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In Matlab/GNU Octave a matrix A can be vectorized by A(:). GNU Octave also allows vectorization and half-vectorization with vec(A) and vech(A) respectively. Julia has the vec(A) function as well. In Python NumPy arrays implement the flatten method, [note 1] while in R the desired effect can be achieved via the c() or as.vector() functions.
Several open-source scripts have been developed to facilitate the construction of Python one-liners. Scripts such as pyp or Pyline import commonly used modules and provide more human-readable variables in an attempt to make Python functionality more accessible on the command line. Here is a redo of the above example (printing the last field of ...
Then one can write t = pQ = vMQ, so the matrix product transformation MQ maps v directly to t. Continuing with row vectors, matrix transformations further reconfiguring n-space can be applied to the right of previous outputs. When a column vector is transformed to another column vector under an n × n matrix action, the operation occurs to the ...
In array languages, operations are generalized to apply to both scalars and arrays. Thus, a+b expresses the sum of two scalars if a and b are scalars, or the sum of two arrays if they are arrays. An array language simplifies programming but possibly at a cost known as the abstraction penalty.
The findgen function in the above example returns a one-dimensional array of floating point numbers, with values equal to a series of integers starting at 0.. Note that the operation in the second line applies in a vectorized manner to the whole 100-element array created in the first line, analogous to the way general-purpose array programming languages (such as APL, J or K) would do it.
Additionally, one could create a two-dimensional vector with A.wx or a five-dimensional vector with A.xyzwx. Combining vectors and swizzling can be employed in various ways. This is common in GPGPU applications. [3] In terms of linear algebra, this is equivalent to multiplying by a matrix whose rows are standard basis vectors.
The library supports x86 CPUs and Intel GPUs [2] and is available for Windows and Linux operating systems. [ 5 ] [ 6 ] [ 7 ] Intel oneAPI Math Kernel Library is not to be confused with oneMKL Interfaces, an open-source wrapper library that allows DPC++ applications to call oneMKL routines that can be offloaded to multiple hardware architectures ...
A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [′ ′ ′] = [] [] where = for some point on the plane, or equivalently, + + + =.