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Algorithms
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Recent Advances in Numerical Algorithms and Their Applications
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Recent Advances in Numerical Algorithms and Their Applications

A special issue of Algorithms (ISSN 1999-4893). This special issue belongs to the section "Algorithms for Multidisciplinary Applications".

Deadline for manuscript submissions: 30 November 2026 | Viewed by 3166

Special Issue Editor

Dr. Stoil I. Ivanov

E-Mail Website
Guest Editor
Faculty of Physics and Technology, University of Plovdiv Paisii Hilendarski, 24 Tzar Asen, 4000 Plovdiv, Bulgaria
Interests: iterative methods; numerical algorithms; convergence analysis; polynomial zeros; phase transitions
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

It is well known that the numerical algorithms are amongst the most powerful techniques for solving various kinds of mathematical tasks. The continuous advancement of computer technologies further impels the fast development of numerical algorithms because of their easy computer implementation, which, in turn, makes them highly effective and, therefore, the preferred tool for solving problems in numerous branches of natural sciences, engineering, finance, and education.

The aim of this Special Issue is to provide an advanced forum for high-value scientific studies on numerical algorithms and their numerous applications. In particular, works dedicated to the construction, analysis, interdisciplinary applications, and computer implementations of original numerical algorithms are greatly appreciated. Some expository and survey articles on the topic could be considered for publication as well.

Dr. Stoil I. Ivanov
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Algorithms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • numerical algorithms
  • iterative methods
  • convergence analysis
  • stability analysis
  • dynamical analysis
  • computational efficiency
  • error analysis
  • real-world applications

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Published Papers (6 papers)

