Though the Butterfly Bptimization Algorithm(BOA)has already proved its effectiveness as a robust optimization algorithm,it has certain disadvantages.So,a new variant of BOA,namely mLBOA,is proposed here to improve its...Though the Butterfly Bptimization Algorithm(BOA)has already proved its effectiveness as a robust optimization algorithm,it has certain disadvantages.So,a new variant of BOA,namely mLBOA,is proposed here to improve its performance.The proposed algorithm employs a self-adaptive parameter setting,Lagrange interpolation formula,and a new local search strategy embedded with Levy flight search to enhance its searching ability to make a better trade-off between exploration and exploitation.Also,the fragrance generation scheme of BOA is modified,which leads for exploring the domain effectively for better searching.To evaluate the performance,it has been applied to solve the IEEE CEC 2017 benchmark suite.The results have been compared to that of six state-of-the-art algorithms and five BOA variants.Moreover,various statistical tests,such as the Friedman rank test,Wilcoxon rank test,convergence analysis,and complexity analysis,have been conducted to justify the rank,significance,and complexity of the proposed mLBOA.Finally,the mLBOA has been applied to solve three real-world engineering design problems.From all the analyses,it has been found that the proposed mLBOA is a competitive algorithm compared to other popular state-of-the-art algorithms and BOA variants.展开更多
S.M.Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [-1,1]. In 2000, M. Rever generalized S.M.Lozinskii's result to |x|α(0 <≤...S.M.Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [-1,1]. In 2000, M. Rever generalized S.M.Lozinskii's result to |x|α(0 <≤ α≤ 1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|α1(1 < α < 2)..展开更多
It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to \x\ at e-qually spaced nodes in [-1.1] diverges everywhere. except at zero and the end-points. In this paper we show tha...It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to \x\ at e-qually spaced nodes in [-1.1] diverges everywhere. except at zero and the end-points. In this paper we show that the sequence of Lagrange interpolation polynomials corresponding to the functions which possess better smoothness on equidistant nodes in [-1.1] still diverges every -where in the interval except at zero and the end-points.展开更多
It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to ｜x｜ at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, t...It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to ｜x｜ at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, toe prove that the sequence of Lagrange interpolation polynomials corresponding to ｜x｜^α （2 〈 α 〈 4） on equidistant nodes in [-1, 1] diverges everywhere, except at zero and the end-points.展开更多
In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f（x） = ｜x｜α wit...In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f（x） = ｜x｜α with on equally展开更多
This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the rune tion f（z） =｜x｜^α（1〈α〈2） on [-1,1] can diverge everywhere in the interval except at zero and the end-points.
Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobi polynom...Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobi polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.展开更多
We study the optimal order of approximation for |x|α (0 < α < 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained.
In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is appli...In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.展开更多
In this paper, we investigate the negative extremums of fundamental functions of Lagrange interpolation based on Chebyshev nodes. Moreover, we establish some companion results to the theorem of J. Szabados on the posi...In this paper, we investigate the negative extremums of fundamental functions of Lagrange interpolation based on Chebyshev nodes. Moreover, we establish some companion results to the theorem of J. Szabados on the positive extremum.展开更多
We study some approximation properties of Lagrange interpolation polynomial based on the zeros of （1-x^2）cosnarccosx. By using a decomposition for f（x） ∈ C^τC^τ＋1 we obtain an estimate of ‖f（x） -Ln＋2（f, ...We study some approximation properties of Lagrange interpolation polynomial based on the zeros of （1-x^2）cosnarccosx. By using a decomposition for f（x） ∈ C^τC^τ＋1 we obtain an estimate of ‖f（x） -Ln＋2（f, x）‖ which reflects the influence of the position of the x＇s and ω（f^（r＋1）,δ）j,j = 0, 1,... , s,on the error of approximation.展开更多
This paper investigates the optimal Lagrange interpolation of a class F∞of infinitely differentiable functions on[-1,1]in L_(∞)[-1,1]and weighted spaces L_(p,ω)[-1,1],1≤p<∞withωa continuous integrable weight ...This paper investigates the optimal Lagrange interpolation of a class F∞of infinitely differentiable functions on[-1,1]in L_(∞)[-1,1]and weighted spaces L_(p,ω)[-1,1],1≤p<∞withωa continuous integrable weight function in(-1,1).We proved that the Lagrange interpolation polynomials based on the zeros of polynomials with the leading coefficient 1 of the least deviation from zero in L_(p,ω)[-1,1]are optimal for 1≤p<∞.We also give the optimal Lagrange interpolation nodes when the endpoints are included in the nodes.展开更多
