Detailed Description

Adaptive 1D integration, a generic CLEF integrator wrapper and root-finding routines.

These utilities provide the numerical building blocks used by the experimental Lattice tools (experimental) tools: an adaptive 13-point Kronrod 1D integrator, a generic wrapper that integrates CLEF expressions over a placeholder, and dichotomy/bisection root finding.

Classes
class	triqs::experimental::utility::integrate_1d_adapt< T >
	Adaptive one-dimensional integrator based on a 13-point Gauss-Kronrod-Lobatto rule. More...

Functions
std::pair< double, double >	triqs::experimental::utility::bisection (std::function< double(double)> f, double x_low, double x_high, double y_target, double precision, long max_loops, std::string x_name, std::string y_name, bool verbosity)
	Solve \( f(x) = y \) on a bracketing interval using the bisection method.
std::pair< double, double >	triqs::experimental::utility::dichotomy (std::function< double(double)> f, double x_low, double x_high, double y_target, double precision, long max_loops, std::string x_name, std::string y_name, bool verbosity)
	Solve \( f(x) = y \) on a bracketing interval using the false-position (dichotomy) method.
std::pair< double, double >	triqs::experimental::utility::find_bounds (std::function< double(double)> f, double x_init, double y_value, double delta_x, double precision, long max_loops=1000)
	Find a lower and an upper bound that bracket the solution of \( f(x) = y \).
template<typename D, int I>
auto	triqs::experimental::utility::integrate (auto const &integrator1d, auto expr_to_integrate, nda::clef::pair< I, D > const &p)
	Integrate a CLEF expression over one of its placeholders using an arbitrary one-dimensional integrator.
std::pair< double, double >	triqs::experimental::utility::root_finder (std::string method, std::function< double(double)> f, double x_init, double y_value, double precision, double delta_x, long max_loops=1000, std::string x_name="", std::string y_name="", bool verbosity=false)
	Find the value \( x \) that solves \( f(x) = y \) using the requested root-finding method.

Function Documentation

◆ bisection()

std::pair< double, double > triqs::experimental::utility::bisection	(	std::function< double(double)>	f,
		double	x_low,
		double	x_high,
		double	y_target,
		double	precision,
		long	max_loops,
		std::string	x_name,
		std::string	y_name,
		bool	verbosity )

inline

#include <triqs/experimental/utility/root_finder.hpp>

Solve \( f(x) = y \) on a bracketing interval using the bisection method.

Given an interval \( [x_\mathrm{low}, x_\mathrm{high}] \) that brackets the target value, this method repeatedly halves the interval, keeping the half in which the target value remains bracketed, until the residual drops below the requested precision.

It raises an error if convergence is not reached within max_loops iterations.

Parameters

f	Function \( f(x) : \mathbb{R} \to \mathbb{R} \) whose root is sought.
x_low	Lower bound of the bracketing interval.
x_high	Upper bound of the bracketing interval.
y_target	Target value \( y \) for \( f(x) \).
precision	Absolute precision \( \|f(x) - y\| \) at which the iteration stops.
max_loops	Maximum number of iterations.
x_name	Name of the \( x \) variable, used in the progress log.
y_name	Name of the \( y \) variable, used in the progress log.
verbosity	Whether to print the convergence progress.

Returns: Pair \( (x, f(x)) \) with \( f(x) = y \) within the requested precision.

Definition at line 153 of file root_finder.hpp.

◆ dichotomy()

std::pair< double, double > triqs::experimental::utility::dichotomy	(	std::function< double(double)>	f,
		double	x_low,
		double	x_high,
		double	y_target,
		double	precision,
		long	max_loops,
		std::string	x_name,
		std::string	y_name,
		bool	verbosity )

inline

#include <triqs/experimental/utility/root_finder.hpp>

Solve \( f(x) = y \) on a bracketing interval using the false-position (dichotomy) method.

Given an interval \( [x_\mathrm{low}, x_\mathrm{high}] \) that brackets the target value, this method iteratively replaces one of the bounds with the linear (secant) estimate of the root until the residual drops below the requested precision.

It raises an error if convergence is not reached within max_loops iterations.

