We introduce a unified framework for computing approximately-optimal preconditioners for solving linear and non-linear systems of equations. We demonstrate that the condition number minimization problem, under structured transformations such as diagonal and block-diagonal preconditioners, is geodesically convex with respect to unitarily invariant norms, including the Frobenius and Bombieri--Weyl norms. This allows us to introduce efficient first-order algorithms with precise convergence guarantees. For linear systems, we analyze the action of symmetric Lie subgroups $G \subseteq \GL_m(\CC) \times \GL_n(\CC)$ on the input matrix and prove that the logarithm of the condition number is a smooth geodesically convex function on the associated Riemannian quotient manifold. We obtain explicit gradient formulas, show Lipschitz continuity, and prove convergence rates for computing the optimal Frobenius condition number: $\widetilde{O}(1/\eps^2)$ iterations for general two-sided preconditioners and $\widetilde{O}(κ_F^2 \log(1/\eps))$ for strongly convex cases such as left preconditioning. We extend our framework to consider preconditioning of polynomial systems $\f(x) = 0$, where $\f$ is a system of multivariate polynomials. We analyze the local condition number $μ(\f, ξ)$, at a root $ξ$ and prove that it also admits a geodesically convex formulation under appropriate group actions. We deduce explicit formulas for the Riemannian gradients and present convergence bounds for the corresponding optimization algorithms. To the best of our knowledge, this is the first preconditioning algorithm with theoretical guarantees for polynomial systems.

翻译：我们提出了一个统一框架，用于计算求解线性和非线性方程组的近似最优预条件子。我们证明，在诸如对角和块对角预条件子等结构化变换下，条件数最小化问题对于酉不变范数（包括Frobenius范数和Bombieri–Weyl范数）是测地凸的。这使我们能够引入具有精确收敛保证的高效一阶算法。对于线性系统，我们分析了对称李子群 $G \\subseteq \\GL_m(\\CC) \\times \\GL_n(\\CC)$ 对输入矩阵的作用，并证明条件数的对数在相关的黎曼商流形上是光滑的测地凸函数。我们获得了显式的梯度公式，证明了Lipschitz连续性，并给出了计算最优Frobenius条件数的收敛速率：对于一般双侧预条件子为 $\\widetilde{O}(1/\\eps^2)$ 次迭代，对于强凸情形（如左预条件子）为 $\\widetilde{O}(\\kappa_F^2 \\log(1/\\eps))$。我们将框架扩展到多项式系统 $\\f(x) = 0$ 的预条件化，其中 $\\f$ 是一个多元多项式系统。我们分析了在根 $\\xi$ 处的局部条件数 $\\mu(\\f, \\xi)$，并证明在适当的群作用下，该问题同样具有测地凸形式。我们推导了黎曼梯度的显式公式，并给出了相应优化算法的收敛界。据我们所知，这是首个针对多项式系统具有理论保证的预条件化算法。