Data.Number.ER.RnToRm.UnitDom.Base

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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: varid

variable to integrate by

-> ufb

p(x)

-> ufb

Approximate the integral of p (with 0 at 0) from below.

integrateUp

Arguments

:: varid

variable to integrate by

-> ufb

p(x)

-> ufb

Approximate the integral of p (with 0 at 0) from above.

volumeAboveZero

Arguments

:: [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

Arguments

:: Int

max degree for result

-> ufb

p(x)

-> (ufb, ufb)

Approximate the function max(0,p(x)) from below and from above.

recip

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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.

cos

Arguments

:: 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.

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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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

Arguments

:: 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.)

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)