A study to compare the error propagation of normal approximation and the gamma distribution

One of the most effective statistical models that the Gamma continuous distribution uses is that of error propagation, for which the numerical solvers are responsible, thereby improving the stability, adaptability, and convergence efficiency in computational methods. The conventional solvers, for example, deterministical like Newton’s method or Runge-Kutta methods or even Monte Carlo simulations, often come across the obstacles such as error accumulation, oscillations, and instability, parted with less accuracy, especially for complex numerical landscapes. This research delves into the topic of how Gamma-distributed step-size variations can be used to probabilistic-ally correct the errors, thereby preventing the solver from diverging and therefore enhancing the performance of the numerical analysis.\r\nUsing computational simulations and statistical modeling, five key numerical solvers: Runge-Kutta methods for step-size variations, finite difference methods for truncation error propagation, Monte Carlo simulations for statistical variability, differential equation solvers for accumulated rounding errors, and numerical integration schemes for step-size variations in quadrature methods world have been analyzed. Moreover, Newton’s method has been used with Gamma-distributed step-size variations that introduce uncertainty by random numbers in the algorithm. The research reveals that Gamma-distributed step-size has emerged as the most powerful way to toughen solvers in certain functions with multiple roots, steep gradients, or uncertain function evaluations.\r\nThe probabilistic approach that introduces uncertainty in the root-finding process via Gamma-distributed step-size adjusts the convergence of the algorithm, reducing the probability of overshooting. It is good for the performance of the computational physics, machine learning optimizers, and real-time numerical simulations. The conclusion of this paper calls the readers\' attention to the possibility of Gamma-distributed models to be the main factor influencing the success of numerical accuracy together with the improvement of solver efficiency, therefore they are valuable to engineering, scientific computing, as well as data-driven applications. There are prospects for Gamma-based solvers to be combined with machine learning and thus providing a solution to their adaptability in the field of numerical methods.\r\n