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+
+\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}