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

Data.Number.ER.RnToRm.UnitDom.Base

Description

Classes abstracting function arithmetic with directed rounding. Instances are used to describe a boundary for an approximation to a real function on the interval [-1,1]^n.

To be imported qualified, usually with the synonym UFB.

Documentation

class (ERRealBase b, ERIntApprox ra, Ord ufb, DomainBox boxb varid b, DomainIntBox boxra varid ra) => ERUnitFnBase boxb boxra varid b ra ufb | ufb -> boxb boxra varid b ra whereSource

Methods

initialiseBaseArithmetic :: ufb -> IO ()Source

This should be evaluated before using any of the following operations.

raEndpoints Source

Arguments

:: ufb	this parameter is not used except for type checking
-> ra
-> (b, b)

Convert from the associated interval type to the base type. (The types are determined by the given example function.)

raFromEndpoints Source

Arguments

:: ufb	this parameter is not used except for type checking
-> (b, b)
-> ra

Convert from the base type to the associated interval type. (The types are determined by the given example function.)

compareApprox :: ufb -> ufb -> Ordering Source

A linear ordering on basic functions, which can be syntactic and rather arbitrary.

showDiGrCmp Source

Arguments

:: Int	number of decimal digits to show
-> Bool	whether to show granularity
-> Bool	whether to show internal structure
-> ufb
-> String

isValid :: ufb -> Bool Source

Check internal consistency of the basic function, typically absence of NaN.

check Source

Arguments

:: String	indentification of caller location for easier debugging
-> ufb
-> ufb

Check internal consistency of the basic function and report problem if any.

getGranularity :: ufb -> Granularity Source

Get the granularity of the coefficients inside this basic function.

setMinGranularity :: Granularity -> ufb -> ufbSource

setGranularity :: Granularity -> ufb -> ufbSource

getDegree :: ufb -> Int Source

Get the degree of this basic function.

If the function is a polynomial, this function should return its degree.

reduceDegreeUp :: Int -> ufb -> ufbSource

Decrease the degree of a basic function, rounding pointwise upwards.

getSize :: ufb -> Int Source

Get the term size of this basic function.

If the function is a polynomial, this function should return the number of terms in the polynomial.

reduceSizeUp :: Int -> ufb -> ufbSource

Decrease the size of this basic function, rounding pointwise upwards.

getVariables :: ufb -> [varid]Source

Get a list of all variables featured in this basic function.

const :: b -> ufbSource

Construct a constant basic function.

affine Source

Arguments

:: b	value at 0
-> Map varid b	ascent of each base vector
-> ufb

Construct an affine basic function.

bounds :: EffortIndex -> ufb -> (b, b)Source

Find an upper bound of a basic function over [-1,1]^n.

upperBound :: EffortIndex -> ufb -> bSource

Find an upper bound of a basic function over [-1,1]^n.

upperBoundPrecise :: EffortIndex -> ufb -> bSource

Find an upper bound of a basic function over [-1,1]^n.

maxUp Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	f1
-> ufb	f2
-> ufb

Approximate the function max(f1,f2) from above.

minUp Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	f1
-> ufb	f2
-> ufb

Approximate the function min(f1,f2) from above.

maxDown Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	f1
-> ufb	f2
-> ufb

Approximate the function max(f1,f2) from below.

minDown Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	f1
-> ufb	f2
-> ufb

Approximate the function min(f1,f2) from below.

neg :: ufb -> ufbSource

Pointwise exact negation of a basic function

addConstUp :: b -> ufb -> ufbSource

Add a scalar to a basic function, rounding upwards.

scaleUp :: b -> ufb -> ufbSource

Multiply a basic function by a scalar, rounding upwards.

scaleApproxUp Source

Arguments

:: Int	maximum polynomial degree
-> Int	maximum term count
-> ra
-> ufb
-> ufb

Multiply a basic function by an approximation of a scalar, rounding upwards.

(+^) :: ufb -> ufb -> ufbSource

Pointwise upwards rounded addition

(-^) :: ufb -> ufb -> ufbSource

Pointwise upwards rounded subtraction

(*^) :: ufb -> ufb -> ufbSource

Pointwise upwards rounded multiplication

recipUp :: Int -> Int -> EffortIndex -> ufb -> ufbSource

Approximate the function 1/f from above, assuming f does not hit zero in the unit domain.

evalUp :: boxb -> ufb -> bSource

Evaluate a basic function at a point rounding upwards using a basic number for both the point and the result.

evalApprox :: boxra -> ufb -> raSource

Evaluate a basic function at a point rounding downwards using a basic number for both the point and the result.

Safely evaluate a basic function at a point using a real number approximation for both the point and the result.

partialEvalApproxUp :: boxra -> ufb -> ufbSource

Partially evaluate a basic function at a lower-dimensional point given using a real number approximation. Approximate the resulting function from above.

composeUp Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	function `f`
-> varid	variable `v` to substitute in `f`
-> ufb	function `f_v` to substitute for `v` that maps `[-1,1]` into `[-1,1]`
-> ufb	pointwise upper bound of `f[v \|-> f_v]`

Compose two basic functions, rounding upwards, assuming f_v ranges within the domain [-1,1].

composeManyUp Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	function `f`
-> Map varid ufb	variables to substitute and for each variable `v`, function `f_v` to substitute for `v` that maps `[-1,1]` into `[-1,1]`
-> ufb	pointwise upper bound of `f[v \|-> f_v]`

Substitute several variables in a basic function with other basic functions, rounding upwards, assuming each f_v ranges within the domain [-1,1].

composeDown Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	function `f`
-> varid	variable `v` to substitute in `f`
-> ufb	function `f_v` to substitute for `v` that maps `[-1,1]` into `[-1,1]`
-> ufb	pointwise lower bound of `f[v \|-> f_v]`

Compose two basic functions, rounding downwards, assuming f_v ranges within the domain [-1,1].

composeManyDown Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	function `f`
-> Map varid ufb	variables to substitute and for each variable `v`, function `f_v` to substitute for `v` that maps `[-1,1]` into `[-1,1]`
-> ufb	pointwise lower bound of `f[v \|-> f_v]`

Substitute several variables in a basic function with other basic functions, rounding downwards, assuming each f_v ranges within the domain [-1,1].

integrate Source

Arguments

:: varid	variable to integrate by
-> ufb	f
-> (ufb, ufb)

Approximate the primitive function of f from below and from above.

differentiate Source

Arguments

:: varid	variable to differentiate by
-> ufb	f
-> (ufb, ufb)

Approximate the derivative of f from below and from above.

volumeAboveZeroUp Source

Arguments

:: [varid]	dimensions to include in the measuring domain; have to include all those present in `f`
-> ufb	f
-> b

Measure the volume between a function and the zero hyperplane on the domain [-1,1]^n.

Instances

(ERRealBase rb, RealFrac rb, DomainBox box varid Int, Ord box, Show varid, DomainBoxMappable boxb boxras varid rb [ERInterval rb], DomainBoxMappable boxra boxras varid (ERInterval rb) [ERInterval rb], DomainIntBox boxra varid (ERInterval rb)) => ERUnitFnBase boxb boxra varid rb (ERInterval rb) (ERChebPoly box rb)

class ERUnitFnBase boxb boxra varid b ra ufb => ERUnitFnBaseEncl boxb boxra varid b ra ufb | ufb -> boxb boxra varid b ra whereSource

Methods

boundsEncl :: EffortIndex -> (ufb, ufb) -> (b, b)Source

constEncl :: (b, b) -> (ufb, ufb)Source

Construct a constant basic enclosure (negated lower bound fn, upper bound fn) from bounds given as coeffients (lower bound, upper bound).

evalEncl :: boxra -> (ufb, ufb) -> raSource

evalEnclInner :: boxra -> (ufb, ufb) -> raSource

addConstEncl Source

Arguments

:: Int	maximum polynomial degree
-> Int	maximum term count
-> b
-> (ufb, ufb)
-> (ufb, ufb)

Enclosure and base constant addition

IMPORTANT: enclosure = (NEGATED lower bound, upper bound)

scaleEncl Source

Arguments

:: Int	maximum polynomial degree
-> Int	maximum term count
-> b
-> (ufb, ufb)
-> (ufb, ufb)