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Research

21 pages, 5112 KB  
Article
A Scalable Framework with Modified Loop-Based Multi-Initial Simulation and Numerical Algorithm for Classifying Brain-Inspired Nonlinear Dynamics with Stability Analysis
by Haseeba Sajjad, Adil Jhangeer and Lubomír Říha
Algorithms 2025, 18(12), 805; https://doi.org/10.3390/a18120805 - 18 Dec 2025
Viewed by 64
Abstract
The principal problem with the analysis of nonlinear dynamical systems is that it is repetitive and inefficient to simulate every initial condition and parameter configuration individually. This not only raises the cost of computation but also constrains scalability in the exploration of a [...] Read more.
The principal problem with the analysis of nonlinear dynamical systems is that it is repetitive and inefficient to simulate every initial condition and parameter configuration individually. This not only raises the cost of computation but also constrains scalability in the exploration of a large parameter space. To solve this, we restructured and extended the computational framework so that variation in the parameters and initial conditions can be automatically explored in a unified structure. This strategy is implemented in the brain-inspired nonlinear dynamical model that has three parameters and multiple coupling strengths. The framework enables detailed categorization of the system responses through statistical analysis and through eigenvalue-based assessment of the stability by considering multiple initial states of the system. These results reveal clear differences between periodic, divergent, and non-divergent behavior and show the extent to which the strength of the coupling kij can drive transitions to stable periodic behavior under all conditions examined. This method makes the analysis process easier, less redundant, and provides a scalable tool to study nonlinear dynamics. In addition to its computational benefits, the framework provides a general method that can be generalized to models with more parameters or more complicated network structures. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Algorithms and Their Applications)
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23 pages, 1181 KB  
Article
Robust Regularized Recursive Least-Squares Algorithm Based on Third-Order Tensor Decomposition
by Radu-Andrei Otopeleanu, Constantin Paleologu, Jacob Benesty, Cristian-Lucian Stanciu, Laura-Maria Dogariu and Ruxandra-Liana Costea
Algorithms 2025, 18(12), 768; https://doi.org/10.3390/a18120768 - 5 Dec 2025
Viewed by 201
Abstract
The decomposition-based adaptive filtering algorithms have recently gained increasing interest due to their capability to reduce the parameter space. In this context, the third-order tensor (TOT) decomposition technique reformulates the conventional approach that involves a single (usually long) adaptive filter by using a [...] Read more.
The decomposition-based adaptive filtering algorithms have recently gained increasing interest due to their capability to reduce the parameter space. In this context, the third-order tensor (TOT) decomposition technique reformulates the conventional approach that involves a single (usually long) adaptive filter by using a combination of three shorter filters via the Kronecker product. This leads to a twofold gain in terms of both performance and complexity. Thus, it can be applied efficiently when operating with more complex algorithms, like the recursive least-squares (RLS) approach. In this paper, we develop an RLS-TOT algorithm with improved robustness features due to a novel regularization method that considers the contribution of the external noise and the so-called model uncertainties (which are related to the system). Simulation results obtained in the framework of echo cancelation support the performance of the proposed algorithm, which outperforms the existing RLS-TOT counterparts, as well as the conventional RLS algorithm that uses the specific regularization technique. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Algorithms and Their Applications)
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42 pages, 3632 KB  
Article
Logistic Biplots for Ordinal Variables Based on Alternating Gradient Descent on the Cumulative Probabilities, with an Application to Survey Data
by Julio C. Hernández-Sánchez, Laura Vicente-González, Elisa Frutos-Bernal and José L. Vicente-Villardón
Algorithms 2025, 18(11), 718; https://doi.org/10.3390/a18110718 - 14 Nov 2025
Viewed by 263
Abstract
Biplot methods provide a framework for the simultaneous graphical representation of both rows and columns of a data matrix. Classical biplots were originally developed for continuous data in conjunction with principal component analysis (PCA). In recent years, several extensions have been proposed for [...] Read more.
Biplot methods provide a framework for the simultaneous graphical representation of both rows and columns of a data matrix. Classical biplots were originally developed for continuous data in conjunction with principal component analysis (PCA). In recent years, several extensions have been proposed for binary and nominal data. These variants, referred to as logistic biplots (LBs), are based on logistic rather than linear response models. However, existing formulations remain insufficient for analyzing ordinal data, which are common in many social and behavioral research contexts. In this study, we extend the biplot methodology to ordinal data and introduce the ordinal logistic biplot (OLB). The proposed method estimates row scores that generate ordinal logistic responses along latent dimensions, whereas column parameters define logistic response surfaces. When these surfaces are projected onto the space defined by the row scores, they form a linear biplot representation. The model is based on a framework, leading to a multidimensional structure analogous to the graded response model used in Item Response Theory (IRT). We further examine the geometric properties of this representation and develop computational algorithms—based on an alternating gradient descent procedure—for parameter estimation and computation of prediction directions to facilitate visualization. The OLB method can be viewed as an extension of multidimensional IRT models, incorporating a graphical representation that enhances interpretability and exploratory power. Its primary goal is to reveal meaningful patterns and relationships within ordinal datasets. To illustrate its usefulness, we apply the methodology to the analysis of job satisfaction among PhD holders in Spain. The results reveal two dominant latent dimensions: one associated with intellectual satisfaction and another related to job-related aspects such as salary and benefits. Comparative analyses with alternative techniques indicate that the proposed approach achieves superior discriminatory power across variables. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Algorithms and Their Applications)
