Class container for the scalar scalar root-finder. More...

#include <ScalarRootFinder.hh>

Inherits Optimist::Solver< Real, 1, 1, DerivedSolver >.

Inherited by Optimist::ScalarRootFinder::Bracketing< Real, DerivedSolver >.

Public Types
using	FunctionWrapper = typename Solver<Real, 1, 1, DerivedSolver>::FunctionWrapper

using	FirstDerivativeWrapper = typename Solver<Real, 1, 1, DerivedSolver>::FirstDerivativeWrapper

using	SecondDerivativeWrapper = typename Solver<Real, 1, 1, DerivedSolver>::SecondDerivativeWrapper

Public Member Functions
	ScalarRootFinder ()

std::string	name () const

bool	solve (FunctionWrapper function, Real x_ini, Real &x_sol)

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

bool	solve (FunctionWrapper function, FirstDerivativeWrapper first_derivative, SecondDerivativeWrapper second_derivate, Real x_ini, Real &x_sol)

Public Member Functions inherited from Optimist::Solver< Real, 1, 1, DerivedSolver >
	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

Public Attributes
friend	Solver< Real, 1, 1, ScalarRootFinder< Real, DerivedSolver > >

Static Public Attributes
static constexpr bool	is_rootfinder {true}

static constexpr bool	is_optimizer {false}

static constexpr bool	requires_function {DerivedSolver::requires_function}

static constexpr bool	requires_first_derivative {DerivedSolver::requires_first_derivative}

static constexpr bool	requires_second_derivative {DerivedSolver::requires_second_derivative}

Additional Inherited Members
Protected Types inherited from Optimist::Solver< Real, 1, 1, DerivedSolver >
using	InputType

using	OutputType

using	TraceType

using	FirstDerivativeType

using	SecondDerivativeType

using	FunctionWrapper

using	FirstDerivativeWrapper

using	SecondDerivativeWrapper

Protected Member Functions inherited from Optimist::Solver< Real, 1, 1, DerivedSolver >
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, 1, 1, DerivedSolver >
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, typename DerivedSolver>
class Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >

Scalar root finder

This section describes the scalar root-finders implemented in Optimist. The available root-finders are derivative and non-derivative methods. Derivative methods employ the function's derivative to find the root with high accuracy, while non-derivative methods approximate the derivative for improved efficiency in certain scenarios.

Here, the solvers are implemented for solving problems of the form

\[ f(x) = 0 \text{,} \quad \text{with} \quad f: \mathbb{R} \rightarrow \mathbb{R} \text{,} \]

which consist in finding the root of the function \(f\) by iteratively updating the current iterate \(x_k\) until convergence is achieved. The solvers require the function \(f\) and its first derivative \(f^{\prime}(x)\) to be provided by the user. Alternatively, the derivative can be approximated numerically using finite differences, depending on the problem's complexity and the user's preference.

Affine-invariant step

By default, the derivative scalar root-finders in Optimist use optionally an affine-invariant advancing step to ensure convergence. The generic advancing step \(h_k\) is computed as

\[ x_{k+1} = x_k + \alpha_k h_k \text{,} \]

where \(\alpha_k\) is a damping coefficient that ensures convergence by satisfying

\[ \left\|f^{\prime}_{x}(x_k)^{-1} f(x_{k+1})\right\| \leq \left(1 - \displaystyle\frac{\alpha_k}{2}\right) \left\|f^{\prime}_{x}(x_k)^{-1} f(x_k)\right\| = \left(1 - \displaystyle\frac{\alpha_k}{2} \right) \left\|h_k\right\| \text{.} \]

For detailed information on the affine-invariant Newton's method, refer to this link, or the works by P. Deuflhard.

Template Parameters

Real	Scalar number type.
DerivedSolver	Derived solver class.

