AERN2.MP.Enclosure

Enclosure classes and operations.

Documentation

class IsBall t where

Minimal complete definition

centre, centreAsBallAndRadius, updateRadius

Associated Types

type CentreType t

Methods

centre :: t -> CentreType t

centreAsBallAndRadius :: t -> (t, ErrorBound)

centreAsBall :: t -> t

radius :: t -> ErrorBound

updateRadius :: (ErrorBound -> ErrorBound) -> t -> t

makeExactCentre :: t -> t

When the radius of the ball is implicitly contributed to by imprecision in the centre (eg if the centre is a polynomial with inexact coefficients), move all that imprecision to the explicit radius, making the centre exact. This may lose some information, but as a ball is equivalent to the original. For MPBall this function is pointless because it is equivalent to the identity.

Instances

IsBall MPBall

Defined in AERN2.MP.Ball.Type

Associated Types

type CentreType MPBall

Methods

centre :: MPBall -> CentreType MPBall

centreAsBallAndRadius :: MPBall -> (MPBall, ErrorBound)

centreAsBall :: MPBall -> MPBall

radius :: MPBall -> ErrorBound

updateRadius :: (ErrorBound -> ErrorBound) -> MPBall -> MPBall

makeExactCentre :: MPBall -> MPBall

IsBall t => IsBall (CN t)

Defined in AERN2.MP.Enclosure

Associated Types

type CentreType (CN t)

Methods

centre :: CN t -> CentreType (CN t)

centreAsBallAndRadius :: CN t -> (CN t, ErrorBound)

centreAsBall :: CN t -> CN t

radius :: CN t -> ErrorBound

updateRadius :: (ErrorBound -> ErrorBound) -> CN t -> CN t

makeExactCentre :: CN t -> CN t

ballFunctionUsingLipschitz

Arguments

:: (IsBall t, HasEqCertainly t t)

=> (t -> t)

fThin: a version of f that works well on thin balls

-> (t -> ErrorBound)

fLip: a Lipschitz function of f over large balls

-> t -> t

f on *large* balls

Computes a ball function f on the centre and updating the error bound using a Lipschitz constant.

class IsInterval i where

Associated Types

type IntervalEndpoint i

Methods

endpoints :: i -> (IntervalEndpoint i, IntervalEndpoint i)

fromEndpoints :: IntervalEndpoint i -> IntervalEndpoint i -> i

Instances

IsInterval MPBall

Defined in AERN2.MP.Ball.Type

Associated Types

type IntervalEndpoint MPBall

Methods

endpoints :: MPBall -> (IntervalEndpoint MPBall, IntervalEndpoint MPBall)

fromEndpoints :: IntervalEndpoint MPBall -> IntervalEndpoint MPBall -> MPBall

IsInterval t => IsInterval (CN t)

Defined in AERN2.MP.Enclosure

Associated Types

type IntervalEndpoint (CN t)

Methods

endpoints :: CN t -> (IntervalEndpoint (CN t), IntervalEndpoint (CN t))

fromEndpoints :: IntervalEndpoint (CN t) -> IntervalEndpoint (CN t) -> CN t

endpointL :: IsInterval i => i -> IntervalEndpoint i

endpointR :: IsInterval i => i -> IntervalEndpoint i

fromEndpointsAsIntervals :: (IsInterval i, CanMinMaxSameType (IntervalEndpoint i)) => i -> i -> i

endpointsAsIntervals :: IsInterval i => i -> (i, i)

endpointLAsInterval :: IsInterval i => i -> i

endpointRAsInterval :: IsInterval i => i -> i

intervalFunctionByEndpoints

Arguments

:: (IsInterval t, CanMinMaxSameType (IntervalEndpoint t), HasEqCertainly t t)

=> (t -> t)

fThin: a version of f that works well on thin intervals

-> t -> t

f on *large* intervals

Computes a *monotone* ball function f on intervals using the interval endpoints.

intervalFunctionByEndpointsUpDown

Arguments

:: IsInterval t

=> (IntervalEndpoint t -> IntervalEndpoint t)

fDown: a version of f working on endpoints, rounded down

-> (IntervalEndpoint t -> IntervalEndpoint t)

fUp: a version of f working on endpoints, rounded up

-> t -> t

f on intervals rounding *outwards*

Computes a *monotone* ball function f on intervals using the interval endpoints.

