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+{-# LANGUAGE CPP #-}
+{-# LANGUAGE TypeFamilies #-}
+{-# LANGUAGE FlexibleContexts #-}
+{-# LANGUAGE FlexibleInstances #-}
+{-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE UndecidableInstances #-}
+
+-----------------------------------------------------------------------------
+-- |
+-- Module      :  Data.Packed.Internal.Numeric
+-- Copyright   :  (c) Alberto Ruiz 2010-14
+-- License     :  BSD3
+-- Maintainer  :  Alberto Ruiz
+-- Stability   :  provisional
+--
+-----------------------------------------------------------------------------
+
+module Data.Packed.Internal.Numeric (
+    -- * Basic functions
+    ident, diag, ctrans,
+    -- * Generic operations
+    Container(..),
+    -- * Matrix product and related functions
+    Product(..), udot,
+    mXm,mXv,vXm,
+    outer, kronecker,
+    -- * Element conversion
+    Convert(..),
+    Complexable(),
+    RealElement(),
+
+    RealOf, ComplexOf, SingleOf, DoubleOf,
+
+    IndexOf,
+    module Data.Complex
+) where
+
+import Data.Packed
+import Data.Packed.ST as ST
+import Numeric.Conversion
+import Data.Packed.Development
+import Numeric.Vectorized
+import Data.Complex
+import Control.Applicative((<*>))
+
+import Numeric.LinearAlgebra.LAPACK(multiplyR,multiplyC,multiplyF,multiplyQ)
+
+-------------------------------------------------------------------
+
+type family IndexOf (c :: * -> *)
+
+type instance IndexOf Vector = Int
+type instance IndexOf Matrix = (Int,Int)
+
+type family ArgOf (c :: * -> *) a
+
+type instance ArgOf Vector a = a -> a
+type instance ArgOf Matrix a = a -> a -> a
+
+-------------------------------------------------------------------
+
+-- | Basic element-by-element functions for numeric containers
+class (Complexable c, Fractional e, Element e) => Container c e where
+    -- | create a structure with a single element
+    --
+    -- >>> let v = fromList [1..3::Double]
+    -- >>> v / scalar (norm2 v)
+    -- fromList [0.2672612419124244,0.5345224838248488,0.8017837257372732]
+    --
+    scalar      :: e -> c e
+    -- | complex conjugate
+    conj        :: c e -> c e
+    scale       :: e -> c e -> c e
+    -- | scale the element by element reciprocal of the object:
+    --
+    -- @scaleRecip 2 (fromList [5,i]) == 2 |> [0.4 :+ 0.0,0.0 :+ (-2.0)]@
+    scaleRecip  :: e -> c e -> c e
+    addConstant :: e -> c e -> c e
+    add         :: c e -> c e -> c e
+    sub         :: c e -> c e -> c e
+    -- | element by element multiplication
+    mul         :: c e -> c e -> c e
+    -- | element by element division
+    divide      :: c e -> c e -> c e
+    equal       :: c e -> c e -> Bool
+    --
+    -- element by element inverse tangent
+    arctan2     :: c e -> c e -> c e
+    --
+    -- | cannot implement instance Functor because of Element class constraint
+    cmap        :: (Element b) => (e -> b) -> c e -> c b
+    -- | constant structure of given size
+    konst'      :: e -> IndexOf c -> c e
+    -- | create a structure using a function
+    --
+    -- Hilbert matrix of order N:
+    --
+    -- @hilb n = build' (n,n) (\\i j -> 1/(i+j+1))@
+    build'       :: IndexOf c -> (ArgOf c e) -> c e
+    -- | indexing function
+    atIndex     :: c e -> IndexOf c -> e
+    -- | index of min element
+    minIndex    :: c e -> IndexOf c
+    -- | index of max element
+    maxIndex    :: c e -> IndexOf c
+    -- | value of min element
+    minElement  :: c e -> e
+    -- | value of max element
+    maxElement  :: c e -> e
+    -- the C functions sumX/prodX are twice as fast as using foldVector
+    -- | the sum of elements (faster than using @fold@)
+    sumElements :: c e -> e
+    -- | the product of elements (faster than using @fold@)
+    prodElements :: c e -> e
+
+    -- | A more efficient implementation of @cmap (\\x -> if x>0 then 1 else 0)@
+    --
+    -- >>> step $ linspace 5 (-1,1::Double)
+    -- 5 |> [0.0,0.0,0.0,1.0,1.0]
+    --
+    
+    step :: RealElement e => c e -> c e
+
+    -- | Element by element version of @case compare a b of {LT -> l; EQ -> e; GT -> g}@.
+    --
+    -- Arguments with any dimension = 1 are automatically expanded: 
+    --
+    -- >>> cond ((1><4)[1..]) ((3><1)[1..]) 0 100 ((3><4)[1..]) :: Matrix Double
+    -- (3><4)
+    -- [ 100.0,   2.0,   3.0,  4.0
+    -- ,   0.0, 100.0,   7.0,  8.0
+    -- ,   0.0,   0.0, 100.0, 12.0 ]
+    --
+    
+    cond :: RealElement e 
+         => c e -- ^ a
+         -> c e -- ^ b
+         -> c e -- ^ l 
+         -> c e -- ^ e
+         -> c e -- ^ g
+         -> c e -- ^ result
+
+    -- | Find index of elements which satisfy a predicate
+    --
+    -- >>> find (>0) (ident 3 :: Matrix Double)
+    -- [(0,0),(1,1),(2,2)]
+    --
+
+    find :: (e -> Bool) -> c e -> [IndexOf c]
+
+    -- | Create a structure from an association list
+    --
+    -- >>> assoc 5 0 [(3,7),(1,4)] :: Vector Double
+    -- fromList [0.0,4.0,0.0,7.0,0.0]
+    --
+    -- >>> assoc (2,3) 0 [((0,2),7),((1,0),2*i-3)] :: Matrix (Complex Double)
+    -- (2><3)
+    --  [    0.0 :+ 0.0, 0.0 :+ 0.0, 7.0 :+ 0.0
+    --  , (-3.0) :+ 2.0, 0.0 :+ 0.0, 0.0 :+ 0.0 ]
+    --
+    assoc :: IndexOf c        -- ^ size
+          -> e                -- ^ default value
+          -> [(IndexOf c, e)] -- ^ association list
+          -> c e              -- ^ result
+
+    -- | Modify a structure using an update function
+    --
+    -- >>> accum (ident 5) (+) [((1,1),5),((0,3),3)] :: Matrix Double
+    -- (5><5)
+    --  [ 1.0, 0.0, 0.0, 3.0, 0.0
+    --  , 0.0, 6.0, 0.0, 0.0, 0.0
+    --  , 0.0, 0.0, 1.0, 0.0, 0.0
+    --  , 0.0, 0.0, 0.0, 1.0, 0.0
+    --  , 0.0, 0.0, 0.0, 0.0, 1.0 ]
+    --
+    -- computation of histogram:
+    --
+    -- >>> accum (konst 0 7) (+) (map (flip (,) 1) [4,5,4,1,5,2,5]) :: Vector Double
+    -- fromList [0.0,1.0,1.0,0.0,2.0,3.0,0.0]
+    --
+    
+    accum :: c e              -- ^ initial structure
+          -> (e -> e -> e)    -- ^ update function
+          -> [(IndexOf c, e)] -- ^ association list
+          -> c e              -- ^ result
+
+--------------------------------------------------------------------------
+
+instance Container Vector Float where
+    scale = vectorMapValF Scale
+    scaleRecip = vectorMapValF Recip
+    addConstant = vectorMapValF AddConstant
+    add = vectorZipF Add
+    sub = vectorZipF Sub
+    mul = vectorZipF Mul
+    divide = vectorZipF Div
+    equal u v = dim u == dim v && maxElement (vectorMapF Abs (sub u v)) == 0.0
+    arctan2 = vectorZipF ATan2
+    scalar x = fromList [x]
+    konst' = constant
+    build' = buildV
+    conj = id
+    cmap = mapVector
+    atIndex = (@>)
+    minIndex     = emptyErrorV "minIndex"   (round . toScalarF MinIdx)
+    maxIndex     = emptyErrorV "maxIndex"   (round . toScalarF MaxIdx)
+    minElement   = emptyErrorV "minElement" (toScalarF Min)
+    maxElement   = emptyErrorV "maxElement" (toScalarF Max)
+    sumElements  = sumF
+    prodElements = prodF
+    step = stepF
+    find = findV
+    assoc = assocV
+    accum = accumV
+    cond = condV condF
+
+instance Container Vector Double where
+    scale = vectorMapValR Scale
+    scaleRecip = vectorMapValR Recip
