The $ω$-Condition Number: Applications to Optimal Preconditioning and Low Rank Generalized Jacobian Updating

TL;DR

The paper introduces the $\omega$-condition number, defined as $\omega(A)=\dfrac{\frac{1}{n}\operatorname{trace}(A)}{\det(A)^{1/n}}$ for positive definite matrices, and argues that it yields a more representative conditioning measure than the classical $\kappa$ by leveraging all eigenvalues. It develops explicit $\omega$-optimal preconditioners (diagonal, block-diagonal, and incomplete triangular forms) and shows how to compute them efficiently from Cholesky or LU factorizations, enabling exact evaluation of $\omega$ without eigenvalue decompositions. The authors extend the framework to low-rank updates, deriving closed-form $\omega$-optimal update parameters for rank-one and rank-$t$ perturbations, with practical approximations when exact computation is costly. Numerical experiments on SPD systems and generalized Jacobians demonstrate that $\omega$-based preconditioning and updates often improve eigenvalue clustering and reduce iteration counts, offering a robust alternative to $\kappa$-based approaches. The work also connects $\omega$-optimal incomplete factorizations to sparse Cholesky-type preconditioners and discusses data-driven and algorithmic implications for optimization and linear system solvers.

Abstract

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 $ω$-condition number, $ω$ for short. $ω$ is the ratio of the arithmetic and geometric means of the singular values, rather than largest and smallest. Moreover, unlike the latter classical $κ$ condition number, $ω$ 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 $ω$ allow one to exploit optimality conditions and derive explicit formulae for $ω$-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 $ω$-condition number compared to the classical $κ$-condition number.

Figures (6)

• Figure 2.1: Comparing accuracy under perturbations for calculating $\omega,\kappa$.
• Figure 2.2: Linear regression models (LRM) between $\mathop{\mathrm{cond}}\nolimits$ and: $\kappa,\, \omega,\, \omega^{-2}$, respectively; uniformly distributed eigenvalues.
• Figure 2.3: Linear regression models (LRM) between $\mathop{\mathrm{cond}}\nolimits$ and: $\kappa,\, \omega,\, \omega^{-2}$, respectively; normally distributed eigenvalues.
• Figure 2.4: Comparison for clustering of eigenvalues pre-post preconditioning
• Figure 4.1: Iterations and time performance profiles for solving the system with the different choices of preconditioner.

Theorems & Definitions (31)

• Proposition 2.1: DeWo:90KrukDoanW:10
• proof
• Theorem 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Remark 2.5
• Remark 2.6: Recursive formula for the Cholesky decomposition
• Theorem 2.7