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These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques during the numerical factorization.
The methods are based on maximum weighted matchings and improve the quality of the factor in a complementary way to the alternative idea of using more complete pivoting techniques.
They also proved that the bound of this method, in terms of the growth of elements in the reduced matrices, is almost as good as that of the Gaussian elimination with complete pivoting.
Examples of pivoting strategies which simultaneously control the conditioning of both triangular matrices L, U are the complete pivoting (of a computational cost of O([n.
They use symmetric complete pivoting and so they can guarantee that one of the obtained triangular matrices is diagonally dominant and the other one has the off-diagonal elements with absolute value bounded above by the diagonal elements, in contrast to our method, where we can guarantee that both triangular matrices are diagonally dominant.
Complete pivoting strategy is discussed in section 7, and numerical experiments are presented in section 8.
However, for nearly singular matrices even complete pivoting may not reveal the rank correctly.

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