Relaxation Method Research Articles

Wireless power transfer empowered by reconfigurable intelligent surfaces

ABSTRACT Reconfigurable Intelligent Surfaces (RIS) represent a significant advancement in hardware technology, enhancing both wireless energy and spectral efficiency within wireless communication systems. Comprising a programmable metasurface, RIS can adeptly manipulate the wavefronts of incident signals – encompassing parameters such as phase, amplitude, frequency, and polarisation – without necessitating complex signal processing techniques. The integration of RIS into wireless communication networks enables the strategic manipulation of radio waves, thereby mitigating the adverse effects associated with natural wireless propagation. This study examines a multi-user (MU) multiple-input multiple-output (MIMO) wireless power transfer (WPT) system augmented by multiple RISs, wherein the transmitter is equipped with a constant-envelope beamformer. The optimisation of the phase shifts of the RIS and the beamformer of the transmitter is conducted jointly to maximise the total power received by users. An alternating optimisation approach is proposed, utilising the semidefinite relaxation (SDR) method. Additionally, a problem is formulated to maximise the total received power while adhering to constraints on the minimum power received by users, thereby addressing issues of user fairness. To resolve this problem, an iterative algorithm based on the Alternating Direction Method of Multipliers (ADMM) is introduced. Analysis and simulation results indicate a significant enhancement in total received power when employing the proposed methodology compared to existing algorithms documented in the literature. Furthermore, in scenarios involving multiple users, a detailed analysis of the system’s transmission performance, facilitated by RIS, is conducted. The numerical results substantiate the validity of the analysis and demonstrate the efficacy of the proposed algorithms.

Recursive Quantum Relaxation for Combinatorial Optimization Problems

Quantum optimization methods use a continuous degree-of-freedom of quantum states to heuristically solve combinatorial problems, such as the MAX-CUT problem, which can be attributed to various NP-hard combinatorial problems. This paper shows that some existing quantum optimization methods can be unified into a solver to find the binary solution which is most likely measured from the optimal quantum state. Combining this finding with the concept of quantum random access codes (QRACs) for encoding bits into quantum states on fewer qubits, we propose an efficient recursive quantum relaxation method called recursive quantum random access optimization (RQRAO) for MAX-CUT. Experiments on standard benchmark graphs with several hundred nodes in the MAX-CUT problem, conducted in a fully classical manner using a tensor network technique, show that RQRAO not only outperforms the Goemans-Williamson and recursive QAOA methods, but also is comparable to state-of-the-art classical solvers. The code is available at https://github.com/ToyotaCRDL/rqrao.

Modelling heat and mass transfer in gyro-tactic hybrid nanofluid flow through a converging pipe with injection and suction

In the evolving field of nanotechnology, hybrid nanofluids play a crucial role in various engineering applications, including oil recovery, power generation and heat exchangers. This study investigates the enhancement of heat transfer in oil recovery using hybrid gyro-tactic nanofluids, addressing the growing challenge of declining extraction efficiency due to traditional methods. Efficient heat transfer is vital to maintain the mobility of crude oil during extraction and prevent gel formation, ensuring a continuous flow. The flow system, governed by nonlinear partial differential equations for momentum, energy, nanoparticle concentration, and microorganism density, is transformed into ordinary differential equations via similarity transformations. These equations are solved using the Spectral Relaxation Method, implemented in MATLAB to generate profiles for key variables such as velocity, temperature, nanoparticle concentration and microorganism density. Quantitative results show that increasing the nanoparticle volume fraction by 20% enhances heat transfer by 15%, while an inclined magnetic field reduces fluid velocity by 10%, leading to more controlled flow dynamics. The study also demonstrates that higher chemical reaction rates significantly suppress microorganism density, with a reduction of up to 25%. The novelty of this work lies in the combined analysis of several complex factors, including an inclined magnetic field, chemical reactions, variable viscosity, and microorganisms, whose interactions were not previously studied in this context. These findings provide critical insights for optimising heat transfer and improving oil recovery processes, offering solutions beyond existing technologies.

