N-dimensional array type, i.e. Bigarray Genarray type.

``empty m n`` creates an ``m`` by ``n`` matrix without initialising the values of elements in ``x``.

``create m n a`` creates an ``m`` by ``n`` matrix and all the elements of ``x`` are initialised with the value ``a``.

``init m n f`` creates a matrix ``x`` of shape ``m x n``, then using ``f`` to initialise the elements in ``x``. The input of ``f`` is 1-dimensional index of the matrix. You need to explicitly convert it if you need 2D index. The function ``Owl_utils.ind`` can help you.

``init_2d m n f`` s almost the same as ``init`` but ``f`` receives 2D index as input. It is more convenient since you don't have to convert the index by yourself, but this also means ``init_2d`` is slower than ``init``.

``zeros m n`` creates an ``m`` by ``n`` matrix where all the elements are initialised to zeros.

``ones m n`` creates an ``m`` by ``n`` matrix where all the elements are ones.

``eye m`` creates an ``m`` by ``m`` identity matrix.

``complex re im`` constructs a complex ndarray/matrix from ``re`` and ``im``. ``re`` and ``im`` contain the real and imaginary part of ``x`` respectively.

Note that both ``re`` and ``im`` can be complex but must have same type. The real part of ``re`` will be the real part of ``x`` and the imaginary part of ``im`` will be the imaginary part of ``x``.

``complex rho theta`` constructs a complex ndarray/matrix from polar coordinates ``rho`` and ``theta``. ``rho`` contains the magnitudes and ``theta`` contains phase angles. Note that the behaviour is undefined if ``rho`` has negative elelments or ``theta`` has infinity elelments.

``unit_basis k n i`` returns a unit basis vector with ``i``th element set to 1.

``sequential ~a ~step m n`` creates an ``m`` by ``n`` matrix. The elements in ``x`` are initialised sequentiallly from ``~a`` and is increased by ``~step``.

The default value of ``~a`` is zero whilst the default value of ``~step`` is one.

``uniform m n`` creates an ``m`` by ``n`` matrix where all the elements follow a uniform distribution in ``(0,1)`` interval. ``uniform ~scale:a m n`` adjusts the interval to ``(0,a)``.

``gaussian m n`` creates an ``m`` by ``n`` matrix where all the elements in ``x`` follow a Gaussian distribution with specified sigma. By default ``sigma = 1``.

`` semidef n `` returns an random ``n`` by ``n`` positive semi-definite matrix.

``linspace a b n`` linearly divides the interval ``a,b`` into ``n`` pieces by creating an ``m`` by ``1`` row vector. E.g., ``linspace 0. 5. 5`` will create a row vector ``0;1;2;3;4;5``.

``logspace base a b n`` ... the default value of base is ``e``.

``meshgrid a1 b1 a2 b2 n1 n2`` is similar to the ``meshgrid`` function in Matlab. It returns two matrices ``x`` and ``y`` where the row vectors in ``x`` are linearly spaced between ``a1,b1`` by ``n1`` whilst the column vectors in ``y`` are linearly spaced between ``(a2,b2)`` by ``n2``.

``meshup x y`` creates mesh grids by using two row vectors ``x`` and ``y``.

``bernoulli k ~p:0.3 m n``

``diagm k v`` creates a diagonal matrix using the elements in ``v`` as diagonal values. ``k`` specifies the main diagonal index. If ``k > 0`` then it is above the main diagonal, if ``k < 0`` then it is below the main diagonal. This function is the same as the ``diag`` function in Matlab.

``triu k x`` returns the element on and above the ``k``th diagonal of ``x``. ``k = 0`` is the main diagonal, ``k > 0`` is above the main diagonal, and ``k < 0`` is below the main diagonal.

``tril k x`` returns the element on and below the ``k``th diagonal of ``x``. ``k = 0`` is the main diagonal, ``k > 0`` is above the main diagonal, and ``k < 0`` is below the main diagonal.

``symmetric ~upper x`` creates a symmetric matrix using either upper or lower triangular part of ``x``. If ``upper`` is ``true`` then it uses the upper part, if ``upper`` is ``false``, then ``symmetric`` uses the lower part. By default ``upper`` is true.

``hermitian ~upper x`` creates a hermitian matrix based on ``x``. By default, the upper triangular part is used for creating the hermitian matrix, but you use the lower part by setting ``upper=false``

``bidiagonal upper dv ev`` creates a bidiagonal matrix using ``dv`` and ``ev``. Both ``dv`` and ``ev`` are row vectors. ``dv`` is the main diagonal. If ``upper`` is ``true`` then ``ev`` is superdiagonal; if ``upper`` is ``false`` then ``ev`` is subdiagonal. By default, ``upper`` is ``true``.

NOTE: because the diagonal elements in a hermitian matrix must be real, the function set the imaginary part of the diagonal elements to zero by default. In other words, if the diagonal elements of ``x`` have non-zero imaginary parts, the imaginary parts will be dropped without a warning.

``toeplitz ~c r`` generates a toeplitz matrix using ``r`` and ``c``. Both ``r`` and ``c`` are row vectors of the same length. If the first elements of ``c`` is different from that of ``r``, ``r``'s first element will be used.

Note: 1) If ``c`` is not passed in, then ``c = r`` will be used. 2) If ``c`` is not passed in and ``r`` is complex, the ``c = conj r`` will be used. 3) If ``r`` and ``c`` have different length, then the result is a rectangular matrix.

