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In theoretical computer science, the computational complexity of matrix multiplication dictates how quickly the operation of matrix multiplication can be performed. Matrix multiplication algorithms are a central subroutine in theoretical and numerical algorithms for numerical linear algebra and optimization, so finding the fastest algorithm for matrix multiplication is of major practical ...
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
MATLAB (an abbreviation of "MATrix LABoratory" [18]) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks.MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages.
For the example below, there are four sides: A, B, C and the final result ABC. A is a 10×30 matrix, B is a 30×5 matrix, C is a 5×60 matrix, and the final result is a 10×60 matrix. The regular polygon for this example is a 4-gon, i.e. a square: The matrix product AB is a 10x5 matrix and BC is a 30x60 matrix.
The elementary functions are constructed by composing arithmetic operations, the exponential function (), the natural logarithm (), trigonometric functions (,), and their inverses. The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's ...
For example, a matrix such that all entries of a row (or a column) are 0 does not have an inverse. If it exists, the inverse of a matrix A is denoted A −1, and, thus verifies = =. A matrix that has an inverse is an invertible matrix.
This reduces the number of matrix additions and subtractions from 18 to 15. The number of matrix multiplications is still 7, and the asymptotic complexity is the same. [6] The algorithm was further optimised in 2017, [7] reducing the number of matrix additions per step to 12 while maintaining the number of matrix multiplications, and again in ...
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.