In this paper, we obtain a properly posed set of nodes for interpolation on a sphere. Moreover it is applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of to...In this paper, we obtain a properly posed set of nodes for interpolation on a sphere. Moreover it is applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of total degree n.展开更多
This paper investigates the optimal recovery of Sobolev spaces W_(1)^(r)[-1,1],r∈N in the space L_(1)[-1,1].They obtain the values of the sampling numbers of W_(1)^(r)[-1,1]in L_(1)[-1,1]and show that the Lagrange in...This paper investigates the optimal recovery of Sobolev spaces W_(1)^(r)[-1,1],r∈N in the space L_(1)[-1,1].They obtain the values of the sampling numbers of W_(1)^(r)[-1,1]in L_(1)[-1,1]and show that the Lagrange interpolation algorithms based on the extreme points of Chebyshev polynomials are optimal algorithms.Meanwhile,they prove that the extreme points of Chebyshev polynomials are optimal Lagrange interpolation nodes.展开更多
For the weighted approximation in Lp-norm, we determine the asymptotic order for the p- average errors of Lagrange interpolation sequence based on the Chebyshev nodes on the Wiener space. We also determine its value i...For the weighted approximation in Lp-norm, we determine the asymptotic order for the p- average errors of Lagrange interpolation sequence based on the Chebyshev nodes on the Wiener space. We also determine its value in some special case.展开更多
In this paper, based on fourth order Ostrowski method, we derive an optimal eighth order iteration scheme for obtaining simple roots of nonlinear equations using Lagrange interpolation and suitable weight functions. T...In this paper, based on fourth order Ostrowski method, we derive an optimal eighth order iteration scheme for obtaining simple roots of nonlinear equations using Lagrange interpolation and suitable weight functions. The scheme requires three evaluations of the function and one evaluation of the first derivative per iteration. Numerical examples are included to confirm the theoretical results and to show the competitive performance of the proposed iteration scheme.展开更多
Accurate simulations of ultra-wideband （UWB） electromagnetic radiation from an antenna were developed based on a time-domain finite element method （TDFEM） based on p-step Lagrange interpolation for the temporal ex...Accurate simulations of ultra-wideband （UWB） electromagnetic radiation from an antenna were developed based on a time-domain finite element method （TDFEM） based on p-step Lagrange interpolation for the temporal expansion. The motivation was to utilize the good interpolation features and straightforward computations for UWB antenna simulations. Numerical results were obtained from the cases of the cavity resonance problem, a bowtie and a Sierpinski bowtie antenna. Comparisons with an existing TDFEM approach employed linear temporal basis functions show good agreement to demonstrate the validity of the present schemes. The TDFEM with 2-step Lagrange interpolation as the temporal basis functions achieves better numerical results with only a small increase to run time and memory use in terms of the relative errors of the resonant frequency in the cavity for the transverse electric mode and the radiation patterns of the bowtie antenna.展开更多
The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies ...The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies the inequality then for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality 丨p^(k)(x)丨≤max{丨q^((k))(x)丨,丨1/k(x^2-1)q^(k+1)(x)+xq^((k))(x)丨}. This estimate leads to the Markov inequality for the higher order derivatives of polynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero. Some other results are established which gives evidence to the conjecture that under the conditions of Theorem 1 the inequality ‖p^((k))‖≤‖q^(k)‖holds.展开更多
This paper focuses on applying the barycentric Lagrange interpolation collocation method(BLICM)for solving 2D time-fractional diffusion-wave equation(TFDWE).In order to obtain the discrete format of the equation,we co...This paper focuses on applying the barycentric Lagrange interpolation collocation method(BLICM)for solving 2D time-fractional diffusion-wave equation(TFDWE).In order to obtain the discrete format of the equation,we construct the multivariate barycentric Lagrange interpolation approximation function and process the integral terms by using the Gauss-Legendre quadrature formula.We provide a detailed error analysis of the discrete format on the second kind of Chebyshev nodes.The efficacy of the proposed method is substantiated by some numerical experiments.The results of these experiments demonstrate that our method can obtain high-precision numerical solutions for fractional partial differential equations.Additionally,the method's capability to achieve high precision with a reduced number of nodes is confirmed.展开更多
