We analyze the convergence of quasi-Newton methods in exact and finite precision arithmetic. In particular, we derive an upper bound for the stagnation level and we show that any sufficiently exact quasi-Newton method will converge quadratically until stagnation. In the absence of sufficient accuracy, we are likely to retain rapid linear convergence. We confirm our analysis by computing square roots and solving bond constraint equations in the context of molecular dynamics. We briefly discuss implications for parallel solvers.

翻译：我们分析了拟牛顿法在精确和有限精度算术下的收敛性。特别地，我们推导了停滞水平的上界，并证明任何足够精确的拟牛顿法将在停滞之前以二次收敛速度收敛。在精度不足的情况下，我们仍可能保持快速的线性收敛。我们通过计算平方根以及在分子动力学背景下求解键约束方程来验证我们的分析。我们简要讨论了这些结果对并行求解器的启示。