Class container for the Algorithm 748. More...

#include <Algo748.hh>

Inherits Optimist::RootFinder::Bracketing< Real, Algo748< Real > >.

Public Types
using	FunctionWrapper = typename Bracketing<Real, Algo748<Real>>::FunctionWrapper
using	FirstDerivativeWrapper = typename Bracketing<Real, Algo748<Real>>::FirstDerivativeWrapper
using	SecondDerivativeWrapper = typename Bracketing<Real, Algo748<Real>>::SecondDerivativeWrapper
Public Types inherited from Optimist::RootFinder::Bracketing< Real, Algo748< Real > >
using	FunctionWrapper
using	FirstDerivativeWrapper
using	SecondDerivativeWrapper
Public Types inherited from Optimist::RootFinder::RootFinder< Real, 1, Algo748< Real > >
using	Vector
using	Matrix
using	Tensor
using	FunctionWrapper
using	JacobianWrapper
using	HessianWrapper

Public Member Functions
	Algo748 ()
std::string	name_impl () const
Real	find_root_impl (FunctionWrapper function)
Public Member Functions inherited from Optimist::RootFinder::Bracketing< Real, Algo748< Real > >
	Bracketing ()
std::string	name_impl () const
void	tolerance_bracketing (Real t_tolerance)
bool	solve_impl (FunctionWrapper function, Real, Real &x_sol)
Real	find_root (FunctionWrapper function)
Public Member Functions inherited from Optimist::RootFinder::RootFinder< Real, 1, Algo748< Real > >
	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, Algo748< Real >, false >
	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 {false}
static constexpr bool	requires_second_derivative {false}
Static Public Attributes inherited from Optimist::RootFinder::Bracketing< Real, Algo748< Real > >
static constexpr bool	requires_function
static constexpr bool	requires_first_derivative
static constexpr bool	requires_second_derivative
Static Public Attributes inherited from Optimist::RootFinder::RootFinder< Real, 1, Algo748< Real > >
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 Member Functions
bool	all_different (Real a, Real b, Real c, Real d) const
bool	bracketing (FunctionWrapper function)
Real	cubic_interpolation ()
Real	newton_quadratic (Integer n)

Additional Inherited Members
Protected Types inherited from Optimist::SolverBase< Real, N, N, Algo748< Real >, false >
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, 1, Algo748< Real > >
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, Algo748< Real >, false >
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::RootFinder::Bracketing< Real, Algo748< Real > >
Real	m_tolerance_bracketing
Real	m_mu
Real	m_interval_shink
Real	m_a
Real	m_fa
Real	m_b
Real	m_fb
Real	m_c
Real	m_fc
Real	m_d
Real	m_fd
Real	m_e
Real	m_fe
Protected Attributes inherited from Optimist::SolverBase< Real, N, N, Algo748< Real >, false >
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>
class Optimist::RootFinder::Algo748< Real >

The algorithm 748 allows to find the roots of a scalar function \(f(x)\) in a given interval \([a, b]\). The algorithm is based on the work of G. Alefeld, F. Potra, and Y. Shi, Algorithm 748: Enclosing Zeros of Continuous Functions, ACM Transactions on Mathematical Software, 21 (1995), pp. 327-344, 10.1145/210089.210111.

Template Parameters

Real	Scalar number type.

Member Typedef Documentation

◆ FirstDerivativeWrapper

template<typename Real>

using Optimist::RootFinder::Algo748< Real >::FirstDerivativeWrapper = typename Bracketing<Real, Algo748<Real>>::FirstDerivativeWrapper

◆ FunctionWrapper

template<typename Real>

using Optimist::RootFinder::Algo748< Real >::FunctionWrapper = typename Bracketing<Real, Algo748<Real>>::FunctionWrapper

◆ SecondDerivativeWrapper

template<typename Real>

using Optimist::RootFinder::Algo748< Real >::SecondDerivativeWrapper = typename Bracketing<Real, Algo748<Real>>::SecondDerivativeWrapper

Constructor & Destructor Documentation

◆ Algo748()

template<typename Real>

Optimist::RootFinder::Algo748< Real >::Algo748 ( )

inline

Class constructor for the Algorithm 748.

