Preconditioning for Generalized Jacobians with the $\omega$-Condition Number

Preconditioning is essential in iterative methods for solving linear systems of equations. We study a nonclassic matrix condition number, the $\omega$-condition number, in the context of optimal conditioning for low rank updating of positive definite matrices. For a positive definite matrix, this condition measure is the ratio of the arithmetic and geometric means of the eigenvalues. In particular, we concentrate on linear systems with low rank updates of positive definite matrices which are close to singular. These systems arise in the contexts of nonsmooth Newton methods using generalized Jacobians. We derive an explicit formula for the optimal $\omega$-preconditioned update in this framework. Evaluating or estimating the classical condition number $\kappa$ can be expensive. We show that the $\omega$-condition number can be evaluated exactly following a Cholesky or LU factorization and it estimates the actual condition of a linear system significantly better. Moreover, our empirical results show a significant decrease in the number of iterations required for a requested accuracy in the residual during an iterative method, i.e., these results confirm the efficacy of using the $\omega$-condition number compared to the classical condition number.