A probabilistic approach to numerical analysis: Gaussian difference continuous distribution

A Gaussian Difference Continuous Distribution provides a process where the effect of numeric errors on scientific computations, engineering simulations, and applied mathematics can be represented in a probabilistic framework. The classical numerical methods like Newton’s method, Runge-Kutta solvers, finite difference approximations, Monte Carlo simulations, differential equation solvers, and numerical integration techniques typically presuppose normal distribution of errors. However, in practice, the computation often portals the errors as a variation that is more accurately depicted by the difference between two Gaussian-distributed processes, therefore, the Gaussian Difference Continuous Distribution is the best choice for error analysis.\r\nThis research spotlights the Gaussian Difference Distribution in analyzing six basic numerical methods: (1) Newton’s method via root-finding errors, (2) Runge-Kutta methods by using step-size variations, (3) Finite difference methods because of truncation errors, (4) Monte Carlo simulations on account of statistical variability, (5) Differential equation solvers when stability is under the spotlight, and (6) Numerical integration (quadrature-based approximations). Using computational simulations and statistical modeling, we examine how Gaussian Difference models contribute to the solver’s stability, step-size adjustments, and convergence efficiency.\r\nIt is obvious from the results that the Gaussian Difference Continuous Distribution model provides a more precise account of the behavior of numerical errors and, especially, in cases where error distributions are asymmetrical, fluctuating variance, and probabilistic step size corrections. This study focuses on the accuracy and robustness of numerical solvers, thus it makes Gaussian Difference-based models useful for machine learning optimizers, scientific computing, and real-time engineering applications. In the future, research should delve into using these probability models with adaptive solvers to better the numerical stability and convergence behavior.\r\n