AI 中文总结
研究如何用混合精度算法提升牛顿法效率，通过对不同不精确源进行误差分析，给出收敛性分析，涵盖拟牛顿和不精确牛顿法并据此提出算法，用数值实验验证结果。
AI 中文摘要
二阶优化方法如牛顿算法虽收敛快精度高，但计算成本限制其应用。本文通过对牛顿法不同不精确源进行误差分析，给出收敛性分析，涵盖拟牛顿和不精确牛顿法，据此提出混合精度算法并通过数值实验验证。
英文摘要
Second-order optimization methods, such as Newton's algorithm, achieve fast local convergence and high accuracy, but their practical use is often limited by high computational costs. To mitigate this issue, variants such as inexact and quasi-Newton methods are widely used. A complementary and promising approach to improve the efficiency of the method is to employ mixed precision arithmetic, using different floating-point precisions for different operations, based on their impact on the convergence and accuracy of the method. In this work, we perform an error analysis of Newton's method accounting for different sources of inexactness, including approximations and rounding errors. We present a convergence analysis for the generated sequence, establishing bounds on the convergence rate and attainable accuracy. This theoretical framework covers quasi-Newton and inexact Newton methods, and is leveraged to propose mixed precision algorithms. We present a wide set of numerical experiments to illustrate our theoretical results and the behavior of Newton's method and its approximate variants in mixed precision floating-point arithmetic.
