On Floating Point Errors in Cholesky

Let H be a symmetric positive de nite matrix. Consider solving the linear system Hx = b using Cholesky, forward and back substitution in the standard way, yielding a computed solution x̂. The usual oating point error analysis says that the relative error kx x̂k2=kx̂k2 = O(") (H), where (H) is the condition number of H . Now write H = DAD, where D is diagonal and A has unit diagonal; then (A) n min ~ D ( ~ DH ~ D) and it may be that (A) (H). We show that the scaled error may be bounded by kD(x x̂)k2=kDx̂k2 = O(") (A). This often provides better error bounds than the standard formula. We show that (A) is the \right" condition number in several senses. First, its reciprocal is approximately the smallest componentwise relative perturbation that makes H singular. Second, it provides a nearly sharp criterion for the successful termination of Cholesky in oating point. Third, the bound on kD(x x̂)k2 is nearly attainable.