Comparision of Romberg and Fixed Tolerance Gaussian Quadrature Over Imaginary Exponent Functions

Abstract This research paper presents an extensive comparative investigation of two numerical integration techniques: Romberg 1 and Fixed Tolerance Gaussian quadrature, 2 when applied to functions with Imaginary exponents. The main objective is to evaluate the accuracy and efficiency of these methods in approximating definite integrals over functions with imaginary powers. The study involves systematic analyses of the computational requirements and precision achieved by each method over multiple intervals. QUADPACK's 3 adaptive quadrature is utilized as a benchmark. Through this study, valuable insights are offered into the performance of these quadrature techniques when dealing with integrals involving Imaginary exponents.