Class container for the Greenstadt's method. More...

#include <Greenstadt.hh>

Inherits Optimist::RootFinder::QuasiNewton< Real, N, Greenstadt< Real, N > >.

Public Types
using	Method = enum class Method : Integer {ONE = 1, TWO = 2}
Public Types inherited from Optimist::RootFinder::QuasiNewton< Real, N, Greenstadt< Real, N > >
using	Method
Public Types inherited from Optimist::RootFinder::RootFinder< Real, N, Greenstadt< Real, N >, true >
using	Vector
using	Matrix
using	Tensor
Public Types inherited from Optimist::SolverBase< Real, N, N, Greenstadt< Real, N >, ForceEigen >
using	InputType
using	OutputType
using	TraceType
using	FirstDerivativeType
using	SecondDerivativeType

Public Member Functions
	Greenstadt ()
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 (Vector const &, Vector const &, Matrix const &jacobian_old, Vector const &delta_x_new, Vector const &delta_function_new, Vector const &function_new, Matrix &jacobian_new)
Public Member Functions inherited from Optimist::RootFinder::QuasiNewton< Real, N, Greenstadt< Real, N > >
	QuasiNewton ()
std::string	name_impl () const
bool	solve_impl (FunctionLambda &&function, JacobianLambda &&jacobian, Vector const &x_ini, Vector &x_sol)
void	update (Vector const &delta_x_old, Vector const &delta_function_old, Matrix const &jacobian_old, Vector const &delta_x_new, Vector const &delta_function_new, Vector const &function_new, Matrix &jacobian_new)
Public Member Functions inherited from Optimist::RootFinder::RootFinder< Real, N, Greenstadt< Real, N >, true >
	RootFinder ()
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, Vector const &x_ini, Vector &x_sol)
Public Member Functions inherited from Optimist::SolverBase< Real, N, N, Greenstadt< Real, N >, ForceEigen >
	SolverBase ()
InputType const &	lower_bound () const
InputType const &	upper_bound () const
void	bounds (InputType const &t_lower_bound, InputType const &t_upper_bound)
constexpr Integer	input_dimension () const
constexpr Integer	output_dimension () const
Integer	function_evaluations () const
void	max_function_evaluations (Integer t_max_function_evaluations)
Integer	iterations () const
Integer	max_iterations () const
Real	alpha () const
Integer	relaxations () const
Integer	max_relaxations () const
Real	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
const TraceType &	trace () const
std::ostream &	ostream () const
bool	solve (FunctionLambda &&function, InputType const &x_ini, InputType &x_sol)
bool	rootfind (FunctionBase< Real, FunInDim, FunOutDim, DerivedFunction, ForceEigen &&FunOutDim==1 > const &function, InputType const &x_ini, InputType &x_sol)
bool	optimize (FunctionBase< Real, FunInDim, FunOutDim, DerivedFunction, ForceEigen &&FunOutDim==1 > const &function, InputType const &x_ini, InputType &x_sol)
std::string	name () const

Static Public Attributes
static constexpr bool	requires_function {true}
static constexpr bool	requires_first_derivative {true}
static constexpr bool	requires_second_derivative {false}
Static Public Attributes inherited from Optimist::RootFinder::RootFinder< Real, N, Greenstadt< Real, N >, true >
static constexpr bool	is_rootfinder
static constexpr bool	is_optimizer

Private Attributes
Method	m_method {Method::ONE}

Additional Inherited Members
Protected Member Functions inherited from Optimist::RootFinder::RootFinder< Real, N, Greenstadt< Real, N >, true >
bool	evaluate_jacobian (JacobianLambda &&jacobian, Vector const &x, Matrix &out)
bool	evaluate_hessian (HessianLambda &&hessian, Vector const &x, Matrix &out)
Protected Member Functions inherited from Optimist::SolverBase< Real, N, N, Greenstadt< Real, N >, ForceEigen >
Integer	first_derivative_evaluations () const
Integer	max_first_derivative_evaluations () const
Integer	second_derivative_evaluations () const
Integer	max_second_derivative_evaluations () const
void	reset ()
bool	evaluate_function (FunctionLambda &&function, InputType const &x, OutputType &out)
bool	evaluate_first_derivative (FirstDerivativeLambda &&function, InputType const &x, FirstDerivativeType &out)
bool	evaluate_second_derivative (SecondDerivativeLambda &&function, InputType const &x, SecondDerivativeType &out)
void	store_trace (InputType const &x)
bool	damp (FunctionLambda &&function, InputType const &x_old, InputType const &function_old, InputType const &step_old, InputType &x_new, InputType &function_new, InputType &step_new)
void	header ()
void	bottom ()
void	info (Real residuals, std::string const &notes="-")
Protected Attributes inherited from Optimist::SolverBase< Real, N, N, Greenstadt< Real, N >, ForceEigen >
InputType	m_lower_bound
InputType	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
Real	m_alpha
Integer	m_relaxations
Integer	m_max_relaxations
Real	m_tolerance
bool	m_verbose
bool	m_damped
std::ostream *	m_ostream
std::string	m_task
bool	m_converged
TraceType	m_trace

Detailed Description

template<typename Real, Integer N>
class Optimist::RootFinder::Greenstadt< Real, N >

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

Real	Scalar number type.
N	Dimension of the root-finding problem.