``hankel ~r c`` generates a hankel matrix using ``r`` and ``c``. ``c`` will be the first column and ``r`` will be the last row of the returned matrix.

Note: 1) If only ``c`` is passed in, the elelments below the anti-diagnoal are zero. 2) If the last element of ``c`` is different from the first element of ``r`` then the first element of ``c`` prevails. 3) ``c`` and ``r`` can have different length, the return will be an rectangular matrix.

``hadamard k n`` constructs a hadamard matrix of order ``n``. For a hadamard ``H``, we have ``H'*H = n*I``. Currrently, this function handles only the cases where ``n``, ``n/12``, or ``n/20`` is a power of 2.

``magic k n`` constructs a ``n x n`` magic square matrix ``x``. The elements in ``x`` are consecutive numbers increasing from ``1`` to ``n^2``. ``n`` must ``n >= 3``.

There are three different algorithms to deal with ``n`` is odd, singly even, and doubly even respectively.

If ``x`` is an ``m`` by ``n`` matrix, ``shape x`` returns ``(m,n)``, i.e., the size of two dimensions of ``x``.

``row_num x`` returns the number of rows in matrix ``x``.

``col_num x`` returns the number of columns in matrix ``x``.

``numel x`` returns the number of elements in matrix ``x``. It is equivalent to ``(row_num x) * (col_num x)``.

``nnz x`` returns the number of non-zero elements in ``x``.

``density x`` returns the percentage of non-zero elements in ``x``.

``size_in_bytes x`` returns the size of ``x`` in bytes in memory.

``same_shape x y`` returns ``true`` if two matrics have the same shape.

Refer to :doc:`owl_dense_ndarray_generic`.

``kind x`` returns the type of matrix ``x``.

``get x i j`` returns the value of element ``(i,j)`` of ``x``. The shorthand for ``get x i j`` is ``x.,j``

``set x i j a`` sets the element ``(i,j)`` of ``x`` to value ``a``. The shorthand for ``set x i j a`` is ``x.,j <- a``

``get_index i x`` returns an array of element values specified by the indices ``i``. The length of array ``i`` equals the number of dimensions of ``x``. The arrays in ``i`` must have the same length, and each represents the indices in that dimension.

E.g., ``| [|1;2|]; [|3;4|] |`` returns the value of elements at position ``(1,3)`` and ``(2,4)`` respectively.

``set_index`` sets the value of elements in ``x`` according to the indices specified by ``i``. The length of array ``i`` equals the number of dimensions of ``x``. The arrays in ``i`` must have the same length, and each represents the indices in that dimension.

``get_fancy s x`` returns a copy of the slice in ``x``. The slice is defined by ``a`` which is an ``int array``. Please refer to the same function in the ``Owl_dense_ndarray_generic`` documentation for more details.

``set_fancy axis x y`` set the slice defined by ``axis`` in ``x`` according to the values in ``y``. ``y`` must have the same shape as the one defined by ``axis``.

About the slice definition of ``axis``, please refer to ``slice`` function.

``get_slice axis x`` aims to provide a simpler version of ``get_fancy``. This function assumes that every list element in the passed in ``in list list`` represents a range, i.e., ``R`` constructor.

E.g., ``[];[0;3];[0]`` is equivalent to ``R []; R [0;3]; R [0]``.

``set_slice axis x y`` aims to provide a simpler version of ``set_slice``. This function assumes that every list element in the passed in ``in list list`` represents a range, i.e., ``R`` constructor.

E.g., ``[];[0;3];[0]`` is equivalent to ``R []; R [0;3]; R [0]``.

``row x i`` returns row ``i`` of ``x``. Note: Unlike ``col``, the return value is simply a view onto the original row in ``x``, so modifying ``row``'s value also alters ``x``.

The function supports nagative indices.

``col x j`` returns column ``j`` of ``x``. Note: Unlike ``row``, the return value is a copy of the original row in ``x``.

The function supports nagative indices.

``rows x a`` returns the rows (defined in an int array ``a``) of ``x``. The returned rows will be combined into a new dense matrix. The order of rows in the new matrix is the same as that in the array ``a``.

The function supports nagative indices.

Similar to ``rows``, ``cols x a`` returns the columns (specified in array ``a``) of x in a new dense matrix.

The function supports nagative indices.

``resize x s`` please refer to the Ndarray document.

``reshape x s`` returns a new ``m`` by ``n`` matrix from the ``m'`` by ``n'`` matrix ``x``. Note that ``(m * n)`` must be equal to ``(m' * n')``, and the returned matrix shares the same memory with the original ``x``.

``flatten x`` reshape ``x`` into a ``1`` by ``n`` row vector without making a copy. Therefore the returned value shares the same memory space with original ``x``.

``reverse x`` reverse the order of all elements in the flattened ``x`` and returns the results in a new matrix. The original ``x`` remains intact.

``flip ~axis x`` flips a matrix/ndarray along ``axis``. By default ``axis = 0``. The result is returned in a new matrix/ndarray, so the original ``x`` remains intact.

``rotate x d`` rotates ``x`` clockwise ``d`` degrees. ``d`` must be multiple times of ``90``, otherwise the function will fail. If ``x`` is an n-dimensional array, then the function rotates the plane formed by the first and second dimensions.

``reset x`` resets all the elements of ``x`` to zero value.

``fill x a`` fills the ``x`` with value ``a``.

``copy x`` returns a copy of matrix ``x``.

``copy_row_to v x i`` copies an ``1`` by ``n`` row vector ``v`` to the ``ith`` row in an ``m`` by ``n`` matrix ``x``.

``copy_col_to v x j`` copies an ``1`` by ``n`` column vector ``v`` to the ``jth`` column in an ``m`` by ``n`` matrix ``x``.

``concat_vertical x y`` concats two matrices ``x`` and ``y`` vertically, therefore their column numbers must be the same.

The associated operator is ``@=``, please refer to :doc:`owl_operator`.

``concat_horizontal x y`` concats two matrices ``x`` and ``y`` horizontally, therefore their row numbers must be the same.

The associated operator is ``@||``, please refer to :doc:`owl_operator`.

``concat_vh`` is used to assemble small parts of matrices into a bigger one. E.g. ``| [|a; b; c|]; [|d; e; f|]; [|g; h; i|] |`` will be concatenated into a big matrix as follows.

Please refer to :doc:`owl_dense_ndarray_generic`. for details.

``concatenate ~axis:1 x`` concatenates an array of matrices along the second dimension. For the matrices in ``x``, they must have the same shape except the dimension specified by ``axis``. The default value of ``axis`` is 0, i.e., the lowest dimension on a marix, i.e., rows.

``split ~axis parts x`` splits an ndarray ``x`` into parts along the specified ``axis``. This function is the inverse operation of ``concatenate``. The elements in ``x`` must sum up to the dimension in the specified axis.

Please refer to :doc:`owl_dense_ndarray_generic`. for details.

``transpose x`` transposes an ``m`` by ``n`` matrix to ``n`` by ``m`` one.

``ctranspose x`` performs conjugate transpose of a complex matrix ``x``. If ``x`` is a real matrix, then ``ctranspose x`` is equivalent to ``transpose x``.

``diag k x`` returns the ``k``th diagonal elements of ``x``. ``k > 0`` means above the main diagonal and ``k < 0`` means the below the main diagonal.

``swap_rows x i i'`` swaps the row ``i`` with row ``i'`` of ``x``.

``swap_cols x j j'`` swaps the column ``j`` with column ``j'`` of ``x``.

``tile x a`` provides the exact behaviour as ``numpy.tile`` function.

``repeat x a`` repeats the elements ``x`` according the repetition specified by ``a``.

``padd ~v:0. [1;1] x``

``dropout ~rate:0.3 x`` drops out 30% of the elements in ``x``, in other words, by setting their values to zeros.

``top x n`` returns the indices of ``n`` greatest values of ``x``. The indices are arranged according to the corresponding element values, from the greatest one to the smallest one.

``bottom x n`` returns the indices of ``n`` smallest values of ``x``. The indices are arranged according to the corresponding element values, from the smallest one to the greatest one.

``sort x`` performs quicksort of the elelments in ``x``. A new copy is returned as result, the original ``x`` remains intact. If you want to perform in-place sorting, please use `sort_` instead.

``argsort x`` returns the indices with which the elements in ``x`` are sorted in increasing order. Note that the returned index ndarray has the same shape as that of ``x``, and the indices are 1D indices.

``iteri f x`` iterates all the elements in ``x`` and applies the user defined function ``f : int -> int -> float -> 'a``. ``f i j v`` takes three parameters, ``i`` and ``j`` are the coordinates of current element, and ``v`` is its value.

``iter f x`` is the same as as ``iteri f x`` except the coordinates of the current element is not passed to the function ``f : float -> 'a``

``mapi f x`` maps each element in ``x`` to a new value by applying ``f : int -> int -> float -> float``. The first two parameters are the coordinates of the element, and the third parameter is the value.

``map f x`` is similar to ``mapi f x`` except the coordinates of the current element is not passed to the function ``f : float -> float``

``foldi ~axis f a x`` folds (or reduces) the elements in ``x`` from left along the specified ``axis`` using passed in function ``f``. ``a`` is the initial element and in ``f i acc b`` ``acc`` is the accumulater and ``b`` is one of the elemets in ``x`` along the same axis. Note that ``i`` is 1d index of ``b``.

Similar to ``foldi``, except that the index of an element is not passed to ``f``.

``scan ~axis f x`` scans the ``x`` along the specified ``axis`` using passed in function ``f``. ``f acc a b`` returns an updated ``acc`` which will be passed in the next call to ``f i acc a``. This function can be used to implement accumulative operations such as ``sum`` and ``prod`` functions. Note that the ``i`` is 1d index of ``a`` in ``x``.

Similar to ``scani``, except that the index of an element is not passed to ``f``.

``filteri f x`` uses ``f : int -> int -> float -> bool`` to filter out certain elements in ``x``. An element will be included if ``f`` returns ``true``. The returned result is a list of coordinates of the selected elements.

Similar to ``filteri``, but the coordinates of the elements are not passed to the function ``f : float -> bool``.

Similar to `iteri` but 2d indices ``(i,j)`` are passed to the user function.

Similar to `mapi` but 2d indices ``(i,j)`` are passed to the user function.

Similar to `foldi` but 2d indices ``(i,j)`` are passed to the user function.

Similar to `scani` but 2d indices ``(i,j)`` are passed to the user function.

Similar to `filteri` but 2d indices ``(i,j)`` are returned.

Similar to `iter2i` but 2d indices ``(i,j)`` are passed to the user function.

Similar to `map2i` but 2d indices ``(i,j)`` are passed to the user function.

Similar to ``iteri`` but applies to two matrices ``x`` and ``y``. Both ``x`` and ``y`` must have the same shape.

Similar to ``iter2i``, except that the index is not passed to ``f``.

