Implements affine arithmetic that drops negligible error terms. More...

#include <thresholded_affine.hpp>

Classes
struct	Threshold
	Tag type used to pass the relative threshold to constructors. More...

Public Types
using	size_type = std::size_t
	Type used for indexing and offsets.
using	value_t = ValueType
	Type used for storing floating point values.
using	error_term_t = typename Affine<ValueType>::error_term_t
	Opaque type used to store error term information.
using	error_terms_t = typename Affine<ValueType>::error_terms_t
	Type used to map error terms to their radii.
using	interval_t = typename Affine<ValueType>::interval_t
	Type of an interval.
using	affine_t = Affine<ValueType>
	Type of the underlying Affine form.

Public Member Functions
	ThresholdedAffine ()
	Constructs an empty ThresholdedAffine.
	ThresholdedAffine (value_t center, Threshold t=default_threshold())
	Constructs a ThresholdedAffine from a center value.
	ThresholdedAffine (value_t lo, value_t hi, Threshold t=default_threshold())
	Constructs a ThresholdedAffine from a lower and upper bound.
	ThresholdedAffine (const interval_t &interval, Threshold t=default_threshold())
	Constructs a ThresholdedAffine from an interval.
	ThresholdedAffine (value_t center, error_terms_t radii, Threshold t=default_threshold())
	Constructs a ThresholdedAffine from a center and error terms.
	ThresholdedAffine (affine_t a, value_t threshold)
	Constructs a ThresholdedAffine by wrapping an Affine form.
	ThresholdedAffine (const ThresholdedAffine &other)=default
	Copy constructor.
	ThresholdedAffine (ThresholdedAffine &&other) noexcept=default
	Move constructor.
ThresholdedAffine &	operator= (const ThresholdedAffine &other)=default
	Copy assignment operator.
ThresholdedAffine &	operator= (ThresholdedAffine &&other) noexcept=default
	Move assignment operator.
interval_t	range () const
	Returns the interval represented by *this.
value_t	center () const
	Returns the center of the affine form.
const error_terms_t &	error_terms () const
	Returns the error terms of the tracked affine form.
value_t	radius () const
	Returns the total radius: the sum of the tracked terms.
void	set_center (value_t c)
	Sets the center of the affine form.
void	add_error_term (error_term_t error_term, value_t r)
	Adds an error term to the affine form and applies thresholding.
bool	contains (value_t v) const
	Returns whether *this contains a scalar value `v`.
bool	contains (const interval_t &i) const
	Returns whether *this contains an interval.
bool	contains (const ThresholdedAffine &other) const
	Returns whether *this contains another ThresholdedAffine's range.
bool	empty () const noexcept
	Returns whether *this is empty.
std::string	print_affine_form () const
	Returns a string of the affine form (tracked terms only).
std::string	print_interval_form () const
	Returns a string of the interval form.
const affine_t &	affine () const
	Returns the underlying Affine form (tracked terms only).
value_t	threshold () const
	Returns the relative threshold.
ThresholdedAffine	operator- () const
	Returns the additive inverse of *this.
ThresholdedAffine &	operator+= (value_t value)
	Adds a scalar to *this.
ThresholdedAffine &	operator+= (const ThresholdedAffine &other)
	Adds another ThresholdedAffine to *this.
ThresholdedAffine	operator+ (value_t value) const
	Returns the sum of *this and a scalar.
ThresholdedAffine	operator+ (const ThresholdedAffine &other) const
	Returns the sum of *this and another ThresholdedAffine.
ThresholdedAffine &	operator-= (value_t value)
	Subtracts a scalar from *this.
ThresholdedAffine &	operator-= (const ThresholdedAffine &other)
	Subtracts another ThresholdedAffine from *this.
ThresholdedAffine	operator- (value_t value) const
	Returns the difference of *this and a scalar.
ThresholdedAffine	operator- (const ThresholdedAffine &other) const
	Returns the difference of *this and another ThresholdedAffine.
ThresholdedAffine &	operator*= (value_t value)
	Multiplies *this by a scalar.
ThresholdedAffine &	operator*= (const ThresholdedAffine &other)
	Multiplies *this by another ThresholdedAffine.
ThresholdedAffine	operator* (value_t value) const
	Multiplies *this by a scalar.
ThresholdedAffine	operator* (const ThresholdedAffine &other) const
	Multiplies *this by another ThresholdedAffine.
ThresholdedAffine &	operator/= (value_t value)
	Divides *this by a scalar.
ThresholdedAffine &	operator/= (const ThresholdedAffine &other)
	Divides *this by another ThresholdedAffine.
ThresholdedAffine	operator/ (value_t value) const
	Divides *this by a scalar.
ThresholdedAffine	operator/ (const ThresholdedAffine &other) const
	Divides *this by another ThresholdedAffine.
bool	operator== (const ThresholdedAffine &other) const
	Checks equality of two ThresholdedAffine forms.
bool	operator!= (const ThresholdedAffine &other) const
	Checks inequality of two ThresholdedAffine forms.

Static Public Member Functions
static constexpr Threshold	default_threshold ()
	Returns the default threshold value.

Related Symbols
(Note that these are not member symbols.)
template<typename ValueType>
std::ostream &	operator<< (std::ostream &os, const ThresholdedAffine< ValueType > &a)
	Outputs the range of a ThresholdedAffine to a stream.
template<typename ValueType>
ThresholdedAffine< ValueType >	operator* (ValueType value, const ThresholdedAffine< ValueType > &a)
	Multiplies a scalar by a ThresholdedAffine.
template<typename T>
ThresholdedAffine< T >	abs (const ThresholdedAffine< T > &a)
	Absolute value of a ThresholdedAffine.
template<typename T>
ThresholdedAffine< T >	fabs (const ThresholdedAffine< T > &a)
	Absolute value of a ThresholdedAffine (alias for abs).
template<typename T>
ThresholdedAffine< T >	sqrt (const ThresholdedAffine< T > &a)
	Square root of a ThresholdedAffine.
template<typename T>
ThresholdedAffine< T >	exp (const ThresholdedAffine< T > &a)
	Exponential of a ThresholdedAffine.
template<typename T>
ThresholdedAffine< T >	log (const ThresholdedAffine< T > &a)
	Natural logarithm of a ThresholdedAffine.
template<typename T, typename U>
ThresholdedAffine< T >	pow (const ThresholdedAffine< T > &a, const U &exp)
	Power of a ThresholdedAffine.

Detailed Description

template<typename ValueType>
class sigma::ThresholdedAffine< ValueType >

Implements affine arithmetic that drops negligible error terms.

Template Parameters

ValueType The numeric type of the center and error term radii.

ThresholdedAffine extends standard affine arithmetic with a mechanism to control the number of error symbols carried in an affine form. Each non-affine operation can introduce new error symbols (e.g. multiplication creates a nonlinearity symbol); without pruning, the symbol count grows unboundedly.

This class reduces symbol count by deleting error terms whose relative contribution falls below the threshold. Dropped terms are simply discarded; no separate "lump" radius is retained to account for them.

Constructors accept a Threshold tag type to distinguish the threshold parameter from ordinary value parameters and avoid constructor ambiguity.

Constructor & Destructor Documentation

◆ ThresholdedAffine() [1/8]

template<typename ValueType>

sigma::ThresholdedAffine< ValueType >::ThresholdedAffine ( )

inline

Constructs an empty ThresholdedAffine.

The resulting ThresholdedAffine represents the empty set and the threshold is set to the default value.

Exceptions

none	No throw guarantee

◆ ThresholdedAffine() [2/8]

template<typename ValueType>

sigma::ThresholdedAffine< ValueType >::ThresholdedAffine	(	value_t	center,
		Threshold	t = default_threshold() )

inlineexplicit

Constructs a ThresholdedAffine from a center value.

