�!�|�|���� ��)�)���j����j����j���w�r�{�a��� �7�2�;�!� � � �R��� $� $� $� $� �"�r�'�M�B��I��7�2�;�!�#�#�#�#� �R��� %� %� %� %� �R��� %� %� %� %� �S�c� "� "� "� "� �R��� %� %� %� %� �2��7�N�C�C��w�r�{�a� � � � � �7�2�;�!� � � �R��� %� %� %� %���R��.�.�C� �3�J�C� �R��� %� %� %� %� �R��� %� %� %� %� �3�J�C� �R��� %� %� %� %� �R��� %� %� %� %� �3�J�C�C��7�2�;�!�#�#�#�#� �R��� %� %� %� %� �S�c� "� "� "� "� �R��� %� %� %� %� �R��� %� %� %� %� �2��7�N�C� �B��E� "� "�"r� �littlec �Z � t | � � } t | � � }d|z }|rt � � }nt |f| z |�� � }t | � � D ]Q\ }}t ||�� � � |d|� |fz ||dz d� z � � }|r||fz }�L|||< �R|S )a� Return an array representing the indices of a grid. Compute an array where the subarrays contain index values 0, 1, ... varying only along the corresponding axis. Parameters ---------- dimensions : sequence of ints The shape of the grid. dtype : dtype, optional Data type of the result. sparse : boolean, optional Return a sparse representation of the grid instead of a dense representation. Default is False. .. versionadded:: 1.17 Returns ------- grid : one ndarray or tuple of ndarrays If sparse is False: Returns one array of grid indices, ``grid.shape = (len(dimensions),) + tuple(dimensions)``. If sparse is True: Returns a tuple of arrays, with ``grid[i].shape = (1, ..., 1, dimensions[i], 1, ..., 1)`` with dimensions[i] in the ith place See Also -------- mgrid, ogrid, meshgrid Notes ----- The output shape in the dense case is obtained by prepending the number of dimensions in front of the tuple of dimensions, i.e. if `dimensions` is a tuple ``(r0, ..., rN-1)`` of length ``N``, the output shape is ``(N, r0, ..., rN-1)``. The subarrays ``grid[k]`` contains the N-D array of indices along the ``k-th`` axis. Explicitly:: grid[k, i0, i1, ..., iN-1] = ik Examples -------- >>> grid = np.indices((2, 3)) >>> grid.shape (2, 2, 3) >>> grid[0] # row indices array([[0, 0, 0], [1, 1, 1]]) >>> grid[1] # column indices array([[0, 1, 2], [0, 1, 2]]) The indices can be used as an index into an array. >>> x = np.arange(20).reshape(5, 4) >>> row, col = np.indices((2, 3)) >>> x[row, col] array([[0, 1, 2], [4, 5, 6]]) Note that it would be more straightforward in the above example to extract the required elements directly with ``x[:2, :3]``. If sparse is set to true, the grid will be returned in a sparse representation. >>> i, j = np.indices((2, 3), sparse=True) >>> i.shape (2, 1) >>> j.shape (1, 3) >>> i # row indices array([[0], [1]]) >>> j # column indices array([[0, 1, 2]]) )r r� Nr )r r� r � enumerater r� ) � dimensionsr �sparse�Nr~ r� �i�dim�idxs r rb rb � s� � �j �z�"�"�J� �J���A� ��F�E� � 2��g�g����Q�D��O�5�1�1�1���J�'�'� � ���3��S��&�&�&�.�.� �"�1�"�I��� ��q��s�t�t�� ,� � �� � ����,�C�C��C��F�F� �Jr� �r r� c �Z � |�t || |fd|i|��S t ||�� � } | |i |��S )as Construct an array by executing a function over each coordinate. The resulting array therefore has a value ``fn(x, y, z)`` at coordinate ``(x, y, z)``. Parameters ---------- function : callable The function is called with N parameters, where N is the rank of `shape`. Each parameter represents the coordinates of the array varying along a specific axis. For example, if `shape` were ``(2, 2)``, then the parameters would be ``array([[0, 0], [1, 1]])`` and ``array([[0, 1], [0, 