The $$\omega $$ ω -condition number: applications to preconditioning and low rank generalized Jacobian updating

Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g., in quasi-Newton methods. We study a nonclassic matrix condition number, the $$\omega $$ ω -condition number, $$\omega $$ ω for short. $$\omega $$ ω is the ratio of: the arithmetic and geometric means of the singular values, rather than the largest and smallest for the classical $$\kappa $$ κ -condition number. The simple functions in $$\omega $$ ω allow one to exploit first order optimality conditions. We use this fact to derive explicit formulae for (i) $$\omega $$ ω -optimal low rank updating of generalized Jacobians arising in the context of nonsmooth Newton methods; and (ii) $$\omega $$ ω -optimal preconditioners of special structure for iterative methods for linear systems. In the latter context, we analyze the benefits of $$\omega $$ ω for (a) improving the clustering of eigenvalues; (b) reducing the number of iterations; and (c) estimating the actual condition of a linear system. Moreover we show strong theoretical connections between the $$\omega $$ ω -optimal preconditioners and incomplete Cholesky factorizations, and highlight the misleading effects arising from the inverse invariance of $$\kappa $$ κ . Our results confirm the efficacy of using the $$\omega $$ ω -condition number compared to the $$\kappa $$ κ -condition number.