```-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Packed.Matrix
-- Copyright   :  (c) Alberto Ruiz 2007
--
-- Maintainer  :  Alberto Ruiz <aruiz@um.es>
-- Stability   :  provisional
-- Portability :  portable
--
-- A Matrix representation suitable for numerical computations using LAPACK and GSL.
--
-----------------------------------------------------------------------------

module Data.Packed.Matrix (
Element,
Matrix,rows,cols,
(><),
trans,
reshape, flatten,
fromLists, toLists, buildMatrix,
(@@>),
asRow, asColumn,
fromRows, toRows, fromColumns, toColumns,
fromBlocks, repmat,
flipud, fliprl,
subMatrix, takeRows, dropRows, takeColumns, dropColumns,
extractRows,
ident, diag, diagRect, takeDiag,
liftMatrix, liftMatrix2,
format, dispf, disps,
) where

import Data.Packed.Internal
import qualified Data.Packed.ST as ST
import Data.Packed.Vector
import Data.List(transpose,intersperse)
import Data.Array
import Text.Printf(printf)

-- | creates a matrix from a vertical list of matrices
joinVert :: Element t => [Matrix t] -> Matrix t
joinVert ms = case common cols ms of
Nothing -> error "(impossible) joinVert on matrices with different number of columns"
Just c  -> reshape c \$ join (map flatten ms)

-- | creates a matrix from a horizontal list of matrices
joinHoriz :: Element t => [Matrix t] -> Matrix t
joinHoriz ms = trans. joinVert . map trans \$ ms

{- | Creates a matrix from blocks given as a list of lists of matrices.

Single row/column components are automatically expanded to match the
corresponding common row and column:

@\> let disp = putStr . dispf 2
\> let vector xs = fromList xs :: Vector Double
\> let diagl = diag . vector
\> let rowm = asRow . vector

\> disp \$ fromBlocks [[ident 5, 7, rowm[10,20]], [3, diagl[1,2,3], 0]]

8x10
1  0  0  0  0  7  7  7  10  20
0  1  0  0  0  7  7  7  10  20
0  0  1  0  0  7  7  7  10  20
0  0  0  1  0  7  7  7  10  20
0  0  0  0  1  7  7  7  10  20
3  3  3  3  3  1  0  0   0   0
3  3  3  3  3  0  2  0   0   0
3  3  3  3  3  0  0  3   0   0@
-}
fromBlocks :: Element t => [[Matrix t]] -> Matrix t

fromBlocksRaw mms = joinVert . map joinHoriz \$ mms

bc = case common length ms of
Just c -> c
Nothing -> error "fromBlocks requires rectangular [[Matrix]]"
rs = map (compatdim . map rows) ms
cs = map (compatdim . map cols) (transpose ms)
szs = sequence [rs,cs]
ms' = partit bc \$ zipWith g szs (concat ms)

g [Just nr,Just nc] m
| nr == r && nc == c = m
| r == 1 && c == 1 = reshape nc (constant x (nr*nc))
| r == 1 = fromRows (replicate nr (flatten m))
| otherwise = fromColumns (replicate nc (flatten m))
where
r = rows m
c = cols m
x = m@@>(0,0)
g _ _ = error "inconsistent dimensions in fromBlocks"

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

-- | Reverse rows
flipud :: Element t => Matrix t -> Matrix t
flipud m = fromRows . reverse . toRows \$ m

-- | Reverse columns
fliprl :: Element t => Matrix t -> Matrix t
fliprl m = fromColumns . reverse . toColumns \$ m

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

-- | Creates a square matrix with a given diagonal.
diag :: Element a => Vector a -> Matrix a
diag v = ST.runSTMatrix \$ do
let d = dim v
m <- ST.newMatrix 0 d d
mapM_ (\k -> ST.writeMatrix m k k (v@>k)) [0..d-1]
return m

{- | creates a rectangular diagonal matrix

@> diagRect (constant 5 3) 3 4 :: Matrix Double
(3><4)
[ 5.0, 0.0, 0.0, 0.0
, 0.0, 5.0, 0.0, 0.0
, 0.0, 0.0, 5.0, 0.0 ]@
-}
diagRect :: (Element t, Num t) => Vector t -> Int -> Int -> Matrix t
diagRect v r c
| dim v < min r c = error "diagRect called with dim v < min r c"
| otherwise = ST.runSTMatrix \$ do
m <- ST.newMatrix 0 r c
let d = min r c
mapM_ (\k -> ST.writeMatrix m k k (v@>k)) [0..d-1]
return m

-- | extracts the diagonal from a rectangular matrix
takeDiag :: (Element t) => Matrix t -> Vector t
takeDiag m = fromList [flatten m `at` (k*cols m+k) | k <- [0 .. min (rows m) (cols m) -1]]

-- | creates the identity matrix of given dimension
ident :: Element a => Int -> Matrix a
ident n = diag (constant 1 n)

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

{- | An easy way to create a matrix:

@\> (2><3)[1..6]
(2><3)
[ 1.0, 2.0, 3.0
, 4.0, 5.0, 6.0 ]@

This is the format produced by the instances of Show (Matrix a), which
can also be used for input.