1]])`` shape : (N,) tuple of ints Shape of the output array, which also determines the shape of the coordinate arrays passed to `function`. dtype : data-type, optional Data-type of the coordinate arrays passed to `function`. By default, `dtype` is float. ${ARRAY_FUNCTION_LIKE} .. versionadded:: 1.20.0 Returns ------- fromfunction : any The result of the call to `function` is passed back directly. Therefore the shape of `fromfunction` is completely determined by `function`. If `function` returns a scalar value, the shape of `fromfunction` would not match the `shape` parameter. See Also -------- indices, meshgrid Notes ----- Keywords other than `dtype` and `like` are passed to `function`. Examples -------- >>> np.fromfunction(lambda i, j: i, (2, 2), dtype=float) array([[0., 0.], [1., 1.]]) >>> np.fromfunction(lambda i, j: j, (2, 2), dtype=float) array([[0., 1.], [0., 1.]]) >>> np.fromfunction(lambda i, j: i == j, (3, 3), dtype=int) array([[ True, False, False], [False, True, False], [False, False, True]]) >>> np.fromfunction(lambda i, j: i + j, (3, 3), dtype=int) array([[0, 1, 2], [1, 2, 3], [2, 3, 4]]) Nr r� )�_fromfunction_with_likerb )�functionr~ r r� �kwargs�argss r rc rc � sb � �@ ��&��h��>� >�-2�>�6<�>� >� >� �5�� &� &� &�D� �8�T� $�V� $� $�$r� c �N � t | |�� � � ||�� � S )Nr� )r| )r r� )�bufr r~ r| s r � _frombufferrO ; s( � � �c�� '� '� '� /� /��U� /� C� C�Cr� c � � t | t � � p/t | � � t v pt | t j � � S )a� Returns True if the type of `element` is a scalar type. Parameters ---------- element : any Input argument, can be of any type and shape. Returns ------- val : bool True if `element` is a scalar type, False if it is not. See Also -------- ndim : Get the number of dimensions of an array Notes ----- If you need a stricter way to identify a *numerical* scalar, use ``isinstance(x, numbers.Number)``, as that returns ``False`` for most non-numerical elements such as strings. In most cases ``np.ndim(x) == 0`` should be used instead of this function, as that will also return true for 0d arrays. This is how numpy overloads functions in the style of the ``dx`` arguments to `gradient` and the ``bins`` argument to `histogram`. Some key differences: +--------------------------------------+---------------+-------------------+ | x |``isscalar(x)``|``np.ndim(x) == 0``| +======================================+===============+===================+ | PEP 3141 numeric objects (including | ``True`` | ``True`` | | builtins) | | | +--------------------------------------+---------------+-------------------+ | builtin string and buffer objects | ``True`` | ``True`` | +--------------------------------------+---------------+-------------------+ | other builtin objects, like | ``False`` | ``True`` | | `pathlib.Path`, `Exception`, | | | | the result of `re.compile` | | | +--------------------------------------+---------------+-------------------+ | third-party objects like | ``False`` | ``True`` | | `matplotlib.figure.Figure` | | | +--------------------------------------+---------------+-------------------+ | zero-dimensional numpy arrays | ``False`` | ``True`` | +--------------------------------------+---------------+-------------------+ | other numpy arrays | ``False`` | ``False`` | +--------------------------------------+---------------+-------------------+ | `list`, `tuple`, and other sequence | ``False`` | ``False`` | | objects | | | +--------------------------------------+---------------+-------------------+ Examples -------- >>> np.isscalar(3.1) True >>> np.isscalar(np.array(3.1)) False >>> np.isscalar([3.1]) False >>> np.isscalar(False) True >>> np.isscalar('numpy') True NumPy supports PEP 3141 numbers: >>> from fractions import Fraction >>> np.isscalar(Fraction(5, 17)) True >>> from numbers import Number >>> np.isscalar(Number()) True )� isinstance�genericr� � ScalarType�numbers�Number)�elements r re re ? s> � �X �w�� (� (� 3��G�}�}� �*� 3��'�7�>�2�2�4r� c �x � d� }t j | � � } | dk rd|pdz S | dk r`t | � � dd� }t |� � }|�|nt j ||� � } |||� � |� |� � S |�dt | � � dd� z S t t | � � dd� � � }d|dz z | k r|dz }d|dz z | z }t |� � dd� }t |� � }t j ||� � } |||� � d||z z |z S ) a� Return the binary representation of the input number as a string. For negative numbers, if width is not given, a minus sign is added to the front. If width is given, the two's complement of the number is returned, with respect to that width. 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 ---------- num : int Only an integer decimal number can be used. width : int, optional The length of the returned string if `num` is positive, or the length of the two's complement if `num` is negative, provided that `width` is at least a sufficient number of bits for `num` to be represented in the designated form. If the `width` value is insufficient, it will be ignored, and `num` will be returned in binary (`num` > 0) or two's complement (`num` < 0) form with its width equal to the minimum number of bits needed to represent the number in the designated form. This behavior is deprecated and will later raise an error. .. deprecated:: 1.12.0 Returns ------- bin : str Binary representation of `num` or two's complement of `num`. See Also -------- base_repr: Return a string representation of a number in the given base system. bin: Python's built-in binary representation generator of an integer. Notes ----- `binary_repr` is equivalent to using `base_repr` with base 2, but about 25x faster. References ---------- .. [1] Wikipedia, "Two's complement", https://en.wikipedia.org/wiki/Two's_complement Examples -------- >>> np.binary_repr(3) '11' >>> np.binary_repr(-3) '-11' >>> np.binary_repr(3, width=4) '0011' The two's complement is returned when the input number is negative and width is specified: >>> np.binary_repr(-3, width=3) '101' >>> np.binary_repr(-3, width=5) '11101' c �V � | �$| |k r t j dt d�� � d S d S d S )NzQInsufficient bit width provided. This behavior will raise an error in the future.r0 )� stacklevel)�warnings�warn�DeprecationWarning)�width�binwidths r �warn_if_insufficientz)binary_repr.<locals>.warn_if_insufficient� sN � � � ���!1�!1� �M�5�6H� � � � � � � � � �!1�!1r� r �0r rH N�-�1)r r �binr� �builtins�max�zfill)�numr] r_ �binaryr^ �outwidth�poswidth�twocomps r rf rf � sq � �P � � � �.�� � �C� �a�x�x��e�j�q�!�!� �q����S���!�"�"� ���v�;�;�� %� �H�H�!� �h��6�6� � � �U�H�-�-�-��|�|�H�%�%�%� �=���c�T���1�2�2� �&� &� �3��t�9�9�Q�R�R�=�)�)�H� �8�a�<� �S�D�(�(��A� ���(�Q�,�'�#�-�G���\�\�!�"�"�%�F��6�{�{�H� �|�H�e�4�4�H� � ��� 1� 1� 1��(�X�-�.��7� 7r� c � � d}|t |� � k rt d� � �|dk rt d� � �t | � � }g }|r%|� |||z � � ||z }|�%|r|� d|z � � | dk r|� d� � d� t |pd� � � � S ) aZ Return a string representation of a number in the given base system. Parameters ---------- number : int The value to convert. Positive and negative values are handled. base : int, optional Convert `number` to the `base` number system. The valid range is 2-36, the default value is 2. padding : int, optional Number of zeros padded on the left. Default is 0 (no padding). Returns ------- out : str String representation of `number` in `base` system. See Also -------- binary_repr : Faster version of `base_repr` for base 2. Examples -------- >>> np.base_repr(5) '101' >>> np.base_repr(6, 5) '11' >>> np.base_repr(7, base=5, padding=3) '00012' >>> np.base_repr(10, base=16) 'A' >>> np.base_repr(32, base=16) '20' �$0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZz/Bases greater than 36 not handled in base_repr.rH z+Bases less than 2 not handled in base_repr.r` r ra � )r� r� �abs�append�join�reversed)�number�base�padding�digitsrg r� s r rg rg s� � �N 4�F� �c�&�k�k����J�K�K�K� �����F�G�G�G� �f�+�+�C� �C� � � � � �6�#��*�%�&�&�&� �� �� � � � "� � � �3��=�!�!�!� ��z�z� � � �3���� �7�7�8�C�J�3�'�'� (� (�(r� c � ��� t | � � }|j ����S ��fd�|j D � � }t |� � S )Nc �H �� g | ] }t �| d �� � �� S r )�_maketup)r� �name�fields�vals ��r r� z _maketup.<locals>.<listcomp>G s+ �� �C�C�C�$�x��t� �Q���-�-�C�C�Cr� )r r{ �namesr )�descrr| �dtr� r{ s ` @r ry ry @ sJ ��� � �u� � �B� �Y�F� �~�� �C�C�C�C�C�"�(�C�C�C���S�z�z�r� c �R � |�t || |�� � S ddlm} || ||�� � S )az Return the identity array. The identity array is a square array with ones on the main diagonal. Parameters ---------- n : int Number of rows (and columns) in `n` x `n` output. dtype : data-type, optional Data-type of the output. Defaults to ``float``. ${ARRAY_FUNCTION_LIKE} .. versionadded:: 1.20.0 Returns ------- out : ndarray `n` x `n` array with its main diagonal set to one, and all other elements 0. Examples -------- >>> np.identity(3) array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]]) Nr� r )�eyerG )�_identity_with_likerN r� )r r r� r� s r ri ri K sI � �B ��"�4��%�8�8�8�8������� �3�q��D� )� )� )�)r� c � � | |fS rx ry �r{ r� �rtol�atol� equal_nans r �_allclose_dispatcherr� v r� r� ��h㈵��>�:�0�yE>c �b � t t | ||||�� � � � }t |� � S )aw Returns True if two arrays are element-wise equal within a tolerance. The tolerance values are positive, typically very small numbers. The relative difference (`rtol` * abs(`b`)) and the absolute difference `atol` are added together to compare against the absolute difference between `a` and `b`. NaNs are treated as equal if they are in the same place and if ``equal_nan=True``. Infs are treated as equal if they are in the same place and of the same sign in both arrays. Parameters ---------- a, b : array_like Input arrays to compare. rtol : float The relative tolerance parameter (see Notes). atol : float The absolute tolerance parameter (see Notes). equal_nan : bool Whether to compare NaN's as equal. If True, NaN's in `a` will be considered equal to NaN's in `b` in the output array. .. versionadded:: 1.10.0 Returns ------- allclose : bool Returns True if the two arrays are equal within the given