Parameters

[in]	center	The center value.
[in]	t	The relative threshold, defaults to default_threshold().

Exceptions

none	No throw guarantee

◆ ThresholdedAffine() [3/8]

template<typename ValueType>

sigma::ThresholdedAffine< ValueType >::ThresholdedAffine	(	value_t	lo,
		value_t	hi,
		Threshold	t = default_threshold() )

inline

Constructs a ThresholdedAffine from a lower and upper bound.

Parameters

[in]	lo	The lower bound.
[in]	hi	The upper bound.
[in]	t	The relative threshold, defaults to default_threshold().

Exceptions

std::bad_alloc If memory allocation for the error term fails.

◆ ThresholdedAffine() [4/8]

template<typename ValueType>

sigma::ThresholdedAffine< ValueType >::ThresholdedAffine	(	const interval_t &	interval,
		Threshold	t = default_threshold() )

inlineexplicit

Constructs a ThresholdedAffine from an interval.

Parameters

[in]	interval	The interval.
[in]	t	The relative threshold, defaults to default_threshold().

Exceptions

std::bad_alloc If memory allocation for the error term fails.

◆ ThresholdedAffine() [5/8]

template<typename ValueType>

sigma::ThresholdedAffine< ValueType >::ThresholdedAffine	(	value_t	center,
		error_terms_t	radii,
		Threshold	t = default_threshold() )

inline

Constructs a ThresholdedAffine from a center and error terms.

Parameters

[in]	center	The center value.
[in]	radii	Map of error symbols to their radii.
[in]	t	The relative threshold, defaults to default_threshold().

Exceptions

none	No throw guarantee

◆ ThresholdedAffine() [6/8]

template<typename ValueType>

sigma::ThresholdedAffine< ValueType >::ThresholdedAffine	(	affine_t	a,
		value_t	threshold )

inline

Constructs a ThresholdedAffine by wrapping an Affine form.

Used internally by operations that produce an Affine result and need to re-wrap it as a ThresholdedAffine.

Parameters

[in]	a	The Affine form to wrap.
[in]	threshold	The relative threshold.

Exceptions

std::bad_alloc If memory allocation for the error term fails. Strong throw guarantee.

◆ ThresholdedAffine() [7/8]

template<typename ValueType>

sigma::ThresholdedAffine< ValueType >::ThresholdedAffine ( const ThresholdedAffine< ValueType > & other )

default

Copy constructor.

Parameters

[in] other The ThresholdedAffine to copy.

Exceptions

std::bad_alloc If memory allocation for the error terms fails. Strong throw guarantee.

◆ ThresholdedAffine() [8/8]

template<typename ValueType>

sigma::ThresholdedAffine< ValueType >::ThresholdedAffine ( ThresholdedAffine< ValueType > && other )

defaultnoexcept

Move constructor.

Parameters

[in] other The ThresholdedAffine to move.

Exceptions

none	No throw guarantee

Member Function Documentation

◆ add_error_term()

template<typename ValueType>

void sigma::ThresholdedAffine< ValueType >::add_error_term	(	error_term_t	error_term,
		value_t	r )

inline

Adds an error term to the affine form and applies thresholding.

Parameters

[in]	error_term	The error symbol.
[in]	r	The radius of the new term.

Exceptions

None	No throw guarantee.

◆ affine()

template<typename ValueType>

const affine_t & sigma::ThresholdedAffine< ValueType >::affine ( ) const

inline

Returns the underlying Affine form (tracked terms only).

Returns: Const reference to the internal Affine form.

Exceptions

none	No throw guarantee

◆ center()

template<typename ValueType>

value_t sigma::ThresholdedAffine< ValueType >::center ( ) const

inline

Returns the center of the affine form.

Returns: The center value.

Exceptions

std::domain_error If *this is empty.

◆ contains() [1/3]

template<typename ValueType>

bool sigma::ThresholdedAffine< ValueType >::contains ( const interval_t & i ) const

inline

Returns whether *this contains an interval.

Parameters

[in] i The interval to test.

Returns: True if every point in i is in range().

Exceptions

none	No throw guarantee

◆ contains() [2/3]

template<typename ValueType>

bool sigma::ThresholdedAffine< ValueType >::contains ( const ThresholdedAffine< ValueType > & other ) const

inline

Returns whether *this contains another ThresholdedAffine's range.

Parameters

[in] other The ThresholdedAffine to test.

Returns: True if every point in other.range() is in range().

Exceptions

none	No throw guarantee

◆ contains() [3/3]

template<typename ValueType>

bool sigma::ThresholdedAffine< ValueType >::contains ( value_t v ) const

inline

Returns whether *this contains a scalar value v.

Parameters

[in] v The value to test.

Returns: True if the value is in range().

Exceptions

none	No throw guarantee

◆ default_threshold()

template<typename ValueType>

constexpr Threshold sigma::ThresholdedAffine< ValueType >::default_threshold ( )

inlinestaticconstexpr

Returns the default threshold value.

Returns: The default threshold (0.1%).

Exceptions

none	No throw guarantee

◆ empty()

template<typename ValueType>

bool sigma::ThresholdedAffine< ValueType >::empty ( ) const

inlinenoexcept

Returns whether *this is empty.

Returns: True if *this represents the empty set.

Exceptions

none	No throw guarantee

◆ error_terms()

template<typename ValueType>

const error_terms_t & sigma::ThresholdedAffine< ValueType >::error_terms ( ) const

inline

Returns the error terms of the tracked affine form.

Returns: Const reference to the error terms map.

Exceptions

None	No throw guarantee.

◆ operator!=()

template<typename ValueType>

bool sigma::ThresholdedAffine< ValueType >::operator!= ( const ThresholdedAffine< ValueType > & other ) const

inline

Checks inequality of two ThresholdedAffine forms.

Parameters

[in] other The form to compare.

Returns: True if not equal.

Exceptions

none	No throw guarantee.

◆ operator*=() [1/2]

template<typename ValueType>

ThresholdedAffine & sigma::ThresholdedAffine< ValueType >::operator*= ( const ThresholdedAffine< ValueType > & other )

inline

Multiplies *this by another ThresholdedAffine.

Multiplication introduces a new nonlinearity error term via affine arithmetic, after which sub-threshold terms are dropped.

Parameters

[in] other The form to multiply by.

Returns: Reference to *this.

Exceptions

None	No throw guarantee.

◆ operator*=() [2/2]

template<typename ValueType>

ThresholdedAffine & sigma::ThresholdedAffine< ValueType >::operator*= ( value_t value )

inline

Multiplies *this by a scalar.

Parameters

[in] value The scalar to multiply by.

Returns: Reference to *this.

Exceptions

None	No throw guarantee.

◆ operator+=() [1/2]

template<typename ValueType>

ThresholdedAffine & sigma::ThresholdedAffine< ValueType >::operator+= ( const ThresholdedAffine< ValueType > & other )

inline

Adds another ThresholdedAffine to *this.

The tracked affine terms are combined with normal affine addition (shared symbols may cancel), after which sub-threshold terms are dropped.

Parameters

[in] other The form to add.

Returns: Reference to *this.

Exceptions

None	No throw guarantee.

◆ operator+=() [2/2]

template<typename ValueType>

ThresholdedAffine & sigma::ThresholdedAffine< ValueType >::operator+= ( value_t value )

inline

Adds a scalar to *this.

Parameters

[in] value The scalar to add.

Returns: Reference to *this.

Exceptions

None	No throw guarantee.

◆ operator-()

template<typename ValueType>

ThresholdedAffine sigma::ThresholdedAffine< ValueType >::operator- ( ) const

inline

Returns the additive inverse of *this.

Negation flips the tracked affine terms.

Returns: The negated ThresholdedAffine.

Exceptions

std::bad_alloc If memory allocation fails.

