## 12.4 algebra_solver, algebra_solver_newton algebraic solvers

These algebraic solver functions have been replaced by those described in:

### 12.4.1 Specifying an algebraic equation as a function

An algebraic system is specified as an ordinary function in Stan within the function block. The algebraic system function must have this signature:

 vector algebra_system(vector y, vector theta,
data array[] real x_r, array[] int x_i)

The algebraic system function should return the value of the algebraic function which goes to 0, when we plug in the solution to the algebraic system.

The argument of this function are:

• y, the unknowns we wish to solve for

• theta, parameter values used to evaluate the algebraic system

• x_r, data values used to evaluate the algebraic system

• x_i, integer data used to evaluate the algebraic system

The algebraic system function separates parameter values, theta, from data values, x_r, for efficiency in propagating the derivatives through the algebraic system.

### 12.4.2 Call to the algebraic solver

vector algebra_solver(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i)
Solves the algebraic system, given an initial guess, using the Powell hybrid algorithm.
Available since 2.17, deprecated in 2.31

vector algebra_solver(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i, data real rel_tol, data real f_tol, int max_steps)
Solves the algebraic system, given an initial guess, using the Powell hybrid algorithm with additional control parameters for the solver.
Available since 2.17, deprecated in 2.31

Note: In future releases, the function algebra_solver will be deprecated and replaced with algebra_solver_powell.

vector algebra_solver_newton(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i)
Solves the algebraic system, given an initial guess, using Newton’s method.
Available since 2.24, deprecated in 2.31

vector algebra_solver_newton(function algebra_system, vector y_guess, vector theta, data array[] real x_r, array[] int x_i, data real rel_tol, data real f_tol, int max_steps)
Solves the algebraic system, given an initial guess, using Newton’s method with additional control parameters for the solver.
Available since 2.24, deprecated in 2.31

#### 12.4.2.1 Arguments to the algebraic solver

The arguments to the algebraic solvers are as follows:

• algebra_system: function literal referring to a function specifying the system of algebraic equations with signature (vector, vector, array[] real, array[] int):vector. The arguments represent (1) unknowns, (2) parameters, (3) real data, and (4) integer data, and the return value contains the value of the algebraic function, which goes to 0 when we plug in the solution to the algebraic system,

• y_guess: initial guess for the solution, type vector,

• theta: parameters only, type vector,

• x_r: real data only, type array[] real, and

• x_i: integer data only, type array[] int.

For more fine-grained control of the algebraic solver, these parameters can also be provided:

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

• function_tol: function tolerance for the algebraic solver, type real, data only,

• max_num_steps: maximum number of steps to take in the algebraic solver, type int, data only.

#### 12.4.2.2 Return value

The return value for the algebraic solver is an object of type vector, with values which, when plugged in as y make the algebraic function go to 0.

#### 12.4.2.3 Sizes and parallel arrays

Certain sizes have to be consistent. The initial guess, return value of the solver, and return value of the algebraic function must all be the same size.

The parameters, real data, and integer data will be passed from the solver directly to the system function.