tolerance; False otherwise. See Also -------- isclose, all, any, equal Notes ----- If the following equation is element-wise True, then allclose returns True. absolute(`a` - `b`) <= (`atol` + `rtol` * absolute(`b`)) The above equation is not symmetric in `a` and `b`, so that ``allclose(a, b)`` might be different from ``allclose(b, a)`` in some rare cases. The comparison of `a` and `b` uses standard broadcasting, which means that `a` and `b` need not have the same shape in order for ``allclose(a, b)`` to evaluate to True. The same is true for `equal` but not `array_equal`. `allclose` is not defined for non-numeric data types. `bool` is considered a numeric data-type for this purpose. Examples -------- >>> np.allclose([1e10,1e-7], [1.00001e10,1e-8]) False >>> np.allclose([1e10,1e-8], [1.00001e10,1e-9]) True >>> np.allclose([1e10,1e-8], [1.0001e10,1e-9]) False >>> np.allclose([1.0, np.nan], [1.0, np.nan]) False >>> np.allclose([1.0, np.nan], [1.0, np.nan], equal_nan=True) True )r� r� r� )�allrd �bool)r{ r� r� r� r� r� s r rj rj z s1 � �N �g�a���D�I�F�F�F� G� G�C� ��9�9�r� c � � | |fS rx ry r� s r �_isclose_dispatcherr� � r� r� c � � d� }t | � � }t |� � }|j j dk r&t j |d� � }t ||�� � }t |� � } t |� � } t | � � r t | � � r |||||� � S | | z } t | d�� � } |t | � � z }|t | � � z } ||| || ||� � | | < || || k | | < |r*t |� � t |� � z } | | | | < | d S )aW Returns a boolean array where two arrays are element-wise equal within a tolerance. The tolerance values are positive, typically very small numbers. The relative difference (`rtol` * abs(`b`)) and the absolute difference `atol` are added together to compare against the absolute difference between `a` and `b`. .. warning:: The default `atol` is not appropriate for comparing numbers that are much smaller than one (see Notes). Parameters ---------- a, b : array_like Input arrays to compare. rtol : float The relative tolerance parameter (see Notes). atol : float The absolute tolerance parameter (see Notes). equal_nan : bool Whether to compare NaN's as equal. If True, NaN's in `a` will be considered equal to NaN's in `b` in the output array. Returns ------- y : array_like Returns a boolean array of where `a` and `b` are equal within the given tolerance. If both `a` and `b` are scalars, returns a single boolean value. See Also -------- allclose math.isclose Notes ----- .. versionadded:: 1.7.0 For finite values, isclose uses the following equation to test whether two floating point values are equivalent. absolute(`a` - `b`) <= (`atol` + `rtol` * absolute(`b`)) Unlike the built-in `math.isclose`, the above equation is not symmetric in `a` and `b` -- it assumes `b` is the reference value -- so that `isclose(a, b)` might be different from `isclose(b, a)`. Furthermore, the default value of atol is not zero, and is used to determine what small values should be considered close to zero. The default value is appropriate for expected values of order unity: if the expected values are significantly smaller than one, it can result in false positives. `atol` should be carefully selected for the use case at hand. A zero value for `atol` will result in `False` if either `a` or `b` is zero. `isclose` is not defined for non-numeric data types. `bool` is considered a numeric data-type