The convergence properties and limiting behavior of several real sequences are studied by analytical means. Some remarkable properties of these sequences are established.
用解析方法研究了若干实序列的收敛性和极限行为。建立了这些序列的一些显著性质。
{"title":"Asymptotic properties and behavior of some nontrivial sequences","authors":"P. Bracken","doi":"10.33993/jnaat492-1223","DOIUrl":"https://doi.org/10.33993/jnaat492-1223","url":null,"abstract":"The convergence properties and limiting behavior of several real sequences are studied by analytical means.\u0000Some remarkable properties of these sequences are established.\u0000 ","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132868501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let (X) be a Banach space, (G$) be a closed subset of (X), and ((Omega,Sigma,mu )) be a (sigma)-finite measure space. In this paper we present some results on coproximinality (pointwise coproximinality) of (L^{p}(mu,G)), (1leq pleq infty), in (L^{p}(mu,X).
{"title":"Pointwise best coapproximation in the space of Bochner integrable functions","authors":"Eyad Abu-Sirhan","doi":"10.33993/jnaat492-1206","DOIUrl":"https://doi.org/10.33993/jnaat492-1206","url":null,"abstract":"Let (X) be a Banach space, (G$) be a closed subset of (X), and ((Omega,Sigma,mu )) be a (sigma)-finite measure space. In this paper we present some results on coproximinality (pointwise coproximinality) of (L^{p}(mu,G)), (1leq pleq infty), in (L^{p}(mu,X).","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130929222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By using the concept of nonlinear Choquet integral with respect to a capacity and as a generalization of the Poisson-Cauchy-Choquet operators, we introduce the nonlinear Angheluta-Choquet singular integrals with respect to a family of submodular set functions. Quantitative approximation results in terms of the modulus of continuity are obtained with respect to some particular possibility measures and with respect to the Choquet measure (mu(A)=sqrt{M(A)}), where (M) represents the Lebesgue measure. For some subclasses of functions we prove that these Choquet type operators can have essentially better approximation properties than their classical correspondents. The paper ends with the important, independent remark that for Choquet-type operators which are comonotone additive too, like Kantorovich-Choquet operators, Szasz-Mirakjan-Kantorovich-Choquet operators and Baskakov-Kantorovich-Choquet operators studied in previous papers, the approximation results remain identically valid not only for non-negative functions, but also for all functions which take negative values too, if they are lower bounded.
{"title":"Quantitative approximation by nonlinear Angheluta-Choquet singular integrals","authors":"S. Gal, Ionut T. Iancu","doi":"10.33993/jnaat491-1217","DOIUrl":"https://doi.org/10.33993/jnaat491-1217","url":null,"abstract":"By using the concept of nonlinear Choquet integral with respect to a capacity and as a generalization of the Poisson-Cauchy-Choquet operators, we introduce the nonlinear Angheluta-Choquet singular integrals with respect to a family of submodular set functions. Quantitative approximation results in terms of the modulus of continuity are obtained with respect to some particular possibility measures and with respect to the Choquet measure (mu(A)=sqrt{M(A)}), where (M) represents the Lebesgue measure. For some subclasses of functions we prove that these Choquet type operators can have essentially better approximation properties than their classical correspondents. The paper ends with the important, independent remark that for Choquet-type operators which are comonotone additive too, like Kantorovich-Choquet operators, Szasz-Mirakjan-Kantorovich-Choquet operators and Baskakov-Kantorovich-Choquet operators studied in previous papers, the approximation results remain identically valid not only for non-negative functions, but also for all functions which take negative values too, if they are lower bounded.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125630446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the NP-hard absolute value equations (AVE), (Ax-B|x| =b), where (A,B) are given symmetric matrices in (mathbb{R}^{ntimes n}, bin mathbb{R}^{n}).By reformulating the AVE as an equivalent unconstrained convex quadratic optimization, we prove that the unique solution of the AVE is the unique minimum of the corresponding quadratic optimization. Then across the latter, we adopt the preconditioned conjugate gradient methods to determining an approximate solution of the AVE.The computational results show the efficiency of these approaches in dealing with the AVE.
