{-# LANGUAGE TypeFamilies #-} {-# LANGUAGE FlexibleContexts #-} {-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE CPP #-} ----------------------------------------------------------------------------- -- | -- Module : Data.Packed.Matrix -- Copyright : (c) Alberto Ruiz 2007-10 -- License : GPL -- -- Maintainer : Alberto Ruiz -- Stability : provisional -- -- A Matrix representation suitable for numerical computations using LAPACK and GSL. -- -- This module provides basic functions for manipulation of structure. ----------------------------------------------------------------------------- module Data.Packed.Matrix ( Matrix, Element, rows,cols, (><), trans, reshape, flatten, fromLists, toLists, buildMatrix, (@@>), asRow, asColumn, fromRows, toRows, fromColumns, toColumns, fromBlocks, toBlocks, toBlocksEvery, repmat, flipud, fliprl, subMatrix, takeRows, dropRows, takeColumns, dropColumns, extractRows, diagRect, takeDiag, mapMatrix, mapMatrixWithIndex, mapMatrixWithIndexM, mapMatrixWithIndexM_, liftMatrix, liftMatrix2, liftMatrix2Auto,fromArray2D ) where import Data.Packed.Internal import qualified Data.Packed.ST as ST import Data.Array import Data.List(transpose,intersperse) import Foreign.Storable(Storable) import Control.Arrow((***)) ------------------------------------------------------------------- #ifdef BINARY import Data.Binary import Control.Monad(replicateM) instance (Binary a, Element a, Storable a) => Binary (Matrix a) where put m = do let r = rows m let c = cols m put r put c mapM_ (\i -> mapM_ (\j -> put $ m @@> (i,j)) [0..(c-1)]) [0..(r-1)] get = do r <- get c <- get xs <- replicateM r $ replicateM c get return $ fromLists xs #endif ------------------------------------------------------------------- instance (Show a, Element a) => (Show (Matrix a)) where show m = (sizes++) . dsp . map (map show) . toLists $ m where sizes = "("++show (rows m)++"><"++show (cols m)++")\n" dsp as = (++" ]") . (" ["++) . init . drop 2 . unlines . map (" , "++) . 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 ", " ------------------------------------------------------------------ instance (Element a, Read a) => Read (Matrix a) where readsPrec _ s = [((rs>' $ dims breakAt c l = (a++[c],tail b) where (a,b) = break (==c) l ------------------------------------------------------------------ -- | 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 fromBlocks = fromBlocksRaw . adaptBlocks fromBlocksRaw mms = joinVert . map joinHoriz $ mms adaptBlocks ms = ms' where 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' = splitEvery bc $ zipWith g szs (concat ms) g [Just nr,Just nc] m | nr == r && nc == c = m | r == 1 && c == 1 = reshape nc (constantD 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 rectangular diagonal matrix: @> diagRect 7 (fromList [10,20,30]) 4 5 :: Matrix Double (4><5) [ 10.0, 7.0, 7.0, 7.0, 7.0 , 7.0, 20.0, 7.0, 7.0, 7.0 , 7.0, 7.0, 30.0, 7.0, 7.0 , 7.0, 7.0, 7.0, 7.0, 7.0 ]@ -} diagRect :: (Storable t) => t -> Vector t -> Int -> Int -> Matrix t diagRect z v r c = ST.runSTMatrix $ do m <- ST.newMatrix z r c let d = min r c `min` (dim v) 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]] ------------------------------------------------------------ {- | 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 ]@ -} (><) :: (Storable 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 :: Storable a => Vector a -> Matrix a asRow v = reshape (dim v) v -- | creates a 1-column matrix from a vector asColumn :: Storable 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]@ Hilbert matrix of order N: @hilb n = buildMatrix n n (\(i,j)->1/(fromIntegral i + fromIntegral