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{\tt U} with a structurally zero-free diagonal.
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Unless the matrix ill-conditioned or
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poorly scaled, factorizing {\tt A'} in this case will guarantee that both
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as to the size of the diagonal entries of {\tt U}; UMFPACK does not do a
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rank-revealing factorization. Here's how you can factorize {\tt A'}
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and get the factors of {\tt A} instead: