论文标题
线性代数方程的NXN系统的解决方案：1-重新审视的最陡峭下降方法
Solution of a NxN System of Linear algebraic Equations: 1 -- The Steepest Descent Method Revisited
论文作者
论文摘要
这是一系列论文中的第一篇，该论文介绍了新方法的开发，该方法求解了比现有算法低的时间复杂性的线性代数方程系统。线性方程式AX = B的NXN系统通常是通过使用众所周知的优化技术最小化相应的二次形式的迭代求解的。其中最简单的是最陡峭的下降方法，其溶液的方法通常在开始时很快，但经过几次迭代后会大大减慢。本文调查了可能减少或避免减速的方法。这里使用的两种方法涉及迭代之间的点随机运动，以及迭代之间可能的矩阵变换。本文报告了计算实验的结果，并显示了可能的最陡下降方法的性能的显着改善。此处描述的方法不会立即提供实用算法。但是，他们为仅基于SD开发实用算法奠定了基础，该算法将在以后的出版物中介绍。
This is the first in a series of papers which deal with the development of novel methods for solving a system of linear algebraic equations with a time complexity lower than existing algorithms. The NxN system of linear equations, Ax = b, is often solved iteratively by minimizing the corresponding quadratic form using well known optimization techniques. The simplest of these is the steepest descent method, whose approach to the solution is usually quite rapid at the beginning but slows down drastically after a few iterations. This paper investigates possible approaches which can reduce or avoid this slowing down. The two approaches used here involve random movement of the point between iterations, and possible matrix transformations between iterations. This paper reports the results of computational experiments and shows the remarkable improvement in performance of the steepest descent method that is possible. The approaches described here do not give a practical algorithm right away. However, they set the stage for developing practical algorithms based on SD alone, which will be presented in later publications.
