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

#include <Newton.hh>

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

Public Types
using	Vector = typename RootFinder<Real, N, Newton<Real, N>>::Vector

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

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

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

Public Types inherited from Optimist::RootFinder::RootFinder< Real, N, Newton< Real, N > >
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 > >
	RootFinder ()

std::string	name () const

Integer	jacobian_evaluations () const

Integer	max_jacobian_evaluations () const

void	max_jacobian_evaluations (Integer t_jacobian_evaluations)

Integer	hessian_evaluations () const

Integer	max_hessian_evaluations () const

void	max_hessian_evaluations (Integer t_hessian_evaluations)

bool	solve (FunctionWrapper function, Vector const &x_ini, Vector &x_sol)

bool	solve (FunctionWrapper function, JacobianWrapper jacobian, Vector const &x_ini, Vector &x_sol)

bool	solve (FunctionWrapper function, JacobianWrapper jacobian, HessianWrapper hessian, Vector const &x_ini, Vector &x_sol)

Public Member Functions inherited from Optimist::Solver< Real, N, N, Newton< Real, N > >
	Solver ()

	Solver (FunctionWrapper function, const InputType &x_ini, InputType &x_sol)

	Solver (FunctionWrapper function, FirstDerivativeWrapper first_derivative, const InputType &x_ini, InputType &x_sol)

	Solver (FunctionWrapper function, FirstDerivativeWrapper first_derivative, SecondDerivativeWrapper second_derivative, const InputType &x_ini, InputType &x_sol)

const InputType &	lower_bound () const

void	lower_bound (const InputType &t_lower_bound)

const InputType &	upper_bound () const

void	upper_bound (const InputType &t_upper_bound)

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	max_function_evaluations () const

Integer	iterations () const

Integer	max_iterations () const

void	max_iterations (Integer t_max_iterations)

Real	alpha () const

void	alpha (Real t_alpha)

Integer	relaxations () const

Integer	max_relaxations () const

void	max_relaxations (Integer t_max_relaxations)

Real	tolerance () const

void	tolerance (Real t_tolerance)

void	verbose_mode (bool t_verbose)

bool	verbose_mode () const

void	enable_verbose_mode ()

void	disable_verbose_mode ()

void	damped_mode (bool t_damped)

bool	damped_mode () const

void	enable_damped_mode ()

void	disable_damped_mode ()

std::string	task () const

void	task (std::string t_task)

bool	converged () const

const TraceType &	trace () const

std::ostream &	ostream () const

void	ostream (std::ostream &t_ostream)

bool	solve (FunctionWrapper function, const InputType &x_ini, InputType &x_sol)

bool	solve (FunctionWrapper function, FirstDerivativeWrapper first_derivative, const InputType &x_ini, InputType &x_sol)

bool	solve (FunctionWrapper function, FirstDerivativeWrapper first_derivative, SecondDerivativeWrapper second_derivative, const InputType &x_ini, InputType &x_sol)

bool	rootfind (Function< Real, FunInDim, FunOutDim, DerivedFunction > const &function, const InputType &x_ini, InputType &x_sol)

bool	optimize (Function< Real, FunInDim, FunOutDim, DerivedFunction > 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 > >
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
Public Attributes inherited from Optimist::RootFinder::RootFinder< Real, N, Newton< Real, N > >
friend	Solver< Real, N, N, RootFinder< Real, N, DerivedSolver > >

Protected Types inherited from Optimist::Solver< Real, N, N, Newton< Real, N > >
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 > >
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::Solver< Real, N, N, Newton< Real, N > >
Integer	first_derivative_evaluations () const

Integer	max_first_derivative_evaluations () const

void	max_first_derivative_evaluations (Integer first_derivative_evaluations)

Integer	second_derivative_evaluations () const

Integer	max_second_derivative_evaluations () const

void	max_second_derivative_evaluations (Integer second_derivative_evaluations)

bool	solve (Function< Real, FunInDim, FunOutDim, DerivedFunction > const &function, const InputType &x_ini, InputType &x_sol, bool is_optimization)

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::Solver< Real, N, N, Newton< Real, N > >
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>>::FunctionWrapper

◆ JacobianWrapper

template<typename Real, Integer N>

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

◆ Matrix

template<typename Real, Integer N>

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

◆ Vector

template<typename Real, Integer N>

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