content: 
  \begin{code}
  module Arr (
    module Data.Array,
  
    safezipWith, safezip,
    row,
    sum1, map2, map3,
    mapat, mapat2, mapat3,
    mapindexed, mapindexed2, mapindexed3,
  --  zipArr, sumArr, scaleArr,
    arraySize,
  
    matvec, inner, 
    outerVector,
    
    Vector (Vector), toVector, fromVector, listVector, vectorList, vector, 
    zipVector, scaleVector, sumVector, vectorNorm2, vectorSize,
    
    Matrix (Matrix), toMatrix, fromMatrix, listMatrix, matrixList, matrix, 
    zipMatrix, scaleMatrix, sumMatrix,
  
    augment,
    trMatrix,
  
  --   showsVector,
  --   showsMatrix,
  -- showsVecList, showsMatList
  --  spy,
  ) where
  import Data.Array
  import Numeric
  --import Trace
  --import IOExtensions(unsafePerformIO)
  \end{code}
  
  @Vector@ and @Matrix@ are 1-based arrays with read/show in form of Lists.
  
  \begin{code}
  data Vector a = Vector (Array Int a) deriving (Eq) --, Show)
  
  toVector :: Array Int a -> Vector a
  toVector x = Vector x
  
  fromVector :: Vector a -> Array Int a
  fromVector (Vector x) = x
  
  instance Functor (Vector) where
    fmap fn x = toVector (fmap fn (fromVector x))    
  
  {-instance Eq a => Eq (Vector a) where
  --  (Vector x) == (Vector y) = x == y
  -}
  
  instance Show a => Show (Vector a) where
    showsPrec p x = showsPrec p (elems (fromVector x))
  
  instance Read a => Read (Vector a) where
    readsPrec p = readParen False 
                    (\r -> [(listVector s, t) | (s, t) <- reads r])
  
  instance Num b => Num (Vector b) where
    (+) = zipVector "+" (+)
    (-) = zipVector "-" (-)
    negate = fmap negate
    abs = fmap abs
    signum = fmap signum
  --   (*) = matMult -- works only for matrices!
  --  fromInteger = fmap fromInteger
  \end{code}
  
  
  Convert a list to 1-based vector.
  
  \begin{code}
  listVector :: [a] -> Vector a
  listVector x = toVector (listArray (1,length x) x)
  
  vectorList :: Vector a -> [a]
  vectorList = elems . fromVector
  
  vector (l,u) x | l == 1 = toVector (array (l,u) x)
                 | otherwise = error "vector: l != 1"
                 
  zipVector :: String -> (b -> c -> d) -> Vector b -> Vector c -> Vector d
  zipVector s f (Vector a) (Vector b) 
    | bounds a == bounds b = vector (bounds a) [(i, f (a!i) (b!i)) | i <- indices a]
    | otherwise            = error ("zipVector: " ++ s ++ ": unconformable arrays")
  
  scaleVector :: Num a => a -> Vector a -> Vector a
  scaleVector a = fmap (* a)
  
  sumVector :: Num a => Vector a -> a
  sumVector = sum . elems . fromVector
  
  vectorNorm2 :: Num a => Vector a -> a
  vectorNorm2 x = inner x x
  
  vectorSize :: Vector a -> Int
  vectorSize (Vector x) = rangeSize (bounds x)
  
  \end{code}
  
  ==============
  
  \begin{code}
  data Matrix a = Matrix (Array (Int, Int) a) deriving Eq
  
  toMatrix :: Array (Int, Int) a -> Matrix a
  toMatrix x = Matrix x
  
  fromMatrix :: Matrix a -> Array (Int, Int) a
  fromMatrix (Matrix x) = x
  
  instance Functor (Matrix) where
    fmap fn x = toMatrix (fmap fn (fromMatrix x))    
  
