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In mathematics and numerical analysis, an adaptive step size is used in some methods for the numerical solution of ordinary differential equations (including the special case of numerical integration) in order to control the errors of the method and to ensure stability properties such as A-stability. Using an adaptive stepsize is of particular ...
It has an embedded second-order method which can be used to implement adaptive step size. The Bogacki–Shampine method is implemented in the ode3 for fixed step solver and ode23 for a variable step solver function in MATLAB (Shampine & Reichelt 1997).
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]
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.
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 ...
This can be contrasted with implicit linear multistep methods (the other big family of methods for ODEs): an implicit s-step linear multistep method needs to solve a system of algebraic equations with only m components, so the size of the system does not increase as the number of steps increases. [27]
The lower-order step is given by ... The coefficients here allow for an adaptive stepsize to ... Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary ...