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A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the ...
Core math functions include BLAS, LAPACK, ScaLAPACK, sparse solvers, fast Fourier transforms, and vector math. Intel IPP is a multi-threaded software library of functions for multimedia and data processing applications. OpenBLAS is an open source implementation of the BLAS API with many hand-crafted optimizations for specific processor types ...
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.
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract vector space does not possess a magnitude. A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space. [8] The norm of a vector v in a normed vector space can be considered to be the magnitude of v.
The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field.
The transfer function coefficients can also be used to construct another type of canonical form ˙ = [] + [] () = [] (). This state-space realization is called observable canonical form because the resulting model is guaranteed to be observable (i.e., because the output exits from a chain of integrators, every state has an effect on the output).