The numerical integration of ordinary differential equations
The numerical integration of ordinary differential equations
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DOI：
10.1090/s0025-5718-1967-0225494-5
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发表时间：
1967-05
影响因子：
2
通讯作者：
C. Gear
中科院分区：
文献类型：
--
作者：
C. Gear
用矩阵形式表示了求解初值问题的多步法。研究了这种方法在高阶方程中的应用，结果发现了一阶和高阶方程的新方法，认为高阶方程的直接方法具有速度和精度的优点，并给出了一些数值证据。应用于一阶方程的新方法是传统多步法的一个微小的推广，避免了Dahlquist [2]稳定性定理，即这些新步法是有序的，但仍是收敛的。引入的矩阵形式主义提供了一个简单的机制来检查Descloux [3]引入的方法的等价性。指出了新的一阶分步法、亚当斯分步法和Nordsieck [7]的带分量法是等价的。事实上，所有讨论的方法都可以放在等价类中，这样定理只需要对每个类的一个成员证明。一个类的成员之间的选择可以根据舍入误差和计算量。由于Nordsieck方法的速度和直接适用于高阶问题，因此，我们赞成将Nordsieck方法推广到一般情况。给出了保证收敛性的定理和误差的渐近形式。证据可以在一份被引用的报告中找到。引用
Multistep methods for initial value problems are expressed in a matrix form. The application of such methods to higher-order equations is studied with the result that new techniques for both first-and higher-order equations are found. The direct approach to higher-order equations is believed to offer speed and accuracy advantages; some numerical evidence is presented. The new technique applied to first-order equations is a slight extension of the conventional multistep method and avoids the Dahlquist [2] stability theorem, that is, these new-step methods are of orderand yet convergent. The matrix formalism introduced provides an easy mechanism for examining the equivalence of methods as introduced by Descloux [3]. It is pointed out that the new first-order method on-steps, Adams’ method on-steps and Nordsieck’s [7] method withcomponents are equivalent to each other. In fact, all methods discussed can be placed in equivalence classes so that theorems need only be proved for one member of each class. The choice between the members of a class can be made on the basis of round-off errors and amount of computation only. Arguments are given in favor of the extension of Nordsieck’s method for general use because of its speed and applicability to higher order problems directly. The theorems ensuring convergence and giving the asymptotic form of the error are stated. The proofs can be found in a cited report. References