AERN-RnToRm-0.5.0.1: polynomial function enclosures (PFEs) approximating exact real functions

Portability portable experimental mik@konecny.aow.cz

Data.Number.ER.RnToRm.UnitDom.Approx

Description

Approximation of continuous real functions defined on the unit rectangle domain of a certain dimension.

To be imported qualified, usually with the synonym UFA.

Synopsis

# Documentation

class ERFnApprox box varid domra ranra fa => ERUnitFnApprox box varid domra ranra fa | fa -> box varid domra ranra whereSource

This class extends `ERFnApprox` by:

• assuming that the domain of the function enclosures is always `[-1,1]^n` for some `n`;
• allowing the construction of basic function enclosures where the domain has to be known.

Methods

bottomApprox :: faSource

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

const :: [ranra] -> faSource

Construct a constant enclosure for a tuple of functions.

Arguments

 :: [ranra] values at 0 -> Map varid [ranra] ascents of each base vector -> fa

Construct the exact enclosure of an affine function on `[-1,1]^n`.

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.

volume :: [varid] -> 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.

Explicitly specify the variables to identify the dimension of the domain.

Arguments

 :: EffortIndex -> [varid] -> fa -> fa -> (fa, ranra) enclosure intersection and measurement of improvement analogous to the one returned by the 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.)

Explicitly specify the variables to identify the dimension of the domain.

Arguments

 :: EffortIndex how hard to try -> fa function to integrate -> varid `x` = variable to integrate by -> domra origin in terms of `x`; this has to be exact! -> fa values at origin -> fa

Safely integrate a `[-1,1]^n -> R^m` function enclosure with some initial condition (origin and function at origin).

Instances

 (ERUnitFnBaseElementary boxb boxra varid b ra fb, Show varid, Show boxra) => ERUnitFnApprox boxra varid ra ra (ERFnInterval fb) (ERUnitFnBaseElementary boxb boxra varid b ra fb, ERUnitFnBaseIElementary boxb boxra varid b ra fb, Show varid, Show boxra) => ERUnitFnApprox boxra varid ra ra (ERFnIntervalOI fb)

Arguments

 :: ERUnitFnApprox 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.

Arguments

 :: ERUnitFnApprox 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.