Dormand–Prince method
The Dormand–Prince method (often called DOPRI) is an embedded Runge–Kutta method for solving ordinary differential equations (ODEs). It produces two solutions at each step: a fifth-order accurate one and a fourth-order accurate one, by using the same internal stages. The difference between these two results serves as an error estimate for adaptive step sizing.
The method has seven stages, but it uses the First-Same-As-Last (FSAL) property, so only six function evaluations are needed per step. The last stage of one step is at the same point as the first stage of the next step, which saves computation.
Dormand and Prince designed the coefficients to minimize the error of the fifth-order solution. This makes the method especially good when the higher-order result is used to continue the integration (local extrapolation).
In short, the Dormand–Prince method is an efficient, accurate, adaptive solver for ODEs that provides an embedded error estimate and is widely used in many software packages. It is the default choice in MATLAB’s ode45, GNU Octave, and Simulink’s model explorer solver, and is available in Python’s SciPy, Julia’s ODE solvers, and implementations across Fortran, Java, C++, and Rust.
This page was last edited on 2 February 2026, at 07:40 (CET).