◆ operator-=() [1/2]

template<typename ValueType>

ThresholdedAffine & sigma::ThresholdedAffine< ValueType >::operator-= ( const ThresholdedAffine< ValueType > & other )

inline

Subtracts another ThresholdedAffine from *this.

Tracked affine terms are combined with normal affine subtraction (shared symbols cancel, capturing dependency), after which sub-threshold terms are dropped.

Parameters

[in] other The form to subtract.

Returns: Reference to *this.

Exceptions

None	No throw guarantee.

◆ operator-=() [2/2]

template<typename ValueType>

ThresholdedAffine & sigma::ThresholdedAffine< ValueType >::operator-= ( value_t value )

inline

Subtracts a scalar from *this.

Parameters

[in] value The scalar to subtract.

Returns: Reference to *this.

Exceptions

None	No throw guarantee.

◆ operator/=() [1/2]

template<typename ValueType>

ThresholdedAffine & sigma::ThresholdedAffine< ValueType >::operator/= ( const ThresholdedAffine< ValueType > & other )

inline

Divides *this by another ThresholdedAffine.

Division is performed by affine arithmetic, after which sub-threshold terms are dropped. The divisor's interval must not contain zero.

Parameters

[in] other The form to divide by.

Returns: Reference to *this.

Exceptions

std::domain_error If other is empty or its interval contains zero.

◆ operator/=() [2/2]

template<typename ValueType>

ThresholdedAffine & sigma::ThresholdedAffine< ValueType >::operator/= ( value_t value )

inline

Divides *this by a scalar.

Parameters

[in] value The scalar to divide by.

Returns: Reference to *this.

Exceptions

std::domain_error If value is zero.

◆ operator=() [1/2]

template<typename ValueType>

ThresholdedAffine & sigma::ThresholdedAffine< ValueType >::operator= ( const ThresholdedAffine< ValueType > & other )

default

Copy assignment operator.

Parameters

[in] other The ThresholdedAffine to copy.

Returns: Reference to this ThresholdedAffine after assignment.

Exceptions

std::bad_alloc If memory allocation for the error terms fails. Strong throw guarantee.

◆ operator=() [2/2]

template<typename ValueType>

ThresholdedAffine & sigma::ThresholdedAffine< ValueType >::operator= ( ThresholdedAffine< ValueType > && other )

defaultnoexcept

Move assignment operator.

Parameters

[in] other The ThresholdedAffine to move.

Returns: Reference to this ThresholdedAffine after assignment.

Exceptions

none	No throw guarantee

◆ operator==()

template<typename ValueType>

bool sigma::ThresholdedAffine< ValueType >::operator== ( const ThresholdedAffine< ValueType > & other ) const

inline

Checks equality of two ThresholdedAffine forms.

Parameters

[in] other The form to compare.

Returns: True if equal.

Exceptions

none	No throw guarantee.

◆ print_affine_form()

template<typename ValueType>

std::string sigma::ThresholdedAffine< ValueType >::print_affine_form ( ) const

inline

Returns a string of the affine form (tracked terms only).

Returns: Affine form string.

Exceptions

std::bad_alloc If memory allocation for the string fails. Strong throw guarantee.

◆ print_interval_form()

template<typename ValueType>

std::string sigma::ThresholdedAffine< ValueType >::print_interval_form ( ) const

inline

Returns a string of the interval form.

Returns: Interval form string, e.g. [lo, hi].

Exceptions

std::bad_alloc

◆ radius()

template<typename ValueType>

value_t sigma::ThresholdedAffine< ValueType >::radius ( ) const

inline

Returns the total radius: the sum of the tracked terms.

Returns: The total radius.

Exceptions

std::domain_error If *this is empty.

◆ range()

template<typename ValueType>

interval_t sigma::ThresholdedAffine< ValueType >::range ( ) const

inline

Returns the interval represented by *this.

The interval is [center - radius(), center + radius()], where radius() is the sum of the tracked affine terms.

Returns: The interval.