The traditional train positioning methods suffer from inadequate accuracy and high maintenance costs,rendering them unsuitable for the development requirements of lightweight and intelligent train positioning technolo...The traditional train positioning methods suffer from inadequate accuracy and high maintenance costs,rendering them unsuitable for the development requirements of lightweight and intelligent train positioning technology.To address these restraints,the BeiDou navigation satellite system/strapdown inertial navigation system(BDS/SINS)integrated train positioning system based on an adaptive unscented Kalman filter(AUKF)is proposed.Firstly,the combined denoising algorithm(CDA)and Lagrange interpolation algorithm are introduced to preprocess the original data,effectively eliminating the influence of noise signals and abnormal measurements on the train positioning system.Secondly,the innovation theory is incorporated into the unscented Kalman filter(UKF)to derive the AUKF,which accomplishes an adaptive update of the measurement noise covariance.Finally,the positioning performance of the proposed AUKF is contrasted with that of conventional algorithms in various operation scenes.Simulation results demonstrate that the average value of error calculated by AUKF is less than 1.5 m,and the success rate of positioning touches 95.0%.Compared to Kalman filter(KF)and UKF,AUKF exhibits superior accuracy and stability in train positioning.Consequently,the proposed AUKF is well-suited for providing precise positioning services in variable operating environments for trains.展开更多
文摘Though the Butterfly Bptimization Algorithm(BOA)has already proved its effectiveness as a robust optimization algorithm,it has certain disadvantages.So,a new variant of BOA,namely mLBOA,is proposed here to improve its performance.The proposed algorithm employs a self-adaptive parameter setting,Lagrange interpolation formula,and a new local search strategy embedded with Levy flight search to enhance its searching ability to make a better trade-off between exploration and exploitation.Also,the fragrance generation scheme of BOA is modified,which leads for exploring the domain effectively for better searching.To evaluate the performance,it has been applied to solve the IEEE CEC 2017 benchmark suite.The results have been compared to that of six state-of-the-art algorithms and five BOA variants.Moreover,various statistical tests,such as the Friedman rank test,Wilcoxon rank test,convergence analysis,and complexity analysis,have been conducted to justify the rank,significance,and complexity of the proposed mLBOA.Finally,the mLBOA has been applied to solve three real-world engineering design problems.From all the analyses,it has been found that the proposed mLBOA is a competitive algorithm compared to other popular state-of-the-art algorithms and BOA variants.
文摘S.M.Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [-1,1]. In 2000, M. Rever generalized S.M.Lozinskii's result to |x|α(0 <≤ α≤ 1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|α1(1 < α < 2)..
文摘It is a classical result of Bernstein that the sequence of Lagrange interpolation polynomials to \x\ at e-qually spaced nodes in [-1.1] diverges everywhere. except at zero and the end-points. In this paper we show that the sequence of Lagrange interpolation polynomials corresponding to the functions which possess better smoothness on equidistant nodes in [-1.1] still diverges every -where in the interval except at zero and the end-points.
文摘It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to ｜x｜ at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, toe prove that the sequence of Lagrange interpolation polynomials corresponding to ｜x｜^α （2 〈 α 〈 4） on equidistant nodes in [-1, 1] diverges everywhere, except at zero and the end-points.
文摘In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f（x） = ｜x｜α with on equally
文摘This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the rune tion f（z） =｜x｜^α（1〈α〈2） on [-1,1] can diverge everywhere in the interval except at zero and the end-points.
文摘Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobi polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.
文摘We study the optimal order of approximation for |x|α (0 < α < 1) by Lagrange interpolation polynomials based on Chebyshev nodes of the first kind. It is proved that the Jackson order of approximation is attained.
基金partially supported by National Natural Science Foundation of China(11772165,11961054,11902170)Key Research and Development Program of Ningxia(2018BEE03007)+1 种基金National Natural Science Foundation of Ningxia(2018AAC02003,2020AAC03059)Major Innovation Projects for Building First-class Universities in China’s Western Region(Grant No.ZKZD2017009).
文摘In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.
文摘In this paper, we investigate the negative extremums of fundamental functions of Lagrange interpolation based on Chebyshev nodes. Moreover, we establish some companion results to the theorem of J. Szabados on the positive extremum.
基金Supported by the National Nature Science Foundation.
文摘We study some approximation properties of Lagrange interpolation polynomial based on the zeros of （1-x^2）cosnarccosx. By using a decomposition for f（x） ∈ C^τC^τ＋1 we obtain an estimate of ‖f（x） -Ln＋2（f, x）‖ which reflects the influence of the position of the x＇s and ω（f^（r＋1）,δ）j,j = 0, 1,... , s,on the error of approximation.
基金Supported by the National Natural Science Foundation of China(Grant No.11871006).
文摘This paper investigates the optimal Lagrange interpolation of a class F∞of infinitely differentiable functions on[-1,1]in L_(∞)[-1,1]and weighted spaces L_(p,ω)[-1,1],1≤p<∞withωa continuous integrable weight function in(-1,1).We proved that the Lagrange interpolation polynomials based on the zeros of polynomials with the leading coefficient 1 of the least deviation from zero in L_(p,ω)[-1,1]are optimal for 1≤p<∞.We also give the optimal Lagrange interpolation nodes when the endpoints are included in the nodes.
文摘In this paper, we obtain a properly posed set of nodes for interpolation on a sphere. Moreover it is applied to construct properly posed set of nodes for Lagrange interpolation on the trivariate polynomial space of total degree n.
