In several fields of scientific computing such as scientific simulations, machine learning, data analysis and optimization, large-scale sparse linear systems with huge numbers of variables and equations are becoming increasingly common. This study aims to introduce an innovative and unified iterative approach for efficiently solving systems of linear equations of the type . The method is designed to reduce computational time, memory usage and improve scalability for solving such large-scale, sparse problems.
A new iterative method is proposed with an emphasis on optimal parameter selection to achieve fast convergence and accurate solutions. Numerical experiments and graphical analysis are conducted to evaluate accuracy, convergence and efficiency. Comparative studies with existing techniques are performed to validate the performance of the proposed approach. Theoretical convergence criteria are also established through detailed mathematical analysis.
The proposed iterative technique demonstrates improved accuracy and faster convergence compared to conventional methods. Numerical results confirm the method’s efficiency, while graphical analysis illustrates its scalability and reduced computational cost. The comparative study highlights the effectiveness of the approach across different large-scale sparse linear systems.
The method’s performance has been tested on benchmark problems and simulated datasets. Further investigations may be required to assess robustness in extremely large-scale or ill-conditioned systems, as well as applications to more diverse domains in engineering and computational sciences.
This study presents a unified and efficient iterative framework for solving large sparse linear systems, which has potential applications in multiple disciplines, including civil engineering (structural analysis and fluid dynamics), aerospace engineering (aerodynamics and structural analysis), chemical engineering (process control and transport phenomena), computer graphics, 3D modeling, animation and climate modeling. The method combines theoretical rigor and computational efficiency, offering a novel solution for large-scale scientific computing challenges.
