In the present paper we define a q-analogue of the modified a-Bernstein operators introduced by Kajla and Acar (Ann. Funct. Anal. 10 (4) 2019, 570-582). We study uniform convergence theorem and the Voronovskaja type asymptotic theorem. We determine the estimate of error in the approximation by these operators by virtue of second order modulus of continuity via the approach of Steklov means and the technique of Peetre's (K)-functional. Next, we investigate the Gruss- Voronovskaya type theorem. Further, we define a bivariate tensor product of these operatos and derive the convergence estimates by utilizing the partial and total moduli of continuity. The approximation degree by means of Peetre's K- functional , the Voronovskaja and Gruss Voronovskaja type theorems are also investigated. Lastly, we construct the associated GBS (Generalized Boolean Sum) operator and examine its convergence behavior by virtue of the mixed modulus of smoothness.
{"title":"On the rate of convergence of modified (alpha)-Bernstein operators based on q-integers","authors":"P. Agrawal, Dharmendra Kumar, Behar Baxhaku","doi":"10.33993/jnaat511-1244","DOIUrl":"https://doi.org/10.33993/jnaat511-1244","url":null,"abstract":"In the present paper we define a q-analogue of the modified a-Bernstein operators introduced by Kajla and Acar (Ann. Funct. Anal. 10 (4) 2019, 570-582). We study uniform convergence theorem and the Voronovskaja type asymptotic theorem. We determine the estimate of error in the approximation by these operators by virtue of second order modulus of continuity via the approach of Steklov means and the technique of Peetre's (K)-functional. Next, we investigate the Gruss- Voronovskaya type theorem. Further, we define a bivariate tensor product of these operatos and derive the convergence estimates by utilizing the partial and total moduli of continuity. The approximation degree by means of Peetre's K- functional , the Voronovskaja and Gruss Voronovskaja type theorems are also investigated. Lastly, we construct the associated GBS (Generalized Boolean Sum) operator and examine its convergence behavior by virtue of the mixed modulus of smoothness.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114262060","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 revisit Popovski’s family of methods for simple roots. We compare several members using basins of attraction visually and qualitatively by comparing the run-time on several examples, the average number of iterations and the number of divergent points. We chose 5 different members of the family. We also develop an equivalent family of methods for multiple roots and compare several members on six different numerical examples.
{"title":"Basins of attraction for family of Popovski’s methods and their extension to multiple roots","authors":"B. Neta","doi":"10.33993/jnaat511-1248","DOIUrl":"https://doi.org/10.33993/jnaat511-1248","url":null,"abstract":"In this paper we revisit Popovski’s family of methods for simple roots. We compare several members using basins of attraction visually and qualitatively by comparing the run-time on several examples, the average number of iterations and the number of divergent points. We chose 5 different members of the family. We also develop an equivalent family of methods for multiple roots and compare several members on six different numerical examples.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133251091","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 spectral radius condition [rho (vert A^{-1} vertcdot vert B vert)<1]for the unique solvability of generalized absolute value matrix equation (GAVME) [AX + B vert X vert = D] is provided. For some instances, our condition is superior to the earlier published singular values conditions (sigma_{max}(vert B vert)
给出了广义绝对值矩阵方程(GAVME)唯一可解的谱半径条件[rho (vert A^{-1} vertcdot vert B vert)<1][AX + B vert X vert = D]。在某些情况下，我们的条件优于先前发表的奇异值条件(sigma_{max}(vert B vert)
{"title":"A note on the unique solvability condition for generalized absolute value matrix equation","authors":"Shubham Kumar, .. Deepmala","doi":"10.33993/jnaat511-1263","DOIUrl":"https://doi.org/10.33993/jnaat511-1263","url":null,"abstract":"The spectral radius condition [rho (vert A^{-1} vertcdot vert B vert)<1]for the unique solvability of generalized absolute value matrix equation (GAVME) [AX + B vert X vert = D] is provided. For some instances, our condition is superior to the earlier published singular values conditions (sigma_{max}(vert B vert)<sigma_{min}(A)) [M. Dehghan, 2020] and (sigma_{max}(B)<sigma_{min}(A)) [Kai Xie, 2021]. For the validity of our condition, we also provided an example.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121713478","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 introduce an explicit form of the connection coefficients for Appell polynomial sequences via Toeplitz-Hessenberg matrix determinants. Generalizing, we give an explicit form of the connection coefficients for arbitrary polynomial sequences and constitute the combinatorial meaning of both constants in terms of integer composition.
{"title":"Integer composition, connection Appell constants and Bell polynomials","authors":"N. Luno","doi":"10.33993/jnaat502-1245","DOIUrl":"https://doi.org/10.33993/jnaat502-1245","url":null,"abstract":"We introduce an explicit form of the connection coefficients for Appell polynomial sequences via Toeplitz-Hessenberg matrix determinants.\u0000Generalizing, we give an explicit form of the connection coefficients for arbitrary polynomial sequences and constitute the combinatorial meaning of both constants in terms of integer composition.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124897236","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 extend the solvability of equations dened on a Banach space using numerically ecient secant-type methods. The convergence domain of these methods is enlarged using our new idea of restricted convergence region. By using this approach, we obtain a more precise location where the iterates lie than in earlier studies leading to tighter Lipschitz constants. This way the semi-local convergence produces weaker sucient convergence criteria and tighter error bounds than in earlier works. These improvements are also obtained under the same computational eort, since the new Lipschitz constants are special cases of the old ones.
