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The following algorithm is essentially a modified form of Gaussian elimination. Computing an LU decomposition using this algorithm requires floating-point operations, ignoring lower-order terms. Partial pivoting adds only a quadratic term; this is not the case for full pivoting. [13]
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].
This system has the exact solution of x 1 = 10.00 and x 2 = 1.000, but when the elimination algorithm and backwards substitution are performed using four-digit arithmetic, the small value of a 11 causes small round-off errors to be propagated. The algorithm without pivoting yields the approximation of x 1 ≈ 9873.3 and x 2 ≈ 4.
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 solution method called "Fang Cheng Shi" is best known today as Gaussian elimination. Among the eighteen problems listed in the Fang Cheng chapter, some are equivalent to simultaneous linear equations with two unknowns, some are equivalent to simultaneous linear equations with 3 unknowns, and the most complex example analyzes the solution to ...
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.
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
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