j +1))@ -} buildMatrix :: Element a => Int -> Int -> ((Int, Int) -> a) -> Matrix a buildMatrix rc cc f = fromLists $ map (map f) $ map (\ ri -> map (\ ci -> (ri, ci)) [0 .. (cc - 1)]) [0 .. (rc - 1)] ----------------------------------------------------- fromArray2D :: (Storable e) => Array (Int, Int) e -> Matrix e fromArray2D m = (r> [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 $ splitEvery c $ replicate (r*c) m -- | A version of 'liftMatrix2' which automatically adapt matrices with a single row or column to match the dimensions of the other matrix. liftMatrix2Auto :: (Element t, Element a, Element b) => (Vector a -> Vector b -> Vector t) -> Matrix a -> Matrix b -> Matrix t liftMatrix2Auto f m1 m2 | compat' m1 m2 = lM f m1 m2 | ok = lM f m1' m2' | otherwise = error $ "nonconformable matrices in liftMatrix2Auto: " ++ shSize m1 ++ ", " ++ shSize m2 where (r1,c1) = size m1 (r2,c2) = size m2 r = max r1 r2 c = max c1 c2 r0 = min r1 r2 c0 = min c1 c2 ok = r0 == 1 || r1 == r2 && c0 == 1 || c1 == c2 m1' = conformMTo (r,c) m1 m2' = conformMTo (r,c) m2 lM f m1 m2 = reshape (max (cols m1) (cols m2)) (f (flatten m1) (flatten m2)) compat' :: Matrix a -> Matrix b -> Bool compat' m1 m2 = s1 == (1,1) || s2 == (1,1) || s1 == s2 where s1 = size m1 s2 = size m2 ------------------------------------------------------------ toBlockRows [r] m | r == rows m = [m] toBlockRows rs m = map (reshape (cols m)) (takesV szs (flatten m)) where szs = map (* cols m) rs toBlockCols [c] m | c == cols m = [m] toBlockCols cs m = map trans . toBlockRows cs . trans $ m -- | Partition a matrix into blocks with the given numbers of rows and columns. -- The remaining rows and columns are discarded. toBlocks :: (Element t) => [Int] -> [Int] -> Matrix t -> [[Matrix t]] toBlocks rs cs m = map (toBlockCols cs) . toBlockRows rs $ m -- | Fully partition a matrix into blocks of the same size. If the dimensions are not -- a multiple of the given size the last blocks will be smaller. toBlocksEvery :: (Element t) => Int -> Int -> Matrix t -> [[Matrix t]] toBlocksEvery r c m = toBlocks rs cs m where (qr,rr) = rows m `divMod` r (qc,rc) = cols m `divMod` c rs = replicate qr r ++ if rr > 0 then [rr] else [] cs = replicate qc c ++ if rc > 0 then [rc] else [] ------------------------------------------------------------------- mk c g = \k v -> g ((fromIntegral *** fromIntegral) (divMod k c)) v {- | @ghci> mapMatrixWithIndexM_ (\\(i,j) v -> printf \"m[%.0f,%.0f] = %.f\\n\" i j v :: IO()) ((2><3)[1 :: Double ..]) m[0,0] = 1 m[0,1] = 2 m[0,2] = 3 m[1,0] = 4 m[1,1] = 5 m[1,2] = 6@ -} mapMatrixWithIndexM_ :: (Element a, Num a, Functor f, Monad f) => ((a, a) -> a -> f ()) -> Matrix a -> f () mapMatrixWithIndexM_ g m = mapVectorWithIndexM_ (mk c g) . flatten $ m where c = cols m {- | @ghci> mapMatrixWithIndexM (\\(i,j) v -> Just $ 100*v + 10*i + j) (ident 3:: Matrix Double) Just (3><3) [ 100.0, 1.0, 2.0 , 10.0, 111.0, 12.0 , 20.0, 21.0, 122.0 ]@ -} mapMatrixWithIndexM :: (Foreign.Storable.Storable t, Element a, Num a, Functor f, Monad f) => ((a, a) -> a -> f t) -> Matrix a -> f (Matrix t) mapMatrixWithIndexM g m = fmap (reshape c) . mapVectorWithIndexM (mk c g) . flatten $ m where c = cols m {- | @ghci> mapMatrixWithIndex (\\(i,j) v -> 100*v + 10*i + j) (ident 3:: Matrix Double) (3><3) [ 100.0, 1.0, 2.0 , 10.0, 111.0, 12.0 , 20.0, 21.0, 122.0 ]@ -} mapMatrixWithIndex :: (Foreign.Storable.Storable t, Element a, Num a) => ((a, a) -> a -> t) -> Matrix a -> Matrix t mapMatrixWithIndex g = head . mapMatrixWithIndexM (\a b -> [g a b]) mapMatrix :: (Storable a, Storable b) => (a -> b) -> Matrix a -> Matrix b mapMatrix f = liftMatrix (mapVector f)