solve_powell_tol() [2/2]

template<typename F , typename T , typename... T_Args, require_eigen_vector_t< T > * = nullptr, require_any_st_var< T_Args... > * = nullptr>

Eigen::Matrix< var, Eigen::Dynamic, 1 > stan::math::solve_powell_tol	(	const F &	f,
		const T &	x,
		const double	relative_tolerance,
		const double	function_tolerance,
		const int64_t	max_num_steps,
		std::ostream *const	msgs,
		const T_Args &...	args
	)

Return the solution to the specified system of algebraic equations given an initial guess, and parameters and data, which get passed into the algebraic system.

Use Powell's dogleg solver.

The user can also specify the relative tolerance (xtol in Eigen's code), the function tolerance, and the maximum number of steps (maxfev in Eigen's code).

This overload handles var parameters.

The Jacobian (J_{xy}) (i.e., Jacobian of unknown (x) w.r.t. the parameter (y)) is calculated given the solution as follows. Since [ f(x, y) = 0, ] we have ((J_{pq}) being the Jacobian matrix (\tfrac {dq} {dq})) [

• J_{fx} J_{xy} = J_{fy}, ] and therefore (J_{xy}) can be solved from system [

J_{fx} J_{xy} = J_{fy}. ] Let (eta) be the adjoint with respect to (x); then to calculate [ \eta J_{xy}, ] we solve [

• (\eta J_{fx}^{-1}) J_{fy}. ]

Template Parameters

F	type of equation system function
T	type of elements in the x vector
Args	types of additional parameters to the equation system functor

Parameters

[in]	f	Functor that evaluates the system of equations.
[in]	x	Vector of starting values (initial guess).
[in]	relative_tolerance	determines the convergence criteria for the solution.
[in]	function_tolerance	determines whether roots are acceptable.
[in]	max_num_steps	maximum number of function evaluations.
[in,out]	msgs	the print stream for warning messages.
[in]	args	Additional parameters to the equation system functor.

Returns

theta Vector of solutions to the system of equations.

Precondition

f returns finite values when passed any value of x and the given args.

Exceptions

<code>std::invalid_argument</code>	if x has size zero.
<code>std::invalid_argument</code>	if x has non-finite elements. elements.
<code>std::invalid_argument</code>	if relative_tolerance is strictly negative.
<code>std::invalid_argument</code>	if function_tolerance is strictly negative.
<code>std::invalid_argument</code>	if max_num_steps is not positive.
<code>std::domain_error</code>	solver exceeds max_num_steps.
<code>std::domain_error</code>	if the norm of the solution exceeds the function tolerance.

Definition at line 325 of file solve_powell.hpp.