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+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE FunctionalDependencies #-}
+{-# LANGUAGE UndecidableInstances #-}
+
+-----------------------------------------------------------------------------
+-- |
+-- Module      :  Internal.Container
+-- Copyright   :  (c) Alberto Ruiz 2010-14
+-- License     :  BSD3
+-- Maintainer  :  Alberto Ruiz
+-- Stability   :  provisional
+--
+-- Basic numeric operations on 'Vector' and 'Matrix', including conversion routines.
+--
+-- The 'Container' class is used to define optimized generic functions which work
+-- on 'Vector' and 'Matrix' with real or complex elements.
+--
+-- Some of these functions are also available in the instances of the standard
+-- numeric Haskell classes provided by "Numeric.LinearAlgebra".
+--
+-----------------------------------------------------------------------------
+
+module Internal.Container where
+
+import Internal.Tools
+import Internal.Vector
+import Internal.Matrix
+import Internal.Element
+import Internal.Numeric
+import Data.Complex
+import Internal.Algorithms(Field,linearSolveSVD)
+import Data.Vector.Storable(fromList)
+
+------------------------------------------------------------------
+
+{- | Creates a real vector containing a range of values:
+
+>>> linspace 5 (-3,7::Double)
+fromList [-3.0,-0.5,2.0,4.5,7.0]@
+
+>>> linspace 5 (8,2+i) :: Vector (Complex Double)
+fromList [8.0 :+ 0.0,6.5 :+ 0.25,5.0 :+ 0.5,3.5 :+ 0.75,2.0 :+ 1.0]
+
+Logarithmic spacing can be defined as follows:
+
+@logspace n (a,b) = 10 ** linspace n (a,b)@
+-}
+linspace :: (Fractional e, Container Vector e) => Int -> (e, e) -> Vector e
+linspace 0 _     = fromList[]
+linspace 1 (a,b) = fromList[(a+b)/2]
+linspace n (a,b) = addConstant a $ scale s $ fromList $ map fromIntegral [0 .. n-1]
+    where s = (b-a)/fromIntegral (n-1)
+
+--------------------------------------------------------------------------------
+
+infixl 7 <.>
+-- | An infix synonym for 'dot'
+(<.>) :: Numeric t => Vector t -> Vector t -> t
+(<.>) = dot
+
+
+infixr 8 <·>, #>
+
+{- | infix synonym for 'dot'
+
+>>> vector [1,2,3,4] <·> vector [-2,0,1,1]
+5.0
+
+>>> let 𝑖 = 0:+1 :: ℂ
+>>> fromList [1+𝑖,1] <·> fromList [1,1+𝑖]
+2.0 :+ 0.0
+
+(the dot symbol "·" is obtained by Alt-Gr .)
+
+-}
+(<·>) :: Numeric t => Vector t -> Vector t -> t
+(<·>) = dot
+
+
+{- | infix synonym for 'app'
+
+>>> let m = (2><3) [1..]
+>>> m
+(2><3)
+ [ 1.0, 2.0, 3.0
+ , 4.0, 5.0, 6.0 ]
+
+>>> let v = vector [10,20,30]
+
+>>> m #> v
+fromList [140.0,320.0]
+
+-}
+(#>) :: Numeric t => Matrix t -> Vector t -> Vector t
+(#>) = mXv
+
+-- | dense matrix-vector product
+app :: Numeric t => Matrix t -> Vector t -> Vector t
+app = (#>)
+
+infixl 8 <#
+-- | dense vector-matrix product
+(<#) :: Numeric t => Vector t -> Matrix t -> Vector t
+(<#) = vXm
+
+--------------------------------------------------------------------------------
+
+class Mul a b c | a b -> c where
+ infixl 7 <>
+ -- | Matrix-matrix, matrix-vector, and vector-matrix products.
+ (<>)  :: Product t => a t -> b t -> c t
+
+instance Mul Matrix Matrix Matrix where
+    (<>) = mXm
+
+instance Mul Matrix Vector Vector where
+    (<>) m v = flatten $ m <> asColumn v
+
+instance Mul Vector Matrix Vector where
+    (<>) v m = flatten $ asRow v <> m
+
+--------------------------------------------------------------------------------
+
+{- | Least squares solution of a linear system, similar to the \\ operator of Matlab\/Octave (based on linearSolveSVD)
+
+@
+a = (3><2)
+ [ 1.0,  2.0
+ , 2.0,  4.0
+ , 2.0, -1.0 ]
+@
+
+@
+v = vector [13.0,27.0,1.0]
+@
+
+>>> let x = a <\> v
+>>> x
+fromList [3.0799999999999996,5.159999999999999]
+
+>>> a #> x
+fromList [13.399999999999999,26.799999999999997,1.0]
+
+It also admits multiple right-hand sides stored as columns in a matrix.
