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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.
After that, in each step the expressions R k ij are computed from the previous ones by R k ij = R k-1 ik (R k-1 kk) * R k-1 kj | R k-1 ij. Another way to understand the operation of the algorithm is as an "elimination method", where the states from 0 to n are successively removed: when state k is removed, the regular expression R k-1
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].
In terms of operations, zeroing/elimination of remaining elements of first column of A involves division of , with , impossible if it is 0. This is a procedural problem. This is a procedural problem. It can be removed by simply reordering the rows of A so that the first element of the permuted matrix is nonzero.
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.
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.
Fraction-free algorithm — uses division to keep the intermediate entries smaller, but due to the Sylvester's Identity the transformation is still integer-preserving (the division has zero remainder). For completeness Bareiss also suggests fraction-producing multiplication-free elimination methods. [2]
Using this value in one of the equations, the same solution as in the previous method is obtained. { x = 2 y = 3. {\displaystyle {\begin{cases}x=2\\y=3.\end{cases}}} This is not the only way to solve this specific system; in this case as well, y could have been solved before x .