我们研究了NP-hard绝对值方程(AVE)， (Ax-B|x| =b)，其中(A,B)是(mathbb{R}^{ntimes n}, bin mathbb{R}^{n})中给定的对称矩阵。通过将AVE重新表示为等价的无约束凸二次优化，我们证明了AVE的唯一解是相应二次优化的唯一最小值。在此基础上，采用预条件共轭梯度法求出AVE的近似解，计算结果表明了这些方法处理AVE的有效性。
{"title":"Preconditioned conjugate gradient methods for absolute value equations","authors":"Nassima Anane, M. Achache","doi":"10.33993/jnaat491-1197","DOIUrl":"https://doi.org/10.33993/jnaat491-1197","url":null,"abstract":"We investigate the NP-hard absolute value equations (AVE), (Ax-B|x| =b), where (A,B) are given symmetric matrices in (mathbb{R}^{ntimes n}, bin mathbb{R}^{n}).By reformulating the AVE as an equivalent unconstrained convex quadratic optimization, we prove that the unique solution of the AVE is the unique minimum of the corresponding quadratic optimization. Then across the latter, we adopt the preconditioned conjugate gradient methods to determining an approximate solution of the AVE.The computational results show the efficiency of these approaches in dealing with the AVE.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116109244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we give a suitable definition for the pairs of adjacent (convergent) sequences of real numbers, we present some two-sided estimations which caracterize the order of convergence to its limits of some of these sequences and we give certain general explanations for its similar orders of convergence.
{"title":"On the convergence rates of pairs of adjacent sequences","authors":"D. Duca, A. Vernescu","doi":"10.33993/jnaat491-1221","DOIUrl":"https://doi.org/10.33993/jnaat491-1221","url":null,"abstract":"In this paper we give a suitable definition for the pairs of adjacent (convergent) sequences of real numbers, we present some two-sided estimations which caracterize the order of convergence to its limits of some of these sequences and we give certain general explanations for its similar orders of convergence.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114223794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we derive closed form expressions for the nearest rank-(k) matrix on canonical subspaces. We start by studying three kinds of subspaces. Let (X) and (Y) be a pair of given matrices. The first subspace contains all the (mtimes n) matrices (A) that satisfy (AX=O). The second subspace contains all the (m times n) matrices (A) that satisfy (Y^TA = O), while the matrices in the third subspace satisfy both (AX =O) and (Y^TA = 0). The second part of the paper considers a subspace that contains all the symmetric matrices (S) that satisfy (SX =O). In this case, in addition to the nearest rank-(k) matrix we also provide the nearest rank-(k) positive approximant on that subspace. A further insight is gained by showing that the related cones of positive semidefinite matrices, and negative semidefinite matrices, constitute a polar decomposition of this subspace. The paper ends with two examples of applications. The first one regards the problem of computing the nearest rank-(k) centered matrix, and adds new insight into the PCA of a matrix. The second application comes from the field of Euclidean distance matrices. The new results on low-rank positive approximants are used to derive an explicit expression for the nearest source matrix. This opens a direct way for computing the related positions matrix.
{"title":"Low-rank matrix approximations over canonical subspaces","authors":"A. Dax","doi":"10.33993/jnaat491-1195","DOIUrl":"https://doi.org/10.33993/jnaat491-1195","url":null,"abstract":"In this paper we derive closed form expressions for the nearest rank-(k) matrix on canonical subspaces. \u0000 \u0000We start by studying three kinds of subspaces. Let (X) and (Y) be a pair of given matrices. The first subspace contains all the (mtimes n) matrices (A) that satisfy (AX=O). The second subspace contains all the (m times n) matrices (A) that satisfy (Y^TA = O), while the matrices in the third subspace satisfy both (AX =O) and (Y^TA = 0). \u0000 \u0000The second part of the paper considers a subspace that contains all the symmetric matrices (S) that satisfy (SX =O). In this case, in addition to the nearest rank-(k) matrix we also provide the nearest rank-(k) positive approximant on that subspace. \u0000 \u0000A further insight is gained by showing that the related cones of positive semidefinite matrices, and negative semidefinite matrices, constitute a polar decomposition of this subspace. \u0000The paper ends with two examples of applications. The first one regards the problem of computing the nearest rank-(k) centered matrix, and adds new insight into the PCA of a matrix. \u0000The second application comes from the field of Euclidean distance matrices. The new results on low-rank positive approximants are used to derive an explicit expression for the nearest source matrix. This opens a direct way for computing the related positions matrix.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126404823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate the existence of infinitely many solutions for the second-order self-adjoint discrete Hamiltonian system$$Deltaleft[p(n)Delta u(n-1)right]-L(n)u(n)+nabla W(n,u(n))=0, tag{*}$$where (ninmathbb{Z}, uinmathbb{R}^{N}, p,L:mathbb{Z}rightarrowmathbb{R}^{Ntimes N}) and (W:mathbb{Z}timesmathbb{R}^{N}rightarrowmathbb{R}) are no periodic in (n). The novelty of this paper is that (L(n)) is bounded in the sense that there two constants (0
