Search results
Results from the WOW.Com Content Network
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].
Elimination theory culminated with the work of Leopold Kronecker, and finally Macaulay, who introduced multivariate resultants and U-resultants, providing complete elimination methods for systems of polynomial equations, which are described in the chapter on Elimination theory in the first editions (1930) of van der Waerden's Moderne Algebra.
The standard algorithm for solving a system of linear equations is based on Gaussian elimination with some modifications. Firstly, it is essential to avoid division by small numbers, which may lead to inaccurate results. This can be done by reordering the equations if necessary, a process known as pivoting.
This is called the Gaussian elimination greedoid because this structure underlies the Gaussian elimination algorithm. It is a greedoid, but not an interval greedoid.
The EM algorithm consists of two steps: the E-step and the M-step. Firstly, the model parameters and the () can be randomly initialized. In the E-step, the algorithm tries to guess the value of () based on the parameters, while in the M-step, the algorithm updates the value of the model parameters based on the guess of () of the E-step.
Euclidean algorithm for polynomial greatest common divisor computation and Gaussian elimination of linear systems are special cases of Buchberger's algorithm when the number of variables or the degrees of the polynomials are respectively equal to one. For other Gröbner basis algorithms, see Gröbner basis § Algorithms and implementations.
Rolle's statement of the method seems not to have been noticed insofar as the lesson for Gaussian elimination that was taught in 18th- and 19th-century algebra textbooks owes more to Newton than to Rolle. Rolle is best known for Rolle's theorem in differential calculus. Rolle had used the result in 1690, and he proved it (by the standards of ...
The algorithm runs in Θ(n 2) time, which is a strong improvement over Gauss–Jordan elimination, which runs in Θ(n 3). The Levinson–Durbin algorithm was proposed first by Norman Levinson in 1947, improved by James Durbin in 1960, and subsequently improved to 4n 2 and then 3n 2 multiplications by W. F. Trench and S. Zohar, respectively.