``map2i f x y`` applies ``f`` to two elements of the same position in both ``x`` and ``y``. Note that 1d index is passed to funciton ``f``.

``map2 f x y`` is similar to ``map2i f x y`` except the index is not passed.

``iteri_rows f x`` iterates every row in ``x`` and applies function ``f : int -> mat -> unit`` to each of them.

Similar to ``iteri_rows`` except row number is not passed to ``f``.

``iter2_rows f x y`` iterates rows of two matrices ``x`` and ```y``.

Similar to ``iter2iter2i_rows`` but without passing in indices.

``iteri_cols f x`` iterates every column in ``x`` and applies function ``f : int -> mat -> unit`` to each of them. Column number is passed to ``f`` as the first parameter.

Similar to ``iteri_cols`` except col number is not passed to ``f``.

``filteri_rows f x`` uses function ``f : int -> mat -> bool`` to check each row in ``x``, then returns an int array containing the indices of those rows which satisfy the function ``f``.

Similar to ``filteri_rows`` except that the row indices are not passed to ``f``.

``filteri_cols f x`` uses function ``f : int -> mat -> bool`` to check each column in ``x``, then returns an int array containing the indices of those columns which satisfy the function ``f``.

Similar to ``filteri_cols`` except that the column indices are not passed to ``f``.

``fold_rows f a x`` folds all the rows in ``x`` using function ``f``. The order of folding is from the first row to the last one.

``fold_cols f a x`` folds all the columns in ``x`` using function ``f``. The order of folding is from the first column to the last one.

``mapi_rows f x`` maps every row in ``x`` to a type ``'a`` value by applying function ``f : int -> mat -> 'a`` to each of them. The results is an array of all the returned values.

Similar to ``mapi_rows`` except row number is not passed to ``f``.

``mapi_cols f x`` maps every column in ``x`` to a type ``'a`` value by applying function ``f : int -> mat -> 'a``.

Similar to ``mapi_cols`` except column number is not passed to ``f``.

``mapi_by_row d f x`` applies ``f`` to each row of a ``m`` by ``n`` matrix ``x``, then uses the returned ``d`` dimensional row vectors to assemble a new ``m`` by ``d`` matrix.

``map_by_row d f x`` is similar to ``mapi_by_row`` except that the row indices are not passed to ``f``.

``mapi_by_col d f x`` applies ``f`` to each column of a ``m`` by ``n`` matrix ``x``, then uses the returned ``d`` dimensional column vectors to assemble a new ``d`` by ``n`` matrix.

``map_by_col d f x`` is similar to ``mapi_by_col`` except that the column indices are not passed to ``f``.

``mapi_at_row f x i`` creates a new matrix by applying function ``f`` only to the ``i``th row in matrix ``x``.

``map_at_row f x i`` is similar to ``mapi_at_row`` except that the coordinates of an element is not passed to ``f``.

``mapi_at_col f x j`` creates a new matrix by applying function ``f`` only to the ``j``th column in matrix ``x``.

``map_at_col f x i`` is similar to ``mapi_at_col`` except that the coordinates of an element is not passed to ``f``.

``exists f x`` checks all the elements in ``x`` using ``f``. If at least one element satisfies ``f`` then the function returns ``true`` otherwise ``false``.

``not_exists f x`` checks all the elements in ``x``, the function returns ``true`` only if all the elements fail to satisfy ``f : float -> bool``.

``for_all f x`` checks all the elements in ``x``, the function returns ``true`` if and only if all the elements pass the check of function ``f``.

``is_zero x`` returns ``true`` if all the elements in ``x`` are zeros.

``is_positive x`` returns ``true`` if all the elements in ``x`` are positive.

``is_negative x`` returns ``true`` if all the elements in ``x`` are negative.

``is_nonpositive`` returns ``true`` if all the elements in ``x`` are non-positive.

``is_nonnegative`` returns ``true`` if all the elements in ``x`` are non-negative.

``is_normal x`` returns ``true`` if all the elelments in ``x`` are normal float numbers, i.e., not ``NaN``, not ``INF``, not ``SUBNORMAL``. Please refer to

https://www.gnu.org/software/libc/manual/html_node/Floating-Point-Classes.html https://www.gnu.org/software/libc/manual/html_node/Infinity-and-NaN.html#Infinity-and-NaN

``not_nan x`` returns ``false`` if there is any ``NaN`` element in ``x``. Otherwise, the function returns ``true`` indicating all the numbers in ``x`` are not ``NaN``.

``not_inf x`` returns ``false`` if there is any positive or negative ``INF`` element in ``x``. Otherwise, the function returns ``true``.

``equal x y`` returns ``true`` if two matrices ``x`` and ``y`` are equal.

``not_equal x y`` returns ``true`` if there is at least one element in ``x`` is not equal to that in ``y``.

``greater x y`` returns ``true`` if all the elements in ``x`` are greater than the corresponding elements in ``y``.

``less x y`` returns ``true`` if all the elements in ``x`` are smaller than the corresponding elements in ``y``.

``greater_equal x y`` returns ``true`` if all the elements in ``x`` are not smaller than the corresponding elements in ``y``.

``less_equal x y`` returns ``true`` if all the elements in ``x`` are not greater than the corresponding elements in ``y``.

``elt_equal x y`` performs element-wise ``=`` comparison of ``x`` and ``y``. Assume that ``a`` is from ``x`` and ``b`` is the corresponding element of ``a`` from ``y`` of the same position. The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` indicates ``a = b``.