{"title":"Infinitely homoclinic solutions in discrete hamiltonian systems without coercive conditions","authors":"F. Khelifi","doi":"10.33993/jnaat491-1204","DOIUrl":"https://doi.org/10.33993/jnaat491-1204","url":null,"abstract":"In this paper, we investigate the existence of infinitely many solutions for the second-order self-adjoint discrete Hamiltonian system$$Deltaleft[p(n)Delta u(n-1)right]-L(n)u(n)+nabla W(n,u(n))=0, tag{*}$$where (ninmathbb{Z}, uinmathbb{R}^{N}, p,L:mathbb{Z}rightarrowmathbb{R}^{Ntimes N}) and (W:mathbb{Z}timesmathbb{R}^{N}rightarrowmathbb{R}) are no periodic in (n). The novelty of this paper is that (L(n)) is bounded in the sense that there two constants (0<tau_1<tau_2<infty) such that$$tau_1left|uright|^{2}<left(L(n)u,uright)<tau_2left|uright|^{2},;forall ninmathbb{Z},; uinmathbb{R}^{N},$$(W(t,u)) satisfies Ambrosetti-Rabinowitz condition and some other reasonable hypotheses, we show that ((*)) has infinitely many homoclinic solutions via the Symmetric Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122431708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the well-known Khattri et al. methods and Zheng et al. methods are identical. In passing, we propose a suitable calculation formula for Khattri et al. methods. We also show that the families of eighth-order derivative-free methods obtained in [8] include some existing methods, among them the above-mentioned ones as particular cases. We also give the sufficient convergence condition of these families. Numerical examples and comparison with some existing methods were made. In addition, the dynamical behavior of methods of these families is analyzed.
{"title":"Comparison of some optimal derivative-free three-point iterations","authors":"Thugal Zhanlav, Kh. Otgondorj","doi":"10.33993/jnaat491-1179","DOIUrl":"https://doi.org/10.33993/jnaat491-1179","url":null,"abstract":"We show that the well-known Khattri et al. methods and Zheng et al. methods are identical. In passing, we propose a suitable calculation formula for Khattri et al. methods. We also show that the families of eighth-order derivative-free methods obtained in [8] include some existing methods, among them the above-mentioned ones as particular cases. We also give the sufficient convergence condition of these families. Numerical examples and comparison with some existing methods were made. In addition, the dynamical behavior of methods of these families is analyzed.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128152976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a comparative numerical study between line search methods and majorant functions to compute the displacement step in barrier logarithmic method for linear programming. This study favorate majorant function on line search which is promoted by numerical experiments.
{"title":"Comparative numerical study between line search methods and majorant functions in barrier logarithmic methods for linear programming","authors":"S. Chaghoub, D. Benterki","doi":"10.33993/jnaat491-1199","DOIUrl":"https://doi.org/10.33993/jnaat491-1199","url":null,"abstract":"This paper presents a comparative numerical study between line search methods and majorant functions to compute the displacement step in barrier logarithmic method for linear programming. This study favorate majorant function on line search which is promoted by numerical experiments.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115691416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to extend the radius of convergence and improve the ratio of convergence for a certain class of Euler-Halley type methods with one parameter in a Banach space. These improvements over earlier works are obtained using the same functions as before but more precise information on the location of the iterates. Special cases and examples are also presented in this study.
{"title":"Extending the radius of convergence for a class of Euler-Halley type methods","authors":"S. George, I. Argyros","doi":"10.33993/jnaat482-1115","DOIUrl":"https://doi.org/10.33993/jnaat482-1115","url":null,"abstract":"The aim of this paper is to extend the radius of convergence and improve the ratio of convergence for a certain class of Euler-Halley type methods with one parameter in a Banach space. These improvements over earlier works are obtained using the same functions as before but more precise information on the location of the iterates. Special cases and examples are also presented in this study.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115521539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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