Dormand–Prince method
Encyclopedia
In numerical analysis
, the Dormand–Prince method, or DOPRI method, is a method for solving ordinary differential equations . The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth-order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg
(RKF) and Cash–Karp (RKCK).
The Dormand–Prince method has seven stages, but it uses only six function evaluations per step because it has the FSAL (First Same As Last) property: the last stage is evaluated at the same point as the first stage of the next step. Dormand and Prince choose the coefficients of their method to minimize the error of the fifth-order solution. This is the main difference with the Fehlberg method, which was constructed so that the fourth-order solution has a small error. For this reason, the Dormand–Prince method is more suitable when the higher-order solution is used to continue the integration, a practice known as local extrapolation .
Dormand–Prince is currently the default method in MATLAB
and GNU Octave
's ode45 solver and is the default choice for the Simulink
's model explorer solver. A Fortran
free software
implementation of the algorithm called DOPRI5 is also available.
The Butcher tableau is:
The first row of b coefficients gives the fourth-order accurate solution, and the second row gives the fifth-order accurate solution.
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
, the Dormand–Prince method, or DOPRI method, is a method for solving ordinary differential equations . The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth-order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg
Runge–Kutta–Fehlberg method
In mathematics, the Runge–Kutta–Fehlberg method is an algorithm of numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German mathematician Erwin Fehlberg and is based on the class of Runge–Kutta methods...
(RKF) and Cash–Karp (RKCK).
The Dormand–Prince method has seven stages, but it uses only six function evaluations per step because it has the FSAL (First Same As Last) property: the last stage is evaluated at the same point as the first stage of the next step. Dormand and Prince choose the coefficients of their method to minimize the error of the fifth-order solution. This is the main difference with the Fehlberg method, which was constructed so that the fourth-order solution has a small error. For this reason, the Dormand–Prince method is more suitable when the higher-order solution is used to continue the integration, a practice known as local extrapolation .
Dormand–Prince is currently the default method in MATLAB
MATLAB
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
and GNU Octave
GNU Octave
GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...
's ode45 solver and is the default choice for the Simulink
Simulink
Simulink, developed by MathWorks, is a commercial tool for modeling, simulating and analyzing multidomain dynamic systems. Its primary interface is a graphical block diagramming tool and a customizable set of block libraries. It offers tight integration with the rest of the MATLAB environment and...
's model explorer solver. A Fortran
Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
free software
Free software
Free software, software libre or libre software is software that can be used, studied, and modified without restriction, and which can be copied and redistributed in modified or unmodified form either without restriction, or with restrictions that only ensure that further recipients can also do...
implementation of the algorithm called DOPRI5 is also available.
The Butcher tableau is:
| 0 | |||||||
| | 1/5 | 1/5 | ||||||
| | 3/10 | 3/40 | 9/40 | |||||
| | 4/5 | 44/45 | −56/15 | 32/9 | ||||
| | 8/9 | 19372/6561 | −25360/2187 | 64448/6561 | −212/729 | |||
| | 1 | 9017/3168 | −355/33 | 46732/5247 | 49/176 | −5103/18656 | ||
| | 1 | 35/384 | 0 | 500/1113 | 125/192 | −2187/6784 | 11/84 | |
| | | 35/384 | 0 | 500/1113 | 125/192 | −2187/6784 | 11/84 | 0 |
| | | 5179/57600 | 0 | 7571/16695 | 393/640 | −92097/339200 | 187/2100 | 1/40 |
The first row of b coefficients gives the fourth-order accurate solution, and the second row gives the fifth-order accurate solution.