for this purpose. Examples -------- >>> np.isclose([1e10,1e-7], [1.00001e10,1e-8]) array([ True, False]) >>> np.isclose([1e10,1e-8], [1.00001e10,1e-9]) array([ True, True]) >>> np.isclose([1e10,1e-8], [1.0001e10,1e-9]) array([False, True]) >>> np.isclose([1.0, np.nan], [1.0, np.nan]) array([ True, False]) >>> np.isclose([1.0, np.nan], [1.0, np.nan], equal_nan=True) array([ True, True]) >>> np.isclose([1e-8, 1e-7], [0.0, 0.0]) array([ True, False]) >>> np.isclose([1e-100, 1e-7], [0.0, 0.0], atol=0.0) array([False, False]) >>> np.isclose([1e-10, 1e-10], [1e-20, 0.0]) array([ True, True]) >>> np.isclose([1e-10, 1e-10], [1e-20, 0.999999e-10], atol=0.0) array([False, True]) c �" � t d�� � 5 t � � 5 t t | |z � � ||t |� � z z � � cd d d � � cd d d � � S # 1 swxY w Y d d d � � d S # 1 swxY w Y d S )N�ignore)�invalid)rL rM � less_equalro )�x�yr� r� s r � within_tolzisclose.<locals>.within_tol s& � � �h� '� '� '� >� >�):�)<�)<� >� >� �c�!�A�#�h�h��t�c�!�f�f�}�(<�=�=� >� >� >� >� >� >� >� >� >� >� >� >� >� >� >� >� >� >� >���� >� >� >� >� >� >� >� >� >� >� >� >� >� >� >� >���� >� >� >� >� >� >s4 �B�3A,� B�,A0 �0B�3A0 �4B�B� B�mg �?r� T)r} ry ) r r �kindr r- �isfiniter� rU rV �isnan)r{ r� r� r� r� r� r� r� r �xfin�yfin�finite�cond�both_nans r rd rd � sW � �b>� >� >� �1� � �A��1� � �A� �w�|�s��� � #�A�r� *� *�� �q�� #� #� #�� �A�;�;�D� �A�;�;�D� �4�y�y� �S��Y�Y� ��z�!�Q��d�+�+�+������&��-�-�-�� � �$��� �� � �$��� ��!�z�!�F�)�Q�v�Y��d�C�C��V� ��F�7��q�&��z�1��f�W� � � 0� �Q�x�x�%��(�(�*�H� &�h� /�D��N��B�x�r� c � � | |fS rx ry )r5 r6 r� s r �_array_equal_dispatcherr� F � � � ��8�Or� c �� � t | � � t |� � }} n# t $ r Y dS w xY w| j |j k rdS |s2t t | |k � � � � � � � S t | � � t |� � }}||k � � � sdS t t | | || k � � � � � � � S )a� True if two arrays have the same shape and elements, False otherwise. Parameters ---------- a1, a2 : array_like Input arrays. equal_nan : bool Whether to compare NaN's as equal. If the dtype of a1 and a2 is complex, values will be considered equal if either the real or the imaginary component of a given value is ``nan``. .. versionadded:: 1.19.0 Returns ------- b : bool Returns True if the arrays are equal. See Also -------- allclose: Returns True if two arrays are element-wise equal within a tolerance. array_equiv: Returns True if input arrays are shape consistent and all elements equal. Examples -------- >>> np.array_equal([1, 2], [1, 2]) True >>> np.array_equal(np.array([1, 2]), np.array([1, 2])) True >>> np.array_equal([1, 2], [1, 2, 3]) False >>> np.array_equal([1, 2], [1, 4]) False >>> a = np.array([1, np.nan]) >>> np.array_equal(a, a) False >>> np.array_equal(a, a, equal_nan=True) True When ``equal_nan`` is True, complex values with nan components are considered equal if either the real *or* the imaginary components are nan. >>> a = np.array([1 + 1j]) >>> b = a.copy() >>> a.real = np.nan >>> b.imag = np.nan >>> np.array_equal(a, b, equal_nan=True) True F)r r� r~ r� r� r� )r5 r6 r� �a1nan�a2nans r r` r` J s� � �l�����g�b�k�k�B���� � � � ��u�u����� �x�2�8����u� � -��G�B�"�H�%�%�)�)�+�+�,�,�,���9�9�e�B�i�i�5�E� �U�N� � � !� !