Browsing by Author "James Nkereuwem Essien"

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A THIRD REFINEMENT OF JACOBI METHOD FOR SOLUTIONS TO SYSTEM OF LINEAR EQUATIONS
(Federal University, Dutsin Ma, Nigeria, 2023-10-15) Khadeejah James Audu; James Nkereuwem Essien; Abraham Baba Zhiri; Aliyu Rasheed Taiwo
Solving linear systems of equations stands as one of the fundamental challenges in linear algebra, given their prevalence across various fields. The demand for an efficient and rapid method capable of addressing diverse linear systems remains evident. In scenarios involving large and sparse systems, iterative techniques come into play to deliver solutions. This research paper contributes by introducing a refinement to the existing Jacobi method, referred to as the "Third Refinement of Jacobi Method." This novel iterative approach exhibits its validity when applied to coefficient matrices exhibiting characteristics such as symmetry, positive definiteness, strict diagonal dominance, and 𝑀 -matrix properties. Importantly, the proposed method significantly reduces the spectral radius, thereby curtailing the number of iterations and substantially enhancing the rate of convergence. Numerical experiments were conducted to assess its performance against the original Jacobi method, the second refinement of Jacobi, and the Gauss-Seidel method. The outcomes underscore the "Third Refinement of Jacobi" method's potential to enhance the efficiency of linear system solving, thereby making it a valuable addition to the toolkit of numerical methodologies in scientific and engineering domains.
An Accelerated Iterative Technique: Third Refinement of Gauss–Seidel Algorithm for Linear Systems
(Multidisciplinary Digital Publishing Institute, Switszerland, 2023-04-28) Khaddejah James Audu; James Nkereuwem Essien
Obtaining an approximation for the majority of sparse linear systems found in engineering and applied sciences requires efficient iteration approaches. Solving such linear systems using iterative techniques is possible, but the number of iterations is high. To acquire approximate solutions with rapid convergence, the need arises to redesign or make changes to the current approaches. In this study, a modified approach, termed the “third refinement” of the Gauss-Seidel algorithm, for solving linear systems is proposed. The primary objective of this research is to optimize for convergence speed by reducing the number of iterations and the spectral radius. Decomposing the coefficient matrix using a standard splitting strategy and performing an interpolation operation on the resulting simpler matrices led to the development of the proposed method. We investigated and established the convergence of the proposed accelerated technique for some classes of matrices. The efficiency of the proposed technique was examined numerically, and the findings revealed a substantial enhancement over its previous modifications
Numerical Assessment of Some Almost Runge-Kutta and Runge-Kutta Methods for First- Order Differential Equation
(Maths Model Research Group. FUT, Minna, Nigeria, 2025-02-20) Khadeejah James Audu; Muideen Taiwo Kharashi; Yusuph Amuda Yahaya; James Nkereuwem Essien; Abraham Ajeolu Oluwasegun
Numerical methods play a critical role in solving first-order Ordinary Differential Equations (ODEs), with their efficiency and accuracy being key considerations. This study conducts a detailed comparative analysis of four numerical schemes: the Almost Runge-Kutta fourth-order scheme (ARK4), the Almost Runge-Kutta third-order fourth-stage scheme (ARK34), the classical Runge- Kutta fourth-order scheme (RK4), and the Runge-Kutta fourth-order fifth-stage scheme (RK45). The methods are evaluated based on their computational accuracy, error behavior, and efficiency. Numerical experiments reveal that all methods deliver highly accurate solutions, with ARK4 emerging as the most effective due to its lower computational complexity. ARK4 demonstrates superior performance in achieving minimal absolute error with reduced computational effort, making it a suitable choice for solving first-order ODEs. This study highlights ARK4 as a viable alternative to conventional Runge-Kutta methods for practical applications.
Refinements of Some Iterative Methods for Solving Linear System of Equations
(Nigerian Women in Mathematics (NWM), 2023-06-20) Khadeejah James Audu; James Nkereuwem Essien
The efficient and accurate solution of linear systems of equations is a fundamental problem in various scientific and engineering fields. In this study, we focus on the refinements of iterative methods for solving linear systems of equations (▁A k=▁b). The research proposes two methods namely, third refinement of Jacobi method (TRJ) and third refinement of Gauss-Seidel (TRGS) method, which minimizes the spectral radius of the iteration matrix significantly when compared to any of the initial refinements of Jacobi and Gauss-Seidel methods. The study explores ways to optimize their convergence behavior by incorporating refinement techniques and adaptive strategies. These refinements exploit the structural properties of the coefficient matrix to achieve faster convergence and improved solution accuracy. To evaluate the effectiveness of the proposed refinements, numerical examples were tested to see the efficiency of the proposed TRJ and TRGS on a diverse set of linear equations. We compare the convergence behavior, computational efficiency, and solution accuracy of the refined iterative methods against their traditional counterparts. The experimental results demonstrate significant improvements in terms of convergence rate and computational efficiency when compared to their initial refinements. The proposed refinements have the potential to contribute to the development of more efficient and reliable solvers for linear systems, benefiting various scientific and engineering applications.