``elt_not_equal x y`` performs element-wise ``!=`` comparison of ``x`` and ``y``. Assume that ``a`` is from ``x`` and ``b`` is the corresponding element of ``a`` from ``y`` of the same position. The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` indicates ``a <> b``.

``elt_less x y`` performs element-wise ``<`` comparison of ``x`` and ``y``. Assume that ``a`` is from ``x`` and ``b`` is the corresponding element of ``a`` from ``y`` of the same position. The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` indicates ``a < b``.

``elt_greater x y`` performs element-wise ``>`` comparison of ``x`` and ``y``. Assume that ``a`` is from ``x`` and ``b`` is the corresponding element of ``a`` from ``y`` of the same position. The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` indicates ``a > b``.

``elt_less_equal x y`` performs element-wise ``<=`` comparison of ``x`` and ``y``. Assume that ``a`` is from ``x`` and ``b`` is the corresponding element of ``a`` from ``y`` of the same position. The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` indicates ``a <= b``.

``elt_greater_equal x y`` performs element-wise ``>=`` comparison of ``x`` and ``y``. Assume that ``a`` is from ``x`` and ``b`` is the corresponding element of ``a`` from ``y`` of the same position. The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` indicates ``a >= b``.

``equal_scalar x a`` checks if all the elements in ``x`` are equal to ``a``. The function returns ``true`` iff for every element ``b`` in ``x``, ``b = a``.

``not_equal_scalar x a`` checks if all the elements in ``x`` are not equal to ``a``. The function returns ``true`` iff for every element ``b`` in ``x``, ``b <> a``.

``less_scalar x a`` checks if all the elements in ``x`` are less than ``a``. The function returns ``true`` iff for every element ``b`` in ``x``, ``b < a``.

``greater_scalar x a`` checks if all the elements in ``x`` are greater than ``a``. The function returns ``true`` iff for every element ``b`` in ``x``, ``b > a``.

``less_equal_scalar x a`` checks if all the elements in ``x`` are less or equal to ``a``. The function returns ``true`` iff for every element ``b`` in ``x``, ``b <= a``.

``greater_equal_scalar x a`` checks if all the elements in ``x`` are greater or equal to ``a``. The function returns ``true`` iff for every element ``b`` in ``x``, ``b >= a``.

``elt_equal_scalar x a`` performs element-wise ``=`` comparison of ``x`` and ``a``. Assume that ``b`` is one element from ``x`` The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` of the corresponding position indicates ``a = b``, otherwise ``0``.

``elt_not_equal_scalar x a`` performs element-wise ``!=`` comparison of ``x`` and ``a``. Assume that ``b`` is one element from ``x`` The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` of the corresponding position indicates ``a <> b``, otherwise ``0``.

``elt_less_scalar x a`` performs element-wise ``<`` comparison of ``x`` and ``a``. Assume that ``b`` is one element from ``x`` The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` of the corresponding position indicates ``a < b``, otherwise ``0``.

``elt_greater_scalar x a`` performs element-wise ``>`` comparison of ``x`` and ``a``. Assume that ``b`` is one element from ``x`` The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` of the corresponding position indicates ``a > b``, otherwise ``0``.

``elt_less_equal_scalar x a`` performs element-wise ``<=`` comparison of ``x`` and ``a``. Assume that ``b`` is one element from ``x`` The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` of the corresponding position indicates ``a <= b``, otherwise ``0``.

``elt_greater_equal_scalar x a`` performs element-wise ``>=`` comparison of ``x`` and ``a``. Assume that ``b`` is one element from ``x`` The function returns another binary (``0`` and ``1``) ndarray/matrix wherein ``1`` of the corresponding position indicates ``a >= b``, otherwise ``0``.

``approx_equal ~eps x y`` returns ``true`` if ``x`` and ``y`` are approximately equal, i.e., for any two elements ``a`` from ``x`` and ``b`` from ``y``, we have ``abs (a - b) < eps``.

Note: the threshold check is exclusive for passed in ``eps``.

``approx_equal_scalar ~eps x a`` returns ``true`` all the elements in ``x`` are approximately equal to ``a``, i.e., ``abs (x - a) < eps``. For complex numbers, the ``eps`` applies to both real and imaginary part.

Note: the threshold check is exclusive for the passed in ``eps``.

``approx_elt_equal ~eps x y`` compares the element-wise equality of ``x`` and ``y``, then returns another binary (i.e., ``0`` and ``1``) ndarray/matrix wherein ``1`` indicates that two corresponding elements ``a`` from ``x`` and ``b`` from ``y`` are considered as approximately equal, namely ``abs (a - b) < eps``.

``approx_elt_equal_scalar ~eps x a`` compares all the elements of ``x`` to a scalar value ``a``, then returns another binary (i.e., ``0`` and ``1``) ndarray/matrix wherein ``1`` indicates that the element ``b`` from ``x`` is considered as approximately equal to ``a``, namely ``abs (a - b) < eps``.

``draw_rows x m`` draws ``m`` rows randomly from ``x``. The row indices are also returned in an int array along with the selected rows. The parameter ``replacement`` indicates whether the drawing is by replacement or not.

``draw_cols x m`` draws ``m`` cols randomly from ``x``. The column indices are also returned in an int array along with the selected columns. The parameter ``replacement`` indicates whether the drawing is by replacement or not.

``draw_rows2 x y c`` is similar to ``draw_rows`` but applies to two matrices.

``draw_col2 x y c`` is similar to ``draw_cols`` but applies to two matrices.

