Efficient and stable generation of higher-order pseudospectral integration matrices

Tang, Xiaojun

doi:10.1016/j.amc.2015.03.090

Cited by 6 publications

(3 citation statements)

References 8 publications

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“…It can be easily seen that conventional pseudospectral integration matrices [56][57][58] in the sense of integer order are recovered from FPIMs when γ ∈ N. Therefore, relevant results of this work are extended naturally to integer differential, integral, and integral-differential equations.…”

Section: Computation Of Fpims Via Explicit Expressionsmentioning

confidence: 95%

Fractional pseudospectral integration matrices for solving fractional differential, integral, and integro-differential equations

Tang

2016

Communications in Nonlinear Science and Numerical Simulation

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Section: Computation Of Fpims Via Explicit Expressionsmentioning

confidence: 95%

Fractional pseudospectral integration matrices for solving fractional differential, integral, and integro-differential equations

Tang

2016

Communications in Nonlinear Science and Numerical Simulation

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“…In the simplest case when the vertices are independent and uniformly distributed on the circle, it is known that the semiperimeter S n and area A n of such random inscribed polygons, and the semiperimeter (or area) S ′ n of the random circumscribing polygons all converge to π w.p.1 as n → ∞ and their distributions are also asymptotically Gaussian [4,24]. Furthermore, by using extrapolation techniques [12,15] originating exactly from the famous Archimedean approximations of π based on regular polygons [3,13,19], it has been shown [22,23,25] that simple weighted averages such as 4 3 S n − 1 3 A n , 2 3 S n + 1 3 S ′ n and 16 15 S n − 1 5 A n + 2 15 S ′ n , etc., provide much more accurate approximations of π, and at the same time also satisfy similar central limit theorems as n → ∞. We note that extrapolation methods are useful in many important applications such as numerical evaluation of integrals, numerical solution of differential equations, and polynomial interpolations, etc.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear random extrapolation estimates of \(\pi\) under Dirichlet distributions

Wang,

Li,

2023

J. Numer. Anal. Approx. Theory

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We construct optimal nonlinear extrapolation estimates of \(\pi\) based on random cyclic polygons generated from symmetric Dirichlet distributions. While the semiperimeter \( S_n \) and the area \( A_n \) of such random inscribed polygons and the semiperimeter (and area) \( S_n' \) of the corresponding random circumscribing polygons are known to converge to \( \pi \) w.p.\(1\) and their distributions are also asymptotically normal as \( n \to \infty \), we study in this paper nonlinear extrapolations of the forms \( \mathcal{W}_n = S_n^{\alpha} A_n^{\beta} S_n'^{\, \gamma} \) and \( \mathcal{W}_n (p) = ( \alpha S_n^p + \beta A_n^p + \gamma S_n'^{\, p} )^{1/p} \) where \( \alpha + \beta + \gamma = 1 \) and \( p \neq 0 \). By deriving probabilistic asymptotic expansions with carefully controlled error estimates, we show that \( \mathcal{W}_n \) and \( \mathcal{W}_n (p) \) also converge to \( \pi \) w.p.\(1\) and are asymptotically normal. Furthermore, to minimize the approximation error associated with \( \mathcal{W}_n \) and \( \mathcal{W}_n (p) \), the parameters must satisfy the optimality condition \( \alpha + 4 \beta - 2 \gamma = 0 \). Our results generalize previous work on nonlinear extrapolations of \( \pi \) which employ inscribed polygons only and the vertices are also assumed to be independently and uniformly distributed on the unit circle.

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“…, Elgindy and Smith-Miles (2013b), Françolin, Benson, Hager, and Rao (2014), Elgindy and Smith-Miles (2013c), Tang (2015), Coutsias, Hagstrom, and Torres (1996), , Viswanath (2015), Driscoll (2010), Olver and Townsend (2013), El-Gendi (1969)]. Perhaps one of the reasons that laid the foundation of this methodology appears in the well stability and boundedness of numerical integral operators in general whereas numerical differential operators are inherently ill-conditioned; cf.…”

Section: Introductionmentioning

confidence: 99%

High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures

Elgindy

2017

Applied Numerical Mathematics

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SUMMARYThe work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the wellconditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by [Elgindy and Smith-Miles (2013a)]. The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.

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Efficient and stable generation of higher-order pseudospectral integration matrices

Cited by 6 publications

References 8 publications

Fractional pseudospectral integration matrices for solving fractional differential, integral, and integro-differential equations

Fractional pseudospectral integration matrices for solving fractional differential, integral, and integro-differential equations

Nonlinear random extrapolation estimates of \(\pi\) under Dirichlet distributions

High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures

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