+    addConstant = vectorMapValR AddConstant
+    add = vectorZipR Add
+    sub = vectorZipR Sub
+    mul = vectorZipR Mul
+    divide = vectorZipR Div
+    equal u v = dim u == dim v && maxElement (vectorMapR Abs (sub u v)) == 0.0
+    arctan2 = vectorZipR ATan2
+    scalar x = fromList [x]
+    konst' = constant
+    build' = buildV
+    conj = id
+    cmap = mapVector
+    atIndex = (@>)
+    minIndex     = emptyErrorV "minIndex"   (round . toScalarR MinIdx)
+    maxIndex     = emptyErrorV "maxIndex"   (round . toScalarR MaxIdx)
+    minElement   = emptyErrorV "minElement" (toScalarR Min)
+    maxElement   = emptyErrorV "maxElement" (toScalarR Max)
+    sumElements  = sumR
+    prodElements = prodR
+    step = stepD
+    find = findV
+    assoc = assocV
+    accum = accumV
+    cond = condV condD
+
+instance Container Vector (Complex Double) where
+    scale = vectorMapValC Scale
+    scaleRecip = vectorMapValC Recip
+    addConstant = vectorMapValC AddConstant
+    add = vectorZipC Add
+    sub = vectorZipC Sub
+    mul = vectorZipC Mul
+    divide = vectorZipC Div
+    equal u v = dim u == dim v && maxElement (mapVector magnitude (sub u v)) == 0.0
+    arctan2 = vectorZipC ATan2
+    scalar x = fromList [x]
+    konst' = constant
+    build' = buildV
+    conj = conjugateC
+    cmap = mapVector
+    atIndex = (@>)
+    minIndex     = emptyErrorV "minIndex" (minIndex . fst . fromComplex . (mul <*> conj))
+    maxIndex     = emptyErrorV "maxIndex" (maxIndex . fst . fromComplex . (mul <*> conj))
+    minElement   = emptyErrorV "minElement" (atIndex <*> minIndex)
+    maxElement   = emptyErrorV "maxElement" (atIndex <*> maxIndex)
+    sumElements  = sumC
+    prodElements = prodC
+    step = undefined -- cannot match
+    find = findV
+    assoc = assocV
+    accum = accumV
+    cond = undefined -- cannot match
+
+instance Container Vector (Complex Float) where
+    scale = vectorMapValQ Scale
+    scaleRecip = vectorMapValQ Recip
+    addConstant = vectorMapValQ AddConstant
+    add = vectorZipQ Add
+    sub = vectorZipQ Sub
+    mul = vectorZipQ Mul
+    divide = vectorZipQ Div
+    equal u v = dim u == dim v && maxElement (mapVector magnitude (sub u v)) == 0.0
+    arctan2 = vectorZipQ ATan2
+    scalar x = fromList [x]
+    konst' = constant
+    build' = buildV
+    conj = conjugateQ
+    cmap = mapVector
+    atIndex = (@>)
+    minIndex     = emptyErrorV "minIndex" (minIndex . fst . fromComplex . (mul <*> conj))
+    maxIndex     = emptyErrorV "maxIndex" (maxIndex . fst . fromComplex . (mul <*> conj))
+    minElement   = emptyErrorV "minElement" (atIndex <*> minIndex)
+    maxElement   = emptyErrorV "maxElement" (atIndex <*> maxIndex)
+    sumElements  = sumQ
+    prodElements = prodQ
+    step = undefined -- cannot match
+    find = findV
+    assoc = assocV
+    accum = accumV
+    cond = undefined -- cannot match
+
+---------------------------------------------------------------
+
+instance (Container Vector a) => Container Matrix a where
+    scale x = liftMatrix (scale x)
+    scaleRecip x = liftMatrix (scaleRecip x)
+    addConstant x = liftMatrix (addConstant x)