``shuffle_rows x`` shuffles all the rows in matrix ``x``.

``shuffle_cols x`` shuffles all the columns in matrix ``x``.

``shuffle x`` shuffles all the elements in ``x`` by first shuffling along the rows then shuffling along columns. It is equivalent to ``shuffle_cols (shuffle_rows x)``.

``to_array x`` flattens an ``m`` by ``n`` matrix ``x`` then returns ``x`` as an float array of length ``(numel x)``.

``of_array x m n`` converts a float array ``x`` into an ``m`` by ``n`` matrix. Note the length of ``x`` must be equal to ``(m * n)``.

Similar to ``reshape`` function, you can pass in one negative index to let Owl automatically infer its dimension.

``to arrays x`` returns an array of float arrays, wherein each row in ``x`` becomes an array in the result.

``of_arrays x`` converts an array of ``m`` float arrays (of length ``n``) in to an ``m`` by ``n`` matrix.

``print x`` pretty prints matrix ``x`` without headings.

``save x f`` saves the matrix ``x`` to a file with the name ``f``. The format is binary by using ``Marshal`` module to serialise the matrix.

``load f`` loads a matrix from file ``f``. The file must be previously saved by using ``save`` function.

``save_txt ~sep ~append x f`` saves the matrix ``x`` into a text file ``f`` delimited by the specified string ``sep`` (default: tab). If ``append`` is ``false`` (it is by default), an existing file will be truncated and overwritten. If ``append`` is ``true`` and the file exists, new rows will be appended to it. Files are created, if necessary, with the AND of 0o644 and the user's umask value. Note that the operation can be very time consuming.

``load_txt ~sep k f`` load a text file ``f`` into a matrix of type ``k``. The delimitor is specified by ``sep`` which can be a regular expression.

``re_c2s x`` returns all the real components of ``x`` in a new ndarray of same shape.

``re_d2z x`` returns all the real components of ``x`` in a new ndarray of same shape.

``im_c2s x`` returns all the imaginary components of ``x`` in a new ndarray of same shape.

``im_d2z x`` returns all the imaginary components of ``x`` in a new ndarray of same shape.

``min x`` returns the minimum of all elements in ``x`` along specified ``axis``. If no axis is specified, ``x`` will be flattened and the minimum of all the elements will be returned. For two complex numbers, the one with the smaller magnitude will be selected. If two magnitudes are the same, the one with the smaller phase will be selected.

``min' x`` is similar to ``min`` but returns the minimum of all elements in ``x`` in scalar value.

``max x`` returns the maximum of all elements in ``x`` along specified ``axis``. If no axis is specified, ``x`` will be flattened and the maximum of all the elements will be returned. For two complex numbers, the one with the greater magnitude will be selected. If two magnitudes are the same, the one with the greater phase will be selected.

``max' x`` is similar to ``max`` but returns the maximum of all elements in ``x`` in scalar value.

``minmax' x`` returns ``(min_v, max_v)``, ``min_v`` is the minimum value in ``x`` while ``max_v`` is the maximum.

``minmax' x`` returns ``(min_v, max_v)``, ``min_v`` is the minimum value in ``x`` while ``max_v`` is the maximum.

``min_i x`` returns the minimum of all elements in ``x`` as well as its index.

``max_i x`` returns the maximum of all elements in ``x`` as well as its index.

``minmax_i x`` returns ``((min_v,min_i), (max_v,max_i))`` where ``(min_v,min_i)`` is the minimum value in ``x`` along with its index while ``(max_v,max_i)`` is the maximum value along its index.

``trace x`` returns the sum of diagonal elements in ``x``.

``sum_ axis x`` sums the elements in ``x`` along specified ``axis``.

``sum x`` returns the summation of all the elements in ``x``.

``prod_ axis x`` multiplies the elements in ``x`` along specified ``axis``.

``prod x`` returns the product of all the elements in ``x``.

``mean ~axis x`` calculates the mean along specified ``axis``.

``mean' x`` calculates the mean of all the elements in ``x``.

``var ~axis x`` calculates the variance along specified ``axis``.

``var' x`` calculates the variance of all the elements in ``x``.

``std ~axis`` calculates the standard deviation along specified ``axis``.

``std' x`` calculates the standard deviation of all the elements in ``x``.

``sum_rows x`` returns the summation of all the row vectors in ``x``.

``sum_cols`` returns the summation of all the column vectors in ``x``.

``mean_rows x`` returns the mean value of all row vectors in ``x``. It is equivalent to ``div_scalar (sum_rows x) (float_of_int (row_num x))``.

``mean_cols x`` returns the mean value of all column vectors in ``x``. It is equivalent to ``div_scalar (sum_cols x) (float_of_int (col_num x))``.

``min_rows x`` returns the minimum value in each row along with their coordinates.

``min_cols x`` returns the minimum value in each column along with their coordinates.

``max_rows x`` returns the maximum value in each row along with their coordinates.

``max_cols x`` returns the maximum value in each column along with their coordinates.

``abs x`` returns the absolute value of all elements in ``x`` in a new matrix.

``abs_c2s x`` is similar to ``abs`` but takes ``complex32`` as input.

``abs_z2d x`` is similar to ``abs`` but takes ``complex64`` as input.

``abs2 x`` returns the square of absolute value of all elements in ``x`` in a new ndarray.

``abs2_c2s x`` is similar to ``abs2`` but takes ``complex32`` as input.

``abs2_z2d x`` is similar to ``abs2`` but takes ``complex64`` as input.

