{-| Module : Data.Number.ER.Real.Approx Description : classes abstracting exact reals Copyright : (c) Michal Konecny License : BSD3 Maintainer : mik@konecny.aow.cz Stability : experimental Portability : portable Definitions of classes that describe what is required from arbitrary precision approximations of exact real numbers. We introduce two levels of abstraction for these approximations: * 'ERApprox' = a *set* of approximated numbers whose size is measured using some fixed measure * 'ERIntApprox' = an *interval* of real numbers with finitely representable endpoints To be imported qualified, usually with the synonym RA. -} module Data.Number.ER.Real.Approx ( ERApprox(..), ERIntApprox(..), bounds2ira, effIx2ra, splitIRA, -- checkShrinking, -- eqSingletons, exactMiddle, maxExtensionR2R ) where import Data.Number.ER.BasicTypes import qualified Data.Number.ER.ExtendedInteger as EI import Data.Typeable {-| A type whose elements represent sets that can be used to approximate a single extended real number with arbitrary precision. -} class (Fractional ra, Ord ra) => ERApprox ra where getPrecision :: ra -> Precision {-^ Precision is a measure of the set size. The default interpretation: * If the diameter of the set is d, then the precision should be near floor(- log_2 d). -} getGranularity :: ra -> Granularity -- ^ the lower the granularity the bigger the rounding errors setGranularity :: Granularity -> ra -> ra -- ^ increase or safely decrease granularity setMinGranularity :: Granularity -> ra -> ra -- ^ ensure granularity is not below the first arg isEmpty :: ra -> Bool -- ^ true if this represents a computational error isBottom :: ra -> Bool -- ^ true if this holds no information isExact :: ra -> Bool -- ^ true if this is a singleton isDisjoint :: ra -> ra -> Bool isDisjoint a b = isEmpty $ a /\ b isBounded :: ra -> Bool -- ^ true if the approximation excludes infinity bottomApprox :: ra -- ^ the bottom element - any number emptyApprox :: ra -- ^ the top element - error refines :: ra -> ra -> Bool -- ^ first arg is a subset of the second arg (/\) :: ra -> ra -> ra -- ^ join; combining two approximations of the same number intersectMeasureImprovement :: EffortIndex -> ra -> ra -> (ra, ra) {-^ Like intersection but the second component: * measures improvement of the intersection relative to the first of the two approximations * is a positive number: 1 means no improvement, 2 means doubled precision, etc. -} equalReals :: ra -> ra -> Maybe Bool -- ^ nothing if overlapping and not singletons compareReals :: ra -> ra -> Maybe Ordering -- ^ nothing if overlapping and not singletons leqReals :: ra -> ra -> Maybe Bool -- ^ nothing if overlapping on interior or by a wrong endpoint equalApprox :: ra -> ra -> Bool -- ^ syntactic comparison compareApprox :: ra -> ra -> Ordering -- ^ syntactic linear ordering double2ra :: Double -> ra showApprox :: Int {-^ number of relevant decimals to show -} -> Bool {-^ should show granularity -} -> Bool {-^ should show internal representation details -} -> ra {-^ the approximation to show -} -> String {-| Assuming the arguments are singletons, equality is decidable. -} eqSingletons :: (ERApprox ra) => ra -> ra -> Bool eqSingletons s1 s2 = case equalReals s1 s2 of Just b -> b _ -> False {-| Assuming the arguments are singletons, @<=@ is decidable. -} leqSingletons :: (ERApprox ra) => ra -> ra -> Bool leqSingletons s1 s2 = case compareReals s1 s2 of Just EQ -> True Just LT -> True _ -> False {-| Assuming the arguments are singletons, @<@ is decidable. -} ltSingletons :: (ERApprox ra) => ra -> ra -> Bool ltSingletons s1 s2 = case compareReals s1 s2 of Just LT -> True _ -> False {-| For a finite sequence of real approximations, determine whether it is a shrinking sequence. -} checkShrinking :: (ERApprox ra) => [ra] -> Maybe (ra, ra) checkShrinking [] = Nothing checkShrinking [_] = Nothing checkShrinking (a : b : rest) | b `refines` a = checkShrinking (b : rest) | otherwise = Just (a,b) {-| A type whose elements represent sets that can be used to approximate a recursive set of closed extended real number intervals with arbitrary precision. -} --class (ERApprox sra) => SetOfRealsApprox sra where -- (\/) :: sra -> sra -> sra -- ^ union; either approximation