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.

翻译：预条件技术是迭代法求解线性方程组的关键环节。本文在正定矩阵低秩更新的最优条件化框架下，研究非经典矩阵条件数——$\omega$条件数。对于正定矩阵，该条件度量定义为特征值的算术平均值与几何平均值之比。我们重点关注源于使用广义雅可比矩阵的非光滑牛顿法中、接近奇异的正定矩阵低秩更新线性系统。推导了该框架下最优$\omega$预条件更新的显式表达式。经典条件数$\kappa$的计算或估计成本较高，而我们的研究表明，在完成Cholesky或LU分解后，$\omega$条件数可被精确计算，且能显著更优地估计线性系统的实际条件数。此外，实验结果表明，在迭代法中，采用$\omega$条件数可在达到指定残差精度时显著减少迭代次数，从而验证了其相较于经典条件数的有效性。