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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.
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).
The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows: In the first equation, solve for one of the variables in terms of the others. Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown.
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
LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.
Elimination theory, the theory of the methods to eliminate variables between polynomial equations. Disjunctive syllogism, a rule of inference; Gaussian elimination, a method of solving systems of linear equations; Fourier–Motzkin elimination, an algorithm for reducing systems of linear inequalities
Gaussian elimination for solving systems of linear equations Gauss's algorithm for Determination of the day of the week Gauss's method for preliminary orbit determination
In the absence of rounding errors, direct methods would deliver an exact solution (for example, solving a linear system of equations = by Gaussian elimination). Iterative methods are often the only choice for nonlinear equations. However, iterative methods are often useful even for linear problems involving many variables (sometimes on the ...
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