Data.Number.ER.RnToRm.UnitDom.Base

AERN-RnToRm-0.4: polynomial function enclosures (PFEs) approximating exact real functions

Data.Number.ER.RnToRm.UnitDom.Base

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

Description

A class abstracting function arithmetic with directed rounding. It is 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, Fractional ufb, Ord ufb, DomainBox boxb varid b, DomainIntBox boxra varid ra) => ERUnitFnBase boxb boxra varid b ra ufb | ufb -> boxb boxra varid b ra where

Methods

check

:: String	indentification of caller location for easier debugging
-> ufb
-> ufb
Check internal consistency of the function and report problem if any.

getGranularity :: ufb -> Granularity

setMinGranularity :: Granularity -> ufb -> ufb

setGranularity :: Granularity -> ufb -> ufb

const :: b -> ufb

Construct a constant function.

affine

:: b	value at 0
-> Map varid b	ascent of each base vector
-> ufb
Construct an affine function.

scale :: b -> ufb -> (ufb, ufb)

Multiply a function by a scalar, rounding downwards and upwards.

scaleApprox :: ra -> ufb -> (ufb, ufb)

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

scaleApproxDown :: ra -> ufb -> ufb

Multiply a function by an approximation of a scalar, rounding downwards.

scaleApproxUp :: ra -> ufb -> ufb

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

getDegree :: ufb -> Int

Get the degree of this particular function.

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

reduceDegree :: Int -> ufb -> (ufb, ufb)

Decrease the degree of function approximation, rounding pointwise downwards and upwards.

reduceDegreeDown :: Int -> ufb -> ufb

Decrease the degree of function approximation, rounding pointwise downwards.

reduceDegreeUp :: Int -> ufb -> ufb

Decrease the degree of function approximation, rounding pointwise upwards.

integrate

:: varid	variable to integrate by
-> ufb	p(x)
-> (ufb, ufb)
Approximate the integral of p (with 0 at 0) from below and from above.

integrateDown

:: varid	variable to integrate by
-> ufb	p(x)
-> ufb
Approximate the integral of p (with 0 at 0) from below.

integrateUp

:: varid	variable to integrate by
-> ufb	p(x)
-> ufb
Approximate the integral of p (with 0 at 0) from above.

volumeAboveZero

:: [varid]	axes to include in the measuring domain
-> ufb
-> (b, b)
Measure the volume between a function and the zero hyperplane on the domain `[-1,1]^n`.

upperBound :: EffortIndex -> ufb -> b

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

lowerBound :: EffortIndex -> ufb -> b

Find a lower bound of the function over [-1,1]^n.

nonneg

:: Int	max degree for result
-> ufb	p(x)
-> (ufb, ufb)
Approximate the function max(0,p(x)) from below and from above.

recip

:: Int	max degree for result
-> EffortIndex
-> ufb	p(x)
-> (ufb, ufb)
Approximate the function 1/p(x) from below and from above.

recipDown :: Int -> EffortIndex -> ufb -> ufb

Approximate the function 1/p(x) from below.

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

Approximate the function 1/p(x) from above.

max

:: Int	max degree for result
-> ufb	p_1(x)
-> ufb	p_2(x)
-> (ufb, ufb)
Approximate the function max(p_1(x),p_2(x)) from below and from above.

maxDown

:: Int	max degree for result
-> ufb	p_1(x)
-> ufb	p_2(x)
-> ufb
Approximate the function max(p_1(x),p_2(x)) from below.

maxUp

:: Int	max degree for result
-> ufb	p_1(x)
-> ufb	p_2(x)
-> ufb
Approximate the function max(p_1(x),p_2(x)) from above.

min

:: Int	max degree for result
-> ufb	p_1(x)
-> ufb	p_2(x)
-> (ufb, ufb)
Approximate the function min(p_1(x),p_2(x)) from below and from above.

minDown

:: Int	max degree for result
-> ufb	p_1(x)
-> ufb	p_2(x)
-> ufb
Approximate the function min(p_1(x),p_2(x)) from below.

minUp

:: Int	max degree for result
-> ufb	p_1(x)
-> ufb	p_2(x)
-> ufb
Approximate the function min(p_1(x),p_2(x)) from above.

sqrt

:: Int	max degree for result
-> EffortIndex	how hard to try when approximating exp as a polynomial
-> ufb	p(x)
-> (ufb, ufb)
Approximate `sqrt(p(x))` from below and from above.

exp

:: Int	max degree for result
-> EffortIndex	how hard to try when approximating exp as a polynomial
-> ufb	p(x)
-> (ufb, ufb)
Approximate `exp(p(x))` from below and from above.

log

:: Int	max degree for result
-> EffortIndex	how hard to try when approximating log as a polynomial
-> ufb	p(x)
-> (ufb, ufb)
Approximate `log(p(x))` from below and from above.

sin

:: Int	max degree for result
-> EffortIndex	how hard to try when approximating sin as a polynomial
-> ufb	p(x)
-> (ufb, ufb)
Approximate `sin(p(x))` from below and from above, assuming the range of p is within [-pi2,pi2].

cos

:: Int	max degree for result
-> EffortIndex	how hard to try when approximating cos as a polynomial
-> ufb	p(x)
-> (ufb, ufb)
Approximate `cos(p(x))` from below and from above, assuming the range of p is within [-pi2,pi2].

atan

:: Int	max degree for result
-> EffortIndex	how hard to try when approximating cos as a polynomial
-> ufb	p(x)
-> (ufb, ufb)
Approximate `atan(p(x))` from below and from above.

eval :: boxb -> ufb -> (b, b)

Evaluate at a point, rounding upwards and downwards.

evalDown :: boxb -> ufb -> b

Evaluate at a point, rounding downwards.

evalUp :: boxb -> ufb -> b

Evaluate at a point, rounding downwards.

evalApprox :: boxra -> ufb -> ra

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

partialEvalApprox :: boxra -> ufb -> (ufb, ufb)

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

partialEvalApproxDown :: boxra -> ufb -> ufb

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

partialEvalApproxUp :: boxra -> ufb -> ufb

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

compose

:: Int	max degree 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, ufb)	upper and lower bounds of `f[v \|-> f_v]`
Compose two functions, rounding upwards and downwards provided each `f_v` ranges within the domain `[-1,1]`.

composeDown

:: Int	max degree for result
-> ufb	function `f1`
-> 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	a lower bound of `f1.f2`
Compose two functions, rounding downwards provided each `f_v` ranges within the domain `[-1,1]`.

composeUp

:: Int	max degree for result
-> ufb	function `f1`
-> 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	an upper bound of `f1.f2`
Compose two functions, rounding upwards provided each `f_v` ranges within the domain `[-1,1]`.

raEndpoints

:: ufb	this parameter is not used except for type checking
-> ra
-> (b, b)
Convert from the interval type to the base type. (The types are determined by the given example function.)

raFromEndpoints

:: ufb	this parameter is not used except for type checking
-> (b, b)
-> ra
Convert from the base type to the interval type. (The types are determined by the given example function.)

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Instances

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

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

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