STABILITY OF GENERAL LINEAR METHODS FOR SYSTEMS OF FUNCTIONAL-DIFFERENTIAL AND FUNCTIONAL EQUATIONS

This paper is concerned with the numerical solution of functional-differential and func-tional equations which include functional-differential equations of neutral type as special cases. The adaptation of general linear methods is considered. It is proved that A-stable general linear methods can inherit the asymptotic stability of underlying linear systems.Some general results of numerical stability are also given.     

一般线性方法的正则性

１．引言数值方法的动力特征近年引起了人们的广泛关注。其中之一就是系统的平衡态和数值方法的平衡态相一致的问题,即用一个数值方法沿定步长求解系统时,是否会出现伪平衡。不可能出现伪平衡的方法称为是正则的。ＲＫ方法和线性多步法的正则性已被众多的文献研究［２,３,４］,其它方法的正则性显然是一亟待研究的问题。本文讨论较ＲＫ方法和线性多步法远为广泛的一般线性方法的正则性。设ｆ：Ｒ~（Ｎ）→Ｒ~（Ｎ）是一充分光滑的映射,考虑求解初值问题：的一般线性方法［１］：其中步长逼近于逼近于关于微分方程真解ｙ（ｔ）在第ｎ层…     

HILBERT空间中散逸动力系统一般线性方法的散逸稳定性

１．引言 １９９４年,Ｓｔｕａｒｔ与 Ｈｕｍｐｈｒｉｅｓ［４,５］首先考察了用 Ｒｕｎｇｅ－Ｋｕｔｔａ方法求解 Ｒｍ中的散逸动力系统（２．１）－（２．２）时数值解是否继承真解具有的散逸稳定性,并表明代数稳定且不可约的 Ｒｕｎｇｅ－Ｋｕｔｔａ方法是散逸稳定的且有一有界吸引集．１９９６年,本文作者［１］把这一工作推广到了两类特殊的一般线性方法．１９９７年,Ｈｉｌｌ在［３］中证明了Ａ－稳定是单支方法散逸稳定的充要条件,在［２］中又把文［４,５］的工作推广到了 Ｈｉｌｂｅｒｔ空间中的散逸动力系统（２．１）－（…     

B-CONVERGENCE PROPERTIES OF GENERAL LINEAR METHODS

In this paper, for general linear methods applied to strictly dissipative initial value problem in Hilbert spaces, we prove that algebraic stability implies B-convergence, which extends and improves the existing results on Runge-Kutta methods. Specializing our results for the case of multi-step Runge-Kutta methods, a series of B-convergence results are obtained.     

一般线性方法的线性与非线性稳定性间的关系

１．引言１９６３年,Ｄａｈｌｑｕｉｓｔ以一类线性问题为模型提出了Ａ－稳定性概念,此后有关如何判断方法是否Ａ－稳定或确定其稳定域的研究十分活跃,袁兆鼎等~［１］中对此有详细的讨论．Ｂｕｒｒａｇｅ与Ｂｕｔｃｈｅｒ［２］以一类非线性问题为模型,就一般线性方法引入了代数稳定性概念．Ｂｕｔｃｈｅｒ［４］探讨了代数稳定性与Ａ－稳定性间的内在联系．为确保代数稳定性蕴涵Ａ－稳定性,Ｂｕｔｃｈｅｒ［５］进一步要求代数稳定性定义中涉及的矩阵Ｇ是正定的。然而这样一来,正如李寿佛［６］中指出的那样,许多ＡＮ稳定且按［４…     

二次损失下带线性等式约束的线性模型中的线性预测的可容许性

本文针对带线性等式约束的线性模型 ,在二次损失下研究了线性预测的可容许性 ,得到了条件线性可预测变量的线性预测 Lys(Lys+ a)是可容许线性预测的充要条件。     

二次损失下线性预测的可容许性

本文在二次损失下研究了任意秩有限总体中线性预测的可容性，得到了线性预测Lys(Lys a)是可容许线性预测的充要条件．     

USING THE SKEW-SYMMETRIC ITERATIVE METHODS FOR SOLUTION OF AN INDEFINITE NONSYMMETRIC LINEAR SYSTEMS

The concept of the field of value to localize the spectrum of the iteration matrices of the skew-symmetric iterative methods is further exploited. Obtained formulas are derived to relate the fields of values of the original matrix and the iteration matrix. This allows us to determine theoretically that indefinite nonsymmetric linear systems can be solved by this class of iterative methods.     

