Analytic expressions for finding fill-in, the number of nonzero elements that change in value, and the number of “long operations” during each step of the L U decomposition are given. A new optimal pivot ordering algorithm is proposed which leads to a reduction of the overall fill-in and long operation count. Comparison is made with two other known algorithms.

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Journal IEEE Transactions on Circuits and Systems
Nakhla, M.S, Singhal, K., & Vlach, J. (1974). An Optimal Pivoting Order for the Solution of Sparse Systems of Equations. IEEE Transactions on Circuits and Systems, CAS-21(2), 222–225. doi:10.1109/TCS.1974.1083827