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Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination

Erwin H. Bareiss
Mathematics of Computation
Vol. 22, No. 103 (Jul., 1968), pp. 565-578
DOI: 10.2307/2004533
Stable URL: http://www.jstor.org/stable/2004533
Page Count: 14
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Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination
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Abstract

A method is developed which permits integer-preserving elimination in systems of linear equations, AX = B, such that (a) the magnitudes of the coefficients in the transformed matrices are minimized, and (b) the computational efficiency is considerably increased in comparison with the corresponding ordinary (single-step) Gaussian elimination. The algorithms presented can also be used for the efficient evaluation of determinants and their leading minors. Explicit algorithms and flow charts are given for the two-step method. The method should also prove superior to the widely used fraction-producing Gaussian elimination when A is nearly singular.

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