Lyapunov不等式的最佳解与一般线性方法的可行性

１引言在刚性常微分方程初值问题数值解法理论中,常要求数值方法中的某个系数矩阵Ｂ具有性质：使Ｌｙａｐｕｎｏｖ不等式有对角正定解Ｄ存在,即存在对角正定矩阵Ｄ,使ＤＢ＋ＢＴＤ为正定矩阵·在许多情形,这种解Ｄ是存在的,例如Ｇ－Ｌ-、ＩＡ-和ⅡＡ－ＲＫ方法［３］．然而也存在一些方法,它是Ａ－稳定、Ｌ－稳定,甚至是代数稳定的,但这种Ｄ却不存在．例如ⅢＣ（ｓ≥３）ＲＫ方法［３］和当θ＝１时的块θ－方法［５］（至少对ｒ＝２,３,４是如此）．前者是Ｌ－稳定的,也是代数稳定的,后者也是Ｌ－稳定的．这意味着,可能有不少…     

ORDER PROPERTIES AND CONSTRUCTION OF SYMPLECTIC RUNGE-KUTTA METHODS

1. IntroductionFOr a given s stage Runge-Kutta methodwith A = [ail], p = [pl, PZt... 5 P.]T and ac = [afl, ry23... ) %]T / 0, we introduce thefollowing simplifying conditions as in Butcher [1]and make the notational convensionwhere 1 5 m? pi(x), i ~ 1, 2, 3,' ? are arbitrarily given i--th polynomials with the property that pi(0) = 0,Note that B(P), C(P) and D(P) are equivalent to BI,. = 0, CI,P = 0 and DI,. = 0respectively. We shall always denote BI,., CI,., DI,. and VI,. by B, …     

ORDER RESULTS OF GENERAL LINEAR METHODS FOR MULTIPLY STIFF SINGULAR PERTURBATION PROBLEMS

In this paper we analyze the error behavior of general linear methods applied to some classes of one-parameter multiply stiff singularly perturbed problems. We obtain the global error estimate of algebraically and diagonally stable general linear methods. The main result of this paper can be viewed as an extension of that obtained by Xiao [13] for the case of Runge-Kutta methods.     

BACKWARD ERROR ANALYSIS OF SYMPLECTIC INTEGRATORS FOR LINEAR SEPARABLE HAMILTONIAN SYSTEMS

Symplecticness, stability, and asymptotic properties of Runge-Kutta, partitioned Runge-Kutta, and Runge-Kutta-Nystrom methods applied to the simple Hamiltonian system p = -vg, q = kp are studied. Some new results in connection with P-stability are presented. The main part is focused on backward error analysis. The numerical solution produced by a symplectic method with an appropriate stepsize is the exact solution of a perturbed Hamiltonian system at discrete points. This system is studied in detail and new results are derived. Numerical examples are presented.     

Volterra泛函微分方程一般线性方法的稳定性

本文研究Volterra泛函微分方程(k,p,q)-代数稳定的一般线性方法的稳定性,获得了该类方法的一系列新的稳定性结果.     

非线性中立型延迟积分微分方程一般线性方法的稳定性分析

本文研究求解R(α,β₁,β₂,γ)类非线性中立型延迟积分微分方程的一般线性方法的数值稳定性,获得了代数稳定的一般线性方法稳定及渐近稳定的条件,最后的数值试验验证了所获理论的正确性.       

Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints

We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with holonomic constraints. The methods are symplectic for any fixed scalar parameter $\alpha$, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when $\alpha=0$. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the $\alpha$-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values $(p_{0}, q_0)$ and small step size $h>0$, it is proved that there exists $\alpha^*=\alpha(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving $\alpha$-PRK methods. These $\alpha$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.     

二次损失下一般Gauss-Markov模型中可估函数的线性Minimax估计

设Y是具有均值Xβ和协方差阵σ~2V的n维随机向量,Sβ是线性可估函数,这里X,S和V≥0是已知矩阵,β∈R~p和σ~2>0是未知参数。本文在二次损失下研究了线性估计的Minimax性。在适当的假设下,得到了Sβ的唯一线性Minimax估计(有关唯一性在几乎处处意义下理解)     

广义线性模型的充分性及变换观察值的影响

线性充分性概念由Baksalary and Kala^[1],Drygas^[2,3]和Mueller^[4,5],引广言语线性模型E（Y）＝Xβ,Cov(Y)=V≥0,本文拓广了这一概念,提出关于任一给定可估函数c′β的线性充分性概念,得到了线性充分性理论的一些进一步结果。本文还分析了观测值的一般线性变换对广义线性模型的影响,得到了由变换给可估函数的BLUE和BLUE之方差变化的一般结果。     

BLOCK BIDIAGONALIZATION METHODS FOR MULTIPLE NONSYMMETRIC LINEAR SYSTEMS

1　IntroductionMany applications require the solution of large nonsymmetric linear equations withmultiple right-hand sidesAX =B ( 1 )where A is a real nonsymmetric matrix ofordern,and X=[x(1 ) ,… ,x(s) ] and B=[b(1 ) ,… ,b(s) ] are rectangular matrices of dimension n×s with s n.Most of iterative methods forthe solution of nonsymmetric linear systems with a single right-hand side may be used tosolve( 1 ) by solving the s linear systems individually.But iterative methods,such asKrylov subs…     

一类线性抛物型方程组的交替方向多步法及其理论分析

Efficient multistep procedure for time-stepping Galerkin method in which we use an alternating direction preconditioned iterative methods for approximately solving the linear equations arising at each timestep in a discrete Galerkin method for a class of linear parabolic systems is derived and analyzed. The optimal order error estimate is obtained. Numerical experiments show that the method has the characteristics of high efficiency and high accuracy.     

二阶非齐次线性微分方程解的平方可积性与有界性

本文借助于辅助函数，得到了二阶非齐次线性微分方程所有解均平方可积及所有解都有界的判定准则．