Member Typedef Documentation

◆ FirstDerivativeWrapper

template<typename Real, typename DerivedSolver>

using Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::FirstDerivativeWrapper = typename Solver<Real, 1, 1, DerivedSolver>::FirstDerivativeWrapper

◆ FunctionWrapper

template<typename Real, typename DerivedSolver>

using Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::FunctionWrapper = typename Solver<Real, 1, 1, DerivedSolver>::FunctionWrapper

◆ SecondDerivativeWrapper

template<typename Real, typename DerivedSolver>

using Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::SecondDerivativeWrapper = typename Solver<Real, 1, 1, DerivedSolver>::SecondDerivativeWrapper

Constructor & Destructor Documentation

◆ ScalarRootFinder()

template<typename Real, typename DerivedSolver>

Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::ScalarRootFinder ( )

inline

Class constructor for the scalar root-finder.

Member Function Documentation

◆ name()

template<typename Real, typename DerivedSolver>

std::string Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::name ( ) const

inline

Get the solver name.

Returns: The solver name.

◆ solve() [1/3]

template<typename Real, typename DerivedSolver>

bool Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::solve	(	FunctionWrapper	function,
		FirstDerivativeWrapper	first_derivative,
		Real	x_ini,
		Real &	x_sol )

inline

Solve the root-finding problem given the function, and its first derivative.

Parameters

[in]	function	Function wrapper.
[in]	first_derivative	First derivative wrapper.
[in]	x_ini	Initialization point.
[out]	x_sol	Solution point.

Returns: The convergence boolean flag.

◆ solve() [2/3]

template<typename Real, typename DerivedSolver>

bool Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::solve	(	FunctionWrapper	function,
		FirstDerivativeWrapper	first_derivative,
		SecondDerivativeWrapper	second_derivate,
		Real	x_ini,
		Real &	x_sol )

inline

Solve the root-finding problem given the function, and its first and second derivatives.

Parameters

[in]	function	Function wrapper.
[in]	first_derivative	First derivative wrapper.
[in]	x_ini	Initialization point.
[out]	x_sol	Solution point.

Returns: The convergence boolean flag.

◆ solve() [3/3]

template<typename Real, typename DerivedSolver>

bool Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::solve	(	FunctionWrapper	function,
		Real	x_ini,
		Real &	x_sol )

inline

Solve the root-finding problem given the function, and without derivatives.

Parameters

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

Returns: The convergence boolean flag.

Member Data Documentation

◆ is_optimizer

template<typename Real, typename DerivedSolver>

bool Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::is_optimizer {false}

staticconstexpr

◆ is_rootfinder

template<typename Real, typename DerivedSolver>

bool Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::is_rootfinder {true}

staticconstexpr

◆ requires_first_derivative

template<typename Real, typename DerivedSolver>

bool Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::requires_first_derivative {DerivedSolver::requires_first_derivative}

staticconstexpr

◆ requires_function

template<typename Real, typename DerivedSolver>

bool Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::requires_function {DerivedSolver::requires_function}

staticconstexpr

◆ requires_second_derivative

template<typename Real, typename DerivedSolver>

bool Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::requires_second_derivative {DerivedSolver::requires_second_derivative}

staticconstexpr

Basic constants.

◆ Solver< Real, 1, 1, ScalarRootFinder< Real, DerivedSolver > >

template<typename Real, typename DerivedSolver>

friend Optimist::ScalarRootFinder::ScalarRootFinder< Real, DerivedSolver >::Solver< Real, 1, 1, ScalarRootFinder< Real, DerivedSolver > >

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

include/Optimist/ScalarRootFinder.hh

Public Types

Public Member Functions

Public Attributes

Static Public Attributes

Additional Inherited Members

Detailed Description

Scalar root finder

Affine-invariant step

Member Typedef Documentation

◆ FirstDerivativeWrapper

◆ FunctionWrapper

◆ SecondDerivativeWrapper

Constructor & Destructor Documentation

◆ ScalarRootFinder()

Member Function Documentation

◆ name()

◆ solve() [1/3]

◆ solve() [2/3]

◆ solve() [3/3]

Member Data Documentation

◆ is_optimizer

◆ is_rootfinder

◆ requires_first_derivative

◆ requires_function

◆ requires_second_derivative

◆ Solver< Real, 1, 1, ScalarRootFinder< Real, DerivedSolver > >