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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.
In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations. Classical elimination theory culminated with the work of Francis Macaulay on multivariate resultants, as ...
The Harrow–Hassidim–Lloyd algorithm or HHL algorithm is a quantum algorithm for numerically solving a system of linear equations, designed by Aram Harrow, Avinatan Hassidim, and Seth Lloyd. The algorithm estimates the result of a scalar measurement on the solution vector to a given linear system of equations.
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
Cramer's rule, implemented in a naive way, is computationally inefficient for systems of more than two or three equations. [7] In the case of n equations in n unknowns, it requires computation of n + 1 determinants, while Gaussian elimination produces the result with the same computational complexity as the computation of a single determinant.
First, we solve the equation = for y. Second, we solve the equation U x = y {\textstyle U\mathbf {x} =\mathbf {y} } for x . In both cases we are dealing with triangular matrices ( L and U ), which can be solved directly by forward and backward substitution without using the Gaussian elimination process (however we do need this process or ...
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
More precisely, the system of equations defines an algebraic set which may have several irreducible components, and one must remove the components on which the degeneracy conditions are everywhere zero. This is done by saturating the equations by the degeneracy conditions, which may be done via the elimination property of Gröbner bases.
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