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 /.(/-factorization of A. When A is exactly singular, we show conditions under which a pivoting strategy will produce a zero nth pivot. When A is not singular, we show conditions under which a pivoting strategy will produce an nth pivot that is 0(an) or 0(k˜\A)), where o 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 /f_1 in general but reduce to conditions on the elements of the singular vectors corresponding to o 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 nth pivot for nearly singular matrices. As an example, we exhibit an ¿¿/factorization of the n by n upper triangular matrix 7 = that has an nth pivot equal to 2 ' 2.