The Growth-Factor Bound for the Bunch-Kaufman Factorization Is Tight

We show that the growth-factor bound in the Bunch–Kaufman factorization method is essentially tight. The method factors a symmetric matrix A into A=PTLDLTP, where P is a permutation matrix, L is lower triangular, and D is block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The method uses one of several partial pivoting rules that ensure bounded in the elements of the reduced matrix and the factor D (growth in L is not bounded). We show that the exponential bound is essentially tight, thereby solving a question that has been open since 1977.