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Using an adaptive stepsize is of particular importance when there is a large variation in the size of the derivative. For example, when modeling the motion of a satellite about the earth as a standard Kepler orbit, a fixed time-stepping method such as the Euler method may be sufficient.
Numerous adaptive step size schemes have been proposed throughout the literature. [ 1 ] [ 4 ] [ 11 ] [ 12 ] Applications of these schemes [ 2 ] [ 13 ] suggest that these can offer substantial improvement in number of iterations required for fixed point convergence.
On the other hand, + is a third-order approximation, so the difference between + and + can be used to adapt the step size. The FSAL—first same as last—property is that the stage value k 4 {\displaystyle k_{4}} in one step equals k 1 {\displaystyle k_{1}} in the next step; thus, only three function evaluations are needed per step.
If , then the step is completed. Replace h {\textstyle h} with h new {\textstyle h_{\text{new}}} for the next step. The coefficients found by Fehlberg for Formula 2 (derivation with his parameter α 2 = 3/8) are given in the table below, using array indexing of base 1 instead of base 0 to be compatible with most computer languages:
Adaptive Step Size Random Search (ASSRS) by Schumer and Steiglitz [6] attempts to heuristically adapt the hypersphere's radius: two new candidate solutions are generated, one with the current nominal step size and one with a larger step-size. The larger step size becomes the new nominal step size if and only if it leads to a larger improvement ...
Adaptive quadrature is a numerical integration method in which the integral of a function is approximated using static quadrature rules on adaptively refined subintervals of the region of integration. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for "well behaved" integrands, but are also ...
The step size is denoted by (sometimes called the learning rate in machine learning) and here ":=" denotes the update of a variable in the algorithm. In many cases, the summand functions have a simple form that enables inexpensive evaluations of the sum-function and the sum gradient.
For example, if the objective is assumed to be strongly convex and lipschitz smooth, then gradient descent converges linearly with a fixed step size. [1] Looser assumptions lead to either weaker convergence guarantees or require a more sophisticated step size selection. [33]