Overview

odeint is a library for solving initial value problems (IVP) of ordinary differential equations. Mathematically, these problems are formulated as follows:

x'(t) = f(x,t), x(0) = x0.

x and f can be vectors and the solution is some function x(t) fulfilling both equations above. In the following we will refer to x'(t) also dxdt which is also our notation for the derivative in the source code.

Ordinary differential equations occur nearly everywhere in natural sciences. For example, the whole Newtonian mechanics are described by second order differential equations. Be sure, you will find them in every discipline. They also occur if partial differential equations (PDEs) are discretized. Then, a system of coupled ordinary differential occurs, sometimes also referred as lattices ODEs.

Numerical approximations for the solution x(t) are calculated iteratively. The easiest algorithm is the Euler scheme, where starting at x(0) one finds x(dt) = x(0) + dt f(x(0),0). Now one can use x(dt) and obtain x(2dt) in a similar way and so on. The Euler method is of order 1, that means the error at each step is ~ dt². This is, of course, not very satisfying, which is why the Euler method is rarely used for real life problems and serves just as illustrative example.

The main focus of odeint is to provide numerical methods implemented in a way where the algorithm is completely independent on the data structure used to represent the state x. In doing so, odeint is applicable for a broad variety of situations and it can be used with many other libraries. Besides the usual case where the state is defined as a std::vector or a boost::array, we provide native support for the following libraries:

Boost.uBLAS
Thrust, making odeint naturally running on CUDA devices
gsl_vector for compatibility with the many numerical function in the GSL
Boost.Range
Boost.Fusion (the state type can be a fusion vector)
Boost.Units
Intel Math Kernel Library for maximum performance
VexCL for OpenCL
Boost.Graph (still experimentally)

In odeint, the following algorithms are implemented:

Table 1.1. Stepper Algorithms

Algorithm	Class	Concept	System Concept	Order	Error Estimation	Dense Output	Internal state	Remarks
Explicit Euler	`euler`	Dense Output Stepper	System	1	No	Yes	No	Very simple, only for demonstrating purpose
Modified Midpoint	`modified_midpoint`	Stepper	System	configurable (2)	No	No	No	Used in Bulirsch-Stoer implementation
Runge-Kutta 4	`runge_kutta4`	Stepper	System	4	No	No	No	The classical Runge-Kutta scheme, good general scheme without error control
Cash-Karp	`runge_kutta_cash_karp54`	Error Stepper	System	5	Yes (4)	No	No	Good general scheme with error estimation, to be used in controlled_error_stepper
Dormand-Prince 5	`runge_kutta_dopri5`	Error Stepper	System	5	Yes (4)	Yes	Yes	Standard method with error control and dense output, to be used in controlled_error_stepper and in dense_output_controlled_explicit_fsal.
Fehlberg 78	`runge_kutta_fehlberg78`	Error Stepper	System	8	Yes (7)	No	No	Good high order method with error estimation, to be used in controlled_error_stepper.
Adams Bashforth	`adams_bashforth`	Stepper	System	configurable	No	No	Yes	Multistep method
Adams Bashforth Moulton	`adams_bashforth_moulton`	Stepper	System	configurable	No	No	Yes	Combined multistep method
Controlled Runge-Kutta	`controlled_runge_kutta`	Controlled Stepper	System	depends	Yes	No	depends	Error control for Error Stepper. Requires an Error Stepper from above. Order depends on the given ErrorStepper
Dense Output Runge-Kutta	`dense_output_runge_kutta`	Dense Output Stepper	System	depends	No	Yes	Yes	Dense output for Stepper and Error Stepper from above if they provide dense output functionality (like `euler` and `runge_kutta_dopri5`). Order depends on the given stepper.
Bulirsch-Stoer	`bulirsch_stoer`	Controlled Stepper	System	variable	Yes	No	No	Stepper with step size and order control. Very good if high precision is required.
Bulirsch-Stoer Dense Output	`bulirsch_stoer_dense_out`	Dense Output Stepper	System	variable	Yes	Yes	No	Stepper with step size and order control as well as dense output. Very good if high precision and dense output is required.
Implicit Euler	`implicit_euler`	Stepper	Implicit System	1	No	No	No	Basic implicit routine. Requires the Jacobian. Works only with Boost.uBLAS vectors as state types.
Rosenbrock 4	`rosenbrock4`	Error Stepper	Implicit System	4	Yes	Yes	No	Good for stiff systems. Works only with Boost.uBLAS vectors as state types.
Controlled Rosenbrock 4	`rosenbrock4_controller`	Controlled Stepper	Implicit System	4	Yes	Yes	No	Rosenbrock 4 with error control. Works only with Boost.uBLAS vectors as state types.
Dense Output Rosenbrock 4	`rosenbrock4_dense_output`	Dense Output Stepper	Implicit System	4	Yes	Yes	No	Controlled Rosenbrock 4 with dense output. Works only with Boost.uBLAS vectors as state types.
Symplectic Euler	`symplectic_euler`	Stepper	Symplectic System Simple Symplectic System	1	No	No	No	Basic symplectic solver for separable Hamiltonian system
Symplectic RKN McLachlan	`symplectic_rkn_sb3a_mclachlan`	Stepper	Symplectic System Simple Symplectic System	4	No	No	No	Symplectic solver for separable Hamiltonian system with 6 stages and order 4.
Symplectic RKN McLachlan	`symplectic_rkn_sb3a_m4_mclachlan`	Stepper	Symplectic System Simple Symplectic System	4	No	No	No	Symplectic solver with 5 stages and order 4, can be used with arbitrary precision types.
Velocity Verlet	`velocity_verlet`	Stepper	Second Order System	1	No	No	Yes	Velocity verlet method suitable for molecular dynamics simulation.