基金supported by the National Natural Science Foundation of China(Nos.11871006,11671271)。
文摘This paper investigates the optimal recovery of Sobolev spaces W_(1)^(r)[-1,1],r∈N in the space L_(1)[-1,1].They obtain the values of the sampling numbers of W_(1)^(r)[-1,1]in L_(1)[-1,1]and show that the Lagrange interpolation algorithms based on the extreme points of Chebyshev polynomials are optimal algorithms.Meanwhile,they prove that the extreme points of Chebyshev polynomials are optimal Lagrange interpolation nodes.
基金Supported by National Natural Science Foundation of China(Grant No.10471010)
文摘For the weighted approximation in Lp-norm, we determine the asymptotic order for the p- average errors of Lagrange interpolation sequence based on the Chebyshev nodes on the Wiener space. We also determine its value in some special case.
基金the I.K. Gujral Punjab Technical University, Kapurthala for providing research support
文摘In this paper, based on fourth order Ostrowski method, we derive an optimal eighth order iteration scheme for obtaining simple roots of nonlinear equations using Lagrange interpolation and suitable weight functions. The scheme requires three evaluations of the function and one evaluation of the first derivative per iteration. Numerical examples are included to confirm the theoretical results and to show the competitive performance of the proposed iteration scheme.
文摘Accurate simulations of ultra-wideband （UWB） electromagnetic radiation from an antenna were developed based on a time-domain finite element method （TDFEM） based on p-step Lagrange interpolation for the temporal expansion. The motivation was to utilize the good interpolation features and straightforward computations for UWB antenna simulations. Numerical results were obtained from the cases of the cavity resonance problem, a bowtie and a Sierpinski bowtie antenna. Comparisons with an existing TDFEM approach employed linear temporal basis functions show good agreement to demonstrate the validity of the present schemes. The TDFEM with 2-step Lagrange interpolation as the temporal basis functions achieves better numerical results with only a small increase to run time and memory use in terms of the relative errors of the resonant frequency in the cavity for the transverse electric mode and the radiation patterns of the bowtie antenna.
文摘The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies the inequality then for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality 丨p^(k)(x)丨≤max{丨q^((k))(x)丨,丨1/k(x^2-1)q^(k+1)(x)+xq^((k))(x)丨}. This estimate leads to the Markov inequality for the higher order derivatives of polynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero. Some other results are established which gives evidence to the conjecture that under the conditions of Theorem 1 the inequality ‖p^((k))‖≤‖q^(k)‖holds.
基金Supported by the Scientific Research Foundation for Talents Introduced of Guizhou University of Finance and Economics(Grant No.2023YJ16)the Institute of Complexity Science,Henan University of Technology(Grant No.CSKFJJ-2025-33)the International Science and Technology Cooperation Project of Henan Province(Grant No.252102520007).
文摘This paper focuses on applying the barycentric Lagrange interpolation collocation method(BLICM)for solving 2D time-fractional diffusion-wave equation(TFDWE).In order to obtain the discrete format of the equation,we construct the multivariate barycentric Lagrange interpolation approximation function and process the integral terms by using the Gauss-Legendre quadrature formula.We provide a detailed error analysis of the discrete format on the second kind of Chebyshev nodes.The efficacy of the proposed method is substantiated by some numerical experiments.The results of these experiments demonstrate that our method can obtain high-precision numerical solutions for fractional partial differential equations.Additionally,the method's capability to achieve high precision with a reduced number of nodes is confirmed.
基金supported by Project Fund of China National Railway Group Co.,Ltd.(No.N2022G012)Natonal Natural Science Foundation of China(No.61661027)。
文摘The traditional train positioning methods suffer from inadequate accuracy and high maintenance costs,rendering them unsuitable for the development requirements of lightweight and intelligent train positioning technology.To address these restraints,the BeiDou navigation satellite system/strapdown inertial navigation system(BDS/SINS)integrated train positioning system based on an adaptive unscented Kalman filter(AUKF)is proposed.Firstly,the combined denoising algorithm(CDA)and Lagrange interpolation algorithm are introduced to preprocess the original data,effectively eliminating the influence of noise signals and abnormal measurements on the train positioning system.Secondly,the innovation theory is incorporated into the unscented Kalman filter(UKF)to derive the AUKF,which accomplishes an adaptive update of the measurement noise covariance.Finally,the positioning performance of the proposed AUKF is contrasted with that of conventional algorithms in various operation scenes.Simulation results demonstrate that the average value of error calculated by AUKF is less than 1.5 m,and the success rate of positioning touches 95.0%.Compared to Kalman filter(KF)and UKF,AUKF exhibits superior accuracy and stability in train positioning.Consequently,the proposed AUKF is well-suited for providing precise positioning services in variable operating environments for trains.