The input list is explicitly truncated, so that it can
safely be used with lists that are too long (like infinite lists).

Example:

@\> (2>|<3)[1..]
(2><3)
[ 1.0, 2.0, 3.0
, 4.0, 5.0, 6.0 ]@

-}
(><) :: (Element a) => Int -> Int -> [a] -> Matrix a
r >< c = f where
f l | dim v == r*c = matrixFromVector RowMajor c v
| otherwise    = error \$ "inconsistent list size = "
++show (dim v) ++" in ("++show r++"><"++show c++")"
where v = fromList \$ take (r*c) l

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

-- | Creates a matrix with the first n rows of another matrix
takeRows :: Element t => Int -> Matrix t -> Matrix t
takeRows n mt = subMatrix (0,0) (n, cols mt) mt
-- | Creates a copy of a matrix without the first n rows
dropRows :: Element t => Int -> Matrix t -> Matrix t
dropRows n mt = subMatrix (n,0) (rows mt - n, cols mt) mt
-- |Creates a matrix with the first n columns of another matrix
takeColumns :: Element t => Int -> Matrix t -> Matrix t
takeColumns n mt = subMatrix (0,0) (rows mt, n) mt
-- | Creates a copy of a matrix without the first n columns
dropColumns :: Element t => Int -> Matrix t -> Matrix t
dropColumns n mt = subMatrix (0,n) (rows mt, cols mt - n) mt

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

{- | Creates a 'Matrix' from a list of lists (considered as rows).

@\> fromLists [[1,2],[3,4],[5,6]]
(3><2)
[ 1.0, 2.0
, 3.0, 4.0
, 5.0, 6.0 ]@
-}
fromLists :: Element t => [[t]] -> Matrix t
fromLists = fromRows . map fromList

-- | creates a 1-row matrix from a vector
asRow :: Element a => Vector a -> Matrix a
asRow v = reshape (dim v) v

-- | creates a 1-column matrix from a vector
asColumn :: Element a => Vector a -> Matrix a
asColumn v = reshape 1 v

{- | creates a Matrix of the specified size using the supplied function to
to map the row/column position to the value at that row/column position.

@> buildMatrix 3 4 (\ (r,c) -> fromIntegral r * fromIntegral c)
(3><4)
[ 0.0, 0.0, 0.0, 0.0, 0.0
, 0.0, 1.0, 2.0, 3.0, 4.0
, 0.0, 2.0, 4.0, 6.0, 8.0]@
-}
buildMatrix :: Element a => Int -> Int -> ((Int, Int) -> a) -> Matrix a
buildMatrix rc cc f =
fromLists \$ map (\x -> map f x)
\$ map (\ ri -> map (\ ci -> (ri, ci)) [0 .. (cc - 1)]) [0 .. (rc - 1)]

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

fromArray2D :: (Element e) => Array (Int, Int) e -> Matrix e
fromArray2D m = (r><c) (elems m)
where ((r0,c0),(r1,c1)) = bounds m
r = r1-r0+1
c = c1-c0+1

------------------------------------------------------
{-
-- shows a Double with n digits after the decimal point
shf :: (RealFloat a) => Int -> a -> String
shf dec n | abs n < 1e-10 = "0."
| abs (n - (fromIntegral.round \$ n)) < 1e-10 = show (round n) ++"."
| otherwise = showGFloat (Just dec) n ""
-- shows a Complex Double as a pair, with n digits after the decimal point
shfc n z@ (a:+b)
| magnitude z <1e-10 = "0."
| abs b < 1e-10 = shf n a
| abs a < 1e-10 = shf n b ++"i"
| b > 0         = shf n a ++"+"++shf n b ++"i"
| otherwise     = shf n a ++shf n b ++"i"

-}

dsp' :: String -> [[String]] -> String
dsp' sep as = unlines . map unwords' \$ transpose mtp where
mt = transpose as
longs = map (maximum . map length) mt
mtp = zipWith (\a b -> map (pad a) b) longs mt
pad n str = replicate (n - length str) ' ' ++ str
unwords' = concat . intersperse sep

{- | Creates a string from a matrix given a separator and a function to show each entry. Using
this function the user can easily define any desired display function:

@import Text.Printf(printf)@

@disp = putStr . format \"  \" (printf \"%.2f\")@

-}
format :: (Element t) => String -> (t -> String) -> Matrix t -> String
format sep f m = dsp' sep . map (map f) . toLists \$ m

{-
disp m f = putStrLn \$ "matrix ("++show (rows m) ++"x"++ show (cols m) ++")\n"++format " | " f m

dispR :: Int -> Matrix Double -> IO ()
dispR d m = disp m (shf d)

dispC :: Int -> Matrix (Complex Double) -> IO ()
dispC d m = disp m (shfc d)
-}

-------------------------------------------------------------------
-- display utilities

{- | Show a matrix with \"autoscaling\" and a given number of decimal places.

@disp = putStr . disps 2

\> disp \$ 120 * (3><4) [1..]
3x4  E3
0.12  0.24  0.36  0.48
0.60  0.72  0.84  0.96
1.08  1.20  1.32  1.44
@
-}
disps :: Int -> Matrix Double -> String
disps d x = sdims x ++ "  " ++ formatScaled d x

{- | Show a matrix with a given number of decimal places.

@disp = putStr . dispf 3

\> disp (1/3 + ident 4)
4x4
1.333  0.333  0.333  0.333
0.333  1.333  0.333  0.333
0.333  0.333  1.333  0.333
0.333  0.333  0.333  1.333
@
-}
dispf :: Int -> Matrix Double -> String
dispf d x = sdims x ++ "\n" ++ formatFixed (if isInt x then 0 else d) x

sdims x = show (rows x) ++ "x" ++ show (cols x)

formatFixed d x = format "  " (printf ("%."++show d++"f")) \$ x

isInt = all lookslikeInt . toList . flatten where
lookslikeInt x = show (round x :: Int) ++".0" == shx || "-0.0" == shx
where shx = show x

formatScaled dec t = "E"++show o++"\n" ++ ss
where ss = format " " (printf fmt. g) t
g x | o >= 0    = x/10^(o::Int)
| otherwise = x*10^(-o)
o = floor \$ maximum \$ map (logBase 10 . abs) \$ toList \$ flatten t
fmt = '%':show (dec+3) ++ '.':show dec ++"f"

{- | Show a vector using a function for showing matrices.

@disp = putStr . vecdisp (dispf 2)

\> disp (linspace 10 (0,1))
10 |> 0.00  0.11  0.22  0.33  0.44  0.56  0.67  0.78  0.89  1.00
@
-}
vecdisp :: (Element t) => (Matrix t -> String) -> Vector t -> String
vecdisp f v
= ((show (dim v) ++ " |> ") ++) . (++"\n")
. unwords . lines .  tail . dropWhile (not . (`elem` " \n"))
. f . trans . reshape 1
\$ v

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

-- | reads a matrix from a string containing a table of numbers.
readMatrix :: String -> Matrix Double
readMatrix = fromLists . map (map read). map words . filter (not.null) . lines

{- |  obtains the number of rows and columns in an ASCII data file
(provisionally using unix's wc).
-}
fileDimensions :: FilePath -> IO (Int,Int)
fileDimensions fname = do
wcres <- readProcess "wc" ["-w",fname] ""
c   = length . head . dropWhile null . map words . lines \$ contents
if tot > 0
then return (tot `div` c, c)
else return (0,0)

-- | Loads a matrix from an ASCII file formatted as a 2D table.
loadMatrix :: FilePath -> IO (Matrix Double)
loadMatrix file = fromFile file =<< fileDimensions file

-- | Loads a matrix from an ASCII file (the number of rows and columns must be known in advance).
fromFile :: FilePath -> (Int,Int) -> IO (Matrix Double)
fromFile filename (r,c) = reshape c `fmap` fscanfVector filename (r*c)

-- | rearranges the rows of a matrix according to the order given in a list of integers.
extractRows :: Element t => [Int] -> Matrix t -> Matrix t
extractRows l m = fromRows \$ extract (toRows \$ m) l
where extract l' is = [l'!!i |i<-is]

{- | creates matrix by repetition of a matrix a given number of rows and columns

@> repmat (ident 2) 2 3 :: Matrix Double
(4><6)
[ 1.0, 0.0, 1.0, 0.0, 1.0, 0.0
, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0
, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0
, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0 ]@

-}
repmat :: (Element t) => Matrix t -> Int -> Int -> Matrix t
repmat m r c = fromBlocks \$ partit c \$ replicate (r*c) m
```