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    $\begingroup$ Might try changing the sums to dot products. Also Table probably should be Do. For the machine real case you could run it through Compile. $\endgroup$ Commented Mar 31, 2015 at 16:05
  • $\begingroup$ The oneStep function that I implemented is C-style(like procedural paradigm) . So my main difficult is how to store and organize the intermediate result in right way if I'd like to implement this algorithm with functional or rule-based paradigm. $\endgroup$ Commented Apr 3, 2015 at 5:21
  • $\begingroup$ Might want to check the notebook here. Dated a bit but still quite relevant to the question at hand. $\endgroup$ Commented Apr 4, 2015 at 13:20
  • $\begingroup$ I've just seen this. I believe you know, but just to remind, algorithms of this sort are usually made with a pivoting strategy in place (e.g. partial, total, or rook); otherwise, there are a lot of nonsingular matrices that do not have a decomposition. $\endgroup$ Commented Oct 24, 2015 at 10:56
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    $\begingroup$ With respect to the counterexample, try Reverse[IdentityMatrix[4]]. As for pivoting: did you notice the difference between your method and LUDecomposition[]? The latter one outputs a permutation list, which encodes the (partial) pivoting (interchanges of rows) done in the course of decomposing the matrix. Any robust implementation has to have a pivoting strategy like that. $\endgroup$ Commented Oct 24, 2015 at 11:55