Module Owl_dense_matrix_generic

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._index_1d_nd can help you.

init_nd 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_nd 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.

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 kth 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 kth 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.

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.{i,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.{i,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.

slice 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_slice 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_simple axis x aims to provide a simpler version of get_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]] .

set_slice_simple 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 the row i of x.

col x j returns the column j of x.

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.

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

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.

clone x returns a copy of matrix x.

copy_to x y copies the elements of x to y. x and y must have the same demensions.

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.

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

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

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 kth 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 ~axis x a repeats the elements along ~axis for a times.

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 elelment 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 elelment values, from the smallest one to the greatest one.

sort x performs in-place quicksort of the elelments in x.

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

fold f a x folds all the elements in x with the function f : 'a -> float -> 'a. For an m by n matrix x, the order of folding is from (0,0) to (m-1,n-1), row by row.

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.

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.

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 ith 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 jth 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).

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 sparse matrix from file f. The file must be previously saved by using save function.

save_txt x f save the matrix x into a text file f. The operation can be very time consuming.

load_txt f load a text file f into a matrix.

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. 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.

max x returns the maximum of all elements in x. 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.

minmax x returns both the minimum and minimum values in x.

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.

inv x returns the inverse of a square matrix x.

trace x returns the sum of diagonal elements in x.

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

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

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

average x returns the average value of all the elements in x. It is equivalent to calculate sum x divided by numel 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.

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

average_cols x returns the average 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 in x and returns the result in a new array.

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 all the element in x.

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

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.

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.

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.

var x computes the variance of x along the axis dimension. By default, axis is set to the lowest dimension.

std x computes the standard deviation of x along the axis dimension. By default, axis is set to the lowest dimension.

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.

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.

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 mod 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.

cast_s2d x casts x from float32 to float64.

cast_d2s x casts x from float64 to float32.

cast_c2z x casts x from complex32 to complex64.

cast_z2c x casts x from complex64 to complex32.

cast_s2c x casts x from float32 to complex32.

cast_d2z x casts x from float64 to complex64.

cast_s2z x casts x from float32 to complex64.

cast_d2c x casts x from float64 to complex32.