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The equal coefficients quadrature rules and their numerical improvement

APPLIED MATHEMATICS AND COMPUTATION 2005年第2期171卷 1331-1351页

作者： Eslahchi, MR Dehghan, M Masjed-Jamei, M Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran Iran KN Toosi Univ Technol Dept Math Tehran Iran Minist Sci Res & Technol Sanjesh Org Ctr Res & Studies Tehran Iran

One of the quadrature rules is the "Equal coefficients quadrature rules" represented by integral(h)(a) w(x)f(x)dx similar or equal to C-''Sigma(n)(i=1)f(x(i)) a where C-n is a constant number and w(x) is a weight function on [a, b]. In this work, we show that the precisian degree of above formula can be increased by taking the upper and lower bounds of the integration formula as unknowns. This causes to numerically be extended the monomial space {1, x,..., x(n)} to {1, x,..., x(n+2).} We use a matrix proof to show that the resulting nonlinear system for the basis f(x) = x(j), j = 0,...,n + 2 has no analytic solution. Thus, we solve this system numerically to find unknowns x(1),x(2),x(n), C-n, a and b. Finally, some examples will be given to show the numerical superiority of the new developed method. (c) 2005 Elsevier Inc. All rights reserved.

关键词： equal coefficient quadrature rules numerical integration methods precision degree the method of undetermined coefficient the method of solving nonlinear system

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On numerical improvement of closed Newton-Cotes quadrature rules

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APPLIED MATHEMATICS AND COMPUTATION 2005年第2期165卷 251-260页

作者： Dehghan, M Masjed-Jamei, M Eslahchi, MR Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran 15914 Iran Minist Sci Res & Technol Sanjesh Org Ctr Res & Studies Tehran Iran

This paper discusses on numerical improvement of the Newton-Cotes integration rules, which are in forms of: integral(b=a+nh)(a) f(x) dx similar or equal to Sigma(n)(k=0) B-k((n)) integral(a + kh). It is known that the precision degree of above formula is n + 1 for even n's and is n for odd n's. However, if its bounds are considered as two additional variables (i.e. a and h in fact) we reach a nonlinear system that numerically improves the precision degree of above integration formula up to degree n + 2. In this way, some numerical examples are given to show the numerical superiority of our approach with respect to usual Newton-Cotes integration formulas. (c) 2004 Elsevier Inc. All rights reserved.

关键词： Newton-Cotes formula numerical integration methods degree of accuracy the method of undetermined coefficient the method of solving nonlinear systems

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The first kind Chebyshev-Lobatto quadrature rule and its numerical improvement

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APPLIED MATHEMATICS AND COMPUTATION 2005年第2期171卷 1104-1118页

作者： Masjed-Jamei, M Hashemiparast, SM Eslahchi, MR Dehghan, M Sanjesh Org Minist Sci Res & Technol Ctr Res & Studies Tehran Iran KN Toosi Univ Technol Dept Math Tehran Iran Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran Iran Management & Planning Org Tehran Iran

One of the integration methods is the first kind Chebyshev-Lobatto quadrature rule, denoted by integral(1)(-1) (f(x))(root 1-x2) dx similar or equal to (pi)(n+1) (r=1)Sigma(n)f (cos(((2k-1)pi)(2n))) + (pi)(2(n+1)) f(-1) + (pi)(2(n+1)) f(1). According to this rule, the precision degree of above formula is the highest, i.e. 2n + 1. Hence, it is not possible to increase the precision degree of Chebyshev-Lobatto integration formulas anymore, we will present a matrix proof for this matter. But, on the other hand, we claim that one can improve the above formula numerically. To do this, we consider the integral bounds as two new unknown variables. This causes to numerically be extended the monomial space f(x) = x(j) from j = 0, 1,...,2n + 1 to j = 0, 1,...,2n + 3 (two monomials more than the first kind Chebyshev-Lobatto integration method). In other words, we present an approximate formula as integral(b)(a) (f(x))(root 1-x2) dx similar or equal to (i=1)Sigma(n) w(i)f(x(i)) + nu(1)f(a) + nu(2)f(b), in which a, b and w(1),w(2),..,w(n) and x(1),x(2),...,x(n) are all unknowns and the formula is almost exact for the monomial basis f(x) = x(j), j = 0,1,..., 2n + 3. Several tests are finally given to show the excellent superiority of the proposed nodes and weights with respect to the usual first kind Chebyshev-Lobatto nodes and weights. Let us add that in this part we have also some wonderful 5-point formulas that are comparable with 100002-point formulas of the first kind Chebyshev Lobatto quadrature rules in average. (c) 2005 Published by Elsevier Inc.

关键词： first kind Chebyshev-Lobatto quadrature formula numerical integration methods precision degree the method of undetermined coefficient the method of solving nonlinear systems

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Stability preserving integration of index-2 DAEs

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APPLIED numerical MATHEMATICS 2003年第2-3期45卷 201-229页

作者： Higueras, I März, R Tischendorf, C Humboldt Univ Inst Math D-10099 Berlin Germany Univ Publ Navarra Dept Matemat & Informat Pamplona 31006 Spain

For linear index-2 DAEs with properly stated leading term we characterize contractive and dissipative flows. We study under which conditions the qualitative properties of the DAE solutions are reflected by the numerical approximations. This is the case if the discretization and the decoupling processes commute. Commutativity is achieved if two subspaces associated with the index-2 DAE are constant;in this case we say that the index-2 DAE is numerically qualified. If both subspaces are time dependent, the problem should be reformulated;in, order to avoid numerically equivalent reformulations, a criterion is given. If only one of these subspaces is constant, the problem is in some sense close to a numerically qualified one and thus depending on the problem context, no reformulations are needed. Contractive and dissipative flows induced by the DAE are characterized and results on qualitative properties of the numerical solution are given. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.

