Convergence Acceleration Algorithm via an Equation Related to the Lattice Boussinesq Equation

He, Yi; Hu, Xing−Biao; Sun, Jianqing; Weniger, Ernst Joachim

doi:10.1137/100808757

Cited by 25 publications

(30 citation statements)

References 31 publications

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“…By interchanging rows and columns in the p × p determinant D, it is always possible to assume that i 1 = j 1 = 1 and that i 2 = j 2 = p, which transforms the Jacobi's identity into the Sylvester's one. Thus, the determinantal identities proved in this paper, or in [27] and [29], can be obtained by any of these two identities.…”

Section: If We Setmentioning

confidence: 75%

“…Obviously, when m = 1, the algorithm (1), with the initial conditions (2), reduces to the ε-algorithm, and the transformation (3) is exactly the Shanks'transformation [28]. When m = 2, (1)-(2) is the algorithm obtained from the Boussinesq equation in [29]. The corresponding confluent form was given in [30], and it can be viewed as a continuous prediction algorithm for computing the limit of a given function when the variable is tending to infinity.…”

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confidence: 99%

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Confluent Form of the Multistep ɛ‐Algorithm, and the Relevant Integrable System

Brezinski¹,

et al. 2011

Stud Appl Math

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In this paper, the confluent form of the multistep ε-algorithm is proposed. The molecule solution of this system is derived by using determinantal identities. A new continuous prediction algorithm based on this confluent form is constructed. It also shows that this algorithm is a special case of the extended Lotka-Volterra system. IntroductionLet f be a function such that lim t→∞ f (t) = S. If f tends slowly to S, it can be transformed, by a function transformation, into a set of new functions {T k (t)} converging to S either when t tends to infinity for a fixed value of k, or when k tends to infinity and t ≥ T is fixed, and such that, under some proper assumptions, 1, 2, . . . , and/or limIf ρ = 0, we speak of convergence acceleration while, if 0 ≤ |ρ| < 1, we speak of convergence improvement. On these topics, consult [1, Chap. 5].Convergence acceleration or improvement could be difficult to achieve because it is an asymptotic property. However, a function transformation could

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Section: If We Setmentioning

confidence: 75%

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confidence: 99%

Confluent Form of the Multistep ɛ‐Algorithm, and the Relevant Integrable System

Brezinski¹,

et al. 2011

Stud Appl Math

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“…It can be implemented via the ε-algorithm of Wynn [45]. Recently, a new recursive algorithm for accelerating the convergence of sequences was derived by He, Hu, Sun and Weniger [14] from the lattice Boussinesq equation. This algorithm resembles to the ε-algorithm, and it was proved that the quantities it computes can be expressed as ratios of determinants, thus extending the Shanks' sequence transformation.…”

Section: The Scenerymentioning

confidence: 99%

Multistep $𝜖$–algorithm, Shanks’ transformation, and the Lotka–Volterra system by Hirota’s method

Brezinski¹,

et al. 2012

Math. Comp.

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In this paper, we give a multistep extension of the ε-algorithm of Wynn, and we show that it implements a multistep extension of the Shanks' sequence transformation which is defined by ratios of determinants. Reciprocally, the quantities defined in this transformation can be recursively computed by the multistep ε-algorithm. The multistep ε-algorithm and the multistep Shanks' transformation are related to an extended discrete Lotka-Volterra system. These results are obtained by using the Hirota's bilinear method, a procedure quite useful in the solution of nonlinear partial differential and difference equations.

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“…For example, Wynn's epsilon [101] and rho [102] algorithm can be viewed to be just fully discrete integrable systems [58,64]. New sequence transformations were also derived via this connection with dynamical systems [31,35,51,77].…”

Section: Introductionmentioning

confidence: 99%

Construction of new generalizations of Wynn’s epsilon and rho algorithm by solving finite difference equations in the transformation order

Chang

et al. 2019

Numer Algor

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We construct new sequence transformations based on Wynn's epsilon and rho algorithms. The recursions of the new algorithms include the recursions of Wynn's epsilon and rho algorithm and of Osada's generalized rho algorithm as special cases. We demonstrate the performance of our algorithms numerically by applying them to some linearly and logarithmically convergent sequences as well as some divergent series.

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Convergence Acceleration Algorithm via an Equation Related to the Lattice Boussinesq Equation

Cited by 25 publications

References 31 publications

Confluent Form of the Multistep ɛ‐Algorithm, and the Relevant Integrable System

Confluent Form of the Multistep ɛ‐Algorithm, and the Relevant Integrable System

Multistep $𝜖$–algorithm, Shanks’ transformation, and the Lotka–Volterra system by Hirota’s method

Construction of new generalizations of Wynn’s epsilon and rho algorithm by solving finite difference equations in the transformation order

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