In this paper, preconditioners for the conjugate gradient method are studied to solve the Newton system with symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of Symmetric Rank one (SR1) and Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) low‐rank updates. We develop conditions under which the SR1 update maintains the preconditioner symmetric positive definite. Spectral analysis of the SR1 preconditioned Jacobians shows an improved eigenvalue distribution as the Newton iteration proceeds. A compact matrix formulation of the preconditioner update is developed which reduces the cost of its application and is more suitable to parallel implementation. Some notes on the implementation of the corresponding Inexact Newton method are given and some numerical results on a number of model problems illustrate the efficiency of the proposed preconditioners.

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对称正定线性系统的紧凑拟牛顿预处理器

本文研究了共轭梯度法的预处理器，以求解对称正定Jacobian的牛顿系统。特别是，我们定义了一系列通过对称等级1（SR1）和Broyden-Fletcher-Goldfarb-Shanno（BFGS）低等级更新构建的预处理器。我们开发了在其中SR1更新保持预处理器对称正定的条件。SR1预处理的Jacobian光谱分析表明，随着牛顿迭代的进行，特征值分布得到了改善。开发了紧凑型矩阵的预处理器更新，可降低其应用成本，更适合并行执行。