|
| | Greenstadt () |
| constexpr std::string | name_impl () const |
| Method | method () const |
| void | method (Method t_method) |
| void | enable_one_method () |
| void | enable_two_method () |
| void | set_method (Method t_method) |
| void | update_impl (const Vector &, const Vector &, const FirstDerivative &jacobian_old, const Vector &delta_x_new, const Vector &delta_function_new, const Vector &function_new, FirstDerivative &jacobian_new) |
| | QuasiNewton () |
| constexpr std::string | name_impl () const |
| bool | solve_impl (FunctionLambda &&function, JacobianLambda &&jacobian, const Vector &x_ini, Vector &x_sol) |
| void | update (const Vector &delta_x_old, const Vector &delta_function_old, const FirstDerivative &jacobian_old, const Vector &delta_x_new, const Vector &delta_function_new, const Vector &function_new, FirstDerivative &jacobian_new) |
| | RootFinder () |
| constexpr std::string | name () const |
| Integer | jacobian_evaluations () const |
| Integer | max_jacobian_evaluations () const |
| Integer | hessian_evaluations () const |
| Integer | max_hessian_evaluations () const |
| bool | solve (FunctionLambda &&function, const Input &x_ini, Output &x_sol) |
| | SolverBase () |
| void | reset_bounds (const Integer n=InputTrait::IsDynamic ? 0 :InputTrait::Dimension) |
| const Vector & | lower_bound () const |
| const Vector & | upper_bound () const |
| void | bounds (const Vector &t_lower_bound, const Vector &t_upper_bound) |
| constexpr Integer | input_dimension () const |
| constexpr Integer | output_dimension () const |
| Integer | function_evaluations () const |
| void | max_function_evaluations (const Integer t_max_function_evaluations) |
| Integer | iterations () const |
| Integer | max_iterations () const |
| Scalar | alpha () const |
| Integer | relaxations () const |
| Integer | max_relaxations () const |
| Scalar | tolerance () const |
| void | verbose_mode (bool t_verbose) |
| void | enable_verbose_mode () |
| void | disable_verbose_mode () |
| void | damped_mode (bool t_damped) |
| void | enable_damped_mode () |
| void | disable_damped_mode () |
| std::string | task () const |
| bool | converged () const |
| std::ostream & | ostream () const |
| bool | solve (FunctionLambda &&function, const Vector &x_ini, Vector &x_sol) |
| bool | rootfind (const FunctionBase< FunctionInput, FunctionOutput, DerivedFunction > &function, const Vector &x_ini, Vector &x_sol) |
| bool | optimize (const FunctionBase< FunctionInput, FunctionOutput, DerivedFunction > &function, const Vector &x_ini, Vector &x_sol) |
| constexpr std::string | name () const |
|
| bool | evaluate_jacobian (JacobianLambda &&jacobian, const Input &x, FirstDerivative &out) |
| bool | evaluate_hessian (HessianLambda &&hessian, const Input &x, SecondDerivative &out) |
| Integer | first_derivative_evaluations () const |
| Integer | max_first_derivative_evaluations () const |
| Integer | second_derivative_evaluations () const |
| Integer | max_second_derivative_evaluations () const |
| void | reset_counters () |
| bool | evaluate_function (FunctionLambda &&function, const Vector &x, Vector &out) |
| bool | evaluate_first_derivative (FirstDerivativeLambda &&function, const Vector &x, FirstDerivative &out) |
| bool | evaluate_second_derivative (SecondDerivativeLambda &&function, const Vector &x, SecondDerivative &out) |
| bool | damp (FunctionLambda &&function, const Vector &x_old, const Vector &function_old, const Vector &step_old, Vector &x_new, Vector &function_new, Vector &step_new) |
| void | header () |
| void | bottom () |
| void | info (Scalar residuals, const std::string ¬es="-") |
| Vector | m_lower_bound |
| Vector | m_upper_bound |
| Integer | m_function_evaluations |
| Integer | m_first_derivative_evaluations |
| Integer | m_second_derivative_evaluations |
| Integer | m_max_function_evaluations |
| Integer | m_max_first_derivative_evaluations |
| Integer | m_max_second_derivative_evaluations |
| Integer | m_iterations |
| Integer | m_max_iterations |
| Scalar | m_alpha |
| Integer | m_relaxations |
| Integer | m_max_relaxations |
| Scalar | m_tolerance |
| bool | m_verbose |
| bool | m_damped |
| std::ostream * | m_ostream |
| std::string | m_task |
| bool | m_converged |
template<typename Vector>
requires
TypeTrait<Vector>::IsEigen && (!
TypeTrait<Vector>::IsFixed ||
TypeTrait<Vector>::Dimension > 0)
class Optimist::RootFinder::Greenstadt< Vector >
Greenstadt's method
Similarly to the Broyden's methods, Greenstadt's methods are quasi-Newton methods that approximates the Jacobian matrix \(\mathbf{Jf}_{\mathbf{x}}\) an update rule. The generic Greenstadt's method is defined as
\[ \mathbf{H}_k(\mathbf{x}_k) \mathbf{h}_k = -\mathbf{f}(\mathbf{x}_k) \text{,}
\]
where \(\mathbf{H}_k\) is the (inverse) Jacobian approximation at the \(k\)-th iteration.
Based on the update rule, Greenstadt's method is classified into two methods: trivially method 1 and method 2. The update of the Jacobian approximation is performed as
\[ \mathbf{H}_{k+1}^{-1} = \mathbf{H}_k^{-1} - \displaystyle\frac{\mathbf{H}_k^{-1} \Delta\mathbf{f}_k - \Delta\mathbf{x}_k}{\mathbf{g} \Delta\mathbf{f}_k} \mathbf{g} \text{,}
\]
where \(\Delta\mathbf{x}_k = \mathbf{x}_k - \mathbf{x}_{k-1}\), and \(\Delta\mathbf{f}_k = \mathbf{f}(\mathbf{x}_k) - \mathbf{f}(\mathbf{x}_{k-1})\). The quantity \(\mathbf{g}\) is \(\mathbf{g} = \mathbf{f}_k^\top\) for method 1 and \(\mathbf{g} = \mathbf{H}_{k\top}^{-1} \mathbf{H}_k^{-1} \Delta\mathbf{f}_k^{\top}\) for method 2. The choice of the method is based on the convergence history, switching between the methods to adapt to the problem's behavior.
> For more details on the Greenstadt's methods refer to the reference: Spedicato E., Greenstadt J. On some classes of variationally derived Quasi-Newton methods for systems of nonlinear algebraic equations, Numerische Mathematik, 29 (1978), pp. 363-380.
- Template Parameters
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1.16.1