The General Theory of Relaxation Methods Applied to Linear Systems
The General Theory of Relaxation Methods Applied to Linear Systems
复制标题
应用于线性系统的弛豫方法的一般理论
DOI：
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发表时间：
1939
期刊：
影响因子：
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通讯作者：
G. Temple
中科院分区：
文献类型：
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作者：
G. Temple
在过去的几年里，索斯韦尔和他的同事们开发了一种新的方法，用于数学物理和工程中一类非常普遍的问题的数值解。该方法最初是为确定框架中的应力而设计的，但它已被证明可直接适用于任何可简化为具有有限数量未知变量的非齐次线性联立代数方程组的解的问题。* Southwell的“松弛m法”是一种逐次逼近法，为了完成之前对该方法的研究，有必要证明逐次逼近确实收敛于精确解。这个形式证明在§4中给出。Southwell的“松弛”方法不能直接适用于连续系统，其中未知变量的数量是无限的，但这里表明，对松弛方法的简单扩展和修改使其适用于离散或连续系统（§5）。然后根据线性算子理论（§7）发展了松弛方法的一般理论，并规定了近似过程收敛的充分条件（§8）。然后将这些一般方法应用于求解非齐次线性积分方程（§10）和非齐次线性微分方程（§11,12）。
During the last few years Southwell and his fellow-workers have developed a new method for the numerical solution of a very general type of problem in mathematical physics and engineering. The method was originally devised for the determination of stresses in frameworks, but it has proved to be directly applicable to any problem which is reducible to the solution of a system of non-homogeneous, linear, simultaneous algebraic equations in a finite number of unknown variables.* Southwell’s “ relaxation m ethod” is one of successive approximation and, in order to complete the previous investigations of this method, it is necessary to prove that the successive approximations do actually converge towards the exact solutions. This formal proof is given in § 4. Southwell’s “ relaxation” methods are not directly applicable to continuous systems, where the number of unknown variables is infinite, but it is shown here that simple extensions and modifications of the relaxation method render it suitable for application to either discrete or continuous systems (§ 5). The general theory of relaxation methods is then developed in terms of the theory of linear operators (§7) and sufficient conditions are prescribed for the convergence of the process of approximation (§ 8). These general methods are then applied to the solution of non-homogeneous, linear integral equations (§ 10) and to the solution of nonhomogeneous, linear differential equations (§§ 11, 12).