For each pivot column $k$, select a nonzero pivot (swap rows if needed), compute multipliers $\ell_{ik}=a_{ik}/a_{kk}$, and subtract $\ell_{ik}$ times row $k$ from lower rows. The resulting upper triangular matrix is $U$; stored multipliers form $L$. If pivoting is used, previously stored multiplier columns must be swapped consistently. Check the result with $PA=LU$.
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