+    add = liftMatrix2 add
+    sub = liftMatrix2 sub
+    mul = liftMatrix2 mul
+    divide = liftMatrix2 divide
+    equal a b = cols a == cols b && flatten a `equal` flatten b
+    arctan2 = liftMatrix2 arctan2
+    scalar x = (1><1) [x]
+    konst' v (r,c) = matrixFromVector RowMajor r c (konst' v (r*c))
+    build' = buildM
+    conj = liftMatrix conj
+    cmap f = liftMatrix (mapVector f)
+    atIndex = (@@>)
+    minIndex = emptyErrorM "minIndex of Matrix" $
+                \m -> divMod (minIndex $ flatten m) (cols m)
+    maxIndex = emptyErrorM "maxIndex of Matrix" $
+                \m -> divMod (maxIndex $ flatten m) (cols m)
+    minElement = emptyErrorM "minElement of Matrix" (atIndex <*> minIndex)
+    maxElement = emptyErrorM "maxElement of Matrix" (atIndex <*> maxIndex)
+    sumElements = sumElements . flatten
+    prodElements = prodElements . flatten
+    step = liftMatrix step
+    find = findM
+    assoc = assocM
+    accum = accumM
+    cond = condM
+
+
+emptyErrorV msg f v =
+    if dim v > 0
+        then f v
+        else error $ msg ++ " of Vector with dim = 0"
+
+emptyErrorM msg f m =
+    if rows m > 0 && cols m > 0
+        then f m
+        else error $ msg++" "++shSize m
+
+----------------------------------------------------
+
+-- | Matrix product and related functions
+class (Num e, Element e) => Product e where
+    -- | matrix product
+    multiply :: Matrix e -> Matrix e -> Matrix e
+    -- | sum of absolute value of elements (differs in complex case from @norm1@)
+    absSum     :: Vector e -> RealOf e
+    -- | sum of absolute value of elements
+    norm1      :: Vector e -> RealOf e
+    -- | euclidean norm
+    norm2      :: Vector e -> RealOf e
+    -- | element of maximum magnitude
+    normInf    :: Vector e -> RealOf e
+
+instance Product Float where
+    norm2      = emptyVal (toScalarF Norm2)
+    absSum     = emptyVal (toScalarF AbsSum)
+    norm1      = emptyVal (toScalarF AbsSum)
+    normInf    = emptyVal (maxElement . vectorMapF Abs)
+    multiply   = emptyMul multiplyF
+
+instance Product Double where
+    norm2      = emptyVal (toScalarR Norm2)
+    absSum     = emptyVal (toScalarR AbsSum)
+    norm1      = emptyVal (toScalarR AbsSum)
+    normInf    = emptyVal (maxElement . vectorMapR Abs)
+    multiply   = emptyMul multiplyR
+
+instance Product (Complex Float) where
+    norm2      = emptyVal (toScalarQ Norm2)
+    absSum     = emptyVal (toScalarQ AbsSum)
+    norm1      = emptyVal (sumElements . fst . fromComplex . vectorMapQ Abs)
+    normInf    = emptyVal (maxElement . fst . fromComplex . vectorMapQ Abs)
+    multiply   = emptyMul multiplyQ
+
+instance Product (Complex Double) where
+    norm2      = emptyVal (toScalarC Norm2)
+    absSum     = emptyVal (toScalarC AbsSum)
+    norm1      = emptyVal (sumElements . fst . fromComplex . vectorMapC Abs)
+    normInf    = emptyVal (maxElement . fst . fromComplex . vectorMapC Abs)
+    multiply   = emptyMul multiplyC
+
+emptyMul m a b
+    | x1 == 0 && x2 == 0 || r == 0 || c == 0 = konst' 0 (r,c)
+    | otherwise = m a b
