Multidimensional Differential Equations: Integration and Optimization through Matrix Based Algorithms

Abstract In this paper, we present the development of a novel algorithms employing matrix-based methods for the integration of dependent and independent first order multidimensional functions. Conventional methods often face limitations when dealing with the complexity and computational demands of such high-dimensional problems. Our approach leverages advanced matrix techniques to efficiently manage and solve these intricate equations, significantly reducing computational execution time and providing a robust framework for a wide range of applications in science, technology, and data science. Building on this foundational work, we introduce a pioneering branching matrix method for optimization. This method utilizes the structural advantages of matrix operations to navigate the solution space effectively, enabling the optimization of complicated systems with high-dimensional dependencies. The novel branching matrix method offers significant improvements in both speed and accuracy over conventional techniques, making it particularly suitable for large-scale optimization problems encountered in fields such as science, technology, and data science.