{"title":"Extending the solvability of equations using secant-type methods in Banach space","authors":"I. Argyros, S. George","doi":"10.33993/jnaat502-1134","DOIUrl":"https://doi.org/10.33993/jnaat502-1134","url":null,"abstract":"We extend the solvability of equations dened on a Banach space using numerically ecient secant-type methods. The convergence domain of these methods is enlarged using our new idea of restricted convergence region. By using this approach, we obtain a more precise location where the iterates lie than in earlier studies leading to tighter Lipschitz constants. This way the semi-local convergence produces weaker sucient convergence criteria and tighter error bounds than in earlier works. These improvements are also obtained under the same computational eort, since the new Lipschitz constants are special cases of the old ones.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132219719","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 develop a new families of optimal eight--order methods for solving nonlinear equations. We also extend some classes of optimal methods for any suitable choice of iteration parameter. Such development and extension was made using sufficient convergence conditions given in [14]. Numerical examples are considered to check the convergence order of new families and extensions of some well-known methods.
{"title":"On the development and extensions of some classes of optimal three-point iterations for solving nonlinear equations","authors":"T. Zhanlav, Otgondorj Khuder","doi":"10.33993/jnaat502-1238","DOIUrl":"https://doi.org/10.33993/jnaat502-1238","url":null,"abstract":"We develop a new families of optimal eight--order methods for solving nonlinear equations. We also extend some classes of optimal methods for any suitable choice of iteration parameter. Such development and extension was made using sufficient convergence conditions given in [14]. Numerical examples are considered to check the convergence order of new families and extensions of some well-known methods.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125219741","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 catalogue of mathematical formulas and iterative algorithms for evaluating the mathematical constant (pi), ranging from Archimedes' 2200-year-old iteration to some formulas that were discovered only in the past few decades. Computer implementations and timing results for these formulas and algorithms are also included. In particular, timings are presented for evaluations of various infinite series formulas to approximately 10,000-digit precision, for evaluations of various integral formulas to approximately 4,000-digit precision, and for evaluations of several iterative algorithms to approximately 100,000-digit precision, all based on carefully designed comparative computer runs.
{"title":"A catalogue of mathematical formulas involving (pi), with analysis","authors":"D. Bailey","doi":"10.33993/jnaat502-1259","DOIUrl":"https://doi.org/10.33993/jnaat502-1259","url":null,"abstract":"This paper presents a catalogue of mathematical formulas and iterative algorithms for evaluating the mathematical constant (pi), ranging from Archimedes' 2200-year-old iteration to some formulas that were discovered only in the past few decades. Computer implementations and timing results for these formulas and algorithms are also included. In particular, timings are presented for evaluations of various infinite series formulas to approximately 10,000-digit precision, for evaluations of various integral formulas to approximately 4,000-digit precision, and for evaluations of several iterative algorithms to approximately 100,000-digit precision, all based on carefully designed comparative computer runs.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117035717","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 the present paper, we introduce a modification of Ismail-May operators having weights of Baskakov basis functions. We estimate weighted Korovkin's theorem and difference estimates between two operators and establish a Voronovskaja type asymptotic formula in simultaneous approximation.
{"title":"Asymptotic formula in simultaneous approximation for certain Ismail-May-Baskakov operators","authors":"Vijay Gupta, M. Rassias","doi":"10.33993/jnaat502-1235","DOIUrl":"https://doi.org/10.33993/jnaat502-1235","url":null,"abstract":"In the present paper, we introduce a modification of Ismail-May operators having weights of Baskakov basis functions. We estimate weighted Korovkin's theorem and difference estimates between two operators and establish a Voronovskaja type asymptotic formula in simultaneous approximation.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127147363","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 introduce a Stancu type operator and we prove inequalities of Rașa's type.
本文引入了一个Stancu型算子，并证明了Rașa型的不等式。
{"title":"Some inequalities for a Stancu type operator via (1,1) box convex functions","authors":"I. Gavrea, Daniel Ianoşi","doi":"10.33993/jnaat501-1242","DOIUrl":"https://doi.org/10.33993/jnaat501-1242","url":null,"abstract":"In this paper we introduce a Stancu type operator and we prove inequalities of Rașa's type.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"132 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124262333","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 work the Lipschitz subclass of the generalized grand Lebesgue space with variable exponent is defined and the error of approximation by matrix transforms in this subclass is estimated.
{"title":"Approximation by matrix transform in generalized grand Lebesgue spaces with variable exponent","authors":"A. Testici, D. Israfilov","doi":"10.33993/jnaat501-1234","DOIUrl":"https://doi.org/10.33993/jnaat501-1234","url":null,"abstract":"In this work the Lipschitz subclass of the generalized grand Lebesgue space with variable exponent is defined and the error of approximation by matrix transforms in this subclass is estimated.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127023783","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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