+  where
+    r  = rows a
+    x1 = cols a
+    x2 = rows b
+    c  = cols b
+
+emptyVal f v =
+    if dim v > 0
+        then f v
+        else 0
+
+-- FIXME remove unused C wrappers
+-- | unconjugated dot product
+udot :: Product e => Vector e -> Vector e -> e
+udot u v
+    | dim u == dim v = val (asRow u `multiply` asColumn v)
+    | otherwise = error $ "different dimensions "++show (dim u)++" and "++show (dim v)++" in dot product"
+  where
+    val m | dim u > 0 = m@@>(0,0)
+          | otherwise = 0
+
+----------------------------------------------------------
+
+-- synonym for matrix product
+mXm :: Product t => Matrix t -> Matrix t -> Matrix t
+mXm = multiply
+
+-- matrix - vector product
+mXv :: Product t => Matrix t -> Vector t -> Vector t
+mXv m v = flatten $ m `mXm` (asColumn v)
+
+-- vector - matrix product
+vXm :: Product t => Vector t -> Matrix t -> Vector t
+vXm v m = flatten $ (asRow v) `mXm` m
+
+{- | Outer product of two vectors.
+
+>>> fromList [1,2,3] `outer` fromList [5,2,3]
+(3><3)
+ [  5.0, 2.0, 3.0
+ , 10.0, 4.0, 6.0
+ , 15.0, 6.0, 9.0 ]
+
+-}
+outer :: (Product t) => Vector t -> Vector t -> Matrix t
+outer u v = asColumn u `multiply` asRow v
+
+{- | Kronecker product of two matrices.
+
+@m1=(2><3)
+ [ 1.0,  2.0, 0.0
+ , 0.0, -1.0, 3.0 ]
+m2=(4><3)
+ [  1.0,  2.0,  3.0
+ ,  4.0,  5.0,  6.0
+ ,  7.0,  8.0,  9.0
+ , 10.0, 11.0, 12.0 ]@
+
+>>> kronecker m1 m2
+(8><9)
+ [  1.0,  2.0,  3.0,   2.0,   4.0,   6.0,  0.0,  0.0,  0.0
+ ,  4.0,  5.0,  6.0,   8.0,  10.0,  12.0,  0.0,  0.0,  0.0
+ ,  7.0,  8.0,  9.0,  14.0,  16.0,  18.0,  0.0,  0.0,  0.0
+ , 10.0, 11.0, 12.0,  20.0,  22.0,  24.0,  0.0,  0.0,  0.0
+ ,  0.0,  0.0,  0.0,  -1.0,  -2.0,  -3.0,  3.0,  6.0,  9.0
+ ,  0.0,  0.0,  0.0,  -4.0,  -5.0,  -6.0, 12.0, 15.0, 18.0
+ ,  0.0,  0.0,  0.0,  -7.0,  -8.0,  -9.0, 21.0, 24.0, 27.0
+ ,  0.0,  0.0,  0.0, -10.0, -11.0, -12.0, 30.0, 33.0, 36.0 ]
+
+-}
+kronecker :: (Product t) => Matrix t -> Matrix t -> Matrix t
+kronecker a b = fromBlocks
+              . splitEvery (cols a)
+              . map (reshape (cols b))
+              . toRows
+              $ flatten a `outer` flatten b
+
+-------------------------------------------------------------------
+
+
+class Convert t where
+    real    :: Container c t => c (RealOf t) -> c t
+    complex :: Container c t => c t -> c (ComplexOf t)
+    single  :: Container c t => c t -> c (SingleOf t)
+    double  :: Container c t => c t -> c (DoubleOf t)
+    toComplex   :: (Container c t, RealElement t) => (c t, c t) -> c (Complex t)
+    fromComplex :: (Container c t, RealElement t) => c (Complex t) -> (c t, c t)
+
+
+instance Convert Double where
+    real = id
+    complex = comp'
+    single = single'
+    double = id
+    toComplex = toComplex'
+    fromComplex = fromComplex'
+
+instance Convert Float where
+    real = id
+    complex = comp'
+    single = id
+    double = double'
+    toComplex = toComplex'
+    fromComplex = fromComplex'
+
+instance Convert (Complex Double) where
+    real = comp'
+    complex = id
+    single = single'
+    double = id
+    toComplex = toComplex'
