Advances in Linear Recurrence System

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: 15 May 2024 | Viewed by 4994

Special Issue Editors


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Guest Editor
Institute of Doctoral Studies, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
Interests: online system for evaluation on chemistry topic optimized binomial confidence intervals Szeged polynomial families of molecular descriptors evolution supervised by genetic algorithms agreement between observation and model multiplicative effect of factors
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Guest Editor
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
Interests: computer science; Mathematics

Special Issue Information

Dear Colleagues,

This Special Issue, "Advances in Linear Recurrence System", welcomes submissions from a broad interdisciplinary area. Typical interdisciplinary uses of recurrence relations are to describe the kinetics of physical, chemical, and biological processes.

We should also note that through the characteristic polynomial of the recurrence, recurrence relations are connected to the characteristic polynomials, eigenvector, eigenvalue, and eigenproblem. Many common examples of well-known concepts fall into the category of recurrence relations: binomial coefficients, factorial, Fibonacci numbers, and logistic maps. When solving an ordinary differential equation numerically, one typically encounters a recurrence relation.

In biology, some of the best-known difference equations originated from the attempt to model population dynamics. Coupled difference equations are often used to model the interaction of two or more populations, such as the Nicholson–Bailey model. Integrodifference equations are a form of recurrence relation important to spatial ecology.

In computer science, recurrence relations are also of fundamental importance in the analysis of algorithms, while in digital signal processing, recurrence relations can model feedback in a system, where outputs at one time point become inputs for a future time point.

Furthermore, recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics.

In the terms of MSC classification, recurrences appear in number theory, topological dynamics, and in numerical analysis.

Letters, short communications, original articles, and reviews covering the subject of linear recurrences are welcome.

Prof. Dr. Lorentz Jäntschi
Dr. Virginia Niculescu
Guest Editors

Manuscript Submission Information

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Keywords

  • circuit design
  • continued fraction
  • EG-eliminations
  • first order difference equations
  • functional system
  • general boundary conditions
  • heptadiagonal matrix
  • kernel methods
  • linear homogeneous recurrences
  • linear non-homogeneous recurrences
  • N-fraction
  • pentadiagonal matrix
  • recurrent models
  • recurrent double sequences
  • task graphs
  • toeplitz matrix
  • tridiagonal matrix
  • nonlinear equarions
  • derivative-free methods
  • convergence
  • multiple roots

Published Papers (4 papers)

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Research

13 pages, 514 KiB  
Article
Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications
by Ekta Sharma, Sunil Panday, Shubham Kumar Mittal, Dan-Marian Joița, Lavinia Lorena Pruteanu and Lorentz Jäntschi
Mathematics 2023, 11(21), 4512; https://doi.org/10.3390/math11214512 - 01 Nov 2023
Cited by 4 | Viewed by 707
Abstract
In this paper, we propose a new fifth-order family of derivative-free iterative methods for solving nonlinear equations. Numerous iterative schemes found in the existing literature either exhibit divergence or fail to work when the function derivative is zero. However, the proposed family of [...] Read more.
In this paper, we propose a new fifth-order family of derivative-free iterative methods for solving nonlinear equations. Numerous iterative schemes found in the existing literature either exhibit divergence or fail to work when the function derivative is zero. However, the proposed family of methods successfully works even in such scenarios. We extended this idea to memory-based iterative methods by utilizing self-accelerating parameters derived from the current and previous approximations. As a result, we increased the convergence order from five to ten without requiring additional function evaluations. Analytical proofs of the proposed family of derivative-free methods, both with and without memory, are provided. Furthermore, numerical experimentation on diverse problems reveals the effectiveness and good performance of the proposed methods when compared with well-known existing methods. Full article
(This article belongs to the Special Issue Advances in Linear Recurrence System)
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18 pages, 4302 KiB  
Article
A Cubic Class of Iterative Procedures for Finding the Generalized Inverses
by Munish Kansal, Manpreet Kaur, Litika Rani and Lorentz Jäntschi
Mathematics 2023, 11(13), 3031; https://doi.org/10.3390/math11133031 - 07 Jul 2023
Viewed by 675
Abstract
This article considers the iterative approach for finding the Moore–Penrose inverse of a matrix. A convergence analysis is presented under certain conditions, demonstrating that the scheme attains third-order convergence. Moreover, theoretical discussions suggest that selecting a particular parameter could further improve the convergence [...] Read more.
This article considers the iterative approach for finding the Moore–Penrose inverse of a matrix. A convergence analysis is presented under certain conditions, demonstrating that the scheme attains third-order convergence. Moreover, theoretical discussions suggest that selecting a particular parameter could further improve the convergence order. The proposed scheme defines the special cases of third-order methods for β=0,1/2, and 1/4. Various large sparse, ill-conditioned, and rectangular matrices obtained from real-life problems were included from the Matrix-Market Library to test the presented scheme. The scheme’s performance was measured on randomly generated complex and real matrices, to verify the theoretical results and demonstrate its superiority over the existing methods. Furthermore, a large number of distinct approaches derived using the proposed family were tested numerically, to determine the optimal parametric value, leading to a successful conclusion. Full article
(This article belongs to the Special Issue Advances in Linear Recurrence System)
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22 pages, 373 KiB  
Article
On Generalizing Divide and Conquer Parallel Programming Pattern
by Virginia Niculescu
Mathematics 2022, 10(21), 3925; https://doi.org/10.3390/math10213925 - 23 Oct 2022
Viewed by 1400
Abstract
(1) Background: Structuring is important in parallel programming in order to master its complexity, and this structuring could be achieved through programming patterns and skeletons. Divide-and-conquer computation is essentially defined by a recurrence relation that links the solution of a problem to the [...] Read more.
(1) Background: Structuring is important in parallel programming in order to master its complexity, and this structuring could be achieved through programming patterns and skeletons. Divide-and-conquer computation is essentially defined by a recurrence relation that links the solution of a problem to the solutions of subproblems of the same type, but of smaller sizes. This pattern allows the specification of different types of computations, and so it is important to provide a general specification that comprises all its cases. We intend to prove that the divide-and-conquer pattern could be generalized such that to comprise many of the other parallel programming patterns, and in order to prove this, we provide a general formulation of it. (2) Methods: Starting from the proposed generalized specification of the divide-and-conquer pattern, the computation of the pattern is analyzed based on its stages: decomposition, base-case and composition. Examples are provided, and different execution models are analyzed. (3) Results: a general functional specification is provided for a divide-and-conquer pattern and based on it, and we prove that this general formulation could be specialized through parameters’ instantiating into other classical parallel programming patterns. Based on the specific stages of the divide-and-conquer, three classes of computations are emphasized. In this context, an equivalent efficient bottom-up computation is formally proved. Associated models of executions are emphasized and analyzed based on the three classes of divide-and-conquer computations. (4) Conclusion: A more general definition of the divide-and-conquer pattern is provided, and this includes an arity list for different decomposition degrees, a level of recursion, and also an alternative solution for the cases that are not trivial but allow other approaches (sequential or parallel) that could lead to better performance. Together with the associated analysis of patterns equivalence and optimized execution models, this provides a general formulation that is useful both at the semantic level and implementation level. Full article
(This article belongs to the Special Issue Advances in Linear Recurrence System)
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12 pages, 286 KiB  
Article
Two Inverse Eigenproblems for Certain Symmetric and Nonsymmetric Pentadiagonal Matrices
by S. Arela-Pérez, Charlie Lozano, Hans Nina and H. Pickmann-Soto
Mathematics 2022, 10(17), 3054; https://doi.org/10.3390/math10173054 - 24 Aug 2022
Viewed by 955
Abstract
In this paper, we give sufficient conditions for the construction of certain symmetric and nonsymmetric pentadiagonal matrices from particular spectral information. The construction of the symmetric pentadiagonal matrix considers the extreme eigenvalues of its leading principal submatrices and a prescribed entry, and the [...] Read more.
In this paper, we give sufficient conditions for the construction of certain symmetric and nonsymmetric pentadiagonal matrices from particular spectral information. The construction of the symmetric pentadiagonal matrix considers the extreme eigenvalues of its leading principal submatrices and a prescribed entry, and the construction of the nonsymmetric pentadiagonal matrix also considers an eigenvector and two prescribed entries. Full article
(This article belongs to the Special Issue Advances in Linear Recurrence System)
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