Методы психологической саморегуляции как средства профилактики неблагоприятных состояний в служебной деятельности

<p>The purpose of the study was to analyze existing methods of self-regulation for the prevention of negative conditions and, on its basis, to select the most effective method of self-regulation and test it for representatives of high-risk professions. The research conducted in 2024, involved 28 reserve officers (average age 25 (+2,5) years; 100% men). The subjects were equally divided into 2 groups: experimental and control randomly. <strong>Methods</strong>: assessment of the state of reduced performance (DORS, the scale of neuropsychic tension (T.N. Nemchin), the questionnaire of the degree of chronic fatigue (A.B. Leonova). <strong>Method of experimental influence</strong>: progressive muscle relaxation (in our modification). Two sessions were conducted, lasting 20 minutes, according to the following scheme: scanning, tension, and then relaxation of certain muscle groups and repeated scanning. <strong>Statistical methods</strong>: comparative analysis by the Mann-Whitney criterion, differential analysis ANOVA (Fisher criterion) (independent variable: muscle relaxation training — we identified 2 levels in it. 1 — without training, 2 — with training; dependent variable — unfavorable mental states), arithmetic mean of all indicators, standard deviation of indicators, comparative analysis by the Wilcoxon criterion. All calculations were performed using the STATISTICA program. <strong>Results</strong>: the most pronounced unfavorable states in reserve officers were: neuropsychic tension, monotony, satiety and stress. After the training, there was a reliable decrease in the level of monotony and the degree of chronic fatigue. <strong>Conclusion</strong>: the method of progressive muscle relaxation (in our modification) has a positive effect on such unfavorable conditions as satiety and the degree of chronic fatigue and can be used as a preventive measure for these negative conditions in reserve officers undergoing training in a military registration specialty.</p>

An Action Research for Implementing Affective Teaching Strategies for Reducing Anxiety and Increasing Motivation in EFL Beginner Learners at Campus Liberia of Universidad Nacional (UNA) in Guanacaste, Costa Rica

This study explores the impact of teacher feedback and behavior on student motivation and performance in English as a Foreign Language (EFL) classrooms. It highlights the negative effects of harsh feedback and emotional abuse, contrasted with the positive outcomes of supportive teacher-student interactions. Using a mixed-methods approach involving surveys and classroom observations of 64 beginner EFL students, the research identifies various self-motivation techniques students use, such as positive self-statements and relaxation methods. The study also evaluates the effectiveness of a pedagogic proposal incorporating five affective teaching strategies, which significantly enhance student engagement and learning. The findings underscore the importance of a positive and empathetic classroom environment, recommending that teachers and educational institutions adopt supportive and innovative teaching methods to improve EFL education outcomes.

IFAL-SLAM: an approach to inertial-centered multi-sensor fusion, factor graph optimization, and adaptive Lagrangian method

Abstract In response to the issues of low accuracy, perception degradation, and poor reliability of single-sensor SLAM technologies in complex environments, this study presents a novel IMU-centered multi-sensor fusion SLAM algorithm (IFAL-SLAM) integrating LiDAR, vision, and IMU, based on factor graph elimination optimization (\textbf{I}MU-centered multi-sensor \textbf{F}usion, \textbf{A}daptive \textbf{L}agrangian methods). The proposed system leverages a multi-factor graph model, centering on the IMU, and applies a covariance matrix to fuse visual-inertial and LiDAR-inertial odometries for bias correction, using loop closure factors for global adjustments. To minimize the optimization costs post-fusion, a sliding window mechanism is incorporated, coupled with a QR decomposition elimination method based on Householder transformation to convert the factor graph into a Bayesian network. Finally, an adaptive Lagrangian relaxation method is proposed, employing matrix-form penalty parameters and adaptive strategies to enhance convergence speed and robustness under high rotational dynamics.Experimental results indicate that the proposed algorithm achieves absolute trajectory errors of approximately 0.58 m and 0.24 m in large and small complex scenes, respectively, surpassing classic algorithms in terms of accuracy and reliability.

Explore the Governance Method of Network Environment from the Stratification of the Internet

Due to the maturity of electronic media communication and the increasingly strong requirements of people for boundaries in real life, more and more people are eager to find their belonging from the Internet, which has led to the formation of today's Internet stratification. Under the influence of the lack of barriers to access to Internet information, the poor network environment has also become a major network chaos that has been criticized by the Internet people. Based on the survey results, the phenomenon of the network circle is studied, the reasons and characteristics of the formation of the network circle are analyzed, and two Internet environment governance methods of relaxation and barrier improvement are proposed.