Enclosure scaling by a base constant

IMPORTANT: enclosure = (NEGATED lower bound, upper bound)

addEncl Source

Arguments

:: Int	maximum polynomial degree
-> Int	maximum term count
-> (ufb, ufb)
-> (ufb, ufb)
-> (ufb, ufb)

Enclosure addition

IMPORTANT: enclosure = (NEGATED lower bound, upper bound)

multiplyEncl Source

Arguments

:: Int	maximum polynomial degree
-> Int	maximum term count
-> (ufb, ufb)
-> (ufb, ufb)
-> (ufb, ufb)

Enclosure multiplication

IMPORTANT: enclosure = (NEGATED lower bound, upper bound)

recipEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex
-> (ufb, ufb)	enclosure of `f`
-> (ufb, ufb)

Approximate the reciprocal of an enclosure, assuming f does not hit zero in the unit domain.

IMPORTANT: enclosure = (negated lower bound, upper bound)

composeEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	function `f`
-> varid	variable `v` to substitute in `f`
-> (ufb, ufb)	enclosure of a function `f_v` to substitute for `v` that maps `[-1,1]` into `[-1,1]`
-> (ufb, ufb)	enclosure of `f[v \|-> f_v]`

Compose two basic functions, rounding downwards and upwards, assuming f_v ranges within the domain [-1,1].

composeManyEncls Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	function `f`
-> Map varid (ufb, ufb)	variables to substitute and for each variable `v`, enclosure of a function `f_v` to substitute for `v` that maps `[-1,1]` into `[-1,1]`
-> (ufb, ufb)	enclosure of `f[v \|-> f_v]`

Substitute several variables in a basic function with other basic functions, rounding downwards and upwards, assuming each f_v ranges within the domain [-1,1].

Instances

(ERRealBase rb, RealFrac rb, DomainBox box varid Int, Ord box, Show varid, DomainBoxMappable boxb boxras varid rb [ERInterval rb], DomainBoxMappable boxra boxras varid (ERInterval rb) [ERInterval rb], DomainIntBox boxra varid (ERInterval rb)) => ERUnitFnBaseEncl boxb boxra varid rb (ERInterval rb) (ERChebPoly box rb)

class ERUnitFnBaseEncl boxb boxra varid b ra ufb => ERUnitFnBaseElementary boxb boxra varid b ra ufb | ufb -> boxb boxra varid b ra whereSource

Methods

sqrtEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating exp as a polynomial
-> (ufb, ufb)	f
-> (ufb, ufb)

Approximate sqrt(f) for enclosures.

expEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating exp as a polynomial
-> (ufb, ufb)	f
-> (ufb, ufb)

Approximate exp(f) for enclosures.

logEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating log as a polynomial
-> (ufb, ufb)	f
-> (ufb, ufb)

Approximate log(f) for enclosures.

sinEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating sin as a polynomial
-> (ufb, ufb)	f
-> (ufb, ufb)

Approximate sin(f) for enclosures, assuming the range of f is within [-pi2,pi2].

cosEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating cos as a polynomial
-> (ufb, ufb)	f
-> (ufb, ufb)

Approximate cos(f) for enclosures, assuming the range of f is within [-pi2,pi2].

atanEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating cos as a polynomial
-> (ufb, ufb)	f
-> (ufb, ufb)

Approximate atan(f) for enclosures.

Instances

(ERRealBase rb, RealFrac rb, DomainBox box varid Int, Ord box, Show varid, DomainBoxMappable boxb boxras varid rb [ERInterval rb], DomainBoxMappable boxra boxras varid (ERInterval rb) [ERInterval rb], DomainIntBox boxra varid (ERInterval rb)) => ERUnitFnBaseElementary boxb boxra varid rb (ERInterval rb) (ERChebPoly box rb)

class ERUnitFnBase boxb boxra varid b ra ufb => ERUnitFnBaseIEncl boxb boxra varid b ra ufb | ufb -> boxb boxra varid b ra whereSource

Methods

constIEncl :: (b, b) -> ((ufb, ufb), Bool)Source

Construct a constant basic inner enclosure (negated lower bound fn, upper bound fn, is enclosure definitely anticonsistent?) from bounds given as coeffients (lower bound, upper bound). An inner enclosure (lnI,hI) is anticonsistent iff hI + lnI <= 0, ie upper bound is never above lower bound.

