On the existence and computation of $LU$-factorizations with small pivots

Let A be an n by n matrix which may be singular with a one-dimensional null space, and consider the LU-factorization of A. When A is exactly singular, we show conditions under which a pivoting strategy will produce a zero n th pivot. When A is not singular, we show conditions under which a pivoting strategy will produce an nth pivot that is O(G,,) or O(K(A)), where ,, is the smallest singular value of A and K(A) is the condition number of A. These conditions are expressed in terms of the elements of A in general but reduce to conditions on the elements of the singular vectors corresponding to ,, when A is nearly or exactly singular. They can be used to build a 2-pass factorization algorithm which is guaranteed to produce a small n th pivot for nearly singular matrices. As an example, we exhibit an LU-factorization of the n by n upper triangular matrix