+    fromComplex = fromComplex'
+
+instance Convert (Complex Float) where
+    real = comp'
+    complex = id
+    single = id
+    double = double'
+    toComplex = toComplex'
+    fromComplex = fromComplex'
+
+-------------------------------------------------------------------
+
+type family RealOf x
+
+type instance RealOf Double = Double
+type instance RealOf (Complex Double) = Double
+
+type instance RealOf Float = Float
+type instance RealOf (Complex Float) = Float
+
+type family ComplexOf x
+
+type instance ComplexOf Double = Complex Double
+type instance ComplexOf (Complex Double) = Complex Double
+
+type instance ComplexOf Float = Complex Float
+type instance ComplexOf (Complex Float) = Complex Float
+
+type family SingleOf x
+
+type instance SingleOf Double = Float
+type instance SingleOf Float  = Float
+
+type instance SingleOf (Complex a) = Complex (SingleOf a)
+
+type family DoubleOf x
+
+type instance DoubleOf Double = Double
+type instance DoubleOf Float  = Double
+
+type instance DoubleOf (Complex a) = Complex (DoubleOf a)
+
+type family ElementOf c
+
+type instance ElementOf (Vector a) = a
+type instance ElementOf (Matrix a) = a
+
+------------------------------------------------------------
+
+buildM (rc,cc) f = fromLists [ [f r c | c <- cs] | r <- rs ]
+    where rs = map fromIntegral [0 .. (rc-1)]
+          cs = map fromIntegral [0 .. (cc-1)]
+
+buildV n f = fromList [f k | k <- ks]
+    where ks = map fromIntegral [0 .. (n-1)]
+
+--------------------------------------------------------
+-- | conjugate transpose
+ctrans :: (Container Vector e, Element e) => Matrix e -> Matrix e
+ctrans = liftMatrix conj . trans
+
+-- | Creates a square matrix with a given diagonal.
+diag :: (Num a, Element a) => Vector a -> Matrix a
+diag v = diagRect 0 v n n where n = dim v
+
+-- | creates the identity matrix of given dimension
+ident :: (Num a, Element a) => Int -> Matrix a
+ident n = diag (constant 1 n)
+
+--------------------------------------------------------
+
+findV p x = foldVectorWithIndex g [] x where
+    g k z l = if p z then k:l else l
+
+findM p x = map ((`divMod` cols x)) $ findV p (flatten x)
+
+assocV n z xs = ST.runSTVector $ do
+        v <- ST.newVector z n
+        mapM_ (\(k,x) -> ST.writeVector v k x) xs
+        return v
+
+assocM (r,c) z xs = ST.runSTMatrix $ do
+        m <- ST.newMatrix z r c
+        mapM_ (\((i,j),x) -> ST.writeMatrix m i j x) xs
+        return m
+
+accumV v0 f xs = ST.runSTVector $ do
+        v <- ST.thawVector v0
+        mapM_ (\(k,x) -> ST.modifyVector v k (f x)) xs
+        return v
+
+accumM m0 f xs = ST.runSTMatrix $ do
+        m <- ST.thawMatrix m0
+        mapM_ (\((i,j),x) -> ST.modifyMatrix m i j (f x)) xs
+        return m
+
+----------------------------------------------------------------------
+
+condM a b l e t = matrixFromVector RowMajor (rows a'') (cols a'') $ cond a' b' l' e' t'
+  where
+    args@(a'':_) = conformMs [a,b,l,e,t]
+    [a', b', l', e', t'] = map flatten args
+
+condV f a b l e t = f a' b' l' e' t'
+  where
+    [a', b', l', e', t'] = conformVs [a,b,l,e,t]
+