The aim of this paper is to study the weak integral convergence of Kergin interpolation. The results of the weighted integral convergence and the weighted (partial) derivatives integral convergence of Kergin interpola...The aim of this paper is to study the weak integral convergence of Kergin interpolation. The results of the weighted integral convergence and the weighted (partial) derivatives integral convergence of Kergin interpolation polynomial for the smooth functions on the unit disk were obtained in the paper. Those generalized Liang's main results were acquired in 1998 to the more extensive situation. At the same time, the estimation of convergence rate of Kergin interpolation polynomial is given by means of introducing a new kind of smooth norm.展开更多
In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only o...In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only one evaluation of its derivative are constructed by the inverse interpolation with the memory on the previous step for solving the simple root of a nonlinear equation. The order and order of convergence of them are proved respectively. Finally, the proposed methods and the basins of attraction are demonstrated by the numerical examples.展开更多
In this paper, an interpolation polynomial operator F n(f; l,x) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈C b [-1,1] ...In this paper, an interpolation polynomial operator F n(f; l,x) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈C b [-1,1] (0≤b≤l) F n(f; l,x) converges to f(x) uniformly, where l is an odd number.展开更多
The results of accurate order of uniform approximation and simultaneous approximation in the early work ＂Jackson Type Theorems on Complex Curves＂ are improved from Fejer points to disturbed Fejer points in this arti...The results of accurate order of uniform approximation and simultaneous approximation in the early work ＂Jackson Type Theorems on Complex Curves＂ are improved from Fejer points to disturbed Fejer points in this article. Furthermore, the stability of convergence of Tn,∈（f,z） with disturbed sample values f（z^＊） ＋ Sk are also proved in this article.展开更多
Let X<sub>n</sub>={x<sub>kn</sub>=cosθ<sub>kn</sub>: θ<sub>kn</sub>=（kπ）/（n+1）, 1≤k≤n}be the node system which consists ofroots of U<sub>n</sub> （x...Let X<sub>n</sub>={x<sub>kn</sub>=cosθ<sub>kn</sub>: θ<sub>kn</sub>=（kπ）/（n+1）, 1≤k≤n}be the node system which consists ofroots of U<sub>n</sub> （x） =（sin（n+1）θ）/（sinθ）（x=cosθ θ∈[0,π]）, the second kind Chebyshevpolynomical. All the symbols below have the same meaning as Ref. [1]if notspecifically defined. We shall consider a kind of new interpolating problem in thisnote. For any non-negative integer q and f∈C[-1, 1], it is well known that thepolynomial Q<sub>nq</sub>（f）∈П<sub>N</sub> （N=2（q+1） （n+1） -1） satisfying the following conditions isuniquely determined:Q<sub>nq</sub>(f, x<sub>kn</sub>) =f(x<sub>kn</sub>), 1≤k≤n; Q<sub>nq</sub>（f,±1）=f（±1）,Q<sub>nq</sub><sup>j</sup>(f,x<sub>kn</sub>)=c<sub>jkn</sub>, 1≤k≤n,1≤j≤2q+1,Q<sub>nq</sub><sup>j</sup>（f,1）=d<sub>jn</sub>, Q<sub>nq</sub><sup>j</sup>（f,-1）=g<sub>jn</sub>, 1≤j≤q,where c<sub>jkn</sub>,d<sub>jn</sub>, g<sub>jn</sub>are any given real numbers. Q<sub>nq</sub>（f）is called the higher orderquasi Hermite-Fejer interpolation of f.We展开更多
In this paper, we derive two higher order multipoint methods for solving nonlinear equations. The methodology is based on Ostrowski’s method and further developed by using cubic interpolation process. The adaptation ...In this paper, we derive two higher order multipoint methods for solving nonlinear equations. The methodology is based on Ostrowski’s method and further developed by using cubic interpolation process. The adaptation of this strategy increases the order of Ostrowski’s method from four to eight and its efficiency index from 1.587 to 1.682. The methods are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified on the examples.展开更多
In this paper,a new third type S.N.Bernstein interpolation polynomial H n(f;x,r) with zeros of the Chebyshev ploynomial of the second kind is constructed. H n(f;x,r) converge uniformly on [-1,1] for any continuous fun...In this paper,a new third type S.N.Bernstein interpolation polynomial H n(f;x,r) with zeros of the Chebyshev ploynomial of the second kind is constructed. H n(f;x,r) converge uniformly on [-1,1] for any continuous function f(x) . The convergence order is the best order if \{f(x)∈C j[-1,1],\}0jr, where r is an odd natural number.展开更多
In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method...In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of nonlinear ODEs are shown as well in the numerical examples.展开更多
文摘The aim of this paper is to study the weak integral convergence of Kergin interpolation. The results of the weighted integral convergence and the weighted (partial) derivatives integral convergence of Kergin interpolation polynomial for the smooth functions on the unit disk were obtained in the paper. Those generalized Liang's main results were acquired in 1998 to the more extensive situation. At the same time, the estimation of convergence rate of Kergin interpolation polynomial is given by means of introducing a new kind of smooth norm.
