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In machine learning, early stopping is a form of regularization used to avoid overfitting when training a model with an iterative method, such as gradient descent. Such methods update the model to make it better fit the training data with each iteration. Up to a point, this improves the model's performance on data outside of the training set (e ...
Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one.
If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x 1 in the basin of attraction of x, and let x n+1 = f(x n) for n ≥ 1, and the sequence {x n} n ≥ 1 will converge to the solution x.
Another use of iteration in mathematics is in iterative methods which are used to produce approximate numerical solutions to certain mathematical problems. Newton's method is an example of an iterative method. Manual calculation of a number's square root is a common use and a well-known example.
The smallest such value of m for a given x is called the period of the orbit. The point x itself is called a periodic point . The cycle detection problem in computer science is the algorithmic problem of finding the first periodic point in an orbit, and the period of the orbit.
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is + = (), =,,, … which gives rise to the sequence,,, … of iterated function applications , (), (()), … which is hoped to converge to a point .
The idea of the potential-reduction method is to modify x such that the potential at each iteration drops by at least a fixed constant X (specifically, X=1/3-ln(4/3)). This implies that, after i iterations, the difference between objective value and the optimal objective value is at most V * exp(- i X / M ), where V is a data-dependent constant.
Methods designed for problems from electromagnetics: Finite-difference time-domain method — a finite-difference method; Rigorous coupled-wave analysis — semi-analytical Fourier-space method based on Floquet's theorem; Transmission-line matrix method (TLM) — based on analogy between electromagnetic field and mesh of transmission lines