12.4 Control Parameters for the Algebraic Solver

The call to the algebraic solver shown above uses the default control settings. The solver allows three additional parameters, all of which must be supplied if any of them is supplied.

  y = algebra_solver(system, y_guess, theta, x_r, x_i,
                     rel_tol, f_tol, max_steps);

The three control arguments are relative tolerance, function tolerance, and maximum number of steps. Both tolerances need to be satisfied. If one of them is not met, the metropolis proposal gets rejected with a warning message explaining which criterion was not satisfied. The default values for the control arguments are respectively 1e-10 (\(10^{-10}\)), 1e-6 (\(10^{-6}\)), and 1e3 (\(10^3\)).

Tolerance

The relative and function tolerances control the accuracy of the solution generated by the solver. Relative tolerances are relative to the solution value. The function tolerance is the norm of the algebraic function, once we plug in the proposed solution. This norm should go to 0 (equivalently, all elements of the vector function are 0). It helps to think about this geometrically. Ideally the output of the algebraic function is at the origin; the norm measures deviations from this ideal. As the length of the return vector increases, a certain function tolerance becomes an increasingly difficult criterion to meet, given each individual element of the vector contribute to the norm.

Smaller relative tolerances produce more accurate solutions but require more computational time.

Sensitivity Analysis

The tolerances should be set low enough that setting them lower does not change the statistical properties of posterior samples generated by the Stan program.

Maximum Number of Steps

The maximum number of steps can be used to stop a runaway simulation. This can arise in MCMC when a bad jump is taken, particularly during warmup. If the limit is hit, the current metropolis proposal gets rejected. Users will see a warning message stating the maximum number of steps has been exceeded.