� ��u� ���E�6� �b�%��j�0�1�1�5�5�7�7� 8� 8�8s � ! � /�/c � � | |fS rx ry �r5 r6 s r �_array_equiv_dispatcherr� � r� r� c � � t | � � t |� � }} n# t $ r Y dS w xY w t j | |� � n# t $ r Y dS w xY wt t | |k � � � � � � � S )a� Returns True if input arrays are shape consistent and all elements equal. Shape consistent means they are either the same shape, or one input array can be broadcasted to create the same shape as the other one. Parameters ---------- a1, a2 : array_like Input arrays. Returns ------- out : bool True if equivalent, False otherwise. Examples -------- >>> np.array_equiv([1, 2], [1, 2]) True >>> np.array_equiv([1, 2], [1, 3]) False Showing the shape equivalence: >>> np.array_equiv([1, 2], [[1, 2], [1, 2]]) True >>> np.array_equiv([1, 2], [[1, 2, 1, 2], [1, 2, 1, 2]]) False >>> np.array_equiv([1, 2], [[1, 2], [1, 3]]) False F)r r� r r r� r� r� s r ra ra � s� � �H�����g�b�k�k�B���� � � � ��u�u������� �R��$�$�$�$�� � � � ��u�u����� ���b��!�!�%�%�'�'� (� (�(s � ! � /�/�A � A�Ac � � t t � � }t | d� � }|D ] }||vrt � |� � �!d S )N�__all__)r r� �getattrrp )rO �existing�mallr{ s r � extend_allr� � sS � ��7�|�|�H� �6�9� %� %�D� � � �� �H� � � �N�N�1� � � �� � r� )�*)� fromnumeric)� arrayprint)�_asarray)� _ufunc_config)NNNN)Nr� TN)Nr� )NNrx )r� )ru )rH r )NF)r� r� r� N)rH r )NNN)r� r� F)F)�� functoolsr r �sysrZ rT rd rN r� rn r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r! r" r# r$ r% r& r' r( r) r* r+ r, r- r. r/ r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 r: r; r< r= r> r? r@ rA rB rC rD rE rF rG � exceptionsrI rJ rK r� rL rM rt r� rQ rP �partial�array_function_dispatchr� r� rU rh r� r� rV r� ru r� r� rv r� rR rT r� rS r� rk r� rW r� rX r� rY r� r^ r� rZ r r[ r r# r\ r. r] � byteorderr_ r� rb �floatrc rI rO re rf rg ry ri r� r� rj r� rd r� r` r� ra rl rm rn ro rp rq rr rs r� r� r� r� ry r� r �<module>r� s� �� � � � � � � � � ���� � � � � ���� ���� ���� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � >� >� >� >� >� >� >� >� 5� 5� 5� 5� 5� 5� 5� 5� 5� 5� 5� 5� 5� 5� � � � � � � G� G� G� G� G� G� G� G� G� G� G� G� G� G� G� G� @� @� @� @� @� @� @� @� @� @� 6� 6� 6� 6� 6� 6� 6� 6�� � ��S� � �� ��+�)�+� �%�g�7� 7� 7� � � � ��*� � � � ��/�0�0�A� A� A� 1�0�A�H � ��G���7 �t� 7 � 7 � 7 � 7 � �� � �7 �t ,�)�)�+�+�D�1�1��� � � � ��.�/�/�?� ?� ?� 0�/�?�D�� � � � � � � ��G���7 �4� 7 � 7 � 7 � 7 � �� � �7 �t ,�)�)�+�+�D�1�1��� � � � ��.�/�/�B� B� B� 0�/�B�J�� � � � � � ��2�3�3�FC�E� FC� FC� FC� FC� 4�3�FC�R ��G���@� @� ��@�F� � � ��-�.�.�.!� .!� /�.�.!�b� � � ��0�1�1�%&� %&� 2�1�%&�P� � � � ��.�/�/�F-� F-� F-� 0�/�F-�R� � � � ��-�.�.�_2� _2� _2� /�.�_2�D� � � � ��*�+�+�SG� SG� SG� ,�+�SG�l� � � � ��.�/�/�}$� }$� }$� 0�/�}$�@� � � � ��)�*�*�g� g� g� +�*�g�T� � � � ��-�.�.�V � V � V � /�.�V �r8� 8� 8� 8�v� � � ��-�.�.�C� C� /�.�C�L� � � � ��*�+�+�F#� F#� F#� ,�+�F#�R � �(�*� � ��G��� !�%� c� c� c� ��c�L � ��G���+0�t� C%� C%� C%� C%� �� � �C%�L 4�1�1�3�3�L�A�A� �D� D� D� ��G���M4� M4� ��M4�` ��G���o8� o8� o8� ��o8�d ��G���5)� 5)� 5)� ��5)�x� � � � ��G���#*�D� #*� #*� #*� #*� �� � �#*�L 0�-�-�/�/��9�9� �� � � � ��-�.�.�G� G� G� /�.�G�T� � � � ��,�-�-�y� y� y� .�-�y�x� � � � ��0�1�1�C9� C9� C9� 2�1�C9�L� � � ��0�1�1�,)� ,)� 2�1�,)�^ $� #�� #�c� #�E�H� ���c� ��u� � �� ��d� � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �;� � � � � �5� � � � � �<� � � � � �:� � � � � �8� � � � � �=� � � � � r�
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