The Experts below are selected from a list of 13119 Experts worldwide ranked by ideXlab platform
Lloyd N. Trefethen - One of the best experts on this subject based on the ideXlab platform.
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The Exponentially Convergent Trapezoidal Rule
Siam Review, 2014Co-Authors: Lloyd N. Trefethen, J. A. C. WeidemanAbstract:It is well known that the Trapezoidal Rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
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A Trapezoidal Rule error bound unifying the Euler–Maclaurin formula and geometric convergence for periodic functions
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2014Co-Authors: Mohsin Javed, Lloyd N. TrefethenAbstract:The error in the Trapezoidal Rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the Trapezoidal Rule for periodic analytic functions in another. Similar results are also given for the midpoint Rule.
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a Trapezoidal Rule error bound unifying the euler maclaurin formula and geometric convergence for periodic functions
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2014Co-Authors: Mohsin Javed, Lloyd N. TrefethenAbstract:The error in the Trapezoidal Rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the Trapezoidal Rule for periodic analytic functions in another. Similar results are also given for the midpoint Rule.
Mohsin Javed - One of the best experts on this subject based on the ideXlab platform.
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A Trapezoidal Rule error bound unifying the Euler–Maclaurin formula and geometric convergence for periodic functions
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2014Co-Authors: Mohsin Javed, Lloyd N. TrefethenAbstract:The error in the Trapezoidal Rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the Trapezoidal Rule for periodic analytic functions in another. Similar results are also given for the midpoint Rule.
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a Trapezoidal Rule error bound unifying the euler maclaurin formula and geometric convergence for periodic functions
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2014Co-Authors: Mohsin Javed, Lloyd N. TrefethenAbstract:The error in the Trapezoidal Rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the Trapezoidal Rule for periodic analytic functions in another. Similar results are also given for the midpoint Rule.
Bum-il Hong - One of the best experts on this subject based on the ideXlab platform.
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on a study of error bounds of Trapezoidal Rule
Honam Mathematical Journal, 2014Co-Authors: Nahmwoo Hahm, Bum-il HongAbstract:In this paper, through a direct computation with subintervals partitioning [0, 1], we compute better a posteriori bounds for the average case error of the difference between the true value of with [0, 1] minus the composite Trapezoidal Rule and the composite Trapezoidal Rule minus the basic Trapezoidal Rule for by using zero mean-Gaussian.
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a study of average error bound of Trapezoidal Rule
Honam Mathematical Journal, 2008Co-Authors: Meehyea Yang, Bum-il HongAbstract:In this paper, to have a better a posteriori error bound of the average case error between the true value of I(f) and the Trapezoidal Rule on subintervals using zero mean-Gaussian, we prove that a new average error between the difference of the true value of I(f) from the composite Trapezoidal Rule and that of the composite Trapezoidal Rule from the simple Trapezoidal Rule is bounded by through direct computation of constants for r 2 under the assumption that we have subintervals (for simplicity equal length h) partitioning [0, 1].
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error bounds of Trapezoidal Rule on subintervals using distribution
Honam Mathematical Journal, 2007Co-Authors: Bum-il Hong, Nahmwoo HahmAbstract:We showed in [2] that if , then the average error between simple Trapezoidal Rule and the composite Trapezoidal Rule on two consecutive subintervals is proportional to using zero mean Gaussian distribution under the assumption that we have subintervals (for simplicity equal length) partitioning and that each subinterval has the length. In this paper, if , we show that zero mean Gaussian distribution of average error between simple Trapezoidal Rule and the composite Trapezoidal Rule on two consecutive subintervals is bounded by .
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an error bound of Trapezoidal Rule on subintervals using zero mean gaussian
The Kips Transactions:parta, 2005Co-Authors: Bum-il Hong, Nahmwoo Hahm, Meehyea YangAbstract:In this paper, we study the average case error of the Trapezoidal Rule using zero mean-Gaussian. Assume that we have n subintervals (for simplicity equal length) partitioning [0,1] and that each subinterval has the length h. Then, for , we show that the average error between simple Trapezoidal Rule and the composite Trapezoidal Rule on two consecutive subintervals is bounded by through direct computation of constants .
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ON AN ERROR OF Trapezoidal Rule
Communications of The Korean Mathematical Society, 1998Co-Authors: Bum-il Hong, Sung-hee Choi, Nahmwoo HahmAbstract:We show that if r 2, the average error of the Trapezoidal Rule is proportional to where n is the number of mesh points on the interval [D, 1]. As a result, we show that the Trapezoidal Rule with equally spaced points is optimal in the average case setting when r 2.
R. Knuppen - One of the best experts on this subject based on the ideXlab platform.
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Weighted serum pools in comparison to the Trapezoidal Rule for estimating AUCs for ethinyl estradiol. The relationship of the variance of the determination to the interindividual variance.
European Journal of Clinical Pharmacology, 1994Co-Authors: T. Louton, W. Kuhnz, L. Dibbelt, R. KnuppenAbstract:The concept of a weighted pool for estimating the area under the curve (AUC) is presented and set in relationship to the Trapezoidal Rule. An example from a pharmacokinetic study on ethinyl estradiol is used to demonstrate the use of variance component analysis for relating the intraindividual variance of the AUC, Trapezoidal Rule and weighted pool to the variance of the determination process. Depending on the sampling times, the theoretical variance of the weighted pool is greater than the theoretical variance of the Trapezoidal Rule. In the example presented, it was shown that this difference is of no importance in relation to the interindividual variance of the AUC, which dominates the total variance. In the example study, routine quality control samples were also determined in each assay, which allowed independent confirmation of the discussed results on the intraindividual variance of the AUCs.
Patrick Smolinski - One of the best experts on this subject based on the ideXlab platform.
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A multi-time step integration algorithm for structural dynamics based on the modified Trapezoidal Rule
Computer Methods in Applied Mechanics and Engineering, 2000Co-Authors: Patrick SmolinskiAbstract:An explicit multi-time step (subcycling) integration method is proposed for solving semi-discretized structural dynamics problems. Based on nodal partition theory, this method is derived from the modified Trapezoidal Rule method (MTM). To achieve numerical stability virtually fixed physical quantities, displacement and velocity, are assumed for the non-updated nodal groups. The energy method is used to analyze the stability of the new algorithm and the critical time step for each nodal group is determined in terms of the maximum frequency of the elements in the associated subdomain. Some example problems are evaluated to numerically examine the accuracy and stability of the algorithm.
