bareiss algorithm

The Bareiss algorithm for an LU decomposition is stable.

Otherwise, Bareiss algorithm may be viewed as a variant of Gaussian elimination and needs roughly the same number of arithmetic operations.

During the execution of Bareiss algorithm, every integer that is computed is the determinant of a submatrix of the input matrix.

If the coefficients of the matrix are exactly given integers numbers, the column echelon form of the matrix may be computed by Bareiss algorithm more efficiently than with Gaussian elimination.

However, there is a variant of Gaussian elimination, called Bareiss algorithm that avoids this exponential growth of the intermediate entries, and, with the same arithmetic complexity of O(n), has a bit complexity of O(n).

In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainder).