numbers-2008.4.20: Various number typesSource codeContentsIndex
Data.Number.Dif
Description

The Data.Number.Dif module contains a data type, Dif, that allows for automatic forward differentiation.

All the ideas are from Jerzy Karczmarczuk's work, see http://users.info.unicaen.fr/~karczma/arpap/diffalg.pdf.

A simple example, if we define 

 foo x = x*x

then the function 

 foo' = deriv foo

will behave as if its body was 2*x.

Synopsis
data Dif a
val :: Dif a -> a
df :: Num a => Dif a -> Dif a
mkDif :: a -> Dif a -> Dif a
dCon :: Num a => a -> Dif a
dVar :: Num a => a -> Dif a
deriv :: (Num a, Num b) => (Dif a -> Dif b) -> (a -> b)
unDif :: Num a => (Dif a -> Dif b) -> (a -> b)
Documentation
data Dif a Source

The Dif type is the type of differentiable numbers. It's an instance of all the usual numeric classes. The computed derivative of a function is is correct except where the function is discontinuous, at these points the derivative should be a Dirac pulse, but it isn't.

The Dif numbers are printed with a trailing ~~ to indicate that there is a "tail" of derivatives.

show/hide Instances
Eq a => Eq (Dif a)
Floating a => Floating (Dif a)
Fractional a => Fractional (Dif a)
Num a => Num (Dif a)
Ord a => Ord (Dif a)
Read a => Read (Dif a)
Real a => Real (Dif a)
RealFloat a => RealFloat (Dif a)
RealFrac a => RealFrac (Dif a)
Show a => Show (Dif a)
val :: Dif a -> aSource
The val function takes a Dif number back to a normal number, thus forgetting about all the derivatives.
df :: Num a => Dif a -> Dif aSource
The df takes a Dif number and returns its first derivative. The function can be iterated to to get higher derivaties.
mkDif :: a -> Dif a -> Dif aSource
The mkDif takes a value and Dif value and makes a Dif number that has the given value as its normal value, and the Dif number as its derivatives.
dCon :: Num a => a -> Dif aSource
The dCon function turns a normal number into a Dif number with the same value. Not that numeric literals do not need an explicit conversion due to the normal Haskell overloading of literals.
dVar :: Num a => a -> Dif aSource
The dVar function turns a number into a variable number. This is the number with with respect to which the derivaticve is computed.
deriv :: (Num a, Num b) => (Dif a -> Dif b) -> (a -> b)Source

The deriv function is a simple utility to take the derivative of a (single argument) function. It is simply defined as 

  deriv f = val . df . f . dVar
unDif :: Num a => (Dif a -> Dif b) -> (a -> b)Source
Convert a Dif function to an ordinary function.
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