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I searched for Gilbert-Peierls algorithm, but I haven't found anything useful (well, I found this, but it's not working as it should). I think the problem is the second part, and also that those lines:

U(1:k, k) = x(1:k);
L(k:N, k) = x(k:N)/U(k, k);

should actually be (according to this example):

U(1:N, k) = x(1:N);
L(k:N, k) = x(k:N)/U(k, k);

Also, in that example, L is the identity matrix, which I find a bit strange. Could someone describe the algorithm, please? (with or without code)

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Why don't you check this out: http://cci.lbl.gov/cctbx_sources/scitbx/sparse/lu_factorization.h

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