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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.
The field of elimination theory was motivated by the need of methods for solving systems of polynomial equations. One of the first results was Bézout's theorem, which bounds the number of solutions (in the case of two polynomials in two variables at Bézout time).
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.
Here z is the free variable, while x and y are dependent on z. Any point in the solution set can be obtained by first choosing a value for z, and then computing the corresponding values for x and y. Each free variable gives the solution space one degree of freedom, the number of which is equal to the dimension of the solution set.
Variable elimination and the simplex algorithm are used for solving linear and polynomial equations and inequalities, and problems containing variables with infinite domain. These are typically solved as optimization problems in which the optimized function is the number of violated constraints.
The Cholesky decomposition is commonly used in the Monte Carlo method for simulating systems with multiple correlated variables. The covariance matrix is decomposed to give the lower-triangular L . Applying this to a vector of uncorrelated observations in a sample u produces a sample vector Lu with the covariance properties of the system being ...
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges.
Variable elimination (VE) is a simple and general exact inference algorithm in probabilistic graphical models, such as Bayesian networks and Markov random fields. [1] It can be used for inference of maximum a posteriori (MAP) state or estimation of conditional or marginal distributions over a subset of variables.