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

#include <Newton.hh>

Inherits Optimist::RootFinder::RootFinder< Real, N, Newton< Real, N >, true >.

Public Types
using	Vector = typename RootFinder<Real, N, Newton<Real, N>, true>::Vector
using	Matrix = typename RootFinder<Real, N, Newton<Real, N>, true>::Matrix
using	FunctionWrapper = typename RootFinder<Real, N, Newton<Real, N>, true>::FunctionWrapper
using	JacobianWrapper = typename RootFinder<Real, N, Newton<Real, N>, true>::JacobianWrapper
Public Types inherited from Optimist::RootFinder::RootFinder< Real, N, Newton< Real, N >, true >
using	Vector
using	Matrix
using	Tensor
using	FunctionWrapper
using	JacobianWrapper
using	HessianWrapper

Public Member Functions
	Newton ()
std::string	name_impl () const
bool	solve_impl (FunctionWrapper function, JacobianWrapper jacobian, Vector const &x_ini, Vector &x_sol)
Public Member Functions inherited from Optimist::RootFinder::RootFinder< Real, N, Newton< 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 (FunctionWrapper function, Vector const &x_ini, Vector &x_sol)
Public Member Functions inherited from Optimist::SolverBase< Real, N, N, Newton< Real, N >, ForceEigen >
	SolverBase ()
const InputType &	lower_bound () const
const InputType &	upper_bound () const
void	bounds (const InputType &t_lower_bound, const InputType &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 (FunctionWrapper function, const InputType &x_ini, InputType &x_sol)
bool	rootfind (FunctionBase< Real, FunInDim, FunOutDim, DerivedFunction, ForceEigen &&FunOutDim==1 > const &function, const InputType &x_ini, InputType &x_sol)
bool	optimize (FunctionBase< Real, FunInDim, FunOutDim, DerivedFunction, ForceEigen &&FunOutDim==1 > const &function, const InputType &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, Newton< Real, N >, true >
static constexpr bool	is_rootfinder
static constexpr bool	is_optimizer
static constexpr bool	requires_function
static constexpr bool	requires_first_derivative
static constexpr bool	requires_second_derivative

Private Attributes
Eigen::FullPivLU< Matrix >	m_lu

Additional Inherited Members
Protected Types inherited from Optimist::SolverBase< Real, N, N, Newton< Real, N >, ForceEigen >
using	InputType
using	OutputType
using	TraceType
using	FirstDerivativeType
using	SecondDerivativeType
using	FunctionWrapper
using	FirstDerivativeWrapper
using	SecondDerivativeWrapper
Protected Member Functions inherited from Optimist::RootFinder::RootFinder< Real, N, Newton< Real, N >, true >
void	evaluate_jacobian (JacobianWrapper jacobian, const Vector &x, Matrix &out)
void	evaluate_hessian (HessianWrapper hessian, const Vector &x, Matrix &out)
Protected Member Functions inherited from Optimist::SolverBase< Real, N, N, Newton< 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 ()
void	evaluate_function (FunctionWrapper function, const InputType &x, OutputType &out)
void	evaluate_first_derivative (FirstDerivativeWrapper function, const InputType &x, FirstDerivativeType &out)
void	evaluate_second_derivative (SecondDerivativeWrapper function, const InputType &x, SecondDerivativeType &out)
void	store_trace (const InputType &x)
bool	damp (FunctionWrapper 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, Newton< 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::Newton< Real, N >

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

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

Member Typedef Documentation

◆ FunctionWrapper

template<typename Real, Integer N>

using Optimist::RootFinder::Newton< Real, N >::FunctionWrapper = typename RootFinder<Real, N, Newton<Real, N>, true>::FunctionWrapper

◆ JacobianWrapper

template<typename Real, Integer N>

using Optimist::RootFinder::Newton< Real, N >::JacobianWrapper = typename RootFinder<Real, N, Newton<Real, N>, true>::JacobianWrapper

◆ Matrix

template<typename Real, Integer N>

using Optimist::RootFinder::Newton< Real, N >::Matrix = typename RootFinder<Real, N, Newton<Real, N>, true>::Matrix

◆ Vector

template<typename Real, Integer N>

using Optimist::RootFinder::Newton< Real, N >::Vector = typename RootFinder<Real, N, Newton<Real, N>, true>::Vector

Constructor & Destructor Documentation

◆ Newton()

template<typename Real, Integer N>

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

inline

Class constructor for the Newton solver.

Member Function Documentation

◆ name_impl()

template<typename Real, Integer N>

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

inline

Get the Newton solver name.

Returns: The Newton solver name.

◆ solve_impl()

template<typename Real, Integer N>

bool Optimist::RootFinder::Newton< Real, N >::solve_impl	(	FunctionWrapper	function,
		JacobianWrapper	jacobian,
		Vector const &	x_ini,
		Vector &	x_sol )

inline

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

Parameters

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

Returns: The convergence boolean flag.

Member Data Documentation

◆ m_lu

template<typename Real, Integer N>

Eigen::FullPivLU<Matrix> Optimist::RootFinder::Newton< Real, N >::m_lu

private

LU decomposition.

◆ requires_first_derivative

template<typename Real, Integer N>

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

staticconstexpr

◆ requires_function

template<typename Real, Integer N>

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

staticconstexpr

◆ requires_second_derivative

template<typename Real, Integer N>

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

staticconstexpr

Basic constants.

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

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

◆ FunctionWrapper

◆ JacobianWrapper

◆ Matrix

◆ Vector

Constructor & Destructor Documentation

◆ Newton()

Member Function Documentation

◆ name_impl()

◆ solve_impl()

Member Data Documentation

◆ m_lu

◆ requires_first_derivative

◆ requires_function

◆ requires_second_derivative