Data.Number.ER.RnToRm.Approx

AERN-RnToRm-0.4.9.1: polynomial function enclosures (PFEs) approximating exact real functions

Data.Number.ER.RnToRm.Approx

Portability	portable
Stability	experimental
Maintainer	mik@konecny.aow.cz

Description

Approximation of a real functions with rectangular domains.

To be imported qualified, usually with the synonym FA.

Synopsis

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

check :: String -> fa -> fa

domra2ranra :: fa -> domra -> ranra

ranra2domra :: fa -> ranra -> domra

getDegree :: fa -> Int

setMaxDegree :: Int -> fa -> fa

getMaxDegree :: fa -> Int

getSize :: fa -> Int

setMaxSize :: Int -> fa -> fa

getMaxSize :: fa -> Int

getVariables :: fa -> [varid]

getBestPrecision :: fa -> Precision

getRangeApprox :: fa -> ranra

tuple :: [fa] -> fa

getTupleSize :: fa -> Int

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

volume :: fa -> ranra

scale :: ranra -> fa -> fa

partialIntersect :: EffortIndex -> box -> fa -> fa -> fa

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

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

partialEval :: box -> fa -> fa

composeNonDecreasing :: fa -> varid -> fa -> fa

composeNonIncreasing :: fa -> varid -> fa -> fa

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

bottomApprox :: box -> Int -> fa

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

proj :: box -> varid -> fa

dom :: fa -> box

bisect :: varid -> Maybe domra -> fa -> (fa, fa)

unBisect :: varid -> (fa, fa) -> fa

integrate :: EffortIndex -> fa -> varid -> box -> domra -> fa -> fa

integrateUnary :: EffortIndex -> fa -> domra -> domra -> [ranra] -> fa

integrateMeasureImprovement :: EffortIndex -> fa -> varid -> box -> domra -> fa -> (fa, fa)

integrateMeasureImprovementUnary :: EffortIndex -> fa -> domra -> domra -> fa -> (fa, fa)

bisectUnbisectDepth :: ERFnDomApprox box varid domra ranra fa => Int -> fa -> fa

keyPointsConsistencyCheck :: ERFnDomApprox box varid domra ranra fa => (box -> ranra) -> fa -> [(box, ranra, ranra)]

keyPointsPointwiseConsistencyCheck :: ERFnDomApprox box varid domra ranra fa => ([ranra] -> ranra) -> [fa] -> fa -> [(box, ranra, ranra)]

Documentation

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

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

check

:: String	indentification of caller location for easier debugging
-> fa
-> fa
Check internal consistency and report problem if any.

domra2ranra

:: fa	this parameter is not used except for type checking
-> domra
-> ranra

ranra2domra

:: fa	this parameter is not used except for type checking
-> ranra
-> domra

getDegree :: fa -> Int

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

setMaxDegree :: Int -> fa -> fa

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.

May also set the maximum size of the approximations to a default based on the degree and the dimension of this enclosure.

getMaxDegree :: fa -> Int

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

getSize :: fa -> Int

Get the internal size of the approximation (usually number of polynomial terms).

setMaxSize :: Int -> fa -> fa

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

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

getMaxSize :: fa -> Int

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

getVariables :: fa -> [varid]

List all variables that are actually used in the approximation.

getBestPrecision :: fa -> Precision

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

getRangeApprox :: fa -> ranra

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

tuple :: [fa] -> fa

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

getTupleSize :: fa -> Int

Reveal how many functions are bundled together.

