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12.3 Calling the Algebraic Solver

Let’s suppose \(\theta = (3, 6)\). To call the algebraic solver, we need to provide an initial guess. This varies on a case-by-case basis, but in general a good guess will speed up the solver and, in pathological cases, even determine whether the solver converges or not. If the solver does not converge, the metropolis proposal gets rejected and a warning message, stating no acceptable solution was found, is issued.

The solver has three tuning parameters to determine convergence: the relative tolerance, the function tolerance, and the maximum number of steps. Their behavior is explained in the section about algebraic solvers with control parameters.

The following code returns the solution to our nonlinear algebraic system:

transformed data {
  vector[2] y_guess = {1, 1};
  real x_r[0];
  int x_i[0];
}

transformed parameters {
  vector[2] theta = {3, 6};
  vector[2] y;

  y = algebra_solver_newton(system, y_guess, theta, x_r, x_i);
}

which returns \(y = (3, -2)\).

Data versus Parameters

The arguments for the real data x_r and the integer data x_i must be expressions that only involve data or transformed data variables. theta, on the other hand, must only involve parameters. Note there are no restrictions on the initial guess, y_guess, which may be a data or a parameter vector.

Length of the Algebraic Function and of the Vector of Unknowns

The Jacobian of the solution with respect to the parameters is computed using the implicit function theorem, which imposes certain restrictions. In particular, the Jacobian of the algebraic function \(f\) with respect to the unknowns \(x\) must be invertible. This requires the Jacobian to be square, meaning \(f(y)\) and \(y\) have the same length or, in other words the number of equations in the system is the same as the number of unknowns.

Pathological Solutions

Certain systems may be degenerate, meaning they have multiple solutions. The algebraic solver will not report these cases, as the algorithm stops once it has found an acceptable solution. The initial guess will often determine which solution gets found first. The degeneracy may be broken by putting additional constraints on the solution. For instance, it might make “physical sense” for a solution to be positive or negative.

On the other hand, a system may not have a solution (for a given point in the parameter space). In that case, the solver will not converge to a solution. When the solver fails to do so, the current metropolis proposal gets rejected.