A new method to generate non-autonomous discrete integrable systems via convergence acceleration algorithms

2015 ◽  
Vol 27 (2) ◽  
pp. 194-212 ◽  
Author(s):  
YI HE ◽  
XING-BIAO HU ◽  
HON-WAH TAM ◽  
YING-NAN ZHANG
Keyword(s):  
Integrable Systems ◽  
Numerical Experiments ◽  
Boussinesq Equation ◽  
Volterra Equation ◽  
Algebraic Method ◽  
Convergence Acceleration ◽  
New Method ◽  
Discrete Integrable Systems ◽  
Convergence Acceleration Algorithm ◽  
Autonomous Version

In this paper, we propose a new algebraic method to construct non-autonomous discrete integrable systems. The method starts from constructing generalizations of convergence acceleration algorithms related to discrete integrable systems. Then the non-autonomous version of the corresponding integrable systems are derived. The molecule solutions of the systems are also obtained. As an example of the application of the method, we propose a generalization of the multistep ϵ-algorithm, and then derive a non-autonomous discrete extended Lotka–Volterra equation. Since the convergence acceleration algorithm from the lattice Boussinesq equation is just a particular case of the multistep ϵ-algorithm, we have therefore arrived at a generalization of this algorithm. Finally, numerical experiments on the new algorithm are presented.

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

2011 ◽  
Vol 33 (3) ◽  
pp. 1234-1245 ◽  
Author(s):  
Yi He ◽  
Xing-Biao Hu ◽  
Jian-Qing Sun ◽  
Ernst Joachim Weniger
Keyword(s):  
Boussinesq Equation ◽  
Convergence Acceleration ◽  
Convergence Acceleration Algorithm

scholarly journals ZN graded discrete Lax pairs and Yang–Baxter maps

2017 ◽  
Vol 473 (2201) ◽  
pp. 20160946 ◽  
Author(s):  
Allan P. Fordy ◽  
Pavlos Xenitidis
Keyword(s):  
Integrable Systems ◽  
Boussinesq Equation ◽  
Lax Pairs ◽  
Discrete Integrable Systems ◽  
New Family ◽  
Modified Boussinesq Equation ◽  
Lower Dimensional

We recently introduced a class of Z N graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In this paper, we introduce the corresponding Yang–Baxter maps. Many well-known examples belong to this scheme for N =2, so, for N ≥3, our systems may be regarded as generalizations of these. In particular, for each N we introduce a class of multi-component Yang–Baxter maps, which include H B III (of Papageorgiou et al. 2010 SIGMA 6, 003 (9 p). (doi:10.3842/SIGMA.2010.033)), when N =2, and that associated with the discrete modified Boussinesq equation, for N =3. For N ≥5 we introduce a new family of Yang–Baxter maps, which have no lower dimensional analogue. We also present new multi-component versions of the Yang–Baxter maps F IV and F V (given in the classification of Adler et al. 2004 Commun. Anal. Geom. 12, 967–1007. (doi:10.4310/CAG.2004.v12.n5.a1)).

scholarly journals Manin Involutions for Elliptic Pencils and Discrete Integrable Systems

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Matteo Petrera ◽  
Yuri B. Suris ◽  
Kangning Wei ◽  
René Zander
Keyword(s):  
Integrable Systems ◽  
Geometric Description ◽  
Discrete Integrable Systems ◽  
Birational Maps ◽  
Base Points ◽  
Special Geometry ◽  
Low Degree ◽  
Geometric Study

AbstractWe contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.

scholarly journals Discrete integrable systems generated by Hermite-Padé approximants

Nonlinearity ◽  
2016 ◽  
Vol 29 (5) ◽  
pp. 1487-1506 ◽  
Author(s):  
Alexander I Aptekarev ◽  
Maxim Derevyagin ◽  
Walter Van Assche
Keyword(s):  
Integrable Systems ◽  
Padé Approximants ◽  
Discrete Integrable Systems ◽  
Pade Approximants

scholarly journals On a Fast Convergence of the Rational-Trigonometric-Polynomial Interpolation

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Arnak Poghosyan
Keyword(s):  
Trigonometric Polynomial ◽  
Numerical Experiments ◽  
Polynomial Interpolation ◽  
Convergence Acceleration ◽  
Fast Convergence ◽  
Exact Constants

We consider the convergence acceleration of the Krylov-Lanczos interpolation by rational correction functions and investigate convergence of the resultant parametric rational-trigonometric-polynomial interpolation. Exact constants of asymptotic errors are obtained in the regions away from discontinuities, and fast convergence of the rational-trigonometric-polynomial interpolation compared to the Krylov-Lanczos interpolation is observed. Results of numerical experiments confirm theoretical estimates and show how the parameters of the interpolations can be determined in practice.

scholarly journals A Simpler GMRES Method for Oscillatory Integrals with Irregular Oscillations

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Qinghua Wu ◽  
Meiying Xiang
Keyword(s):  
Theoretical Analysis ◽  
Matrix Factorization ◽  
Numerical Experiments ◽  
Oscillatory Integral ◽  
Oscillatory Integrals ◽  
New Method ◽  
Gmres Method ◽  
Hessenberg Matrix

A simpler GMRES method for computing oscillatory integral is presented. Theoretical analysis shows that this method is mathematically equivalent to the GMRES method proposed by Olver (2009). Moreover, the simpler GMRES does not require upper Hessenberg matrix factorization, which leads to much simpler program and requires less work. Numerical experiments are conducted to illustrate the performance of the new method and show that in some cases the simpler GMRES method could achieve higher accuracy than GMRES.

Three semi-discrete integrable systems related to orthogonal polynomials and their generalized determinant solutions

Nonlinearity ◽  
2015 ◽  
Vol 28 (7) ◽  
pp. 2279-2306 ◽  
Author(s):  
Xiao-Min Chen ◽  
Xiang-Ke Chang ◽  
Jian-Qing Sun ◽  
Xing-Biao Hu ◽  
Yeong-Nan Yeh
Keyword(s):  
Orthogonal Polynomials ◽  
Integrable Systems ◽  
Discrete Integrable Systems

A Simple Method for Inverse Kinematic Analysis of the General 6R Serial Robot

2006 ◽  
Author(s):  
Zhi Xin Shi ◽  
Yu Feng Luo ◽  
Lu Bing Hang ◽  
Ting Li Yang
Keyword(s):  
Kinematic Analysis ◽  
Inverse Kinematics ◽  
Algebraic Method ◽  
Inverse Kinematic ◽  
New Method ◽  
One Dimension ◽  
Parallel Mechanisms ◽  
Simple Method ◽  
Inverse Kinematics Problem ◽  
Serial Robot

Because the solution to inverse kinematics problem of the general 5R serial robot is unique and its assembly condition has been derived, a simple effective method for inverse kinematics problem of general 6R serial robot or forward kinematics problem of general 7R single-loop mechanism is presented based on one-dimension searching algorithm. The new method has the following features: (1) Using one-dimension searching algorithm, all the real inverse kinematic solutions are obtained and it has higher computing efficiency; (2) Compared with algebraic method, it has evidently reduced the difficulty of deducing formulas. The principle of the new method can be generalized to kinematic analysis of parallel mechanisms.

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