The $ω$-Condition Number: Applications to Optimal 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. Motivated by the latter, we study a nonclassic matrix condition number, the $\omega$-condition number, $\omega$ for short. $\omega$ is the ratio of the arithmetic and geometric means of the singular values, rather than largest and smallest. Moreover, unlike the latter classical $\kappa$ condition number, $\omega$ is not invariant under inversion, an important point that allows one to recall that it is the conditioning of the inverse that is important. Our study is in the context of optimal conditioning for: (i) low rank updating of generalized Jacobians arising in the context of nonsmooth Newton methods; and (ii) iterative methods for linear systems; (iia) clustering of eigenvalues; (iib) convergence rates; and (iic) estimating the actual condition of a linear system. We emphasize that the simple functions in $\omega$ allow one to exploit optimality conditions and derive explicit formulae for $\omega$-optimal preconditioners of special structure. Connections to partial Cholesky type sparse preconditioners are made that modify the iterates of Cholesky decomposition by including the entire diagonal at each iteration. Our results confirm the efficacy of using the $\omega$-condition number compared to the classical $\kappa$-condition number.