class CanPlusMinus t1 t2 where

Associated Types

type PlusMinusType t1 t2

type PlusMinusType t1 t2 = t1

Methods

plusMinus :: t1 -> t2 -> PlusMinusType t1 t2

Operator for constructing or enlarging enclosures such as balls or intervals

Instances

CanBeErrorBound t => CanPlusMinus Int t

Defined in AERN2.MP.Ball.Conversions

Associated Types

type PlusMinusType Int t

Methods

plusMinus :: Int -> t -> PlusMinusType Int t

CanBeErrorBound t => CanPlusMinus Integer t

Defined in AERN2.MP.Ball.Conversions

Associated Types

type PlusMinusType Integer t

Methods

plusMinus :: Integer -> t -> PlusMinusType Integer t

CanBeErrorBound t => CanPlusMinus Rational t

Defined in AERN2.MP.Ball.Conversions

Associated Types

type PlusMinusType Rational t

Methods

plusMinus :: Rational -> t -> PlusMinusType Rational t

CanBeErrorBound t => CanPlusMinus MPFloat t

Defined in AERN2.MP.Ball.Conversions

Associated Types

type PlusMinusType MPFloat t

Methods

plusMinus :: MPFloat -> t -> PlusMinusType MPFloat t

CanBeErrorBound t => CanPlusMinus Dyadic t

Defined in AERN2.MP.Ball.Conversions

Associated Types

type PlusMinusType Dyadic t

Methods

plusMinus :: Dyadic -> t -> PlusMinusType Dyadic t

CanBeErrorBound t => CanPlusMinus MPBall t

Defined in AERN2.MP.Ball.Conversions

Associated Types

type PlusMinusType MPBall t

Methods

plusMinus :: MPBall -> t -> PlusMinusType MPBall t

CanBeErrorBound t => CanPlusMinus (CN MPBall) t

Defined in AERN2.MP.Ball.Conversions

Associated Types

type PlusMinusType (CN MPBall) t

Methods

plusMinus :: CN MPBall -> t -> PlusMinusType (CN MPBall) t

(+-) :: CanPlusMinus t1 t2 => t1 -> t2 -> PlusMinusType t1 t2 infixl 6

Operator for constructing or enlarging enclosures such as balls or intervals

class CanTestContains dom e where

Methods

contains

Arguments

:: dom

dom

-> e

e

-> Bool

Test if e is inside dom.

Instances

CanTestContains MPBall Int

Defined in AERN2.MP.Ball.Type

Methods

contains :: MPBall -> Int -> Bool

CanTestContains MPBall Integer

Defined in AERN2.MP.Ball.Type

Methods

contains :: MPBall -> Integer -> Bool

CanTestContains MPBall Rational

Defined in AERN2.MP.Ball.Type

Methods

contains :: MPBall -> Rational -> Bool

CanTestContains MPBall Dyadic

Defined in AERN2.MP.Ball.Type

Methods

contains :: MPBall -> Dyadic -> Bool

CanTestContains MPBall MPBall

Defined in AERN2.MP.Ball.Type

Methods

contains :: MPBall -> MPBall -> Bool

class CanMapInside dom e where

Methods

mapInside

Arguments

:: dom

dom

-> e

e

-> e

Return some value contained in dom. The returned value does not have to equal the given e even if e is already inside dom. All elements of dom should be covered with roughly the same probability when calling this function for evenly distributed e's.

This function is intended mainly for generating values inside dom for randomised tests.

specCanMapInside :: (CanMapInside d e, CanTestContains d e, Arbitrary d, Arbitrary e, Show d, Show e) => T d -> T e -> Spec

class CanIntersectAsymmetric e1 e2 where

A set intersection (usually partial)

Associated Types

type IntersectionType e1 e2

type IntersectionType e1 e2 = CN e1

Methods

intersect :: e1 -> e2 -> IntersectionType e1 e2

Instances

CanIntersectAsymmetric Bool Bool

Defined in AERN2.MP.Enclosure

Associated Types

type IntersectionType Bool Bool

Methods

intersect :: Bool -> Bool -> IntersectionType Bool Bool

CanIntersectAsymmetric Kleenean Kleenean

Defined in AERN2.MP.Enclosure

Associated Types

type IntersectionType Kleenean Kleenean

Methods

intersect :: Kleenean -> Kleenean -> IntersectionType Kleenean Kleenean

CanIntersectAsymmetric MPBall MPBall

Defined in AERN2.MP.Ball.Comparisons

Associated Types

type IntersectionType MPBall MPBall

Methods

intersect :: MPBall -> MPBall -> IntersectionType MPBall MPBall

CanIntersectAsymmetric (CN Bool) (CN b) => CanIntersectAsymmetric Bool (CN b)

Defined in AERN2.MP.Enclosure

Associated Types

type IntersectionType Bool (CN b)

Methods

intersect :: Bool -> CN b -> IntersectionType Bool (CN b)

CanIntersectAsymmetric (CN Kleenean) (CN b) => CanIntersectAsymmetric Kleenean (CN b)

Defined in AERN2.MP.Enclosure

Associated Types

type IntersectionType Kleenean (CN b)