+
+-}
+infixl 7 <\>
+(<\>) :: (LSDiv c, Field t) => Matrix t -> c t -> c t
+(<\>) = linSolve
+
+class LSDiv c
+  where
+    linSolve :: Field t => Matrix t -> c t -> c t
+
+instance LSDiv Vector
+  where
+    linSolve m v = flatten (linearSolveSVD m (reshape 1 v))
+
+instance LSDiv Matrix
+  where
+    linSolve = linearSolveSVD
+
+--------------------------------------------------------------------------------
+
+class Konst e d c | d -> c, c -> d
+  where
+    -- |
+    -- >>> konst 7 3 :: Vector Float
+    -- fromList [7.0,7.0,7.0]
+    --
+    -- >>> konst i (3::Int,4::Int)
+    -- (3><4)
+    --  [ 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0
+    --  , 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0
+    --  , 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0 ]
+    --
+    konst :: e -> d -> c e
+
+instance Container Vector e => Konst e Int Vector
+  where
+    konst = konst'
+
+instance (Num e, Container Vector e) => Konst e (Int,Int) Matrix
+  where
+    konst = konst'
+
+--------------------------------------------------------------------------------
+
+class Build d f c e | d -> c, c -> d, f -> e, f -> d, f -> c, c e -> f, d e -> f
+  where
+    -- |
+    -- >>> build 5 (**2) :: Vector Double
+    -- fromList [0.0,1.0,4.0,9.0,16.0]
+    --
+    -- Hilbert matrix of order N:
+    --
+    -- >>> let hilb n = build (n,n) (\i j -> 1/(i+j+1)) :: Matrix Double
+    -- >>> putStr . dispf 2 $ hilb 3
+    -- 3x3
+    -- 1.00  0.50  0.33
+    -- 0.50  0.33  0.25
+    -- 0.33  0.25  0.20
+    --
+    build :: d -> f -> c e
+
+instance Container Vector e => Build Int (e -> e) Vector e
+  where
+    build = build'
+
+instance Container Matrix e => Build (Int,Int) (e -> e -> e) Matrix e
+  where
+    build = build'
+
+--------------------------------------------------------------------------------
+
+-- @dot u v = 'udot' ('conj' u) v@
+dot :: (Numeric t) => Vector t -> Vector t -> t
+dot u v = udot (conj u) v
+
+--------------------------------------------------------------------------------
+
+optimiseMult :: Monoid (Matrix t) => [Matrix t] -> Matrix t
+optimiseMult = mconcat
+
+--------------------------------------------------------------------------------
+
+
+{- | Compute mean vector and covariance matrix of the rows of a matrix.
+
+>>> meanCov $ gaussianSample 666 1000 (fromList[4,5]) (diagl[2,3])
+(fromList [4.010341078059521,5.0197204699640405],
+(2><2)
+ [     1.9862461923890056, -1.0127225830525157e-2
+ , -1.0127225830525157e-2,     3.0373954915729318 ])
+
+-}
+meanCov :: Matrix Double -> (Vector Double, Matrix Double)
+meanCov x = (med,cov) where
+    r    = rows x
+    k    = 1 / fromIntegral r
+    med  = konst k r `vXm` x
+    meds = konst 1 r `outer` med
+    xc   = x `sub` meds
+    cov  = scale (recip (fromIntegral (r-1))) (trans xc `mXm` xc)
+
+--------------------------------------------------------------------------------
+
+class ( Container Vector t
+      , Container Matrix t
+      , Konst t Int Vector
+      , Konst t (Int,Int) Matrix
+      , Product t
+      ) => Numeric t
+
+instance Numeric Double
+instance Numeric (Complex Double)
+instance Numeric Float
+instance Numeric (Complex Float)
+instance Numeric I
+
+--------------------------------------------------------------------------------
+
+sortVector :: (Ord t, Element t) => Vector t -> Vector t
+sortVector = sortV
+
+sortIndex :: (Ord t, Element t) => Vector t -> Vector I
+sortIndex = sortI
+
+ccompare :: (Ord t, Container c t) => c t -> c t -> c I
+ccompare = ccompare'
+
+cselect :: (Container c t) => c I -> c t -> c t -> c t -> c t
+cselect = cselect'
+
+remap :: Element t => Matrix I -> Matrix I -> Matrix t -> Matrix t
+remap i j m
+    | minElement i >= 0 && maxElement i < fromIntegral (rows m) &&
+      minElement j >= 0 && maxElement j < fromIntegral (cols m) = remapM i' j' m
+    | otherwise = error $ "out of range index in rmap"
+  where
+    [i',j'] = conformMs [i,j]
+    
+