Simpson vs Adaptive Simpson

Provide a comprehensive comparison between the Simpson's rule and Adaptive Simpson's rule for numerical integration

Simpson's Rule

Simpson's rule is a numerical method for approximating the integral of a function. It works by approximating the function with a series of parabolas. The formula for Simpson's rule over a single interval $[a, b]$ is:

Adaptive Simpson's Rule

Adaptive Simpson's rule improves upon the basic Simpson's rule by recursively subdividing the intervals until the desired accuracy is achieved. It adapts the interval size based on the function's behavior, which allows for more accurate approximations with fewer function evaluations.