Member Function Documentation

◆ all_different()

template<typename Real>

bool Optimist::RootFinder::Algo748< Real >::all_different	(	Real	a,
		Real	b,
		Real	c,
		Real	d ) const

inlineprivate

Check if the input values are all different.

Parameters

[in]	a	The first value.
[in]	b	The second value.
[in]	c	The third value.
[in]	d	The fourth value.

Returns: The boolean flag, true if all the values are different, false otherwise.

◆ bracketing()

template<typename Real>

bool Optimist::RootFinder::Algo748< Real >::bracketing ( FunctionWrapper function )

inlineprivate

Given current enclosing interval \([a, b]\) and \(a\) number \(c\) in \((a, b)\):

if \(f(c) = 0\) then sets the output \(a = c\).
Otherwise determines the new enclosing interval \([a, b] = [a, c]\) or \([a, b] = [c, b]\). also updates the termination criterion corresponding to the new enclosing interval. Adjust \(c\) if \((b-a)\) is very small or if \(c\) is very close to \(a\) or \(b\).
Parameters

[in] function Function wrapper.

Returns
The boolean flag, true if the root is found, false otherwise.

◆ cubic_interpolation()

template<typename Real>

Real Optimist::RootFinder::Algo748< Real >::cubic_interpolation ( )

inlineprivate

Perform the cubic inverse interpolation of \(f(x)\) at \(a\), \(b\), \(d\), and \(e\) to get an approximate root \(r\) ( \(f(r) = 0\)).

Returns: The approximate root \(r\).

◆ find_root_impl()

template<typename Real>

Real Optimist::RootFinder::Algo748< Real >::find_root_impl ( FunctionWrapper function )

inline

Finds either an exact solution or an approximate solution of the equation \(f(x) = 0\) in the interval \([a,b]\). At the beginning of each iteration, the current enclosing interval is recorded as \([a_0, b_0]\). The first iteration is simply a secant step. Starting with the second iteration, three steps are taken in each iteration.

The first two steps are either quadratic interpolation or cubic inverse interpolation.
The third step is a double-size secant step. If the diameter of the enclosing interval obtained after these three steps is larger than \(\mu*(b_0-a_0)\), an additional bisection step will be used to shrink the enclosing interval.
Parameters

[in] function Function wrapper.

Returns
The approximate root.

◆ name_impl()

template<typename Real>

std::string Optimist::RootFinder::Algo748< Real >::name_impl ( ) const

inline

Get the Algorithm 748 solver name.

Returns: The Algorithm 748 solver name.

◆ newton_quadratic()

template<typename Real>

Real Optimist::RootFinder::Algo748< Real >::newton_quadratic ( Integer n )

inlineprivate

Perform \(n\) Newton steps to approximate the zero in \((a, b)\) of the quadratic polynomial interpolating \(f(x)\) at \(a\), \(b\), and \(d\). Safeguard is used to avoid overflow.

Parameters

[in] n The number of steps.

Returns: The approximate root.

Member Data Documentation

◆ requires_first_derivative

template<typename Real>

bool Optimist::RootFinder::Algo748< Real >::requires_first_derivative {false}

staticconstexpr

◆ requires_function

template<typename Real>

bool Optimist::RootFinder::Algo748< Real >::requires_function {true}

staticconstexpr

◆ requires_second_derivative

template<typename Real>

bool Optimist::RootFinder::Algo748< Real >::requires_second_derivative {false}

staticconstexpr

Basic constants.

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

include/Optimist/RootFinder/Algo748.hh

Public Types

Public Member Functions

Static Public Attributes

Private Member Functions

Additional Inherited Members

Detailed Description

Member Typedef Documentation

◆ FirstDerivativeWrapper

◆ FunctionWrapper

◆ SecondDerivativeWrapper

Constructor & Destructor Documentation

◆ Algo748()

Member Function Documentation

◆ all_different()

◆ bracketing()

◆ cubic_interpolation()

◆ find_root_impl()

◆ name_impl()

◆ newton_quadratic()

Member Data Documentation

◆ requires_first_derivative

◆ requires_function

◆ requires_second_derivative