Exceptions

none	No throw guarantee

◆ set_center()

template<typename ValueType>

void sigma::ThresholdedAffine< ValueType >::set_center ( value_t c )

inline

Sets the center of the affine form.

Parameters

[in] c The new center.

Exceptions

None	No throw guarantee.

◆ threshold()

template<typename ValueType>

value_t sigma::ThresholdedAffine< ValueType >::threshold ( ) const

inline

Returns the relative threshold.

Returns: The threshold \(t\).

Exceptions

none	No throw guarantee

◆ abs()

template<typename T>

ThresholdedAffine< T > abs ( const ThresholdedAffine< T > & a )

Absolute value of a ThresholdedAffine.

Template Parameters

T	The value type.

◆ exp()

template<typename T>

ThresholdedAffine< T > exp ( const ThresholdedAffine< T > & a )

Exponential of a ThresholdedAffine.

Template Parameters

T	The value type.

◆ fabs()

template<typename T>

ThresholdedAffine< T > fabs ( const ThresholdedAffine< T > & a )

Absolute value of a ThresholdedAffine (alias for abs).

Template Parameters

T	The value type.

◆ log()

template<typename T>

ThresholdedAffine< T > log ( const ThresholdedAffine< T > & a )

Natural logarithm of a ThresholdedAffine.

Template Parameters

T	The value type.

Exceptions

std::domain_error If the range contains non-positive values.

◆ operator*()

template<typename ValueType>

ThresholdedAffine< ValueType > operator*	(	ValueType	value,
		const ThresholdedAffine< ValueType > &	a )

Multiplies a scalar by a ThresholdedAffine.

Template Parameters

ValueType The value type.

◆ operator<<()

template<typename ValueType>

std::ostream & operator<<	(	std::ostream &	os,
		const ThresholdedAffine< ValueType > &	a )

Outputs the range of a ThresholdedAffine to a stream.

Template Parameters

ValueType The value type.

◆ pow()

template<typename T, typename U>

ThresholdedAffine< T > pow	(	const ThresholdedAffine< T > &	a,
		const U &	exp )

Power of a ThresholdedAffine.

Template Parameters

T	The value type.
U	The exponent type.

Exceptions

std::domain_error See Affine::pow for conditions.

◆ sqrt()

template<typename T>

ThresholdedAffine< T > sqrt ( const ThresholdedAffine< T > & a )

Square root of a ThresholdedAffine.

Template Parameters

T	The value type.

Exceptions

std::domain_error If the range contains negative values.

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

include/sigma/affine/eigen_compat.hpp
include/sigma/affine/thresholded_affine.hpp

Classes

Public Types

Public Member Functions

Static Public Member Functions

Related Symbols

Detailed Description

Constructor & Destructor Documentation

◆ ThresholdedAffine() [1/8]

◆ ThresholdedAffine() [2/8]

◆ ThresholdedAffine() [3/8]

◆ ThresholdedAffine() [4/8]

◆ ThresholdedAffine() [5/8]

◆ ThresholdedAffine() [6/8]

◆ ThresholdedAffine() [7/8]

◆ ThresholdedAffine() [8/8]

Member Function Documentation

◆ add_error_term()

◆ affine()

◆ center()

◆ contains() [1/3]

◆ contains() [2/3]

◆ contains() [3/3]

◆ default_threshold()

◆ empty()

◆ error_terms()

◆ operator!=()

◆ operator*=() [1/2]

◆ operator*=() [2/2]

◆ operator+=() [1/2]

◆ operator+=() [2/2]

◆ operator-()

◆ operator-=() [1/2]

◆ operator-=() [2/2]

◆ operator/=() [1/2]

◆ operator/=() [2/2]

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==()

◆ print_affine_form()

◆ print_interval_form()

◆ radius()

◆ range()

◆ set_center()

◆ threshold()

Friends And Related Symbol Documentation

◆ abs()

◆ exp()

◆ fabs()

◆ log()

◆ operator*()

◆ operator<<()

◆ pow()

◆ sqrt()