evalIEncl :: boxra -> ((ufb, ufb), Bool) -> raSource

addIEncl Source

Arguments

:: Int	maximum polynomial degree
-> Int	maximum term count
-> ((ufb, ufb), Bool)
-> ((ufb, ufb), Bool)
-> ((ufb, ufb), Bool)

Inner enclosure addition.

multiplyIEncl Source

Arguments

:: Int	maximum polynomial degree
-> Int	maximum term count
-> ((ufb, ufb), Bool)
-> ((ufb, ufb), Bool)
-> ((ufb, ufb), Bool)

Inner enclosure multiplication.

recipIEnclPositive Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex
-> ((ufb, ufb), Bool)
-> ((ufb, ufb), Bool)

Approximate the reciprocal of an inner enclosure, assuming f is positive in the unit domain.

composeIEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	function `f`
-> varid	variable `v` to substitute in `f`
-> ((ufb, ufb), Bool)	inverse enclosure of a function `f_v` to substitute for `v` that maps `[-1,1]` into `[-1,1]`
-> ((ufb, ufb), Bool)	inverse enclosure of `f[v \|-> f_v]`

Compose two basic functions, rounding downwards and upwards, assuming f_v ranges within the domain [-1,1].

composeManyIEncls Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> ufb	function `f`
-> Map varid ((ufb, ufb), Bool)	variables to substitute and for each variable `v` inverse enclosure of a function `f_v` to substitute for `v` that maps `[-1,1]` into `[-1,1]`
-> ((ufb, ufb), Bool)	inverse enclosure of `f[v \|-> f_v]`

Substitute several variables in a basic function with other basic functions, rounding downwards and upwards, assuming each f_v ranges within the domain [-1,1].

Instances

(ERRealBase rb, RealFrac rb, DomainBox box varid Int, Ord box, Show varid, DomainBoxMappable boxb boxras varid rb [ERInterval rb], DomainBoxMappable boxra boxras varid (ERInterval rb) [ERInterval rb], DomainIntBox boxra varid (ERInterval rb)) => ERUnitFnBaseIEncl boxb boxra varid rb (ERInterval rb) (ERChebPoly box rb)

class ERUnitFnBaseIEncl boxb boxra varid b ra ufb => ERUnitFnBaseIElementary boxb boxra varid b ra ufb | ufb -> boxb boxra varid b ra whereSource

Methods

sqrtIEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating exp as a polynomial
-> ((ufb, ufb), Bool)	f
-> ((ufb, ufb), Bool)

Approximate sqrt(f) for enclosures.

expIEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating exp as a polynomial
-> ((ufb, ufb), Bool)	f
-> ((ufb, ufb), Bool)

Approximate exp(f) for enclosures.

logIEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating log as a polynomial
-> ((ufb, ufb), Bool)	f
-> ((ufb, ufb), Bool)

Approximate log(f) for enclosures.

sinIEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating sin as a polynomial
-> ((ufb, ufb), Bool)	f
-> ((ufb, ufb), Bool)

Approximate sin(f) for enclosures, assuming the range of f is within [-pi2,pi2].

cosIEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating cos as a polynomial
-> ((ufb, ufb), Bool)	f
-> ((ufb, ufb), Bool)

Approximate cos(f) for enclosures, assuming the range of f is within [-pi2,pi2].

atanIEncl Source

Arguments

:: Int	max degree for result
-> Int	max approx size for result
-> EffortIndex	how hard to try when approximating cos as a polynomial
-> ((ufb, ufb), Bool)	f
-> ((ufb, ufb), Bool)

Approximate atan(f) for enclosures.

Instances

(ERRealBase rb, RealFrac rb, DomainBox box varid Int, Ord box, Show varid, DomainBoxMappable boxb boxras varid rb [ERInterval rb], DomainBoxMappable boxra boxras varid (ERInterval rb) [ERInterval rb], DomainIntBox boxra varid (ERInterval rb)) => ERUnitFnBaseIElementary boxb boxra varid rb (ERInterval rb) (ERChebPoly box rb)