``conj x`` computes the conjugate of the elements in ``x`` and returns the result in a new matrix. If the passed in ``x`` is a real matrix, the function simply returns a copy of the original ``x``.

``neg x`` negates the elements in ``x`` and returns the result in a new matrix.

``reci x`` computes the reciprocal of every elements in ``x`` and returns the result in a new ndarray.

``reci_tol ~tol x`` computes the reciprocal of every element in ``x``. Different from ``reci``, ``reci_tol`` sets the elements whose ``abs`` value smaller than ``tol`` to zeros. If ``tol`` is not specified, the defautl ``Owl_utils.eps Float32`` will be used. For complex numbers, refer to Owl's doc to see how to compare.

``signum`` computes the sign value (``-1`` for negative numbers, ``0`` (or ``-0``) for zero, ``1`` for positive numbers, ``nan`` for ``nan``).

``sqr x`` computes the square of the elements in ``x`` and returns the result in a new matrix.

``sqrt x`` computes the square root of the elements in ``x`` and returns the result in a new matrix.

``cbrt x`` computes the cubic root of the elements in ``x`` and returns the result in a new matrix.

``exp x`` computes the exponential of the elements in ``x`` and returns the result in a new matrix.

``exp2 x`` computes the base-2 exponential of the elements in ``x`` and returns the result in a new matrix.

``exp2 x`` computes the base-10 exponential of the elements in ``x`` and returns the result in a new matrix.

``expm1 x`` computes ``exp x -. 1.`` of the elements in ``x`` and returns the result in a new matrix.

``log x`` computes the logarithm of the elements in ``x`` and returns the result in a new matrix.

``log10 x`` computes the base-10 logarithm of the elements in ``x`` and returns the result in a new matrix.

``log2 x`` computes the base-2 logarithm of the elements in ``x`` and returns the result in a new matrix.

``log1p x`` computes ``log (1 + x)`` of the elements in ``x`` and returns the result in a new matrix.

``sin x`` computes the sine of the elements in ``x`` and returns the result in a new matrix.

``cos x`` computes the cosine of the elements in ``x`` and returns the result in a new matrix.

``tan x`` computes the tangent of the elements in ``x`` and returns the result in a new matrix.

``asin x`` computes the arc sine of the elements in ``x`` and returns the result in a new matrix.

``acos x`` computes the arc cosine of the elements in ``x`` and returns the result in a new matrix.

``atan x`` computes the arc tangent of the elements in ``x`` and returns the result in a new matrix.

``sinh x`` computes the hyperbolic sine of the elements in ``x`` and returns the result in a new matrix.

``cosh x`` computes the hyperbolic cosine of the elements in ``x`` and returns the result in a new matrix.

``tanh x`` computes the hyperbolic tangent of the elements in ``x`` and returns the result in a new matrix.

``asinh x`` computes the hyperbolic arc sine of the elements in ``x`` and returns the result in a new matrix.

``acosh x`` computes the hyperbolic arc cosine of the elements in ``x`` and returns the result in a new matrix.

``atanh x`` computes the hyperbolic arc tangent of the elements in ``x`` and returns the result in a new matrix.

``floor x`` computes the floor of the elements in ``x`` and returns the result in a new matrix.

``ceil x`` computes the ceiling of the elements in ``x`` and returns the result in a new matrix.

``round x`` rounds the elements in ``x`` and returns the result in a new matrix.

``trunc x`` computes the truncation of the elements in ``x`` and returns the result in a new matrix.

``fix x`` rounds each element of ``x`` to the nearest integer toward zero. For positive elements, the behavior is the same as ``floor``. For negative ones, the behavior is the same as ``ceil``.

``modf x`` performs ``modf`` over all the elements in ``x``, the fractal part is saved in the first element of the returned tuple whereas the integer part is saved in the second element.

``erf x`` computes the error function of the elements in ``x`` and returns the result in a new matrix.

``erfc x`` computes the complementary error function of the elements in ``x`` and returns the result in a new matrix.

``logistic x`` computes the logistic function ``1/(1 + exp(-a)`` of the elements in ``x`` and returns the result in a new matrix.

``relu x`` computes the rectified linear unit function ``max(x, 0)`` of the elements in ``x`` and returns the result in a new matrix.

refer to ``Owl_dense_ndarray_generic.elu``

refer to ``Owl_dense_ndarray_generic.leaky_relu``

``softplus x`` computes the softplus function ``log(1 + exp(x)`` of the elements in ``x`` and returns the result in a new matrix.

``softsign x`` computes the softsign function ``x / (1 + abs(x))`` of the elements in ``x`` and returns the result in a new matrix.

``softmax x`` computes the softmax functions ``(exp x) / (sum (exp x))`` of all the elements along the specified ``axis`` in ``x`` and returns the result in a new ndarray.

``sigmoid x`` computes the sigmoid function ``1 / (1 + exp (-x))`` for each element in ``x``.

``log_sum_exp x`` computes the logarithm of the sum of exponentials of all the elements in ``x``.

``l1norm x`` calculates the l1-norm of of ``x`` along specified axis.

``l1norm x`` calculates the l1-norm of all the element in ``x``.

``l2norm x`` calculates the l2-norm of of ``x`` along specified axis.

``l2norm x`` calculates the l2-norm of all the element in ``x``.

``l2norm x`` calculates the square l2-norm of of ``x`` along specified axis.

``l2norm_sqr x`` calculates the square of l2-norm (or l2norm, Euclidean norm) of all elements in ``x``. The function uses conjugate transpose in the product, hence it always returns a float number.

