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## 10.3 Differential-Algebraic equation (DAE) solver

Stan provides two higher order functions for solving initial value problems specified as Differential-Algebraic Equations (DAEs) with index-1 .

Solving an initial value DAE means given a set of residual functions $$r(y'(t, \theta), y(t, \theta), t)$$ and initial conditions $$(y(t_0, \theta), y'(t_0, \theta))$$, solving for $$y$$ at a sequence of times $$t_0 < t_1 \leq t_2, \cdots \leq t_n$$. The residual function $$r(y', y, t, \theta)$$ will be defined as a function with a certain signature and provided along with the initial conditions and output times to one of the DAE solver functions.

Similar to ODE solvers, the DAE solver function takes 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 ..., and the types of these arguments, also represented by ... in the DAE solver call, must match the types of the arguments represented by ... in the user-supplied system function.

### 10.3.1 The DAE solver

array[] vector dae(function residual, vector initial_state, vector initial_state_derivative, real initial_time, array[] real times, ...)
Solves the DAE system using the backward differentiation formula (BDF) method .
Available since 2.29

array[] vector dae_tol(function residual, vector initial_state, vector initial_state_derivative, real initial_time, array[] real times, data real rel_tol, data real abs_tol, int max_num_steps, ...)
Solves the DAE system for the times provided using the backward differentiation formula (BDF) method with additional control parameters for the solver.
Available since 2.29

### 10.3.2 DAE system function

The first argument to the DAE solver is the DAE residual function. The DAE residual function must have a vector return type, and the first three arguments must be a real, vector, and vector, in that order. These three arguments are followed by the variadic arguments that are passed through from the DAE solver function call:

  vector residual(real time, vector state, vector state_derivative, ...)

The DAE residual function should return the residuals 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 DAE system

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

• state_derivative, the time derivatives of the state of the DAE system at the time specified

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

### 10.3.3 Arguments to the DAE solver

The arguments to the DAE solver are

• residual: DAE residual function,

• initial_state: initial state, type vector,

• initial_state_derivative: time derivative of the initial state, type vector,

• initial_time: initial time, type real,

• times: solution times, type array[] real,

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

For dae_tol, the following three parameters must be provided after times and before the ... arguments:

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

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

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

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

#### 10.3.3.1 Consistency of the initial conditions

The user is responsible to ensure the residual function becomes zero at the initial time, t0, when the arguments initial_state and initial_state_derivative are introduced as state and state_derivative, respectively.

#### 10.3.3.2 Return values

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

#### 10.3.3.3 Array and vector sizes

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

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

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

### References

Serban, Radu, Cosmin Petra, Alan C. Hindmarsh, Cody J. Balos, David J. Gardner, Daniel R. Reynolds, and Carol S. Woodward. 2021. “User Documentation for IDAS V5.0.0.” Lawrence Livermore National Laboratory.