could be correct {-| A type whose elements represent real *intervals* that can be used to approximate a single extended real number with arbitrary precision. Sometimes, these types can be used to approximate a closed extended real number interval with arbitrary precision. Nevetheless, this is not guaranteed. -} class (ERApprox ira) => ERIntApprox ira where doubleBounds :: ira -> (Double, Double) floatBounds :: ira -> (Float, Float) integerBounds :: ira -> (EI.ExtendedInteger, EI.ExtendedInteger) bisectDomain :: Maybe ira {-^ point to split at -} -> ira {-^ interval to split -} -> (ira, ira) -- ^ left and right, overlapping on a singleton defaultBisectPt :: ira -> ira -- | returns thin approximations of endpoints, in natural order bounds :: ira -> (ira, ira) {-| meet, usually constructing interval from approximations of its endpoints This does not need to be the meet of the real intervals but it has to be a maximal element in the set of all ira elements that are below the two parameters. -} (\/) :: ira -> ira -> ira bounds2ira :: (ERIntApprox ira) => ira -> ira -> ira bounds2ira = (\/) {- old stuff that will probably never be resurrected: -- It is intended that ra and ira are the same type. -- We distinguish them so that we can conveniently -- switch between two levels of abstraction when -- working with values of this one type. -- -- Given some ra or ira, the other type is determined uniquely. -- -- | coercion to more concrete view (allows a more intentional computation) -- ra2ira :: ra -> ira -- -- | coercion to more abstract view (guarantees certain extensionality and convergence properties) -- ira2ra :: ira -> ra -- -- | coercion -- ira2sra :: ira -> sra -- sraCover :: sra -> ira -- sraAllIntervals :: sra -> [ira] -- ^ disjoint, in natural order -} -- --bounds2ira :: -- (ERIntApprox ira) => -- ra -> -- ra -> -- ira --bounds2ira leftRA rightRA = -- (ra2ira leftRA) \/ (ra2ira rightRA) effIx2ra :: (ERApprox ra) => EffortIndex -> ra effIx2ra = fromInteger . toInteger {-| Split an interval to a sequence of intervals whose union is the original interval using a given sequence of cut points. The cut points are expected to be in increasing order and contained in the given interval. Violations of this rule are tolerated. -} splitIRA :: (ERIntApprox ira) => ira {-^ an interval to be split -} -> [ira] {-^ approximations of the cut points in increasing order -} -> [ira] splitIRA interval splitPoints = doSplit [] end pointsRev where (start, end) = bounds interval pointsRev = reverse $ start : splitPoints doSplit previousSegments nextRight [] = previousSegments doSplit previousSegments nextRight (nextLeft : otherPoints) = doSplit (nextLeft \/ nextRight : previousSegments) nextLeft otherPoints {-| * Return the endpoints of the interval as well as the exact midpoint. * To be able to do this, there may be a need to increase granularity. * All three singleton intervals are set to the same new granularity. -} exactMiddle :: (ERIntApprox ira) => ira -> (ira,ira,ira,Granularity) exactMiddle dom = case isExact domM of True -> (domL, domM, domR, gran) False -> (domLhg, domMhg, domRhg, higherGran) where (domL, domR) = bounds dom gran = max (getGranularity domL) (getGranularity domR) domM = (domL + domR) / 2 higherGran = gran + 1 domLhg = setMinGranularity higherGran domL domRhg = setMinGranularity higherGran domR domMhg = (domLhg + domRhg) / 2 {-| This produces a function that computes the maximal extension of the given function. A maximal extension function has the property: f(I) = { f(x) | x in I }. Here we get this property only for the limit function for ix tending to infinity. -} maxExtensionR2R :: (ERIntApprox ira) => (EffortIndex -> ira -> [ira]) {-^ returns a safe approximation of all extrema within the interval -} -> (EffortIndex -> ira -> ira) {-^ a function behaving well on sequences that intersect to a point -} -> (EffortIndex -> ira -> ira) {- ^ a function behaving well on sequences that intersect to a non-empty interval -} maxExtensionR2R getExtremes f ix x | getPrecision x < effIx2prec ix = (f ix xL) \/ (f ix xR) \/ (foldl (\/) emptyApprox $ getExtremes ix x) -- x is thin enough (?), don't bother evaluating by endpoints and extrema: | otherwise = f ix x where (xL, xR) = bounds x