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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.
The Bogacki–Shampine method is a Runge–Kutta method of order three with four stages with the First Same As Last (FSAL) property, so that it uses approximately three function evaluations per step. It has an embedded second-order method which can be used to implement adaptive step size.
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:
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.
For example, a quadrature of the ... Adaptive quadrature is a numerical integration method in which the integral of a ... as the width of the step size between the ...
For example, [5] the first derivative can be calculated by the complex-step derivative formula: [12] [13] [14] ′ = ((+)) + (),:= The recommended step size to obtain accurate derivatives for a range of conditions is h = 10 − 200 {\displaystyle h=10^{-200}} . [ 6 ]
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]
Dormand–Prince is the default method in the ode45 solver for MATLAB [4] and GNU Octave [5] and is the default choice for the Simulink's model explorer solver. It is an option in Python's SciPy ODE integration library [6] and in Julia's ODE solvers library. [7]