Modified of Roots Finding Algorithm of High Degree Polynomials

Although the Durand-Kerner method is widely used across various fields of computer science, especially in numerical computing, it continues to encounter challenges in locating roots of high-degree polynomials, such as issues with accuracies of roots of the polynomial zeros. Our initial tests and observations on several methods for finding polynomial roots revealed that the roots' accuracy starts to degrade noticeably for polynomials where the degree exceeds 10. Based on considerations of algebraic concepts involving polynomial vector spaces, we introduce an improvement of the Durand-Kerner algorithm aimed at improving root precision. This approach includes targeted refinements in coefficient evaluation, identification of root types, and iterative polishing techniques. We also conducted a comparative evaluation to assess its effectiveness against the original Durand Kerner method and MATLAB's roots() function. Overall, the enhanced algorithm delivers superior accuracy for complex roots—particularly in cases involving multiple zero or integer roots—outperforming both benchmarks, but its execution time increases substantially with polynomial degree.