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A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].
Simplified forms of Gaussian elimination have been developed for these situations. [ 6 ] The textbook Numerical Mathematics by Alfio Quarteroni , Sacco and Saleri, lists a modified version of the algorithm which avoids some of the divisions (using instead multiplications), which is beneficial on some computer architectures.
The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it. The variant of Gaussian elimination that transforms a matrix to reduced row echelon form is sometimes called Gauss–Jordan elimination. A matrix is in column echelon form if its transpose is in row echelon form.
The matrices L and U could be thought to have "encoded" the Gaussian elimination process. The cost of solving a system of linear equations is approximately 2 3 n 3 {\textstyle {\frac {2}{3}}n^{3}} floating-point operations if the matrix A {\textstyle A} has size n {\textstyle n} .
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.
Gaussian algorithm may refer to: Gaussian elimination for solving systems of linear equations; Gauss's algorithm for Determination of the day of the week;
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Gauss–Laguerre quadrature — extension of Gaussian quadrature for integrals with weight exp(−x) on [0, ∞] Gauss–Kronrod quadrature formula — nested rule based on Gaussian quadrature; Gauss–Kronrod rules; Tanh-sinh quadrature — variant of Gaussian quadrature which works well with singularities at the end points