AERN-RnToRm-0.3.0.1: polynomial function enclosures (PFEs) approximating exact real functions

Portabilityportable
Stabilityexperimental
Maintainermik@konecny.aow.cz

Data.Number.ER.RnToRm.Approx

Description

Approximation of a real functions with rectangular domains.

To be imported qualified, usually with the synonym FA.

Synopsis

Documentation

class (ERApprox fa, ERIntApprox domra, ERIntApprox ranra, DomainBox box varid domra) => ERFnApprox box varid domra ranra fa | fa -> box varid domra ranra whereSource

A class of types that approximate first-order real functions R^n -> R^m using some type of graph enclosures. The domains of the functions can be neither specified nor investigated by operations in this class.

This class extends ERApprox so that we could perform point-wise operations on the functions.

This class is associated with:

  • two real number types (instances of ERIntApprox) for working with parts of the function's domain and range;
  • a type of boxes indexed by variables (instance of DomainBox) for working with parts of the function's domain.

Methods

checkSource

Arguments

:: String

indentification of caller location for easier debugging

-> fa 
-> fa 

Check internal consistency and report problem if any.

domra2ranraSource

Arguments

:: fa

this parameter is not used except for type checking

-> domra 
-> ranra 

ranra2domraSource

Arguments

:: fa

this parameter is not used except for type checking

-> ranra 
-> domra 

getDegree :: fa -> IntSource

Get the internal degree of quality (usually polynomial degree) of the approximation.

setMaxDegree :: Int -> fa -> faSource

Set an upper bound on the degree of this function approximation.

This reduces the degree immediately if necessary and also affects all operations performed with this value later.

getMaxDegree :: fa -> IntSource

Get the current uppend bound on the degree associated with this function approximation.

getBestPrecision :: fa -> PrecisionSource

Give a close upper bound of the precision of the range at the best approximated point in the domain.

getRangeApprox :: fa -> ranraSource

Find some upper and lower bounds of the function over [-1,1]^n.

tuple :: [fa] -> faSource

Combine several functions with the same domain into one tuple function.

getTupleSize :: fa -> IntSource

Reveal how many functions are bundled together.

applyTupleFn :: ([fa] -> [fa]) -> fa -> faSource

Modify a tuple of functions in a way that does not treat the tuple elements uniformly.

volume :: fa -> ranraSource

Find close upper and lower bounds of the volume of the entire enclosure. A negative volume means that the enclosure is certainly inconsistent.

scale :: ranra -> fa -> faSource

Multiply a function approximation by a real number approximation.

partialIntersectSource

Arguments

:: EffortIndex 
-> box

the subdomain; defined by clipping the range of some variables

-> fa

function to improve by intersecting its subdomain

-> fa

the enclosure to be used on the subdomain (but defined on the whole domain)

-> fa 

Intersect one enclosure by another but only on a box within its domain.

intersectMeasureImprovementSource

Arguments

:: EffortIndex 
-> fa 
-> fa 
-> (fa, ranra)

enclosure intersection and measurement of improvement analogous to the one returned by pointwise intersectMeasureImprovement

Intersect two enclosures and measure the global improvement as one number.

(Use intersectMeasureImprovement defined in module Data.Number.ER.Real.Approx to measure the improvement using a function enclosure.)

eval :: box -> fa -> [ranra]Source

Evaluate the function at the given point.

partialEval :: box -> fa -> faSource

Fix some variables in the function to the given exact values.

composeThinSource

Arguments

:: fa

enclosure of f

-> Map varid fa

specifies the variables to substitute and for each such variable v, gives an exact enclosure of a function f_v to substitute for v

-> fa

enclosure of f[v |-> f_v]

BEWARE: Enclosure is probably incorrect where values of f_v are outside the domain of v in f.

A simple and limited composition of functions.

It is primarily intended to be used for precomposition with affine functions.