  --instance Eq a => Eq (Matrix a) where
  --  (Matrix x) == (Matrix y) = x == y
  
  instance Show a => Show (Matrix a) where
    showsPrec p x = vertl (matrixList x)
    
  vertl [] = showString "[]"
  vertl (x:xs) = showChar '[' . shows x . showl xs 
      where showl [] = showChar ']'
            showl (x:xs) = showString ",\n" . shows x . showl xs
  
  instance Read a => Read (Matrix a) where
      readsPrec p = readParen False
                    (\r -> [(listMatrix s, t) | (s, t) <- reads r])
  
  instance Num b => Num (Matrix b) where
    (+) = zipMatrix "+" (+)
    (-) = zipMatrix "-" (-)
    negate = fmap negate
    abs = fmap abs
    signum = fmap signum
    x * y = toMatrix (matMult (fromMatrix x) (fromMatrix y)) -- works only for matrices!
  --  fromInteger = fmap fromInteger
  \end{code}
  
  Convert a nested list to a matrix.
  
  \begin{code}
  listMatrix :: [[a]] -> Matrix a
  listMatrix x = Matrix (listArray ((1, 1), (length x, length (x!!0))) (concat x))
  
  matrixList :: Matrix a -> [[a]]
  matrixList (Matrix x) = [ [x!(i,j) | j <- range (l',u')] | i <- range (l,u)]
           where ((l,l'),(u,u')) = bounds x
  
  matrix ((l,l'),(u,u')) x | l == 1 && l' == 1 = toMatrix (array ((l,l'),(u,u')) x)
                           | otherwise = error "matrix: l != 1"
  
  zipMatrix :: String -> (b -> c -> d) -> Matrix b -> Matrix c -> Matrix d
  zipMatrix s f (Matrix a) (Matrix b) 
    | bounds a == bounds b = matrix (bounds a) [(i, f (a!i) (b!i)) | i <- indices a]
    | otherwise            = error ("zipMatrix: " ++ s ++ ": unconformable arrays")
  
  scaleMatrix :: Num a => a -> Matrix a -> Matrix a
  scaleMatrix a = fmap (* a)
  
  sumMatrix :: Num a => Matrix a -> a
  sumMatrix = sum . elems . fromMatrix
  
  \end{code}
  
  
  ============
  
  \begin{code}
  safezipWith :: String -> (a -> b -> c) -> [a] -> [b] -> [c]
  safezipWith _ _ [] [] = []
  safezipWith s f (x:xs) (y:ys) = f x y : safezipWith s f xs ys
  safezipWith s _ _ _ = error ("safezipWith: " ++ s ++ ": unconformable vectors")
  
  safezip :: [a] -> [b] -> [(a,b)]
  safezip = safezipWith "(,)" (,)
  
  trMatrix :: Matrix a -> Matrix a
  trMatrix (Matrix x) = matrix ((l,l'),(u',u)) [((j,i), x!(i,j)) | j <- range (l',u'), i <- range (l,u)]
           where ((l,l'),(u,u')) = bounds x
  
  row :: (Ix a, Ix b) => a -> Array (a,b) c -> Array b c
  row i x = ixmap (l',u') (\j->(i,j)) x where ((l,l'),(u,u')) = bounds x
  
  zipArr :: (Ix a) => String -> (b -> c -> d) -> Array a b -> Array a c -> Array a d
  zipArr s f a b | bounds a == bounds b = array (bounds a) [(i, f (a!i) (b!i)) | i <- indices a]
                 | otherwise            = error ("zipArr: " ++ s ++ ": unconformable arrays")
  \end{code}
  
  Valid only for b -> c -> b functions.
  
  \begin{code}
  zipArr' :: (Ix a) => String -> (b -> c -> b) -> Array a b -> Array a c -> Array a b
  zipArr' s f a b | bounds a == bounds b = accum f a (assocs b)
                  | otherwise            = error ("zipArr': " ++ s ++ ": unconformable arrays")
  \end{code}
  
  Overload arithmetical operators to work on lists.
  
