Implicit Interval Methods for Solving the Initial Value Problem

Marciniak, Andrzej

doi:10.1023/b:numa.0000049471.81341.60

Cited by 21 publications

(29 citation statements)

References 12 publications

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“…As for the method (19), for interval equivalents of (25), it is important to write two coefficients ν * n+1 and ν * * n instead of one ν n+1 = ν * n+1 + ν * * n+1 (see Section 4 for details).…”

Section: Conventional Predictor-corrector Methodsmentioning

confidence: 99%

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On interval predictor-corrector methods

2016

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One can approximate numerically the solution of the initial value problem using single or multistep methods. Linear multistep methods are used very often, especially combinations of explicit and implicit methods. In floating-point arithmetic from an explicit method (a predictor), we can get the first approximation to the solution obtained from an implicit method (a corrector). We can do the same with interval multistep methods. Realizing such interval methods in floating-point interval arithmetic, we compute solutions in the form of intervals which contain all possible errors. In this paper, we propose interval predictor-corrector methods based on conventional Adams-Bashforth-Moulton and Nyström-Milne-Simpson methods. In numerical examples, these methods are compared with interval methods of RungeKutta type and methods based on high-order Taylor series. It appears that the presented methods yield comparable approximations to the solutions.

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Section: Conventional Predictor-corrector Methodsmentioning

confidence: 99%

“…An interval method for ordinary differential equations using interval arithmetic was described first by R. E. Moore in 1965 [32,33]. There are also interval methods based on explicit Runge-Kutta methods [21,28,41] and implicit ones [10,11,25,28,31]. In [41], Yu.…”

Section: Introductionmentioning

confidence: 99%

On interval predictor-corrector methods

2016

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“…The realization of proper interval arithmetic in floating-point arithmetic consists in using downwardly directed rounding when calculating the left endpoint of a resulting interval and upwardly directed rounding when calculating the right endpoint of such an interval (see e.g. [2] or [5]). The realization of directed interval arithmetic in floating-point arithmetic is not too easy, because in every elementary operation we must obtain the resulting interval which contains all possible roundings.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…But in our experiments we have applied our own unit called IntervalArithmetic (see e.g. [5]) written in the Delphi Pascal programming language. This unit takes advantage of the Delphi Pascal floating-point Extended type 1 and makes it possible to represent any input numerical data in the form of machine interval, perform all calculations in floating-point interval arithmetic, use some standard interval functions and give results in the form of proper intervals.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Interval Versions of Central-Difference Method for Solving the Poisson Equation in Proper and Directed Interval Arithmetic

Hoffmann¹,

Marciniak²,

Szyszka³

2013

Foundations of Computing and Decision Sciences

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To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h2 +k2), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained.

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“…The first one was described by R. E. Moore in 1965 [33][34][35]. There are also known interval methods based on high-order Taylor series (see, e.g., [2,3,5,15,21,39,41]), explicit and implicit Runge-Kutta methods [7,8,20,24,27,31,43], explicit and implicit multistep methods [17-19, 25-27, 30, 43]. The last ones concern interval methods based on conventional methods of Adams-Bashforth, Adams-Moulton, Nyström, and Milne-Simpson types.…”

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confidence: 99%

Interval versions of Milne’s multistep methods

Marciniak

Jankowska

2017

Numer Algor

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The paper presents explicit interval multistep methods of Milne type, which may be considered as alternative methods to other known explicit interval multistep methods (of Adams-Bashforth and Nyström). It is proved that enclosures of solutions (in the form of intervals) obtained by these methods contain the exact solutions of the initial value problem. Numerical examples show that the widths of intervals obtained by proposed methods are smaller than those obtained by explicit interval multistep methods known so far.

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Implicit Interval Methods for Solving the Initial Value Problem

Cited by 21 publications

References 12 publications

On interval predictor-corrector methods

On interval predictor-corrector methods

Interval Versions of Central-Difference Method for Solving the Poisson Equation in Proper and Directed Interval Arithmetic

Interval versions of Milne’s multistep methods

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