Two Formulas for Numerical Indefinite Integration

Haber, Seymour

doi:10.2307/2153166

Cited by 8 publications

(16 citation statements)

References 4 publications

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“…Remark 3.1. This paper addresses the indefinite integration formulas based on (3.1) developed by Haber [1]. Haber developed his formula for case 4, but did not develop any formula for cases 1-3.…”

Section: New Error Estimates For the Sinc Indefinite Integration Withmentioning

confidence: 99%

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Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

2013

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The Sinc quadrature and the Sinc indefinite integration are approximation formulas for definite integration and indefinite integration, respectively, which can be applied on any interval by using an appropriate variable transformation. Their convergence rates have been analyzed for typical cases including finite, semi-infinite, and infinite intervals. In addition, for verified automatic integration, more explicit error bounds that are computable have been recently given on a finite interval. In this paper, such explicit error bounds are given in the remaining cases on semi-infinite and infinite intervals.

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Section: New Error Estimates For the Sinc Indefinite Integration Withmentioning

confidence: 99%

“…Sinc indefinite integration is an approximation formula for the indefinite integral [1], expressed as…”

Section: Error Estimates With Explicit Constants For the Sinc Indefin...mentioning

confidence: 99%

Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

2013

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“…In this case, the second-order expansion matching prior π(α) cannot be evaluated in closed form, and numerical method, such as in [18], is needed.…”

Section: Implications and Examplesmentioning

confidence: 99%

Bayesian frequentist hybrid inference

Yuan¹

2009

Ann. Statist.

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Bayesian and frequentist methods differ in many aspects, but share some basic optimality properties. In practice, there are situations in which one of the methods is more preferred by some criteria. We consider the case of inference about a set of multiple parameters, which can be divided into two disjoint subsets. On one set, a frequentist method may be favored and on the other, the Bayesian. This motivates a joint estimation procedure in which some of the parameters are estimated Bayesian, and the rest by the maximum-likelihood estimator in the same parametric model, and thus keep the strengths of both the methods and avoid their weaknesses. Such a hybrid procedure gives us more flexibility in achieving overall inference advantages. We study the consistency and high-order asymptotic behavior of the proposed estimator, and illustrate its application. Also, the results imply a new method for constructing objective prior.Comment: Published in at http://dx.doi.org/10.1214/08-AOS649 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…This formula can be applied in the case of a finite interval (a, b), by combining it with a variable transformation that maps R onto (a, b). Haber [10] employed the SE transformation…”

Section: Sinc Indefinite Integrationmentioning

confidence: 99%

Theoretical analysis of a Sinc-Nyström method for Volterra integro-differential equations and its improvement

Okayama

2018

Applied Mathematics and Computation

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A Sinc-Nyström method for Volterra integro-differential equations was developed by Zarebnia in 2010. The method is quite efficient in the sense that exponential convergence can be obtained even if the given problem has endpoint singularity. However, its exponential convergence has not been proved theoretically. In addition, to implement the method, the regularity of the solution is required, although the solution is an unknown function in practice. This paper reinforces the method by presenting two theoretical results: 1) the regularity of the solution is analyzed, and 2) its convergence rate is rigorously analyzed. Moreover, this paper improves the method so that a much higher convergence rate can be attained, and theoretical results similar to those listed above are provided. Numerical comparisons are also provided.

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Two Formulas for Numerical Indefinite Integration

Cited by 8 publications

References 4 publications

Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration

Bayesian frequentist hybrid inference

Theoretical analysis of a Sinc-Nyström method for Volterra integro-differential equations and its improvement

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