-- WARNING: this module depends on type families working fairly well, and
-- requires ghc version at least 6.9. I didn't find a way to specify that
-- dependency in the .cabal.
--
{-# LANGUAGE TypeOperators, TypeFamilies, UndecidableInstances
, FlexibleInstances, MultiParamTypeClasses
#-}
{-# OPTIONS_GHC -Wall -fno-warn-orphans #-}
----------------------------------------------------------------------
-- |
-- Module : Data.ABasis
-- Copyright : (c) Conal Elliott 2008
-- License : BSD3
--
-- Maintainer : conal@conal.net
-- Stability : experimental
--
-- Basis of a vector space, as an associated type
-- This module requires ghc-6.10 or later
----------------------------------------------------------------------
module Data.ABasis (HasBasis(..), linearCombo, recompose) where
-- import Control.Applicative ((<$>))
import Control.Arrow (first)
import Data.Either
import Data.AVectorSpace
class VectorSpace v => HasBasis v where
-- | Representation of the canonical basis for @v@
type Basis v :: *
-- | Interpret basis rep as a vector
basisValue :: Basis v -> v
-- | Extract coordinates
decompose :: v -> [(Basis v, Scalar v)]
-- | Experimental version. More elegant definitions, and friendly to
-- infinite-dimensional vector spaces.
decompose' :: v -> (Basis v -> Scalar v)
-- Defining property: recompose . decompose == id
-- | Linear combination
linearCombo :: VectorSpace v => [(v,Scalar v)] -> v
linearCombo ps = sumV [s *^ v | (v,s) <- ps]
-- Turn a basis decomposition back into a vector.
recompose :: HasBasis v => [(Basis v, Scalar v)] -> v
recompose = linearCombo . fmap (first basisValue)
-- recompose ps = linearCombo (first basisValue <$> ps)
-- I don't know how to define
--
-- recompose' :: HasBasis v => (Basis v -> Scalar v) -> v
--
-- However, I don't seem to use recompose anywhere.
-- I don't even use basisValue or decompose.
instance HasBasis Float where
type Basis Float = ()
basisValue () = 1
decompose s = [((),s)]
decompose' s = const s
instance HasBasis Double where
type Basis Double = ()
basisValue () = 1
decompose s = [((),s)]
decompose' s = const s
instance ( HasBasis u, s ~ Scalar u
, HasBasis v, s ~ Scalar v )
=> HasBasis (u,v) where
type Basis (u,v) = Basis u `Either` Basis v
basisValue (Left a) = (basisValue a, zeroV)
basisValue (Right b) = (zeroV, basisValue b)
decompose (u,v) = decomp2 Left u ++ decomp2 Right v
decompose' (u,v) = decompose' u `either` decompose' v
decomp2 :: HasBasis w => (Basis w -> b) -> w -> [(b, Scalar w)]
decomp2 inject = fmap (first inject) . decompose
instance ( HasBasis u, s ~ Scalar u
, HasBasis v, s ~ Scalar v
, HasBasis w, s ~ Scalar w )
=> HasBasis (u,v,w) where
type Basis (u,v,w) = Basis (u,(v,w))
basisValue = unnest3 . basisValue
decompose = decompose . nest3
decompose' = decompose' . nest3
unnest3 :: (a,(b,c)) -> (a,b,c)
unnest3 (a,(b,c)) = (a,b,c)
nest3 :: (a,b,c) -> (a,(b,c))
nest3 (a,b,c) = (a,(b,c))
instance (Eq a, HasBasis u) => HasBasis (a -> u) where
type Basis (a -> u) = (a, Basis u)
basisValue (a,b) = f
where f a' | a == a' = bv
| otherwise = zeroV
bv = basisValue b
decompose = error "decompose: not defined on functions"
decompose' g (a,b) = decompose' (g a) b
-- Simpler but less efficient:
--
-- basisValue (a,b) a' | a == a' = basisValue b
-- | otherwise = zeroV
-- Just for pointless perversion points:
--
-- decompose' g = uncurry (\ a b -> decompose' (g a) b)
-- decompose' g = uncurry (\ a -> decompose' (g a))
-- decompose' g = uncurry (decompose' . g)
-- decompose' = uncurry . fmap decompose'
-- decompose' = (fmap uncurry) (fmap decompose')
{-
---- Testing
t1 = basisValue () :: Float
t2 = basisValue () :: Double
t3 = basisValue (Right ()) :: (Float,Double)
t4 = basisValue (Right (Left ())) :: (Float,Double,Float)
-}