Some methods of integrating ordinary differential equations on a multiprocessor computer

Молчанов, И. Н.; Yakovlev, M. F.; Geets, E. G.

doi:10.1007/bf01068753

Cybern Syst Anal

1984

DOI: 10.1007/bf01068753

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Some methods of integrating ordinary differential equations on a multiprocessor computer

И. Н. Молчанов¹,

M. F. Yakovlev²,

E. G. Geets³

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1996

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“…Various aspects of time complexity of numerical integration of ODE have been investigated in [3][4][5][6] and elsewhere. The notion of e-complexity of the problem [7] plays an important role in comparing the speed of algorithms that compute the e-solution of the problem.…”

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Computational complexity of integrating ordinary differential equations

Besarab¹

1996

Cybern Syst Anal

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1. The Cauchy problem for ordinary differential equations (ODE) is widely used as a mathematical model in various fields of research. In some cases, numerical solution of problems may involve a very large volume of computations, and it is therefore relevant to consider the issue of obtaining a numerical solution in acceptable time. In other cases, the problem must be solved in real time, which imposes rigid constraints on the machine time available for its solution.Construction of a numerical solution of the Cauchy problem with specified quality characteristics (accuracy of numerical solution, computer time and memory requirements to obtain an approximate solution) is therefore a highly topical issue [1,2].Various aspects of time complexity of numerical integration of ODE have been investigated in [3-6] and elsewhere. The notion of e-complexity of the problem [7] plays an important role in comparing the speed of algorithms that compute the e-solution of the problem. Here e-complexity is the exact lower bound of time complexity on the set of all methods for computing the solution with accuracy e (given a computational model and the input data specification).The construction of exact lower bounds and corresponding algorithms is a very difficult undertaking, and we therefore often consider approximate forms of the problem to obtain time complexity bounds and examine the possibility of lowering the upper complexity bounds.In this paper, we consider time bounds for algorithms that compute an e-solution by traditional numerical methods, and also by methods that implement a special form of input data and use parallelized computation.2. Consider the Cauchy problem X(O=/(t,X), X(O)=X o, t~ I0, t~where X, fare m-dimensional vectors, f is a sufficiently smooth function of the variables t, X, 0 < t o < oo. Assume that for this problem we have generated by some numerical stepwise method of order p [2] the table x,,,tr l-1 ..... ~,.wheret i E ,x,A:0 < t 1 < t 2... < t N=t~ Table (2) is called a numerical e-solution of problem (1) ifwhere p is some measure of closeness of the exact and the numerical solutions, 9 is a given positive number.The value e i = X(ti) --X i is called the accumulated computational error, and its components ~i " X(t~ -X/, "a == X~ -X t (t i ,= c t + "a)are the method error (et, i) and the rounding error (eri). Here {Xi*} is a table of the form (2) corresponding to exact execution of arithmetic operations (without rounding).

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confidence: 99%

Computational complexity of integrating ordinary differential equations

Besarab¹

1996

Cybern Syst Anal

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Some methods of integrating ordinary differential equations on a multiprocessor computer

Cited by 1 publication

References 4 publications

Computational complexity of integrating ordinary differential equations

Computational complexity of integrating ordinary differential equations

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