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Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination. In this case, the term Gaussian elimination refers to the process until it has reached its upper triangular, or (unreduced) row echelon form. For computational reasons, when solving systems of linear equations, it is sometimes ...
In fact, solving the submodule membership problem is what is commonly called solving the system, and solving the syzygy problem is the computation of the null space of the matrix of a system of linear equations. The basic algorithm for both problems is Gaussian elimination.
The standard algorithm for solving a system of linear equations is based on Gaussian elimination with some modifications. Firstly, it is essential to avoid division by small numbers, which may lead to inaccurate results.
The above procedure can be repeatedly applied to solve the equation multiple times for different b. In this case it is faster (and more convenient) to do an LU decomposition of the matrix A once and then solve the triangular matrices for the different b, rather than using Gaussian elimination each time
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).
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
Process of elimination is a logical method to identify an entity of interest among several ones by excluding all other entities. In educational testing , it is a process of deleting options whereby the possibility of an option being correct is close to zero or significantly lower compared to other options.
Modelling Sudoku as an exact cover problem and using an algorithm such as Knuth's Algorithm X and his Dancing Links technique "is the method of choice for rapid finding [measured in microseconds] of all possible solutions to Sudoku puzzles." [18] An alternative approach is the use of Gauss elimination in combination with column and row striking.