Instances

ERFnDomApprox box varid domra ranra fa => ERFnApprox box varid domra ranra (ERFnTuple fa) 
ERUnitFnBase boxb boxra varid b ra fb => ERFnApprox boxra varid ra ra (ERFnInterval fb ra) 
(ERFnDomApprox box varid domra ranra fa, VariableID varid) => ERFnApprox box varid domra ranra (ERFnDomEdgesApprox varid fa) 
(ERUnitFnApprox box varid domra ranra ufa, DomainIntBox box varid domra, DomainBoxMappable dtrbox box varid (DomTransl domra) domra, DomainBoxMappable box dtrbox varid domra (DomTransl domra), Eq dtrbox) => ERFnApprox box varid domra ranra (ERFnDomTranslApprox dtrbox varid ufa domra) 
(ERFnDomApprox box varid domra ranra fa, ERIntApprox fa, DomainBoxMappable box box varid domra domra, Show box) => ERFnApprox box varid domra ranra (ERFnPiecewise box varid domra fa) 

class (ERFnApprox box varid domra ranra fa, DomainIntBox box varid domra) => ERFnDomApprox box varid domra ranra fa | fa -> box varid domra ranra whereSource

This class extends ERFnApprox by:

  • making the domain of the function enclosure available for inspection;
  • allowing the construction of basic function enclosures where the domain has to be specified.

Methods

bottomApproxSource

Arguments

:: box

the domain of the function

-> Int

how many functions are bundled in this tuple

-> fa 

A function enclosure with no information about the function's values.

const :: box -> [ranra] -> faSource

Construct a constant enclosure for a tuple of functions.

proj :: box -> varid -> faSource

Construct the exact enclosure for a projection function (ie a variable).

dom :: fa -> boxSource

Return the domain of the function enclosure.

bisectSource

Arguments

:: varid

variable (axis) to split on

-> Maybe domra

where exactly to split (this has to be exact)

-> fa 
-> (fa, fa) 

Split the domain into two halves, yoelding two function enclosures.

unBisectSource

Arguments

:: varid

variable (axis) to glue on

-> (fa, fa) 
-> fa 

Merge function enclosures with neighbouring domains.

integrateSource

Arguments

:: EffortIndex

how hard to try

-> fa

function to integrate

-> varid

x = variable to integrate by

-> box

integration range

-> domra

origin in terms of x; this has to be thin!

-> fa

values at origin

-> fa 

Safely integrate a R^n -> R^m function enclosure with some initial condition (origin and function at origin).

integrateUnarySource

Arguments

:: EffortIndex

how hard to try

-> fa

unary function to integrate

-> domra

integration range

-> domra

origin

-> [ranra]

values at origin

-> fa 

Safely integrate a R -> R^m function enclosure.

integrateMeasureImprovementSource

Arguments

:: EffortIndex

how hard to try

-> fa

function to integrate

-> varid

variable to integrate by

-> box

integration domain

-> domra

a sub-domain with relevant new information - either about initial value(s) or about derivative

-> fa

approximation to result, including initial value(s)

-> (fa, fa)

improved result and measurement of improvement analogous to the one returned by pointwise intersectMeasureImprovement

Safely integrate a R^n -> R^m function enclosure intersecting it with a prior enclosure for the result.

The prior enclosure could contains one of more initial value.

integrateMeasureImprovementUnarySource

Arguments

:: EffortIndex

how hard to try

-> fa

unary function to integrate

-> domra

integration domain

-> domra

a sub-domain with relevant new information - either about initial value(s) or about derivative

-> fa

approximation to result, including initial value(s)

-> (fa, fa)

improved result and measurement of improvement analogous to the one returned by pointwise intersectMeasureImprovement

Safely integrate a R -> R^m function enclosure intersecting it with a prior enclosure for the result.

The prior enclosure could contains one of more initial value.

Instances

ERFnDomApprox box varid domra ranra fa => ERFnDomApprox box varid domra ranra (ERFnTuple fa) 
(ERFnDomApprox box varid domra ranra fa, VariableID varid) => ERFnDomApprox box varid domra ranra (ERFnDomEdgesApprox varid fa) 
(ERUnitFnApprox box varid domra ranra ufa, DomainIntBox box varid domra, DomainBoxMappable dtrbox box varid (DomTransl domra) domra, DomainBoxMappable box dtrbox varid domra (DomTransl domra), Eq dtrbox) => ERFnDomApprox box varid domra ranra (ERFnDomTranslApprox dtrbox varid ufa domra) 
(ERFnDomApprox box varid domra ranra fa, ERIntApprox fa, Show box, DomainBoxMappable box box varid domra domra) => ERFnDomApprox box varid domra ranra (ERFnPiecewise box varid domra fa) 

bisectUnbisectDepthSource

Arguments

:: ERFnDomApprox box varid domra ranra fa 
=> Int

required depth of bisection

-> fa 
-> fa 

Recursively perform a number of bisections and then glue the bits back together.

This way we can ensure that a piece-wise enclosure has a partition that goes to at least the given depth.