Parameters

f	Function \( f(x) : \mathbb{R} \to \mathbb{R} \) whose root is sought.
x_low	Lower bound of the bracketing interval.
x_high	Upper bound of the bracketing interval.
y_target	Target value \( y \) for \( f(x) \).
precision	Absolute precision \( \|f(x) - y\| \) at which the iteration stops.
max_loops	Maximum number of iterations.
x_name	Name of the \( x \) variable, used in the progress log.
y_name	Name of the \( y \) variable, used in the progress log.
verbosity	Whether to print the convergence progress.

Returns: Pair \( (x, f(x)) \) with \( f(x) = y \) within the requested precision.

Definition at line 92 of file root_finder.hpp.

◆ find_bounds()

std::pair< double, double > triqs::experimental::utility::find_bounds	(	std::function< double(double)>	f,
		double	x_init,
		double	y_value,
		double	delta_x,
		double	precision,
		long	max_loops = 1000 )

inline

#include <triqs/experimental/utility/root_finder.hpp>

Find a lower and an upper bound that bracket the solution of \( f(x) = y \).

Starting from an initial guess, this function steps along \( x \) in increments of delta_x until the target value \( y \) is bracketed by the function values at the two most recent points (or until the maximum number of steps is reached). The returned bounds are ordered such that the first is smaller than the second.

Parameters

f	Function \( f(x) : \mathbb{R} \to \mathbb{R} \) whose root is sought.
x_init	Initial guess for \( x \).
y_value	Target value \( y = f(x) \).
delta_x	Step size by which \( x \) is incremented.
precision	Absolute precision \( \|f(x) - y\| \) used to stop the search early.
max_loops	Maximum number of steps.

Returns: Pair \( (x_1, x_2) \) with \( x_1 \leq x_2 \) whose function values bracket \( y \).

Definition at line 45 of file root_finder.hpp.

◆ integrate()

template<typename D, int I>

auto triqs::experimental::utility::integrate	(	auto const &	integrator1d,
		auto	expr_to_integrate,
		nda::clef::pair< I, D > const &	p )

#include <triqs/experimental/utility/integrator.hpp>

Integrate a CLEF expression over one of its placeholders using an arbitrary one-dimensional integrator.

This function turns the given expression into a unary CLEF function of the placeholder with index \( I \) and hands it to the supplied one-dimensional integrator together with the integration domain. If the expression still depends on other placeholders, the result is again a lazy CLEF expression; otherwise it is the value of the integral.

Template Parameters

D	Type of the integration domain (e.g. a pair of bounds).
I	Index of the placeholder that is integrated over.

Parameters

integrator1d	One-dimensional integrator that integrates a callable \( f(x) \) over the domain.
expr_to_integrate	CLEF expression to be integrated.
p	Placeholder-to-domain assignment, i.e. a placeholder = domain pair.

Returns: Lazy CLEF expression if other placeholders remain, otherwise the value of the integral.

Definition at line 29 of file integrator.hpp.

◆ root_finder()

std::pair< double, double > triqs::experimental::utility::root_finder	(	std::string	method,
		std::function< double(double)>	f,
		double	x_init,
		double	y_value,
		double	precision,
		double	delta_x,
		long	max_loops = 1000,
		std::string	x_name = "",
		std::string	y_name = "",
		bool	verbosity = false )

inline

#include <triqs/experimental/utility/root_finder.hpp>

Find the value \( x \) that solves \( f(x) = y \) using the requested root-finding method.

This is the main entry point of the root finder. It first brackets the solution and then refines it with either a dichotomy or a bisection step, depending on the chosen method.

It raises an error if the method name is not recognized.

Parameters

method	Root-finding method, either "dichotomy" or "bisection".
f	Function \( f(x) : \mathbb{R} \to \mathbb{R} \) whose root is sought.
x_init	Initial guess for \( x \).
y_value	Target value \( y \) for \( f(x) \).
precision	Absolute precision \( \|f(x) - y\| \) at which the iteration stops.
delta_x	Step size used when bracketing the solution.
max_loops	Maximum number of iterations.
x_name	Name of the \( x \) variable, used in the progress log.
y_name	Name of the \( y = f(x) \) variable, used in the progress log.
verbosity	Whether to print the convergence progress.

Returns: Pair \( (x, f(x)) \) with \( f(x) = y \) within the requested precision.

Definition at line 201 of file root_finder.hpp.

Detailed Description

Classes

Functions

Function Documentation

◆ bisection()

◆ dichotomy()

◆ find_bounds()

◆ integrate()

◆ root_finder()