Investigating the Numerical Stability of Dynamic Relaxation Methods with Automatic Load-Increment Schemes to Improve Snap-Back Prediction

Automatic load incrementation schemes tailor-made for the dynamic relaxation (DR) method, including the minimum residual force (MRF) scheme, the minimum residual energy (MRE) scheme, the minimum displacement increment (MDI) scheme, and the minimum kinetic energy (MKE) scheme, are commonly used to capture snapping phenomena in post-buckling structural analysis. Analogous to DR incorporating arc-length constraints, these combined numerical schemes effectively handle snap-through problems, but their capability to address snap-back behavior remains largely unverified. This study aims to investigate the numerical stability of the DR-MRF, DR-MRE, DR-MDI, and DR-MKE methods. Unexpectedly, all these methods are found to be unconditionally unstable and inadequate for snap-back problems. Key findings include (1) the establishment of a general finite-difference equation in canonical form for these four methods, with numerical stability governed by the spectral radius of the recursive amplification matrix; and (2) the discovery of a special property of the tangent stiffness matrix, where a specific minor becomes negative during snap-back, causing the spectral radius to exceed one and leading to instability. These conclusions are supported by numerical verifications on nonlinear springs, trusses, and frames. Simulations consistently demonstrate the reliability of these findings across different structural configurations.

A Randomized Projection Relaxed Method for Mixing Function Design to Enhance MWC Performance

A Randomized Projection Relaxed Method for Mixing Function Design to Enhance MWC Performance

Effect of the Protic vs. Non-Protic Molecular Environment on the cis to trans Conformation Change of Phototrexate Drug.

The therapeutical applicability of the anticancer drug phototrexate, a photoswitchable derivative of the antimetabolite dihydrofolate reductase inhibitor methotrexate, highly depends on the stability of its bioactive isomer. Considering that only the cis configuration of phototrexate is bioactive, in this work, the effect of the molecular environment on the stability of the cis isomer of this drug has been investigated. UV-vis absorption and fluorescence-based solvent relaxation methods have been used. Protic methanol and non-protic dimethylsulfoxide were used as medium-ranged permittivity solvents. The results showed a decreased rate of cis → trans conversion and enhanced stabilities of the cis isomer in methanol. Temperature-dependent measurements of the isomerization rate reflect the increased activation energy in methanol.

Efficient relaxation scheme for the SIR and related compartmental models

In this paper, we introduce a novel numerical approach for approximating the Susceptible–Infectious–Recovered (SIR) model in epidemiology. Our method enhances the existing linearization procedure by incorporating a suitable relaxation term to tackle the transcendental equation of nonlinear type. Developed within the continuous framework, our relaxation method is explicit and easy to implement, relying on a sequence of linear differential equations. This approach yields accurate approximations in both discrete and analytical forms. Through rigorous analysis, we prove that, with an appropriate choice of the relaxation parameter, our numerical scheme is non-negativity-preserving; moreover, it is strongly convergent to the true solution. We also extend the applicability of our relaxation method to handle some variations of the traditional SIR model. Finally, we present numerical examples using simulated data to demonstrate the effectiveness of our proposed method.

Air-rail intermodal collaborative decision making for aircraft recovery from airport station disruption

During airport disruptions caused by capacity shortages, it is crucial for airlines to have an effective recovery plan to minimize losses and prevent the spread of disruptions and delays. This study proposes an air-rail intermodal Collaborative Decision Making (CDM) approach, which recommends incorporating High-Speed Railway (HSR) transportation into the management of aircraft recovery from airport station disruptions. The structural properties of the proposed model indicate employing a Lagrangian relaxation with subgradient methods to effectively obtain near-optimal solutions. A framework for developing Lagrangian heuristics (heuristics based on Lagrangian relaxation and sub-gradient optimization) is proposed to obtain solutions. Additionally, the study proposes a modified aircraft recovery model considering the downstream effects of flight delays and cancellations during the airport disruption recovery period and introduces slack variables to linearize the model. The computational experiments conducted in this study demonstrate the effectiveness of the proposed air-rail intermodal strategy for managing airport disruptions. Experiments conducted on large-scale datasets demonstrate that the Lagrangian Relaxation method outperforms both the Benders method and the Genetic Algorithm in terms of both computational speed and solution quality.This research provides valuable insights into the management of airport disruptions and offers practical solutions for airlines to mitigate the impact of capacity shortages.

