Comparative Analysis of Runtime Complexity for Four Polynomial Evaluation Algorithms

This article presents a comparative analysis of runtime complexity for four distinct polynomial evaluation algorithms. We implement all algorithms within a unified codebase using a modular programming approach, where each algorithm is encapsulated in separate functions with standardized interfaces. The performance comparison is visualized through a single runtime results graph plotting execution time against polynomial degree. The implementation includes Horner's method (O(n)), naive evaluation (O(n²)), precomputed power tables (O(n) with O(n) space), and iterative accumulation techniques.

First, we detail each algorithm's characteristics and implementation specifics: Horner's method reduces multiplicative operations through nested multiplication, the naive approach successively calculates powers, precomputation trades memory for speed by storing x^n values, and iterative methods optimize through incremental power calculation. We conduct standardized tests using randomized polynomials of varying degrees, measuring execution time with high-resolution timers. Algorithm efficiency is analyzed through complexity calculations and empirical timing data comparison.

Furthermore, we examine scenario-specific applicability: Horner's method excels in general-purpose evaluation, precomputation suits repeated evaluations at fixed x-values, while iterative methods benefit from memory-constrained environments. Practical applications in numerical analysis, computer graphics, and scientific computing are discussed with code integration examples.

Finally, we propose enhancement strategies including parallel computation for independent terms, memory access optimization for precomputed values, and hybrid approaches combining multiple techniques. Through this study, readers will gain comprehensive understanding of polynomial evaluation algorithms and their practical implementation trade-offs.