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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.
The short BB step size is same as a linearized minimum-residual step. BB applies the step sizes upon the forward direction vector for the next iterate, instead of the prior direction vector as if for another line-search step. Barzilai and Borwein proved their method converges R-superlinearly for quadratic minimization in two dimensions.
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.
Fastor is a high performance tensor (fixed multi-dimensional array) library for modern C++. GNU Scientific Library [6] GNU Project C, C++ 1996 2.7.1 / 11.2021 Free GPL: General purpose numerical analysis library. Includes some support for linear algebra. IMSL Numerical Libraries: Rogue Wave Software: C, Java, C#, Fortran, Python 1970 many ...
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, + + + =.
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 ...
Then the indices of the pivot (k, l) must be one of the pairs (,). Also the updating of the index array can be done in O(n) average-case complexity: First, the maximum entry in the updated rows k and l can be found in O(n) steps. In the other rows i, only the entries in columns k and l change.
4 rows by 3 cols matrix created, using rho ⍴ and iota ⍳. The 4 x 3 matrix is then stored in a variable named A. A is now reflected or flipped along its vertical axis as symbol ⌽ visually indicates. A is now reflected using the [1] axis indicator or first dimension modifier. The result is that variable A has been reflected across the ...