A platinum degradation reaction model for the high-speed computation in the integrated fuel cell system simulator and its validation by the accelerated stress test data of the commercial fuel cells

The modeling and numerical methods to estimate the Pt degradation rate were developed for a year-long and system-scale durability simulation in an acceptable accuracy and computational time. The model could reproduce the published voltage cycle results of the MEA of 2nd-generation MIRAI in the relative error of 0.38 % and 180 times faster computational speed than the actual time with a standard laptop. It was confirmed from the voltage cycle simulation that even the small number of the coarse particles considerably accelerated the Pt degradation. Modeling and numerical methods: The reaction rate equations of the reaction rates among Pt, PtO, and Pt2+ in the literature [1][2] were employed and the mass transport equations among the particle sizes discretized at 0.1 nm were added. The reaction rate constant of Pt → Pt2+ + 2e- [1] was optimized to reproduce the experimental results and the other parameters were unchanged from those in the literature [1][2]. The particle size distribution (PSD) of Pt in the cathode catalyst layer measured with the MEA of 2nd generation MIRAI by nano X-ray CT [3] is shown in Fig. 1. The coarse particles with the diameters of 8.7, 13.2, 20.6, 44.1, and 85.1 nm were observed in the PSD. The PSD data were pre-processed to create the initial particle density distribution (PDD) data by the following process: Removal of the particles smaller than 1 nm; cubic interpolation of the PSD data by the 0.1 nm discretized diameter from the 0.5 nm bin width; removal of the particles with less than 1 count; scaling particle numbers at each diameter to derive the PDDs with the ECSA of 90 m2-Pt/m2 [3] and the loading of 0.17 mg-Pt/cm2 [4] in the cathode catalyst layer of 2nd-generation MIRAI. The derived PDDs are shown in Figs. 2(a)–2(f). The PDDs calculated from the ECSA and loading agreed when the particles of 85.1 and 44.1 nm were removed as shown in Fig.2(c). They were adopted as the initial PDD in this study. For high-speed computation in the integrated fuel cell system simulator the current authors had presented [5], the non-iterative solver by the explicit Euler method with the low-pass filter by the discrete first order lag [6] was developed (solver 1). Fig. 3 shows the solutions obtained by the developed solver, the 4th-order Runge Kutta solver (solver 2), and the iterative solver with the implicit Euler method and relaxation method [6] (solver 3). The numerical oscillation was observed in the solution by solver 2. The solution by solver 1 was similar to the converged solution by solver 3. The model was implemented on MATLAB/Simulink environment as the open-loop simulator as shown in Fig. 4. The published voltage cycle conditions of 0.95–0.6 V, 80 °C, 100 % relative humidity [4] were input to the simulator. The impact of the coarse particle was also investigated by setting the PDD in Fig. 2(f), where all of the coarse particles were removed, as the initial value and compared with the case in Fig. 2(c). Results and discussion: Fig. 5 shows the experimental and simulation results of the ECSA fractional retention during the 30,000 voltage cycles. The calculation results by the solvers 1 and 3 agreed in the maximum relative error of 1.3×10-4 %. The maximum relative error between the experimental and calculated ECSA at 500, 1000, 3000, 10000, and 30000 cycles was 0.38 % and the acceptable accuracy was confirmed. The computational time using solver 1 for 180,000 s of the actual experimental time was 1002 s with a standard laptop (Toshiba dynabook G83/HS; Intel Core i7-1165G7 @ 2.80 GHz / 2.80 GHz CPU; 16 GB RAM). In case of the initial PDD of Fig. 2(f), the ECSA fractional retention at 30,000 voltage cycles was improved from 44.7 to 69.4 %. Even the small number of the coarse particles considerably accelerated the Pt degradation.

Efficiency of the dynamic relaxation method in the stabilisation process of bridge and building frame

More and more complex Civil Engineering problems are being considered in computational mechanics with the invention of high-quality computing techniques. In addition, the computational cost and storage requirement for complex and or large structures have increased dramatically, leading to an increased interest in removing the difficulties using any form of parallel computing. The process of applying the preload for parallel computing to any unstable structures is called a stabilising process, such as the Dynamic Relaxation Method (DRM) is one. This method minimises the energy by a simple vector iteration technique, which ultimately leads the structure to a static equilibrium state. The present study aims to highlight the utility of the DRM in the stabilisation process for small structures like building frames and large and or complicated structures such as bridges before actual transient analysis. Therefore, the present manuscript discusses the computational cost, CPU runtime, multiple increases of mass and rigid body displacement of building frames and bridges. The DRM allows an explicit solver to conduct a dynamic analysis by increasing the damping until the kinetic energy drops to a proposed value. The simulation of the DRM starts to find the equilibrium state with minimal dynamic effect, which is required to apply at the beginning of the solution phase to obtain the initial stress and displacement field before the start of the actual analysis.