Methods

intersect :: Kleenean -> CN b -> IntersectionType Kleenean (CN b)

(CanIntersectAsymmetric MPBall b, CanBeErrors es) => CanIntersectAsymmetric MPBall (CollectErrors es b)

Defined in AERN2.MP.Ball.Comparisons

Associated Types

type IntersectionType MPBall (CollectErrors es b)

Methods

intersect :: MPBall -> CollectErrors es b -> IntersectionType MPBall (CollectErrors es b)

CanIntersectAsymmetric (CN a) (CN Kleenean) => CanIntersectAsymmetric (CN a) Kleenean

Defined in AERN2.MP.Enclosure

Associated Types

type IntersectionType (CN a) Kleenean

Methods

intersect :: CN a -> Kleenean -> IntersectionType (CN a) Kleenean

CanIntersectAsymmetric (CN a) (CN Bool) => CanIntersectAsymmetric (CN a) Bool

Defined in AERN2.MP.Enclosure

Associated Types

type IntersectionType (CN a) Bool

Methods

intersect :: CN a -> Bool -> IntersectionType (CN a) Bool

(CanIntersectAsymmetric a b, IntersectionType a b ~ CN c) => CanIntersectAsymmetric (CN a) (CN b)

Defined in AERN2.MP.Enclosure

Associated Types

type IntersectionType (CN a) (CN b)

Methods

intersect :: CN a -> CN b -> IntersectionType (CN a) (CN b)

(CanIntersectAsymmetric a MPBall, CanBeErrors es) => CanIntersectAsymmetric (CollectErrors es a) MPBall

Defined in AERN2.MP.Ball.Comparisons

Associated Types

type IntersectionType (CollectErrors es a) MPBall

Methods

intersect :: CollectErrors es a -> MPBall -> IntersectionType (CollectErrors es a) MPBall

type CanIntersect e1 e2 = (CanIntersectAsymmetric e1 e2, CanIntersectAsymmetric e1 e2, IntersectionType e1 e2 ~ IntersectionType e2 e1)

type CanIntersectBy e1 e2 = (CanIntersect e1 e2, IntersectionType e1 e2 ~ e1)

type CanIntersectSameType e1 = CanIntersectBy e1 e1

type CanIntersectCNBy e1 e2 = (CanIntersect e1 e2, IntersectionType e1 e2 ~ CN e1)

type CanIntersectCNSameType e1 = CanIntersectCNBy e1 e1

class CanUnionAsymmetric e1 e2 where

A set union (usually partial)

Associated Types

type UnionType e1 e2

type UnionType e1 e2 = CN e1

Methods

union :: e1 -> e2 -> UnionType e1 e2

Instances

CanUnionAsymmetric MPBall MPBall

Defined in AERN2.MP.Ball.Comparisons

Associated Types

type UnionType MPBall MPBall

Methods

union :: MPBall -> MPBall -> UnionType MPBall MPBall

(CanUnionAsymmetric MPBall b, CanBeErrors es) => CanUnionAsymmetric MPBall (CollectErrors es b)

Defined in AERN2.MP.Ball.Comparisons

Associated Types

type UnionType MPBall (CollectErrors es b)

Methods

union :: MPBall -> CollectErrors es b -> UnionType MPBall (CollectErrors es b)

(CanUnionAsymmetric a b, UnionType a b ~ CN c) => CanUnionAsymmetric (CN a) (CN b)

Defined in AERN2.MP.Enclosure

Associated Types

type UnionType (CN a) (CN b)

Methods

union :: CN a -> CN b -> UnionType (CN a) (CN b)

(CanUnionAsymmetric a MPBall, CanBeErrors es) => CanUnionAsymmetric (CollectErrors es a) MPBall

Defined in AERN2.MP.Ball.Comparisons

Associated Types

type UnionType (CollectErrors es a) MPBall

Methods

union :: CollectErrors es a -> MPBall -> UnionType (CollectErrors es a) MPBall

type CanUnion e1 e2 = (CanUnionAsymmetric e1 e2, CanUnionAsymmetric e1 e2, UnionType e1 e2 ~ UnionType e2 e1)

type CanUnionBy e1 e2 = (CanUnion e1 e2, UnionType e1 e2 ~ e1)

type CanUnionSameType e1 = CanUnionBy e1 e1

type CanUnionCNBy e1 e2 = (CanUnion e1 e2, UnionType e1 e2 ~ CN e1)

type CanUnionCNSameType e1 = CanUnionCNBy e1 e1

Orphan instances

(CanUnionSameType t, CanTakeCNErrors t) => HasIfThenElse Kleenean t

Associated Types

type IfThenElseType Kleenean t

Methods

ifThenElse :: Kleenean -> t -> t -> IfThenElseType Kleenean t