The method by Newton in the early 1700s is commonly named as Newton Method or Newton-Raphson method. This is a powerful technique for solving equations numerically based on the simple idea of linear approximation. This is a root-finding algorithm that uses the first few terms of the Taylor series of a function in the vicinity of a suspected root. Newton’s method is sometimes also known as Newton’s iteration, although in this work the latter term is reserved to the application of Newton’s method for computing square roots.
The Gaussian method is essentially Newton’s method with a couple of modifications, which it uses the approximation for the Hessian matrix and it relies on the function to be minimized taking a certain form.