Auxiliary relaxation method to derive thermodynamically consistent phase field models with constraints and structure preserving numerical approximations

In this paper, we introduce a novel approach for formulating phase field models with constraints. The main idea is to introduce auxiliary variables that regularize and gradually dissipate constraint deviations of the phase variables, which we name the auxiliary relaxation method. It integrates seamlessly with the energy variational framework to ensure thermodynamic consistency in the resulting phase field models. Unlike traditional penalty methods, which introduce high stiffness due to large penalty parameters to enforce constraints in phase field models, our approach reduces system stiffness, allowing larger time step sizes when solving phase field models with constraints numerically, thus improving numerical accuracy and efficiency. We demonstrate the effectiveness and robustness of the proposed auxiliary relaxation method by applying it across several scenarios to derive thermodynamically consistent phase field models with constraints. Furthermore, we introduce a general second-order implicit-explicit Crank-Nicolson scheme, combining the relaxed scalar auxiliary variable method with a stabilization technique to solve these models. Through extensive numerical tests, we validate the capability of our modeling and numerical framework to reliably simulate complex dynamics governed by phase field equations with constraints.

17O NMR relaxation measurements for investigation of molecular dynamics in static solids using sodium nitrate as a model compound

17O NMR methods are emerging as a powerful tool for determination of structure and dynamics in materials and biological solids. We present experimental and theoretical frameworks for measurements of 17O NMR relaxation times in static solids focusing on the excitation of the central transition of the 17O spin 5/2 system. We employ 17O-enriched NaNO3 as a model compound, in which the nitrate oxygen atoms undergo 3-fold jumps. Rotating frame (T1ρ), transverse (T2) and longitudinal (T1) relaxation times as well as line shapes were measured for the central transition in the 280 to 195 K temperature range at 14.1 and 18.8 T field strengths. We conduct experimental and theoretical comparison between different relaxation methods and demonstrate the advantage of combining data from multiple relaxation time and line shape measurements to obtain a more accurate determination of the dynamics as compared to either of the techniques alone. The computational framework for relaxation of spin 5/2 nuclei is developed using the numerical integration of the Liouville − von Neumann equation.

A pressure-jump EPR system to monitor millisecond conformational exchange rates of spin-labeled proteins.

Site-directed spin labeling electron paramagnetic resonance (SDSL-EPR) using nitroxide spin labels is a well-established technology for mapping site-specific secondary and tertiary structure and for monitoring conformational changes in proteins of any degree of complexity, including membrane proteins, with high sensitivity. SDSL-EPR also provides information on protein dynamics in the timescale of ps-μs using continuous wave lineshape analysis and spin lattice relaxation time methods. However, the functionally important time domain of μs-ms, corresponding to large-scale protein motions, is inaccessible to those methods. To extend SDSL-EPR to the longer time domain, the perturbation method of pressure-jump relaxation is implemented. Here, we describe a complete high-pressure EPR system at Q-band for both static pressure and ms-timescale pressure-jump measurements on spin-labeled proteins. The instrument enables pressure jumps both up and down from any holding pressure, ranging from atmospheric pressure to the maximum pressure capacity of the system components (~3500 bar). To demonstrate the utility of the system, we characterize a local folding-unfolding equilibrium of T4 lysozyme. The results illustrate the ability of the system to measure thermodynamic and kinetic parameters of protein conformational exchange on the ms timescale.

A robust optimization approach for steeling-continuous casting charge batch planning with uncertain slab weight

The volatility of slab weight in steelmaking-continuous casting (SCC) production, attributed to factors such as flexible order demand, is addressed in this paper. A robust optimization mathematical model for charge batch planning (CBP) with uncertain slab weight is established, and a collaborative optimization method using the surrogate Lagrangian relaxation (SLR) framework and improved objective feasibility pump (IOFP) is developed to solve the problem. In the SLR method, new step-size updating conditions are developed, eliminating the need for pre-estimating the optimal dual value. Additionally, only a subset of subproblems that satisfy the optimality conditions of the surrogate needs to be solved to overcome the low optimization efficiency resulting from oscillations in the feasible domain during internal searches in traditional Lagrangian relaxation (LR) methods. The IOFP method is employed to match the structure of the subproblem model of 0–1 mixed integer programming (MIP). During the search for integer solutions, a weighted objective function is added to the auxiliary model to improve the quality of solutions. Furthermore, it combines a variable neighborhood branching method to prevent the algorithm from entering into cycles. Finally, the effectiveness of the proposed model and the performance of the algorithm are validated through simulation experiments.

