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Weights of the exponential fitting multistep algorithms for first-order ODEs

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2001年第1期132卷 83-93页

作者： Ixaru, LG Rizea, M Vanden Berghe, G De Meyer, H Univ Ghent Dept Appl Math & Comp Sci B-9000 Ghent Belgium Inst Phys & Nucl Engn Dept Theoret Phys R-76900 Bucharest Romania Inst Phys & Nucl Engn Ctr Comp R-76900 Bucharest Romania

We describe a numerical method for the calculation of the weights of the linear multistep algorithms for solving first-order differential equations. The main novelties are that (i) we admit nonequidistant mesh points in the partition and (ii) the weights are determined on the basis of the exponential functions exp(lambda (i)x), i = 1,2, 3,... rather than on the power function set, as it is done for the classical weights. In this way the method allows computing not only the weights of the well-established algorithms but also those of new ones. Another novelty consists in the construction of a general scheme for the error analysis of this kind of algorithms. Some relevant numerical illustrations are given. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： exponential fitting multistep algorithms regularization error analysis

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Frequency evaluation in exponential fitting multistep algorithms for ODEs

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2002年第1-2期140卷 423-434页

作者： Ixaru, LG Vanden Berghe, G De Meyer, H State Univ Ghent Dept Appl Math & Comp Sci B-9000 Ghent Belgium Inst Phys & Nucl Engn Dept Theoret Phys R-76900 Bucharest Romania

We consider the linear multistep algorithms for first order ODEs and examine the problem of how the lambda-frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms. We find out that the key of the answer consists in analysing the behaviour of the error. On further investigating the simple case of two-step bdf algorithms we produce formulae for the optimal lambda's and show that, if the optimal lambda's are used, the order of the method is increased by one unit. The reported numerical illustrations suggest that further investigations along these lines deserve a real attention. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： exponential fitting multistep algorithms frequency evaluation error analysis

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Exponentially fitted variable two-step BDF algorithm for first order ODEs

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COMPUTER PHYSICS COMMUNICATIONS 2003年第2期150卷 116-128页

作者： Ixaru, LG Vanden Berghe, G De Meyer, H State Univ Ghent Dept Appl Math & Comp Sci B-9000 Ghent Belgium Inst Phys & Nucl Engn Dept Theoret Phys R-76900 Bucharest Romania

We construct the exponential fitting extension of the classical variable step-size two-step BDF algorithm and examine its properties. We give the expression of the optimal value of the associated frequency and introduce a procedure for choosing the step widths in terms of the required accuracy. The order of the new algorithm is three (that is by one unit higher than for the classical one). On three severe stiff test cases we show that the new two-step algorithm behaves much better than its classical companion and that it is as good as the classical variable three-step algorithm. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： exponential fitting variable steps multistep algorithms frequency evaluation error analysis

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A second-order recursive algorithm with applications to adaptive filtering and subspace tracking

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 1998年第6期46卷 1720-1725页

作者： Andersson, A Broman, H Ericsson Microwave Syst AB Molndal Sweden Chalmers Univ Technol Dept Appl Elect S-41296 Goteborg Sweden

A second-order recursive algorithm for adaptive signal processing is proposed, and a similar algorithm is derived for signal subspace tracking. It is shown that the algorithm encompasses both the RLS and the LMS algorithms as special cases. The computational complexity is the same as for the RLS algorithm, but some extra memory storage is required. The associated ordinary differential equation (ODE) for the autoregressive exogenous (ARX) case algorithm is proven to be globally exponentially stable. Furthermore, it is demonstrated that the proposed algorithm has a higher ability to track time-varying signals than has the RLS algorithm. The proposed algorithm especially handles those situations well where there is a simultaneous system change and decrease of signal power.

关键词： adaptive algorithms multistep algorithms parameter estimation recursive algorithms subspace tracking time-varying systems

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