{-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FunctionalDependencies #-} {-| Module : Data.Number.ER.RnToRm.Approx Description : classes abstracting function approximations Copyright : (c) 2007-2008 Michal Konecny License : BSD3 Maintainer : mik@konecny.aow.cz Stability : experimental Portability : portable Approximation of a real functions with rectangular domains. To be imported qualified, usually with the synonym FA. -} module Data.Number.ER.RnToRm.Approx ( ERFnApprox(..), ERFnDomApprox(..), bisectUnbisectDepth ) where import Prelude hiding (const) import qualified Data.Number.ER.Real.Approx as RA import qualified Data.Number.ER.Real.DomainBox as DBox import Data.Number.ER.Real.DomainBox (VariableID(..), DomainBox, DomainIntBox) import Data.Number.ER.BasicTypes import qualified Data.Map as Map {-| 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 'RA.ERApprox' so that we could perform point-wise operations on the functions. This class is associated with: * two real number types (instances of 'RA.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. -} class (RA.ERApprox fa, RA.ERIntApprox domra, RA.ERIntApprox ranra, DomainBox box varid domra) => ERFnApprox box varid domra ranra fa | fa -> box varid domra ranra where {-| Check internal consistency and report problem if any. -} check :: String {-^ indentification of caller location for easier debugging -} -> fa -> fa 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 {-| Get the internal degree of quality (usually polynomial degree) of the approximation. -} getDegree :: fa -> Int {-| 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. -} setMaxDegree :: Int -> fa -> fa {-| Get the current uppend bound on the degree associated with this function approximation. -} getMaxDegree :: fa -> Int {-| Give a close upper bound of the precision of the range at the best approximated point in the domain. -} getBestPrecision :: fa -> Precision {-| Find some upper and lower bounds of the function over @[-1,1]^n@. -} getRangeApprox :: fa -> ranra {-| Combine several functions with the same domain into one /tuple function/. -} tuple :: [fa] -> fa {-| Reveal how many functions are bundled together. -} getTupleSize :: fa -> Int {-| Modify a tuple of functions in a way that does not treat the tuple elements uniformly. -} applyTupleFn :: -- (ERFnApprox box varid domra ranra fa2) => -- ([fa2] -> [fa2]) -> (fa -> fa) ([fa] -> [fa]) -> (fa -> fa) {-| Find close upper and lower bounds of the volume of the entire enclosure. A negative volume means that the enclosure is certainly inconsistent. -} volume :: fa -> ranra {-| Multiply a function approximation by a real number approximation. -} scale :: ranra -> fa -> fa {-| Intersect one enclosure by another but only on a box within its domain. -} 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 two enclosures and measure the global improvement as one number. (Use 'RA.intersectMeasureImprovement' defined in module "Data.Number.ER.Real.Approx" to measure the improvement using a function enclosure.) -} intersectMeasureImprovement :: EffortIndex -> fa -> fa -> (fa, ranra) {-^ enclosure intersection and measurement of improvement analogous to the one returned by pointwise 'intersectMeasureImprovement' -} {-| Evaluate the function at the given point. -} eval :: box -> fa -> [ranra] {-| Fix some variables in the function to the given exact values. -} partialEval :: box -> fa -> fa {-| A simple and limited composition of functions. It is primarily intended to be used for precomposition with affine functions. -} composeThin :: fa {-^ enclosure of @f@ -} -> Map.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@. -} {-| 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. -} class (ERFnApprox box varid domra ranra fa, DomainIntBox box varid domra) => ERFnDomApprox box varid domra ranra fa | fa -> box varid domra ranra where {-| A function enclosure with no information about the function's values. -} bottomApprox :: box {-^ the domain of the function -} -> Int {-^ how many functions are bundled in this tuple -} -> fa {-| Construct a constant enclosure for a tuple of functions. -} const :: box -> [ranra] -> fa {-| Construct the exact enclosure for a projection function (ie a variable). -} proj :: box -> varid -> fa {-| Return the domain of the function enclosure. -} dom :: fa -> box {-| Split the domain into two halves, yoelding two function enclosures. -} bisect :: varid {-^ variable (axis) to split on -} -> Maybe domra {-^ where exactly to split (this has to be exact) -} -> fa -> (fa, fa) {-| Merge function enclosures with neighbouring domains. -} unBisect :: varid {-^ variable (axis) to glue on -} -> (fa, fa) -> fa {-| Safely integrate a @R^n -> R^m@ function enclosure with some initial condition (origin and function at origin). -} 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 -> R^m@ function enclosure. -} integrateUnary :: EffortIndex {-^ how hard to try -} -> fa {-^ unary function to integrate -} -> domra {-^ integration range -} -> domra {-^ origin -} -> [ranra] {-^ values at origin -} -> fa -- default implementation reduces this to integrateMeasureImprovement: integrateUnary ix fD support origin vals = integrate ix fD defaultVar (DBox.unary support) origin (const (DBox.noinfo) vals) {-| 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. -} 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 -> 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' -} -- default implementation reduces this to integrateMeasureImprovement: integrateMeasureImprovementUnary ix fD support origin fP = integrateMeasureImprovement ix fD defaultVar (DBox.unary support) origin fP {-| 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. -} bisectUnbisectDepth :: (ERFnDomApprox box varid domra ranra fa) => Int {-^ required depth of bisection -} -> fa -> fa bisectUnbisectDepth depth f = aux splitVars depth f where splitVars = concat $ repeat $ DBox.keys $ dom f aux (var : restVars) depthsToGo f | depthsToGo <= 0 = f | otherwise = unBisect var (fLDone, fRDone) where fLDone = aux restVars depthsToGoM1 fL fRDone = aux restVars depthsToGoM1 fR (fL, fR) = bisect var Nothing f depthsToGoM1 = depthsToGo - 1