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30 pages, 1770 KB  
Article
A Hybrid Numerical–Semantic Clustering Algorithm Based on Scalarized Optimization
by Ana-Maria Ifrim and Ionica Oncioiu
Algorithms 2025, 18(10), 607; https://doi.org/10.3390/a18100607 - 27 Sep 2025
Viewed by 715
Abstract
This paper addresses the challenge of segmenting consumer behavior in contexts characterized by both numerical regularities and semantic variability. Traditional models, such as RFM-based segmentation, capture the transactional dimension but neglect the implicit meanings expressed through product descriptions, reviews, and linguistic diversity. To [...] Read more.
This paper addresses the challenge of segmenting consumer behavior in contexts characterized by both numerical regularities and semantic variability. Traditional models, such as RFM-based segmentation, capture the transactional dimension but neglect the implicit meanings expressed through product descriptions, reviews, and linguistic diversity. To overcome this gap, we propose a hybrid clustering algorithm that integrates numerical and semantic distances within a unified scalar framework. The central element is a scalar objective function that combines Euclidean distance in the RFM space with cosine dissimilarity in the semantic embedding space. A continuous parameter λ regulates the relative influence of each component, allowing the model to adapt granularity and balance interpretability across heterogeneous data. Optimization is performed through a dual strategy: gradient descent ensures convergence in the numerical subspace, while genetic operators enable a broader exploration of semantic structures. This combination supports both computational stability and semantic coherence. The method is validated on a large-scale multilingual dataset of transactional records, covering five culturally distinct markets. Results indicate systematic improvements over classical approaches, with higher Silhouette scores, lower Davies–Bouldin values, and stronger intra-cluster semantic consistency. Beyond numerical performance, the proposed framework produces intelligible and culturally adaptable clusters, confirming its relevance for personalized decision-making. The contribution lies in advancing a scalarized formulation and hybrid optimization strategy with wide applicability in scenarios where numerical and textual signals must be analyzed jointly. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Algorithms and Their Applications)
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22 pages, 398 KB  
Article
An Improved Convergence Analysis of a Multi-Step Method with High-Efficiency Indices
by Santhosh George, Manjusree Gopal, Samhitha Bhide and Ioannis K. Argyros
Algorithms 2025, 18(8), 483; https://doi.org/10.3390/a18080483 - 4 Aug 2025
Viewed by 565
Abstract
A multi-step method introduced by Raziyeh and Masoud for solving nonlinear systems with convergence order five has been considered in this paper. The convergence of the method was studied using Taylor series expansion, which requires the function to be six times differentiable. However, [...] Read more.
A multi-step method introduced by Raziyeh and Masoud for solving nonlinear systems with convergence order five has been considered in this paper. The convergence of the method was studied using Taylor series expansion, which requires the function to be six times differentiable. However, our convergence study does not depend on the Taylor series. We use the derivative of F up to two only in our convergence analysis, which is presented in a more general Banach space setting. Semi-local analysis is also discussed, which was not given in earlier studies. Unlike in earlier studies (where two sets of assumptions were used), we used the same set of assumptions for semi-local analysis and local convergence analysis. We discussed the dynamics of the method and also gave some numerical examples to illustrate theoretical findings. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Algorithms and Their Applications)
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22 pages, 346 KB  
Article
Two Extrapolation Techniques on Splitting Iterative Schemes to Accelerate the Convergence Speed for Solving Linear Systems
by Chein-Shan Liu and Botong Li
Algorithms 2025, 18(7), 440; https://doi.org/10.3390/a18070440 - 18 Jul 2025
Viewed by 546
Abstract
For the splitting iterative scheme to solve the system of linear equations, an equivalent form in terms of descent and residual vectors is formulated. We propose an extrapolation technique using the new formulation, such that a new splitting iterative scheme (NSIS) can be [...] Read more.
For the splitting iterative scheme to solve the system of linear equations, an equivalent form in terms of descent and residual vectors is formulated. We propose an extrapolation technique using the new formulation, such that a new splitting iterative scheme (NSIS) can be simply generated from the original one by inserting an acceleration parameter preceding the descent vector. The spectral radius of the NSIS is proven to be smaller than the original one, and so has a faster convergence speed. The orthogonality of consecutive residual vectors is coined into the second NSIS, from which a stepwise varying orthogonalization factor can be derived explicitly. Multiplying the descent vector by the factor, the second NSIS is proven to be absolutely convergent. The modification is based on the maximal reduction of residual vector norm. Two-parameter and three-parameter NSIS are investigated, wherein the optimal value of one parameter is obtained by using the maximization technique. The splitting iterative schemes are unified to have the same iterative form, but endowed with different governing equations for the descent vector. Some examples are examined to exhibit the performance of the proposed extrapolation techniques used in the NSIS. Full article
(This article belongs to the Special Issue Recent Advances in Numerical Algorithms and Their Applications)
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