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Gaussian elimination can be performed over any field, not just the real numbers. Buchberger's algorithm is a generalization of Gaussian elimination to systems of polynomial equations. This generalization depends heavily on the notion of a monomial order. The choice of an ordering on the variables is already implicit in Gaussian elimination ...
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.
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. A tridiagonal system for n unknowns may be written as
For example, for a 3 × 3 matrix A, ... The LU decomposition is related to elimination of linear systems of equations, as e.g. described by Ralston. [18]
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.
A system of linear equations is said to be in row echelon form if its augmented matrix is in row echelon form. Similarly, a system of linear equations is said to be in reduced row echelon form or in canonical form if its augmented matrix is in reduced row echelon form. The canonical form may be viewed as an explicit solution of the linear system.
A linear system in three variables determines a collection of planes. The intersection point is the solution. In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. [1] [2] For example,
Kron reduction is a useful tool to eliminate unused nodes in a Y-parameter matrix. [2] [3] For example, three linear elements linked in series with a port at each end may be easily modeled as a 4X4 nodal admittance matrix of Y-parameters, but only the two port nodes normally need to be considered for modeling and simulation.