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

#include <Newton.hh>

Inherits Optimist::RootFinder::RootFinder< Vector, Newton< Vector > >.

Public Types
using	VectorTrait = TypeTrait<Vector>
using	Scalar = typename TypeTrait<Vector>::Scalar
using	Factorization
Public Types inherited from Optimist::RootFinder::RootFinder< Vector, Newton< Vector > >
using	Scalar
using	Input
using	Output
Public Types inherited from Optimist::SolverBase< Vector, Vector, Newton< Vector > >
using	InputTrait
using	OutputTrait
using	Scalar
using	FirstDerivative
using	SecondDerivative

Public Member Functions
	Newton ()
constexpr std::string	name_impl () const
template<typename FunctionLambda, typename JacobianLambda>
bool	solve_impl (FunctionLambda &&function, JacobianLambda &&jacobian, const Vector &x_ini, Vector &x_sol)
Public Member Functions inherited from Optimist::RootFinder::RootFinder< Vector, Newton< Vector > >
	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)
Public Member Functions inherited from Optimist::SolverBase< Vector, Vector, Newton< Vector > >
	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

Static Public Attributes
static constexpr bool	RequiresFunction {true}
static constexpr bool	RequiresFirstDerivative {true}
static constexpr bool	RequiresSecondDerivative {false}
Static Public Attributes inherited from Optimist::RootFinder::RootFinder< Vector, Newton< Vector > >
static constexpr bool	IsRootFinder
static constexpr bool	IsOptimizer

Private Attributes
Factorization	m_lu

Additional Inherited Members
Protected Member Functions inherited from Optimist::RootFinder::RootFinder< Vector, Newton< Vector > >
bool	evaluate_jacobian (JacobianLambda &&jacobian, const Input &x, FirstDerivative &out)
bool	evaluate_hessian (HessianLambda &&hessian, const Input &x, SecondDerivative &out)
Protected Member Functions inherited from Optimist::SolverBase< Vector, Vector, Newton< Vector > >
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 &notes="-")
Protected Attributes inherited from Optimist::SolverBase< Vector, Vector, Newton< Vector > >
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

Detailed Description

template<typename Vector>
requires TypeTrait<Vector>::IsEigen && (!TypeTrait<Vector>::IsFixed || TypeTrait<Vector>::Dimension > 0)
class Optimist::RootFinder::Newton< Vector >

Newton's method

Newton's method, including its damped variant with an affine-invariant step, is based on the linearization of the function \(\mathbf{f}(\mathbf{x})\) around the current iterate \(\mathbf{x}_k\), which leads to the linear system

\[ \mathbf{Jf}_{\mathbf{x}}(\mathbf{x}_k) \mathbf{h}_k = -\mathbf{f}(\mathbf{x}_k) \text{.} \]

The advancing step \(\mathbf{h}_k\) is then computed as

\[ \mathbf{x}_{k+1} = \mathbf{x}_k + \alpha_k \mathbf{h}_k \text{,} \]

where \(\alpha_k\) is a damping coefficient that ensures affine-invariant criteria is satisfied.

Template Parameters

Vector Eigen vector type.

Member Typedef Documentation

◆ Factorization

template<typename Vector>

using Optimist::RootFinder::Newton< Vector >::Factorization

Initial value:

 
std::conditional_t<VectorTrait::IsSparse,
                   Eigen::SparseLU<FirstDerivative>,
                   Eigen::FullPivLU<FirstDerivative>>

◆ Scalar

template<typename Vector>

using Optimist::RootFinder::Newton< Vector >::Scalar = typename TypeTrait<Vector>::Scalar

◆ VectorTrait

template<typename Vector>

using Optimist::RootFinder::Newton< Vector >::VectorTrait = TypeTrait<Vector>

Constructor & Destructor Documentation

◆ Newton()

template<typename Vector>

Optimist::RootFinder::Newton< Vector >::Newton ( )

inline

Class constructor for the Newton solver.

Member Function Documentation

◆ name_impl()

template<typename Vector>

std::string Optimist::RootFinder::Newton< Vector >::name_impl ( ) const

inlineconstexpr

Get the Newton solver name.

Returns: The Newton solver name.

◆ solve_impl()

template<typename Vector>

template<typename FunctionLambda, typename JacobianLambda>

bool Optimist::RootFinder::Newton< Vector >::solve_impl	(	FunctionLambda &&	function,
		JacobianLambda &&	jacobian,
		const Vector &	x_ini,
		Vector &	x_sol )

inline

Solve the nonlinear system of equations \( \mathbf{f}(\mathbf{x}) = 0 \), with \(\mathbf{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n \).

Template Parameters

FunctionLambda	Function lambda type.
JacobianLambda	Jacobian lambda type.

Parameters

[in]	function	Function lambda.
[in]	jacobian	Jacobian lambda.
[in]	x_ini	Initialization point.
[out]	x_sol	Solution point.

Returns: The convergence boolean flag.

Member Data Documentation

◆ m_lu

template<typename Vector>

Factorization Optimist::RootFinder::Newton< Vector >::m_lu

private

LU decomposition.

◆ RequiresFirstDerivative

template<typename Vector>

bool Optimist::RootFinder::Newton< Vector >::RequiresFirstDerivative {true}

staticconstexpr

◆ RequiresFunction

template<typename Vector>

bool Optimist::RootFinder::Newton< Vector >::RequiresFunction {true}

staticconstexpr

◆ RequiresSecondDerivative

template<typename Vector>

bool Optimist::RootFinder::Newton< Vector >::RequiresSecondDerivative {false}

staticconstexpr

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

core/include/Optimist/RootFinder/Newton.hh

Public Types

Public Member Functions

Static Public Attributes

Private Attributes

Additional Inherited Members

Detailed Description

Newton's method

Member Typedef Documentation

◆ Factorization

◆ Scalar

◆ VectorTrait

Constructor & Destructor Documentation

◆ Newton()

Member Function Documentation

◆ name_impl()

◆ solve_impl()

Member Data Documentation

◆ m_lu

◆ RequiresFirstDerivative

◆ RequiresFunction

◆ RequiresSecondDerivative