文摘In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only one evaluation of its derivative are constructed by the inverse interpolation with the memory on the previous step for solving the simple root of a nonlinear equation. The order and order of convergence of them are proved respectively. Finally, the proposed methods and the basins of attraction are demonstrated by the numerical examples.
文摘In this paper, an interpolation polynomial operator F n(f; l,x) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈C b [-1,1] (0≤b≤l) F n(f; l,x) converges to f(x) uniformly, where l is an odd number.
基金Supported by NSF of Henan Province of China(20001110001)
文摘The results of accurate order of uniform approximation and simultaneous approximation in the early work ＂Jackson Type Theorems on Complex Curves＂ are improved from Fejer points to disturbed Fejer points in this article. Furthermore, the stability of convergence of Tn,∈（f,z） with disturbed sample values f（z^＊） ＋ Sk are also proved in this article.
文摘Let X<sub>n</sub>={x<sub>kn</sub>=cosθ<sub>kn</sub>: θ<sub>kn</sub>=（kπ）/（n+1）, 1≤k≤n}be the node system which consists ofroots of U<sub>n</sub> （x） =（sin（n+1）θ）/（sinθ）（x=cosθ θ∈[0,π]）, the second kind Chebyshevpolynomical. All the symbols below have the same meaning as Ref. [1]if notspecifically defined. We shall consider a kind of new interpolating problem in thisnote. For any non-negative integer q and f∈C[-1, 1], it is well known that thepolynomial Q<sub>nq</sub>（f）∈П<sub>N</sub> （N=2（q+1） （n+1） -1） satisfying the following conditions isuniquely determined:Q<sub>nq</sub>(f, x<sub>kn</sub>) =f(x<sub>kn</sub>), 1≤k≤n; Q<sub>nq</sub>（f,±1）=f（±1）,Q<sub>nq</sub><sup>j</sup>(f,x<sub>kn</sub>)=c<sub>jkn</sub>, 1≤k≤n,1≤j≤2q+1,Q<sub>nq</sub><sup>j</sup>（f,1）=d<sub>jn</sub>, Q<sub>nq</sub><sup>j</sup>（f,-1）=g<sub>jn</sub>, 1≤j≤q,where c<sub>jkn</sub>,d<sub>jn</sub>, g<sub>jn</sub>are any given real numbers. Q<sub>nq</sub>（f）is called the higher orderquasi Hermite-Fejer interpolation of f.We
文摘In this paper, we derive two higher order multipoint methods for solving nonlinear equations. The methodology is based on Ostrowski’s method and further developed by using cubic interpolation process. The adaptation of this strategy increases the order of Ostrowski’s method from four to eight and its efficiency index from 1.587 to 1.682. The methods are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified on the examples.
文摘In this paper,a new third type S.N.Bernstein interpolation polynomial H n(f;x,r) with zeros of the Chebyshev ploynomial of the second kind is constructed. H n(f;x,r) converge uniformly on [-1,1] for any continuous function f(x) . The convergence order is the best order if \{f(x)∈C j[-1,1],\}0jr, where r is an odd natural number.
文摘In this paper, a new two-step Newton-type method with third-order convergence for solving systems of nonlinear equations is proposed. We construct the new method based on the integral interpolation of Newton’s method. Its cubic convergence and error equation are proved theoretically, and demonstrated numerically. Its application to systems of nonlinear equations and boundary-value problems of nonlinear ODEs are shown as well in the numerical examples.
