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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].
The matrix () is the matrix in which the elements below the main diagonal have already been eliminated to 0 through Gaussian elimination for the first columns. Below is a matrix to observe to help us remember the notation (where each ∗ {\displaystyle *} represents any real number in the matrix):
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.
As a rule of thumb, iterative refinement for Gaussian elimination produces a solution correct to working precision if double the working precision is used in the computation of r, e.g. by using quad or double extended precision IEEE 754 floating point, and if A is not too ill-conditioned (and the iteration and the rate of convergence are ...
Gaussian algorithm may refer to: 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; Gauss's Easter algorithm; Gauss separation algorithm
Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy. [5] In 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for womb.
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The original application of chordal completion described in Computers and Intractability involves Gaussian elimination for sparse matrices. During the process of Gaussian elimination, one wishes to minimize fill-in, coefficients of the matrix that were initially zero but later become nonzero, because the need to calculate the values of these ...