Pták的非离散归纳法及其在矩阵迭代中的应用
Pták's nondiscrete induction and its application to matrix iterations
AI总结：
本文基于Pták的非离散归纳法，探讨其在矩阵极分解和矩阵平方根迭代中的应用，分析收敛估计的精确性并讨论方法的局限性与扩展。
摘要：
Vlastimil Pták的非离散归纳法基于在迭代过程分析中应追求收敛速率作为函数而非单纯数值的观点，因为函数可提供在整个迭代过程中紧致的收敛估计而非仅渐近性。本文推导并证明了Potra和Pták最初提出的非离散归纳法定理，并将其应用于计算矩阵极分解和矩阵平方根的牛顿迭代中。我们的目标是展示该方法在有限维数值线性代数中的应用。我们通过分析极分解迭代和平方根迭代的特殊情形，以及一些数值例子，展示了所得收敛估计的精确性。我们还讨论了该方法的一些局限性和可能的扩展。
英文摘要：
Vlastimil Pták's method of nondiscrete induction is based on the idea that in the analysis of iterative processes one should aim at rates of convergence as functions rather than just numbers, because functions may give convergence estimates that are tight throughout the iteration rather than just asymptotically. In this paper we motivate and prove a theorem on nondiscrete induction originally due to Potra and Pták, and we apply it to the Newton iterations for computing the matrix polar decomposition and the matrix square root. Our goal is to illustrate the application of the method of nondiscrete induction in the finite dimensional numerical linear algebra context. We show the sharpness of the resulting convergence estimate analytically for the polar decomposition iteration and for special cases of the square root iteration, as well as on some numerical examples for the square root iteration. We also discuss some of the method's limitations and possible extensions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
