This is an old version, view current version.

## 9.2 Ordinary differential equation (ODE) solvers

Stan provides several higher order functions for solving initial value problems specified as Ordinary Differential Equations (ODEs).

Solving an initial value ODE means given a set of differential equations $$y'(t, \theta) = f(t, y, \theta)$$ and initial conditions $$y(t_0, \theta)$$, solving for $$y$$ at a sequence of times $$t_0 < t_1 \leq t_2, \cdots \leq t_n$$. $$f(t, y, \theta)$$ is referred to here as the ODE system function.

$$f(t, y, \theta)$$ will be defined as a function with a certain signature and provided along with the initial conditions and output times to one of the ODE solver functions.

To make it easier to write ODEs, the solve functions take extra arguments that are passed along unmodified to the user-supplied system function. Because there can be any number of these arguments and they can be of different types, they are denoted below as .... The types of the arguments represented by ... in the ODE solve function call must match the types of the arguments represented by ... in the user-supplied system function.

### 9.2.1 Non-stiff solver

vector[] ode_rk45(function ode, vector initial_state, real initial_time, real[] times, ...)
Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method.

vector[] ode_rk45_tol(function ode, vector initial_state, real initial_time, real[] times, data real rel_tol, data real abs_tol, int max_num_steps, ...)
Solves the ODE system for the times provided using the Dormand-Prince algorithm, a 4th/5th order Runge-Kutta method with additional control parameters for the solver.

vector[] ode_ckrk(function ode, vector initial_state, real initial_time, real[] times, ...)
Solves the ODE system for the times provided using the Cash-Karp algorithm, a 4th/5th order explicit Runge-Kutta method.

vector[] ode_ckrk_tol(function ode, vector initial_state, real initial_time, real[] times, data real rel_tol, data real abs_tol, int max_num_steps, ...)
Solves the ODE system for the times provided using the Cash-Karp algorithm, a 4th/5th order explicit Runge-Kutta method with additional control parameters for the solver.

vector[] ode_adams(function ode, vector initial_state, real initial_time, real[] times, ...)
Solves the ODE system for the times provided using the Adams-Moulton method.

vector[] ode_adams_tol(function ode, vector initial_state, real initial_time, real[] times, data real rel_tol, data real abs_tol, int max_num_steps, ...)
Solves the ODE system for the times provided using the Adams-Moulton method with additional control parameters for the solver.

### 9.2.2 Stiff solver

vector[] ode_bdf(function ode, vector initial_state, real initial_time, real[] times, ...)
Solves the ODE system for the times provided using the backward differentiation formula (BDF) method.

vector[] ode_bdf_tol(function ode, vector initial_state, real initial_time, real[] times, data real rel_tol, data real abs_tol, int max_num_steps, ...)
Solves the ODE system for the times provided using the backward differentiation formula (BDF) method with additional control parameters for the solver.

vector[] ode_adjoint_tol_ctl(function ode, vector initial_state, real initial_time, real[] times, data real rel_tol_forward, data vector abs_tol_forward, data real rel_tol_backward, data vector abs_tol_backward, int max_num_steps, int num_steps_between_checkpoints, int interpolation_polynomial, int solver_forward, int solver_backward, ...)

Solves the ODE system for the times provided using the adjoint ODE solver method from CVODES. The adjoint ODE solver requires a checkpointed forward in time ODE integration, a backwards in time integration that makes uses of an interpolated version of the forward solution, and the solution of a quadrature problem (the number of which depends on the number of parameters passed to the solve). The tolerances and numeric methods used for the forward solve, backward solve, quadratures, and interpolation can all be configured.

### 9.2.4 ODE system function

The first argument to one of the ODE solvers is always the ODE system function. The ODE system function must have a vector return type, and the first two arguments must be a real and vector in that order. These two arguments are followed by the variadic arguments that are passed through from the ODE solve function call:

 vector ode(real time, vector state, ...)

The ODE system function should return the derivative of the state with respect to time at the time and state provided. The length of the returned vector must match the length of the state input into the function.

The arguments to this function are:

• time, the time to evaluate the ODE system

• state, the state of the ODE system at the time specified

• ..., sequence of arguments passed unmodified from the ODE solve function call. The types here must match the types in the ... arguments of the ODE solve function call.

### 9.2.5 Arguments to the ODE solvers

The arguments to the ODE solvers in both the stiff and non-stiff solvers are the same. The arguments to the adjoint ODE solver are different; see Arguments to the adjoint ODE solvers.

• ode: ODE system function,

• initial_state: initial state, type vector,

• initial_time: initial time, type real,

• times: solution times, type real[],

• ...: sequence of arguments that will be passed through unmodified to the ODE system function. The types here must match the types in the ... arguments of the ODE system function.

For the versions of the ode solver functions ending in _tol, these three parameters must be provided after times and before the ... arguments:

• data rel_tol: relative tolerance for the ODE solver, type real, data only,

• data abs_tol: absolute tolerance for the ODE solver, type real, data only, and

• max_num_steps: maximum number of steps to take between output times in the ODE solver, type int, data only.

Because the tolerances are data arguments, they must be defined in either the data or transformed data blocks. They cannot be parameters, transformed parameters or functions of parameters or transformed parameters.

### 9.2.6 Arguments to the adjoint ODE solver

The arguments to the adjoint ODE solver are different from those for the other functions (for those see Arguments to the adjoint ODE solvers).

• ode: ODE system function,

• initial_state: initial state, type vector,

• initial_time: initial time, type real,

• times: solution times, type real[],

• data rel_tol_forward: Relative tolerance for forward solve, type real, data only,

• data abs_tol_forward: Absolute tolerance vector for each state for forward solve, type vector, data only,

• data rel_tol_backward: Relative tolerance for backward solve, type real, data only,

• data abs_tol_backward: Absolute tolerance vector for each state for backward solve, type vector, data only,

• data rel_tol_quadrature: Relative tolerance for backward quadrature, type real, data only,

• data abs_tol_quadrature: Absolute tolerance for backward quadrature, type real, data only,

• data max_num_steps: Maximum number of time-steps to take in integrating the ODE solution between output time points for forward and backward solve, type int, data only,

• num_steps_between_checkpoints: number of steps between checkpointing forward solution, type int, data only,

• interpolation_polynomial: can be 1 for hermite or 2 for polynomial interpolation method of CVODES, type int, data only,

• solver_forward: solver used for forward ODE problem: 1=Adams (non-stiff), 2=BDF (stiff), type int, data only,

• solver_backward: solver used for backward ODE problem: 1=Adams (non-stiff), 2=BDF (stiff), type int, data only.

• ...: sequence of arguments that will be passed through unmodified to the ODE system function. The types here must match the types in the ... arguments of the ODE system function.

Because the tolerances are data arguments, they must be defined in either the data or transformed data blocks. They cannot be parameters, transformed parameters or functions of parameters or transformed parameters.

#### 9.2.6.1 Return values

The return value for the ODE solvers is an array of vectors (type vector[]), one vector representing the state of the system at every time in specified in the times argument.

#### 9.2.6.2 Array and vector sizes

The sizes must match, and in particular, the following groups are of the same size:

• state variables passed into the system function, derivatives returned by the system function, initial state passed into the solver, and length of each vector in the output,

• number of solution times and number of vectors in the output,