```{-# LANGUAGE CPP #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE UndecidableInstances #-}

{-# OPTIONS_GHC -fno-warn-simplifiable-class-constraints #-}

-----------------------------------------------------------------------------
-- |
-- 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.Vector
import Internal.Matrix
import Internal.Element
import Internal.Numeric
import Internal.Algorithms(Field,linearSolveSVD,Herm,mTm)
#if MIN_VERSION_base(4,11,0)
import Prelude hiding ((<>))
#endif
------------------------------------------------------------------

{- | Creates a real vector containing a range of values:

>>> linspace 5 (-3,7::Double)
[-3.0,-0.5,2.0,4.5,7.0]
it :: Vector Double

>>> linspace 5 (8,3:+2) :: Vector (Complex Double)
[8.0 :+ 0.0,6.75 :+ 0.5,5.5 :+ 1.0,4.25 :+ 1.5,3.0 :+ 2.0]
it :: Vector (Complex Double)

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)

--------------------------------------------------------------------------------

infixr 8 <.>
{- | An 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

-}

(<.>) :: Numeric t => Vector t -> Vector t -> t
(<.>) = dot

{- | dense matrix-vector product

>>> 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
[140.0,320.0]
it :: Vector Numeric.LinearAlgebra.Data.R

-}
infixr 8 #>
(#>) :: 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
[3.0799999999999996,5.159999999999999]
it :: Vector Numeric.LinearAlgebra.Data.R

>>> a #> x
[13.399999999999999,26.799999999999997,0.9999999999999991]
it :: Vector Numeric.LinearAlgebra.Data.R

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 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
-- [0.0,1.0,4.0,9.0,16.0]
-- it :: Vector Double
--
-- 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]) (trustSym \$ diagl [2,3])
([3.9933155655086696,5.061409102770331],Herm (2><2)
[    1.9963242906624408, -4.227815571404954e-2
, -4.227815571404954e-2,    3.2003833097832857 ])
it :: (Vector Double, Herm Double)
-}
meanCov :: Matrix Double -> (Vector Double, Herm 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))) (mTm xc)

--------------------------------------------------------------------------------

sortVector :: (Ord t, Element t) => Vector t -> Vector t
sortVector = sortV

{- |

>>> m <- randn 4 10
>>> disp 2 m
4x10
-0.31   0.41   0.43  -0.19  -0.17  -0.23  -0.17  -1.04  -0.07  -1.24
0.26   0.19   0.14   0.83  -1.54  -0.09   0.37  -0.63   0.71  -0.50
-0.11  -0.10  -1.29  -1.40  -1.04  -0.89  -0.68   0.35  -1.46   1.86
1.04  -0.29   0.19  -0.75  -2.20  -0.01   1.06   0.11  -2.09  -1.58

>>> disp 2 \$ m ?? (All, Pos \$ sortIndex (m!1))
4x10
-0.17  -1.04  -1.24  -0.23   0.43   0.41  -0.31  -0.17  -0.07  -0.19
-1.54  -0.63  -0.50  -0.09   0.14   0.19   0.26   0.37   0.71   0.83
-1.04   0.35   1.86  -0.89  -1.29  -0.10  -0.11  -0.68  -1.46  -1.40
-2.20   0.11  -1.58  -0.01   0.19  -0.29   1.04   1.06  -2.09  -0.75

-}
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'

{- | Extract elements from positions given in matrices of rows and columns.

>>> r
(3><3)
[ 1, 1, 1
, 1, 2, 2
, 1, 2, 3 ]
>>> c
(3><3)
[ 0, 1, 5
, 2, 2, 1
, 4, 4, 1 ]
>>> m
(4><6)
[  0,  1,  2,  3,  4,  5
,  6,  7,  8,  9, 10, 11
, 12, 13, 14, 15, 16, 17
, 18, 19, 20, 21, 22, 23 ]

>>> remap r c m
(3><3)
[  6,  7, 11
,  8, 14, 13
, 10, 16, 19 ]

The indexes are autoconformable.

>>> c'
(3><1)
[ 1
, 2
, 4 ]
>>> remap r c' m
(3><3)
[  7,  7,  7
,  8, 14, 14
, 10, 16, 22 ]

-}
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 remap"
where
[i',j'] = conformMs [i,j]
```