关键词： differential algebraic equations numerical integration methods global stability BDF Runge-Kutta

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Frequency warping in time-domain circuit simulation

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-FUNDAMENTAL THEORY AND APPLICATIONS 2003年第7期50卷 904-913页

作者： Brambilla, A Storti-Gajani, G Politecn Milan Dipartimento Elettr & Informaz I-20133 Milan Italy

Time-domain simulation of dynamic circuits and, in general, of any physical model characterized by ordinary differential equations or differential algebraic equations, implies the use of so me numerical integration method to find an approximate solution in a discrete set of time points. Among these methods,,the class known as linear multistep includes many well-known formulas such as the backward Euler method, the trapezoid method, and the implicit backward differentiation formulas used in most circuit simulators. All these methods introduce a very subtle effect that-is, here called the warping error. As shown, it is equivalent to a perturbation of the eigenvalues of the linearized ordinary differential problem. The perturbation introduced depends on the integration time step;it is often very small and in most cases irrelevant or even not noticeable. Nevertheless an exception to this situation is found when simulating high-quality factor circuits where even very small warping errors can lead to qualitatively wrong solutions. In this paper, we demonstrate that higher order linear multistep methods, while characterized by weaker stability properties, introduce less of a warping error and are well suited to the simulation of high-quality factor circuits.

关键词： frequency warping linear multistep methods numerical integration methods

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Analysis of numerical methods for the simulation of deformable models

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VISUAL COMPUTER 2003年第7-8期19卷 581-600页

作者： Hauth, M Etzmuss, O Strasser, W Univ Tubingen WSI GRIS Tubingen Germany

Simulating deformable objects based on physical laws has become the most popular technique for modeling textiles, skin, or volumetric soft objects like human tissue. The physical model leads to an ordinary differential equation. Recently, several approaches to fast algorithms have been proposed. In this work, more profound numerical background about numerical stiffness is provided. Stiff equations impose stability restrictions on a numerical integrator. Some one-step and multistep methods with adequate stability properties are presented. For an efficient implementation, the inexact Newton method is discussed. Applications to 2D and 3D elasticity problems show that the discussed methods are faster and give higher-quality solutions than the commonly used linearized Euler method.

关键词： deformable objects cloth modeling numerical integration methods differential equations

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Stability preserving integration of index-1 DAEs

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APPLIED numerical MATHEMATICS 2003年第2-3期45卷 175-200页

作者： Higueras, I März, R Tischendorf, C Humboldt Univ Math Inst D-10099 Berlin Germany Univ Publ Navarra Dept Matemat & Informat Pamplona 31006 Spain

For index-1 DAEs with properly stated leading term, we characterize dissipative and contractive flows and study how the qualitative properties of the DAE solutions are reflected by numerical approximations. The best situation occurs when the discretization and the decoupling procedure commute. It turns out that this is the case if the relevant part of the inherent regular ODE has a constant state space. Different kinds of reformulations are studied to obtain this property. Those reformulations might be expensive, hence, in order to avoid them, criteria ensuring the given DAE to be numerically equivalent to a numerically qualified representation are proved. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.

关键词： differential algebraic equations numerical integration methods global stability

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Differential algebraic systems anew

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APPLIED numerical MATHEMATICS 2002年第1-3期42卷 315-335页

作者： März, R Humboldt Univ Math Inst D-10099 Berlin Germany

It is proposed to figure out the leading term in differential algebraic systems more precisely. Low index linear systems with those properly stated leading terms are considered in detail. In particular, it is asked whether a numerical integration method applied to the original system reaches the inherent regular ODE without conservation, i.e., whether the discretization and the decoupling commute in some sense. In general one cannot expect this commutativity so that additional difficulties like strong stepsize restrictions may arise. Moreover, abstract differential algebraic equations in infinite-dimensional Hilbert spaces are introduced, and the index notion is generalized to those equations. In particular, partial differential algebraic equations are considered in this abstract formulation. (C) 2001 IMACS. Published by Elsevier Science B.V. All rights reserved.

关键词： differential algebraic equations numerical integration methods abstract differential algebraic equations partial differential algebraic equations

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The phase plane:: A tool to solve the transient stability problem

The phase plane:: A tool to solve the transient stability pr...

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Large Engineering Systems Conference on Power Engineering (LESCOPE 02)

作者： Aromataris, L Rodríguez, G Preidikman, S Univ Nacl Rio Cuarto Fac Ingn Grp Anal Sistemas Elect Potencia GASEP RA-5800 Rio Cuarto Cordoba Argentina

ISBN: (纸本)0780375203

One of the most important problems in the study of transient stability of power systems is the determination of perturbation's maximum time of permanence without losing the synchronism of the generators that feed the network. The problem is generally solved by either the application of the equal-area criterion or through numerical integration methods. In the present work, the phase-plane is proposed as an alternative tool to solve the above-mentioned problem with greater efficiency.

关键词： transient stability swing equation phase plane equal-area criterion numerical integration methods

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Differential algebraic systems anew

Differential algebraic systems anew

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9th Seminar on numerical Solution of Differential and Differential - Algebraic Equations (NUMDIFF-9)

作者： März, R Humboldt Univ Math Inst D-10099 Berlin Germany

关键词： differential algebraic equations numerical integration methods abstract differential algebraic equations partial differential algebraic equations

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