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Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can output real solutions. The algorithm is named after Joseph Fourier [ 1 ] who proposed the method in 1826 and Theodore Motzkin who re-discovered it in 1936.
Animation of Gaussian elimination. Red row eliminates the following rows, green rows change their order. In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of row-wise operations performed on the corresponding matrix of coefficients.
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.
Banachiewicz [4] was the first to consider elimination in terms of matrices and in this way formulated LU decomposition, as demonstrated by his graphic illustration. His calculations follow ordinary matrix ones, yet notation deviates in that he preferred to write one factor transposed, to be able to multiply them mechanically column by column ...
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations.
However, loser-elimination methods often fail monotonicity due to chaotic effects (sensitivity to initial conditions): the order in which candidates are eliminated can create erratic behavior. [1] If the base method passes independence from the weakest alternative, the loser-elimination method is equivalent to the base method. [1]
The runoff method closest to IRV is the exhaustive ballot. In this method—familiar to fans of the television show American Idol—one candidate is eliminated after each round, and many rounds of voting are used, rather than just two. Because holding many rounds of voting on separate days is generally expensive, the exhaustive ballot is not ...
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, [5] as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.