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Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
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 matrix multiplication exponent, usually denoted , is the smallest real number for which any matrix over a field can be multiplied together using + field operations. The current best bound on ω {\displaystyle \omega } is ω < 2.371552 {\displaystyle \omega <2.371552} , by Williams , Xu, Xu, and Zhou.
Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]
The number of additions and multiplications required in the Strassen algorithm can be calculated as follows: let () be the number of operations for a matrix. Then by recursive application of the Strassen algorithm, we see that f ( n ) = 7 f ( n − 1 ) + l 4 n {\displaystyle f(n)=7f(n-1)+l4^{n}} , for some constant l {\displaystyle l} that ...
The straightforward multiplication of a matrix that is X × Y by a matrix that is Y × Z requires XYZ ordinary multiplications and X(Y − 1)Z ordinary additions. In this context, it is typical to use the number of ordinary multiplications as a measure of the runtime complexity. If A is a 10 × 30 matrix, B is a 30 × 5 matrix, and C is a 5 × ...
Condition numbers can also be defined for nonlinear functions, and can be computed using calculus.The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
In linear algebra and numerical analysis, a preconditioner of a matrix is a matrix such that has a smaller condition number than . It is also common to call T = P − 1 {\displaystyle T=P^{-1}} the preconditioner, rather than P {\displaystyle P} , since P {\displaystyle P} itself is rarely explicitly available.