[Haskell-cafe] Partial Derivatives

[Arjen van Weelden]

Hi,

Jerzy Karczmarczuk has a nice paper about "Functional Differentiation of 
Computer Programs", see http://users.info.unicaen.fr/~karczma/arpap/

regards,
	Arjen

Gerhard Navratil wrote:
> Hi,
> 
> I need a library that provides partial derivatives for functions. The
> solution I came up with is based on a datatype using reversed polish
> notation to store the function:
> 
> Type VarName = String
> data Fkt a = Val a |
>              Var VarName |
>              Add (FktElem a) (FktElem a) |
>              Sub (FktElem a) (FktElem a) |
>              Mul (FktElem a) (FktElem a) |
>              ...
> 
> I can then use pattern matching to compute the derivative:
> 
> derivative  :: VarName -> Fkt b -> Fkt b
> derivative name (Var n) = if n == name then Val 1.0 else Val 0.0
> derivative name (Add a b)
>     | (containsVar name a) && (containsVar name b)     = Add (derivative
> name a) (derivative name b)
>     | (containsVar name a) && (not(containsVar name b)) = (derivative
> name a)
>     | (not(containsVar name a)) && (containsVar name b) = (derivative
> name b)
>     | otherwise = Val 0.0
> derivative name (Sub a b)
>     | (containsVar name a) && (containsVar name b)      = Sub
> (derivative name a) (derivative name b)
>     | (containsVar name a) && (not(containsVar name b)) = (derivative
> name a)
>     | (not(containsVar name a)) && (containsVar name b) = (derivative
> name b)
>     | otherwise = Val 0.0
> derivative name (Mul a b)
>     | (containsVar name a) && (containsVar name b)      = Add (Mul
> (derivative name a) b) (Mul a (derivative name b))
>     | (containsVar name a) && (not(containsVar name b)) = Mul
> (derivative name a) b
>     | (not(containsVar name a)) && (containsVar name b) = Mul a
> (derivative name b)
>     | otherwise = Val 0.0
> ...
> 
> Where the function
> 
> containsVar :: VarName -> Fkt a -> Bool
> 
> tests if a function contains the variable.
> 
> The solution works but is not very elegant. The complete module is
> appended to the mail.
> 
> Does anyone have a more elegant solution or is there a package that
> provides derivatives in a similar way?
> 
> Thank you,
> Gerhard
> 
> ========================================
> Gerhard Navratil
> Teaching- And Research-Assistant
> Technical University Vienna, Austria
> Institute of Geoinformation and Cartography
> Gusshausstr. 27-29, 1040 Vienna
> Tel.: ++43 (0) 1 / 58 801 - 12712
> Fax.: ++43 (0) 1 / 58 801 - 12799
> Cel.: ++43 (0) 699 / 197 44 761
> http://www.geoinfo.tuwien.ac.at
> 
> 
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