Safe Haskell	Safe
Language	Haskell98

Data.Number.Dif

Description

The 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

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.

Instances

Eq a => Eq (Dif a) Source #
Methods (==) :: Dif a -> Dif a -> Bool # (/=) :: Dif a -> Dif a -> Bool #
(Floating a, Eq a) => Floating (Dif a) Source #
Methods pi :: Dif a # exp :: Dif a -> Dif a # log :: Dif a -> Dif a # sqrt :: Dif a -> Dif a # (**) :: Dif a -> Dif a -> Dif a # logBase :: Dif a -> Dif a -> Dif a # sin :: Dif a -> Dif a # cos :: Dif a -> Dif a # tan :: Dif a -> Dif a # asin :: Dif a -> Dif a # acos :: Dif a -> Dif a # atan :: Dif a -> Dif a # sinh :: Dif a -> Dif a # cosh :: Dif a -> Dif a # tanh :: Dif a -> Dif a # asinh :: Dif a -> Dif a # acosh :: Dif a -> Dif a # atanh :: Dif a -> Dif a # log1p :: Dif a -> Dif a # expm1 :: Dif a -> Dif a # log1pexp :: Dif a -> Dif a # log1mexp :: Dif a -> Dif a #
(Fractional a, Eq a) => Fractional (Dif a) Source #
Methods (/) :: Dif a -> Dif a -> Dif a # recip :: Dif a -> Dif a # fromRational :: Rational -> Dif a #
(Num a, Eq a) => Num (Dif a) Source #
Methods (+) :: Dif a -> Dif a -> Dif a # (-) :: Dif a -> Dif a -> Dif a # (*) :: Dif a -> Dif a -> Dif a # negate :: Dif a -> Dif a # abs :: Dif a -> Dif a # signum :: Dif a -> Dif a # fromInteger :: Integer -> Dif a #
Ord a => Ord (Dif a) Source #
Methods compare :: Dif a -> Dif a -> Ordering # (<) :: Dif a -> Dif a -> Bool # (<=) :: Dif a -> Dif a -> Bool # (>) :: Dif a -> Dif a -> Bool # (>=) :: Dif a -> Dif a -> Bool # max :: Dif a -> Dif a -> Dif a # min :: Dif a -> Dif a -> Dif a #
Read a => Read (Dif a) Source #
Methods readsPrec :: Int -> ReadS (Dif a) # readList :: ReadS [Dif a] # readPrec :: ReadPrec (Dif a) # readListPrec :: ReadPrec [Dif a] #
Real a => Real (Dif a) Source #
Methods toRational :: Dif a -> Rational #
RealFloat a => RealFloat (Dif a) Source #
Methods floatRadix :: Dif a -> Integer # floatDigits :: Dif a -> Int # floatRange :: Dif a -> (Int, Int) # decodeFloat :: Dif a -> (Integer, Int) # encodeFloat :: Integer -> Int -> Dif a # exponent :: Dif a -> Int # significand :: Dif a -> Dif a # scaleFloat :: Int -> Dif a -> Dif a # isNaN :: Dif a -> Bool # isInfinite :: Dif a -> Bool # isDenormalized :: Dif a -> Bool # isNegativeZero :: Dif a -> Bool # isIEEE :: Dif a -> Bool # atan2 :: Dif a -> Dif a -> Dif a #
RealFrac a => RealFrac (Dif a) Source #
Methods properFraction :: Integral b => Dif a -> (b, Dif a) # truncate :: Integral b => Dif a -> b # round :: Integral b => Dif a -> b # ceiling :: Integral b => Dif a -> b # floor :: Integral b => Dif a -> b #
Show a => Show (Dif a) Source #
Methods showsPrec :: Int -> Dif a -> ShowS # show :: Dif a -> String # showList :: [Dif a] -> ShowS #

val :: Dif a -> a Source #

The val function takes a Dif number back to a normal number, thus forgetting about all the derivatives.

df :: (Num a, Eq a) => Dif a -> Dif a Source #

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 a Source #

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 a Source #

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, Eq a) => a -> Dif a Source #

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, Eq a, Eq 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, Eq a) => (Dif a -> Dif b) -> a -> b Source #

Convert a Dif function to an ordinary function.