论文标题
使用冗余数表示的迭代方法中数字稳定性的条件
Conditions for Digit Stability in Iterative Methods Using the Redundant Number Representation
论文作者
论文摘要
迭代方法在科学和工程应用中起重要作用，其用途从有限元方法的线性系统求解器到模型预测性控制中的优化求解器。最近，Li等人提出了一种称为Architect的迭代方法的新计算策略。在[1]中，使用冗余数表示在迭代的最重要的数字（MSD）中创建“稳定数字”，从而允许将来的迭代假设稳定的MSDS没有改变其价值，从而消除了重新计算它们的需求。在这项工作中，我们通过表明冗余数表示中的Fejer单调序列可以在序列的元素中发展出稳定的MSD来介绍这些“稳定数字”如何在迭代方法中产生的理论分析。 Fejer单调序列的这种属性使我们能够在使用冗余数字表示时扩展已知具有MSD稳定性的迭代方法，以包含Bonscontive Lipschitz连续函数的任何定点迭代。然后，我们表明，这允许数字稳定性的理论保证，不仅是在李等人先前分析的雅各比方法中。在[2]中，以及其他常用方法，例如牛顿的方法。
Iterative methods play an important role in science and engineering applications, with uses ranging from linear system solvers in finite element methods to optimization solvers in model predictive control. Recently, a new computational strategy for iterative methods called ARCHITECT was proposed by Li et al. in [1] that uses the redundant number representation to create "stable digits" in the Most-significant Digits (MSDs) of an iterate, allowing the future iterations to assume the stable MSDs have not changed their value, eliminating the need to recompute them. In this work, we present a theoretical analysis of how these "stable digits" arise in iterative methods by showing that a Fejer monotone sequence in the redundant number representation can develop stable MSDs in the elements of the sequence as the sequence grows in length. This property of Fejer monotone sequences allows us to expand the class of iterative methods known to have MSD stability when using the redundant number representation to include any fixed-point iteration of a contractive Lipschitz continuous function. We then show that this allows for the theoretical guarantee of digit stability not just in the Jacobi method that was previously analyzed by Li et al. in [2], but also in other commonly used methods such as Newton's method.
