Magnus展开及其应用
The Magnus expansion and some of its applications
摘要：
变系数线性微分方程组的近似解是从量子力学到控制论的许多科学和工程领域所共有的经常性问题。当以算子或矩阵形式表示时，马格努斯展开式提供了一个优雅的设置，以建立系统解的近似指数表示。它为相应的指数提供了幂级数展开，有时被称为时间相关指数扰动理论。每一个马格努斯逼近对应于摄动理论的部分再求和的无限项与重要的额外财产保持在任何顺序的某些对称性的精确解。本次审查的目标有三个方面。首先，收集了分散在半个世纪以来关于马格努斯展开的科学文献中的一些发展。它们涉及的方法产生的条款的扩展，估计的半径收敛的系列，推广和相关的非微扰扩展。第二，提供一个桥梁，它的实施作为发电机的特殊用途的数值积分方法，一个领域的激烈活动，在过去的十年。第三，举例说明从马格努斯展开所能得到的结果，并与微扰格式和标准数值积分器的结果进行比较。我们支持这个问题的修订发现在文献中的马格努斯扩张的广泛的物理应用。
英文摘要：
Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.
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。