Refer to :doc:`owl_dense_ndarray_generic`.

Refer to :doc:`owl_dense_ndarray_generic`.

Refer to :doc:`owl_dense_ndarray_generic`.

Refer to :doc:`owl_dense_ndarray_generic`.

``cumsum ~axis x``, refer to the documentation in ``Owl_dense_ndarray_generic``.

``cumprod ~axis x``, refer to the documentation in ``Owl_dense_ndarray_generic``.

``cummin ~axis x`` : performs cumulative ``min`` along ``axis`` dimension.

``cummax ~axis x`` : performs cumulative ``max`` along ``axis`` dimension.

``diff ~axis ~n x`` calculates the ``n``-th difference of ``x`` along the specified ``axis``.

Parameters: * ``axis``: axis to calculate the difference. The default value is the highest dimension. * ``n``: how many times to calculate the difference. The default value is 1.

Return: * The difference ndarray y. Note the shape of ``y`` 1 less than that of ``x`` along specified axis.

``angle x`` calculates the phase angle of all complex numbers in ``x``.

``proj x`` computes the projection on Riemann sphere of all elelments in ``x``.

``mat2gray ~amin ~amax x`` converts the matrix ``x`` to the intensity image. The elements in ``x`` are clipped by ``amin`` and ``amax``, and they will be between ``0.`` and ``1.`` after conversion to represents the intensity.

``lgamma x`` computes the loggamma of the elements in ``x`` and returns the result in a new matrix.

``add x y`` adds all the elements in ``x`` and ``y`` elementwise, and returns the result in a new matrix.

``sub x y`` subtracts all the elements in ``x`` and ``y`` elementwise, and returns the result in a new matrix.

``mul x y`` multiplies all the elements in ``x`` and ``y`` elementwise, and returns the result in a new matrix.

``div x y`` divides all the elements in ``x`` and ``y`` elementwise, and returns the result in a new matrix.

``add_scalar x a`` adds a scalar value ``a`` to each element in ``x``, and returns the result in a new matrix.

``sub_scalar x a`` subtracts a scalar value ``a`` from each element in ``x``, and returns the result in a new matrix.

``mul_scalar x a`` multiplies each element in ``x`` by a scalar value ``a``, and returns the result in a new matrix.

``div_scalar x a`` divides each element in ``x`` by a scalar value ``a``, and returns the result in a new matrix.

``scalar_add a x`` adds a scalar value ``a`` to each element in ``x``, and returns the result in a new matrix.

``scalar_sub a x`` subtracts each element in ``x`` from a scalar value ``a``, and returns the result in a new matrix.

``scalar_mul a x`` multiplies each element in ``x`` by a scalar value ``a``, and returns the result in a new matrix.

``scalar_div a x`` divides a scalar value ``a`` by each element in ``x``, and returns the result in a new matrix.

``dot x y`` returns the matrix product of matrix ``x`` and ``y``.

``add_diag x a`` adds ``a`` to the diagonal elements in ``x``. A new copy of the data is returned.

``pow x y`` computes ``pow(a, b)`` of all the elements in ``x`` and ``y`` elementwise, and returns the result in a new matrix.

``scalar_pow a x``

``pow_scalar x a``

``atan2 x y`` computes ``atan2(a, b)`` of all the elements in ``x`` and ``y`` elementwise, and returns the result in a new matrix.

``scalar_atan2 a x``

``scalar_atan2 x a``

``hypot x y`` computes ``sqrt(x*x + y*y)`` of all the elements in ``x`` and ``y`` elementwise, and returns the result in a new matrix.

``min2 x y`` computes the minimum of all the elements in ``x`` and ``y`` elementwise, and returns the result in a new matrix.

``max2 x y`` computes the maximum of all the elements in ``x`` and ``y`` elementwise, and returns the result in a new matrix.

``fmod x y`` performs float modulus division.

``fmod_scalar x a`` performs mod division between ``x`` and scalar ``a``.

``scalar_fmod x a`` performs mod division between scalar ``a`` and ``x``.

``ssqr x a`` computes the sum of squared differences of all the elements in ``x`` from constant ``a``. This function only computes the square of each element rather than the conjugate transpose as sqr_nrm2 does.

``ssqr_diff x y`` computes the sum of squared differences of every elements in ``x`` and its corresponding element in ``y``.

``cross_entropy x y`` calculates the cross entropy between ``x`` and ``y`` using base ``e``.

``clip_by_value ~amin ~amax x`` clips the elements in ``x`` based on ``amin`` and ``amax``. The elements smaller than ``amin`` will be set to ``amin``, and the elements greater than ``amax`` will be set to ``amax``.

``clip_by_l2norm t x`` clips the ``x`` according to the threshold set by ``t``.

``cov ~a`` calculates the covariance matrix of ``a`` wherein each row represents one observation and each column represents one random variable. ``a`` is normalised by the number of observations-1. If there is only one observation, it is normalised by ``1``.

``cov ~a ~b`` takes two matrices as inputs. The functions flatten ``a`` and ``b`` first then returns a ``2 x 2`` matrix, so two must have the same number of elements.

``kron a b`` calculates the Kronecker product between the matrices ``a`` and ``b``. If ``a`` is an ``m x n`` matrix and ``b`` is a ``p x q`` matrix, then ``kron(a,b)`` is an ``m*p x n*q`` matrix formed by taking all possible products between the elements of ``a`` and the matrix ``b``.