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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].
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.
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.
Euclidean algorithm for polynomial greatest common divisor computation and Gaussian elimination of linear systems are special cases of Buchberger's algorithm when the number of variables or the degrees of the polynomials are respectively equal to one. For other Gröbner basis algorithms, see Gröbner basis § Algorithms and implementations.
The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, [5] as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method.
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.
Otherwise, the Bareiss algorithm may be viewed as a variant of Gaussian elimination and needs roughly the same number of arithmetic operations. It follows that, for an n × n matrix of maximum (absolute) value 2 L for each entry, the Bareiss algorithm runs in O( n 3 ) elementary operations with an O( n n /2 2 nL ) bound on the absolute value of ...
Problems that have the same asymptotic complexity as matrix multiplication include determinant, matrix inversion, Gaussian elimination (see next section). Problems with complexity that is expressible in terms of include characteristic polynomial, eigenvalues (but not eigenvectors), Hermite normal form, and Smith normal form.