newton_correct.hpp File Reference

Implements Newton's method for square systems. More...

#include "bertini2/trackers/amp_criteria.hpp"
#include "bertini2/trackers/config.hpp"
#include "bertini2/system.hpp"

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Functions
template<typename ComplexType, typename RealType>
SuccessCode	bertini::tracking::correct::NewtonLoop (Vec< ComplexType > &next_space, RealType &norm_delta_z, RealType &norm_J, RealType &norm_J_inverse, RealType &condition_number_estimate, System const &S, Vec< ComplexType > const &current_space, ComplexType const &current_time, RealType const &tracking_tolerance, unsigned min_num_newton_iterations, unsigned max_num_newton_iterations, AdaptiveMultiplePrecisionConfig const &AMP_config)
	Run Newton's method in multiple precision.
template<typename ComplexType, typename RealType>
SuccessCode	bertini::tracking::correct::NewtonLoop (Vec< ComplexType > &next_space, System const &S, Vec< ComplexType > const &current_space, ComplexType const &current_time, RealType const &tracking_tolerance, unsigned min_num_newton_iterations, unsigned max_num_newton_iterations)
	Run Newton's method in fixed precision.
template<typename ComplexType, typename RealType>
SuccessCode	bertini::tracking::correct::NewtonLoop (Vec< ComplexType > &next_space, System const &S, Vec< ComplexType > const &current_space, ComplexType const &current_time, RealType const &tracking_tolerance, unsigned min_num_newton_iterations, unsigned max_num_newton_iterations, AdaptiveMultiplePrecisionConfig const &AMP_config)
	Run Newton's method in multiple precision.

Detailed Description

Implements Newton's method for square systems.

Function Documentation

NewtonLoop() [1/3]

template<typename ComplexType, typename RealType>

SuccessCode bertini::tracking::correct::NewtonLoop	(	Vec< ComplexType > &	next_space,
		RealType &	norm_delta_z,
		RealType &	norm_J,
		RealType &	norm_J_inverse,
		RealType &	condition_number_estimate,
		System const &	S,
		Vec< ComplexType > const &	current_space,
		ComplexType const &	current_time,
		RealType const &	tracking_tolerance,
		unsigned	min_num_newton_iterations,
		unsigned	max_num_newton_iterations,
		AdaptiveMultiplePrecisionConfig const &	AMP_config )

Run Newton's method in multiple precision.

Run Newton's method until it converges ( \(\Delta z\) < tol), an AMP criterion (B or C) is violated, or the next point's norm exceeds the path truncation threshold.

Template Parameters

ComplexType	The complex type for arithmetic
RealType	The underlying real number type, used for comparitors.

Parameters

[out]	next_space	The computed next space point.
[out]	norm_delta_z	The norm of the last step size.
[out]	norm_J	The matrix norm of the Jacobian matrix.
[out]	norm_J_inverse	The matrix norm of the inverse of the Jacobian matrix.
[out]	condition_number_estimate	An estimate on the condition number.
	S	The system we are tracking on.
	current_space	The base point for newton correcting.
	current_time	The current time value. Note it is complex.
	tracking_tolerance	The upper threshold for step size. Must iterate correcting until the corrector step is less than this threshold in length.
	path_truncation_threshold	Correcting stops the the norm of the current solution exceeds this number.
	min_num_newton_iterations	The corrector must take at least this many steps. This should be at least 1.
	max_num_newton_iterations	The maximum number of iterations to run Newton's method for.
	AMP_config	Adaptive multiple precision settings. Using this argument is how Bertini2 knows you want to use adaptive precision.

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NewtonLoop() [2/3]

template<typename ComplexType, typename RealType>

SuccessCode bertini::tracking::correct::NewtonLoop	(	Vec< ComplexType > &	next_space,
		System const &	S,
		Vec< ComplexType > const &	current_space,
		ComplexType const &	current_time,
		RealType const &	tracking_tolerance,
		unsigned	min_num_newton_iterations,
		unsigned	max_num_newton_iterations )

Run Newton's method in fixed precision.

Run Newton's method until it converges ( \(\Delta z\) < tol), or the next point's norm exceeds the path truncation threshold.

Returns

The SuccessCode indicating what happened.

Template Parameters

ComplexType	The complex type for arithmetic
RealType	The underlying real number type, used for comparitors.

Parameters

[out]	next_space	The computed next space point.
	S	The system we are tracking on.
	current_space	The base point for newton correcting.
	current_time	The current time value. Note it is complex.
	tracking_tolerance	The upper threshold for step size. Must iterate correcting until the corrector step is less than this threshold in length.
	path_truncation_threshold	Correcting stops the the norm of the current solution exceeds this number.
	min_num_newton_iterations	The corrector must take at least this many steps. This should be at least 1.
	max_num_newton_iterations	The maximum number of iterations to run Newton's method for.

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NewtonLoop() [3/3]

template<typename ComplexType, typename RealType>

SuccessCode bertini::tracking::correct::NewtonLoop	(	Vec< ComplexType > &	next_space,
		System const &	S,
		Vec< ComplexType > const &	current_space,
		ComplexType const &	current_time,
		RealType const &	tracking_tolerance,
		unsigned	min_num_newton_iterations,
		unsigned	max_num_newton_iterations,
		AdaptiveMultiplePrecisionConfig const &	AMP_config )

Run Newton's method in multiple precision.

Run Newton's method until it converges ( \(\Delta z\) < tol), an AMP criterion (B or C) is violated, or the next point's norm exceeds the path truncation threshold.

Template Parameters

ComplexType	The complex type for arithmetic
RealType	The underlying real number type, used for comparitors.

Parameters

[out]	next_space	The computed next space point.
	S	The system we are tracking on.
	current_space	The base point for newton correcting.
	current_time	The current time value. Note it is complex.
	tracking_tolerance	The upper threshold for step size. Must iterate correcting until the corrector step is less than this threshold in length.
	path_truncation_threshold	Correcting stops the the norm of the current solution exceeds this number.
	min_num_newton_iterations	The corrector must take at least this many steps. This should be at least 1.
	max_num_newton_iterations	The maximum number of iterations to run Newton's method for.
	AMP_config	Adaptive multiple precision settings. Using this argument is how Bertini2 knows you want to use adaptive precision.

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