  \begin{code}
  instance Num a => Num [a] where
    (+) = safezipWith "+" (+)
    (-) = safezipWith "-" (-)
    negate = fmap negate
    abs = fmap abs
    signum = fmap signum
  --   (*) = undefined
  --   fromInteger = undefined
  \end{code}
  
  \begin{code}
  sum1 :: (Num a) => [a] -> a
  sum1 = foldl1 (+)
  
  --main = print (sum1 [[4,1,1], [5,1,2], [6,1,3,4]])
  \end{code}
  
  \begin{code}
  map2 f = fmap (fmap f) 
  map3 f = fmap (map2 f) 
  \end{code}
  
  Map function f at position n only.  Out of range indices are silently
  ignored.
  
  \begin{code}
  mapat n f x = mapat1 0 f x where
      mapat1 _ _ [] = []
      mapat1 i f (x:xs) = (if i == n then f x else x) : mapat1 (i + 1) f xs
  
  mapat2 (i,j) = mapat i . mapat j
  mapat3 (i,j,k) = mapat i . mapat j . mapat k
  
  -- main = print (mapat 2 (10+) [1,2,3,4])
  -- main = print (mapat2 (1,0) (1000+) ginp)
  -- main = print (mapat3 (1,0,1) (1000+) gw)
  \end{code}
  
  \begin{code}
  mapindexed f x = mapindexed1 f 0 x where
      mapindexed1 _ _ [] = []
      mapindexed1 f n (x:xs) = f n x : mapindexed1 f (n + 1) xs
  
  mapindexed2 f = mapindexed (\i -> mapindexed (\j -> f (i, j))) 
  mapindexed3 f = mapindexed (\i -> mapindexed (\j -> mapindexed (\k -> f (i, j, k))))
  
  -- main = print (mapindexed (\x y -> mapat (10+) [1,2,3,4] y) [1,2,3,4])
  -- main = print (mapindexed2 (\(i,j) x -> 100*i + 10*j + x) ginp)
  -- main = print (mapindexed3 (\(i,j,k) x -> 1000*i + 100*j + 10*k + x) gw)
  \end{code}
  
  
  
  Overload arithmetical operators to work on arrays.
  
  \begin{code}
  instance (Ix a, Show a, Num b) => Num (Array a b) where
    (+) = zipArr "+" (+)
    (-) = zipArr "-" (-)
    negate = fmap negate
    abs = fmap abs
    signum = fmap signum
  --   (*) = matMult -- works only for matrices!
  --   fromInteger = map fromInteger
  \end{code}
  
  \begin{xcode}
  scaleArr :: (Ix i, Num a) => a -> Array i a -> Array i a
  scaleArr a = fmap (*a)
  
  sumArr :: (Ix i, Num a) => Array i a -> a
  sumArr = sum . elems
  \end{xcode}
  
  \begin{code}
  arraySize :: (Ix i) => Array i a -> Int
  arraySize = rangeSize . bounds
  \end{code}
  
  \begin{code}
  matMult         :: (Ix a, Ix b, Ix c, Num d) =>
                     Array (a,b) d -> Array (b,c) d -> Array (a,c) d
  matMult x y     =  array resultBounds
                           [((i,j), sum [x!(i,k) * y!(k,j) | k <- range (lj,uj)])
                                         | i <- range (li,ui),
                                           j <- range (lj',uj') ]
          where ((li,lj),(ui,uj))         =  bounds x
                ((li',lj'),(ui',uj'))     =  bounds y
                resultBounds
                  | (lj,uj)==(li',ui')    =  ((li,lj'),(ui,uj'))
                  | otherwise             = error "matMult: incompatible bounds"
  \end{code}
  
  
  Inner product of two vectors.
  
  \begin{code}
  inner :: Num a => Vector a -> Vector a -> a
  inner (Vector v) (Vector w) = if b == bounds w
                 then sum [v!i * w!i | i <- range b]
                 else error "nn.inner: inconformable vectors"
              where b = bounds v
  \end{code}
  
