(498) the numerical solution for the roots polynomials - abdel karim, abbas I. (the institute of national planning, 2018)
This study examines the numerical solution of polynomial roots as a fundamental topic in numerical analysis, applied mathematics, and operations research. The study focuses on mathematical techniques used to determine both real and complex roots of polynomial equations of different degrees. It is based on the premise that direct analytical solutions become increasingly difficult or impractical as polynomial degrees increase, making numerical procedures more effective and applicable alternatives. The study begins by presenting the theoretical framework of polynomial equations and the characteristics of their roots, emphasizing the relationship between equation coefficients and the nature of possible solutions. It explains that root determination is not merely a theoretical mathematical exercise but also an essential component in solving practical problems in engineering, economics, physics, and mathematical programming. The analysis further addresses several numerical methods used for root estimation through iterative procedures designed to approximate root values progressively until acceptable accuracy levels are achieved. The study explains convergence mechanisms, error estimation criteria, and the influence of initial approximations on computational efficiency and solution speed. Special attention is given to numerical challenges associated with root calculations, including approximation errors, sensitivity to small coefficient variations, and difficulties related to repeated or closely spaced roots. The study emphasizes that computational efficiency can be improved through algorithms possessing numerical stability and reduced computational requirements. The study concludes that numerical methods provide effective tools for solving high-degree polynomial equations and that solution accuracy depends substantially on selecting appropriate numerical techniques and model specifications. Such approaches contribute significantly to scientific, engineering, and quantitative applications requiring reliable computational procedures