Optimizing Machine Learning Models with Data-level Approximate Computing: The Role of Diverse Sampling, Precision Scaling, Quantization and Feature Selection Strategies

Efficiency, low-power consumption, and real-time processing in embedded machine learning implementations are critical, particularly for models deployed in environments with large-scale data processing and resource-constrained environments. This paper investigates the application of approximate computing techniques as a viable solution to reduce computational complexity and optimize machine learning models, focusing on two widely used supervised machine learning models: k-nearest neighbors (KNN) and support vector machines (SVM). Although many studies compare machine learning classification techniques, the combined use of optimization strategies remains underexplored. Specifically, the combined utilization of feature selection, sampling, quantization, precision scaling, and relaxation methods for the purpose of optimizing and acquiring training and validation data is underexplored, particularly within the context of medical diagnosis datasets. In this paper, we propose a framework that uses data-level approximate computing techniques, including by diverse sampling strategies, precision scaling, quantization, and feature selection methods, to evaluate the impact of these techniques on the computational efficiency and accuracy of KNN and SVM models. Experimental results demonstrate that with careful application of approximate computing strategies, especially in critical applications such as medical diagnosis, it is possible to achieve considerable gains in efficiency while maintaining acceptable levels of accuracy. The combined application of these methods by selecting 3 features and quantizing the data values to 8 levels, then applying random sampling with 30% reductions and scaling the precision at 5 bits resulted in reductions of 87.5% in computation, 76.9% in memory usage, and 17% in delay, without any degradation in accuracy, as validated by tenfold cross-validation, Training Data validation, and full dataset validation. This study confirms the potential of approximate computing to optimize machine learning workflows, making it particularly suitable for applications with limited computational resources. The source code is publicly available online https://github.com/AyadMDalloo/DatalvlAxC.

Exploring Solvents and Sensitivity of Systems of Linear Equations Arising from Real World Phenomena Via Optimal Successive Over-relaxation Method

This study places a significant emphasis on assessing the efficiency of numerical methods, specifically in the context of solving linear equations of the form <i>Ax</i> = <i>b</i>, where <I>A</I> is a square matrix, <i>x</i> is a solvent vector, and <i>b</i> is a column vector representing real-world phenomena. The investigation compares the effectiveness of the Refined Successive Over Relaxation (RSOR) method to the standard Successive Over-relaxation (SOR) method. The core evaluation criteria encompass computational time (in seconds), convergence behavior, and the number of iterations necessary to approximate the solvents of five distinct real-world phenomena: Model Problem 1 (MP1) involving an Electrical Circuit, Model Problem 2 (MP2) focusing on Beam Deflection, Model Problem 3 (MP3) addressing Damped Vibrations of a Stretching Spring, Model Problem 4 (MP4) dealing with Linear Springs and Masses, and Model Problem 5 (MP5) focusing on Temperature Distribution on Heated Plate. The RSOR method generally outperforms the SOR method, particularly with a constant relaxation parameter (<i>ω</i>) in the range 1.0 < <i>ω</i> < 1.2. The RSOR method is favored for its robustness and efficiency with less need for fine-tuning <i>ω</i>, whereas the SOR method can achieve superior performance if the optimal <i>ω</i> is found, although this often requires time-consuming trial and error. Despite the potential for better performance with an optimal <i>ω</i>, the RSOR method’s consistent results make it the more practical choice in many cases. The study also explores the stability of the systems of linear equations arising from these phenomena by calculating their condition numbers (<I>K</I>(<I>A</I>)). More interestingly, the results reveal that all systems MP1 to MP4 exhibit instability when subjected to even modest perturbations, shedding light on potential challenges in their solvents. This research not only underscores the advantages of the RSOR method but also emphasizes the importance of understanding the stability of numerical solvents in the context of real-world problems. Additionally, the results for MP5 demonstrates that tiny changes to the original matrix’s coefficients have no effect on the desired solvent because the perturbed matrix’s condition number is the same as the original matrix’s, making the problem well-structured. The problem becomes ill-conditioned if there is an increase or decrement to the matrix’s coefficients that is bigger than 10<sup>−5</sup>. In summary, Sparse systems are sensitive to perturbations, resulting in instability. If the tolerance |<i>k</i>(<I>A</I><sup>0</sup><sub><i>i</i></sub>) − <i>k</i>(<i>A<sub>i</sub></i>)| > 10<sup>−5</sup> for all positive integers <i>i</i>, then the problem becomes poorly structured.