stepsize

Stepsize, or step size, is the increment by which a variable is advanced in an iterative or discrete process. It is a key parameter in numerical analysis, optimization, and simulation, governing how far the state is updated at each iteration or time step.

In numerical time integration, h denotes the time step. Methods such as Euler’s method update y_{n+1} = y_n + h f(t_n, y_n). A smaller h generally increases local accuracy but requires more steps to reach a given interval, while too large h can lead to instability or significant error.

In optimization and machine learning, the stepsize (often called the learning rate) determines how far along a chosen search direction the algorithm moves at each iteration. A large stepsize can speed up progress but may overshoot or diverge; a small stepsize yields robust but slow convergence. Some algorithms employ line searches or adaptive schemes to adjust the stepsize to satisfy certain conditions (e.g., Armijo or Wolfe) or to balance progress and accuracy.

Adaptive stepsize methods adjust h during computation based on estimated error, allowing larger steps when the solution behaves smoothly and smaller steps near rapid changes. This is common in Runge–Kutta and other integrators, and stability constraints such as the CFL condition influence allowable stepsizes in stiff or wave-dominated problems.

In linear algebra and iterative solvers, the stepsize concept may appear as a scalar multiplier applied to a search direction in gradient methods; in fixed-point iterations, it can affect convergence rate.

Choosing an appropriate stepsize involves considering truncation error, stability, and computational cost, often through analysis or adaptive strategies. See also numerical integration, optimization, line search, and learning rate.