Member Typedef Documentation

◆ Method

template<typename Real, Integer N>

using Optimist::RootFinder::Greenstadt< Real, N >::Method = enum class Method : Integer {ONE = 1, TWO = 2}

Greenstadt solver type enumeration.

Constructor & Destructor Documentation

◆ Greenstadt()

template<typename Real, Integer N>

Optimist::RootFinder::Greenstadt< Real, N >::Greenstadt ( )

inline

Class constructor for the Greenstadt solver.

Member Function Documentation

◆ enable_one_method()

template<typename Real, Integer N>

void Optimist::RootFinder::Greenstadt< Real, N >::enable_one_method ( )

inline

Enable the Greenstadt1 solver method.

◆ enable_two_method()

template<typename Real, Integer N>

void Optimist::RootFinder::Greenstadt< Real, N >::enable_two_method ( )

inline

Enable the Greenstadt2 solver method.

◆ method() [1/2]

template<typename Real, Integer N>

Method Optimist::RootFinder::Greenstadt< Real, N >::method ( ) const

inline

Get the enumeration type of the Greenstadt solver method.

Returns: The Greenstadt solver enumeration type.

◆ method() [2/2]

template<typename Real, Integer N>

void Optimist::RootFinder::Greenstadt< Real, N >::method ( Method t_method )

inline

Set the enumeration type of the Greenstadt solver method.

Parameters

[in] t_method The Greenstadt solver method enumeration type.

◆ name_impl()

template<typename Real, Integer N>

std::string Optimist::RootFinder::Greenstadt< Real, N >::name_impl ( ) const

inline

Get the Greenstadt solver name.

Returns: The Greenstadt solver name.

◆ set_method()

template<typename Real, Integer N>

void Optimist::RootFinder::Greenstadt< Real, N >::set_method ( Method t_method )

inline

Set the Greenstadt solver method.

Parameters

[in] t_method The Greenstadt solver method enumeration type.

◆ update_impl()

template<typename Real, Integer N>

void Optimist::RootFinder::Greenstadt< Real, N >::update_impl	(	Vector const &	,
		Vector const &	,
		Matrix const &	jacobian_old,
		Vector const &	delta_x_new,
		Vector const &	delta_function_new,
		Vector const &	function_new,
		Matrix &	jacobian_new )

inline

Jacobian approximation update rule for the Greenstadt's method.

Parameters

[in]	delta_x_old	Old difference between points.
[in]	delta_function_old	Old difference between function values.
[in]	jacobian_old	Old jacobian approximation.
[in]	delta_x_new	New difference between points.
[in]	delta_function_new	New difference between function values.
[in]	function_new	New function value.
[out]	jacobian_new	New jacobian approximation.

Member Data Documentation

◆ m_method

template<typename Real, Integer N>

Method Optimist::RootFinder::Greenstadt< Real, N >::m_method {Method::ONE}

private

Greenstadt solver type.

◆ requires_first_derivative

template<typename Real, Integer N>

bool Optimist::RootFinder::Greenstadt< Real, N >::requires_first_derivative {true}

staticconstexpr

◆ requires_function

template<typename Real, Integer N>

bool Optimist::RootFinder::Greenstadt< Real, N >::requires_function {true}

staticconstexpr

◆ requires_second_derivative

template<typename Real, Integer N>

bool Optimist::RootFinder::Greenstadt< Real, N >::requires_second_derivative {false}

staticconstexpr

The documentation for this class was generated from the following file:

include/Optimist/RootFinder/Greenstadt.hh

Public Types

Public Member Functions

Static Public Attributes

Private Attributes

Additional Inherited Members

Detailed Description

Greenstadt's method

Member Typedef Documentation

◆ Method

Constructor & Destructor Documentation

◆ Greenstadt()

Member Function Documentation

◆ enable_one_method()

◆ enable_two_method()

◆ method() [1/2]

◆ method() [2/2]

◆ name_impl()

◆ set_method()

◆ update_impl()

Member Data Documentation

◆ m_method

◆ requires_first_derivative

◆ requires_function

◆ requires_second_derivative