  Outer product of two vectors $v \dot w^\mathrm{T}$.
  
  \begin{code}
  outerVector :: Num b => Vector b -> Vector b -> Matrix b
  outerVector (Vector v) (Vector w) = if (l,u) == (l',u')
                 then matrix ((l,l'),(u,u')) [((i,j), v!i * w!j) | i <- range (l,u), j <- range (l',u')]
                 else error "nn.outer: inconformable vectors"
              where ((l,u),(l',u')) = (bounds v, bounds w)
  \end{code}
  
  \begin{code}
  outerArr :: (Ix a, Num b) => Array a b -> Array a b -> Array (a,a) b
  outerArr v w = if (l,u) == (l',u')
                 then array ((l,l'),(u,u')) [((i,j), v!i * w!j) | i <- range (l,u), j <- range (l',u')]
                 else error "nn.outer: inconformable vectors"
              where ((l,u),(l',u')) = (bounds v, bounds w)
  \end{code}
  
  Inner product of a matrix and a vector.
  
  \begin{code}
  matvec :: (Ix a, Num b) => Array (a,a) b -> Array a b -> Array a b
  matvec w x | bounds x == (l',u') =
                  array (l,u) [(i, sum [w!(i,j) * x!j | j <- range (l',u')]) 
                                  | i <- range (l,u)]
             | otherwise           = error "nn.matvec: inconformable arrays"
           where ((l,l'),(u,u')) = bounds w
  \end{code}
  
  Append to a vector.
  
  \begin{code}
  augment :: (Num a) => Vector a -> a -> Vector a
  augment (Vector x) y = Vector (array (a,b') ((b',y) : assocs x))
              where (a,b) = bounds x
                    b' = b + 1
  \end{code}
  
  Older approach (x!!i!!j fails in ghc-2.03).
  
  \begin{code}
  toMatrix' :: [[a]] -> Matrix a
  toMatrix' x = Matrix (array ((1,1),(u,u')) [((i,j), (x!!(i-1))!!(j-1)) 
                               | i <- range (1,u), j <- range (1,u')])
            where (u,u') = (length x,length (x!!0))
  \end{code}
  
  Matrix 2D printout.
  
  \begin{code}
  padleft :: Int -> String -> String
  padleft n x | n <= length x = x
              | otherwise = replicate (n - length x) ' ' ++ x
  \end{code}
  
  \begin{code}
  padMatrix :: RealFloat a => Int -> Matrix a -> Matrix String
  padMatrix n x = let ss = fmap (\a -> showFFloat (Just n) a "") x 
                      maxw = maximum (fmap length (elems (fromMatrix ss)))
                in fmap (padleft maxw) ss
  \end{code}
  
  \begin{xcode}
  showsVector :: (RealFloat a) => Int -> Vector a -> ShowS
  showsVector n x z1 = let x' = padArr n x
                           (l,u) = bounds x' in
                    concat (fmap (\ (i, s) -> if i == u then s ++ "\n" else s ++ " ") (assocs x')) ++ z1
  \end{xcode}
  
  \begin{xcode}
  showsMatrix :: RealFloat a => Int -> Matrix a -> ShowS
  showsMatrix n x z1 = let x' = padMatrix n x
                           ((l,l'),(u,u')) = bounds x' in
                     concat (fmap (\ ((i,j), s) -> if j == u' then s ++ "\n" else s ++ " ") (assocs x')) ++ z1
  \end{xcode}
  
  {-
  showsVecList n x s = foldr (showsVector n) s x
  showsMatList n x s = foldr (showsMatrix n) s x
  -}
  
  
  \begin{code}
  --spy :: Show a => String -> a -> a
  --spy msg x = trace ('<' : msg ++ ": " ++ shows x ">\n") x
  --spy x  = seq (unsafePerformIO (putStr ('<' : shows x ">\n"))) x
  --spy x  = traceShow "z" x
  \end{code}
  