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

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

volume :: fa -> ranra

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 -> fa

Multiply a function approximation by a real number approximation.

partialIntersect

:: EffortIndex
-> box	the subdomain; defined by clipping the range of some variables
-> fa	the enclosure to be used on the subdomain (but defined on the whole domain)
-> fa	function to improve by intersecting its subdomain
-> fa
Intersect one approximation by another but only on a box within its domain.

intersectMeasureImprovement

:: 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]

Evaluate the function at the given point.

partialEval :: box -> fa -> fa

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

composeNonDecreasing

:: fa	enclosure of `f`, `f` is non-decreasing in variable `var`
-> varid	variable `var` to get substituted in `f`
-> fa	enclosure of `f_var`, to be substituted for `var`
-> fa	enclosure of `f[var \|-> f_var]` 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 applicable only when the range-defining function is non-decreasing.

composeNonIncreasing

:: fa	enclosure of `f`, `f` is non-increasing in variable `var`
-> varid	variable `var` to get substituted in `f`
-> fa	enclosure of `f_var`, to be substituted for `var`
-> fa	enclosure of `f[var \|-> f_var]` 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 applicable only when the range-defining function is non-increasing.

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Instances

ERFnDomApprox box varid domra ranra fa => ERFnApprox box varid domra ranra (ERFnTuple fa)

(ERFnDomApprox box varid domra ranra fa, VariableID varid, Show box) => ERFnApprox box varid domra ranra (ERFnDomEdgesApprox varid fa)

(ERUnitFnBase boxb boxra varid b ra fb, Show varid, Show boxra) => ERFnApprox boxra varid ra ra (ERFnInterval fb ra)

(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)

(ERUnitFnApprox box varid domra ranra ufa, DomainBoxMappable dtrbox box varid (DomTransl domra) domra, DomainIntBox box varid domra, Show varid, Show box, DomainBoxMappable box dtrbox varid domra (DomTransl domra), Eq dtrbox, Ord dtrbox) => ERFnApprox box varid domra ranra (ERFnDomTranslApprox dtrbox varid ufa domra)

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

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

bottomApprox

:: 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] -> fa

Construct a constant enclosure for a tuple of functions.

proj :: box -> varid -> fa

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

dom :: fa -> box

Return the domain of the function enclosure.

bisect

:: 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.

unBisect

:: varid	variable (axis) to glue on
-> (fa, fa)
-> fa
Merge function enclosures with neighbouring domains.

integrate

:: 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).

integrateUnary

:: 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.

integrateMeasureImprovement

:: 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.

integrateMeasureImprovementUnary

:: 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.

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Instances

ERFnDomApprox box varid domra ranra fa => ERFnDomApprox box varid domra ranra (ERFnTuple fa)

(ERFnDomApprox box varid domra ranra fa, VariableID varid, Show box) => ERFnDomApprox box varid domra ranra (ERFnDomEdgesApprox varid fa)

(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)

(ERUnitFnApprox box varid domra ranra ufa, DomainIntBox box varid domra, Show varid, Show box, DomainBoxMappable dtrbox box varid (DomTransl domra) domra, DomainBoxMappable box dtrbox varid domra (DomTransl domra), Eq dtrbox, Ord dtrbox) => ERFnDomApprox box varid domra ranra (ERFnDomTranslApprox dtrbox varid ufa domra)

bisectUnbisectDepth

:: 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.

keyPointsConsistencyCheck

:: ERFnDomApprox box varid domra ranra fa
=> box -> ranra	function `G` acting on tuples of real numbers
-> fa	alleged approximation of `G` over a domain box
-> [(box, ranra, ranra)]
Check that a function approximation is consistent with a real function that is meant to compute the same function. The result of this function is the list of points in which the consistency check failed. The result of the operation is also included both for the real number version and the function approximation version.

keyPointsPointwiseConsistencyCheck

:: ERFnDomApprox box varid domra ranra fa
=> [ranra] -> ranra	function `G` acting on real numbers
-> [fa]	approximations of input functions
-> fa	alleged approximation of `G` applied pointwise to the above function approximations
-> [(box, ranra, ranra)]
Check that a pointwise operation previously performed on function approximations is consistent with the same operation performed on selected points in the domain of these functions. The selected points are the centres of all faces of all dimensions, which includes the corners. The result of this function is the list of points in which the consistency check failed. The result of the operation is also included both for the real number version and the function approximation version.

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