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  2. Matrix exponential - Wikipedia

    en.wikipedia.org/wiki/Matrix_exponential

    For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...

  3. Matrix multiplication algorithm - Wikipedia

    en.wikipedia.org/wiki/Matrix_multiplication...

    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.

  4. Computational complexity of matrix multiplication - Wikipedia

    en.wikipedia.org/wiki/Computational_complexity...

    The best known lower bound for matrix-multiplication complexity is Ω(n 2 log(n)), for bounded coefficient arithmetic circuits over the real or complex numbers, and is due to Ran Raz. [32] The exponent ω is defined to be a limit point, in that it is the infimum of the exponent over all matrix multiplication algorithms. It is known that this ...

  5. Computational complexity of mathematical operations - Wikipedia

    en.wikipedia.org/wiki/Computational_complexity...

    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.

  6. Matrix multiplication - Wikipedia

    en.wikipedia.org/wiki/Matrix_multiplication

    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]

  7. Condition number - Wikipedia

    en.wikipedia.org/wiki/Condition_number

    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.

  8. Exponentiation by squaring - Wikipedia

    en.wikipedia.org/wiki/Exponentiation_by_squaring

    The method is based on the observation that, for any integer >, one has: = {() /, /,. If the exponent n is zero then the answer is 1. If the exponent is negative then we can reuse the previous formula by rewriting the value using a positive exponent.

  9. Strassen algorithm - Wikipedia

    en.wikipedia.org/wiki/Strassen_algorithm

    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 ...