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|
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
-----------------------------------------------------------------------------
{- |
Module : Numeric.LinearAlgebra.Util
Copyright : (c) Alberto Ruiz 2013
License : BSD3
Maintainer : Alberto Ruiz
Stability : provisional
-}
-----------------------------------------------------------------------------
{-# OPTIONS_HADDOCK hide #-}
module Numeric.LinearAlgebra.Util(
-- * Convenience functions
vect, mat,
disp,
zeros, ones,
diagl,
row,
col,
(&), (¦), (——), (#),
(?), (¿),
Indexable(..), size,
rand, randn,
cross,
norm,
ℕ,ℤ,ℝ,ℂ,ℝn,ℂn,𝑖,i_C, --ℍ
norm_1, norm_2, norm_0, norm_Inf, norm_Frob, norm_nuclear,
mnorm_1, mnorm_2, mnorm_0, mnorm_Inf,
unitary,
mt,
pairwiseD2,
rowOuters,
null1,
null1sym,
-- * Convolution
-- ** 1D
corr, conv, corrMin,
-- ** 2D
corr2, conv2, separable,
-- * Tools for the Kronecker product
--
-- | (see A. Fusiello, A matter of notation: Several uses of the Kronecker product in
-- 3d computer vision, Pattern Recognition Letters 28 (15) (2007) 2127-2132)
--
-- | @`vec` (a \<> x \<> b) == ('trans' b ` 'kronecker' ` a) \<> 'vec' x@
vec,
vech,
dup,
vtrans
) where
import Data.Packed.Numeric
import Numeric.LinearAlgebra.Algorithms hiding (i)
import Numeric.Matrix()
import Numeric.Vector()
import Numeric.LinearAlgebra.Random
import Numeric.LinearAlgebra.Util.Convolution
type ℝ = Double
type ℕ = Int
type ℤ = Int
type ℂ = Complex Double
type ℝn = Vector ℝ
type ℂn = Vector ℂ
--newtype ℍ m = H m
i_C, 𝑖 :: ℂ
𝑖 = 0:+1
i_C = 𝑖
{- | create a real vector
>>> vect [1..5]
fromList [1.0,2.0,3.0,4.0,5.0]
-}
vect :: [ℝ] -> ℝn
vect = fromList
{- | create a real matrix
>>> mat 5 [1..15]
(3><5)
[ 1.0, 2.0, 3.0, 4.0, 5.0
, 6.0, 7.0, 8.0, 9.0, 10.0
, 11.0, 12.0, 13.0, 14.0, 15.0 ]
-}
mat
:: Int -- ^ columns
-> [ℝ] -- ^ elements
-> Matrix ℝ
mat c = reshape c . fromList
{- | print a real matrix with given number of digits after the decimal point
>>> disp 5 $ ident 2 / 3
2x2
0.33333 0.00000
0.00000 0.33333
-}
disp :: Int -> Matrix Double -> IO ()
disp n = putStrLn . dispf n
{- | create a real diagonal matrix from a list
>>> diagl [1,2,3]
(3><3)
[ 1.0, 0.0, 0.0
, 0.0, 2.0, 0.0
, 0.0, 0.0, 3.0 ]
-}
diagl :: [Double] -> Matrix Double
diagl = diag . fromList
-- | a real matrix of zeros
zeros :: Int -- ^ rows
-> Int -- ^ columns
-> Matrix Double
zeros r c = konst 0 (r,c)
-- | a real matrix of ones
ones :: Int -- ^ rows
-> Int -- ^ columns
-> Matrix Double
ones r c = konst 1 (r,c)
-- | concatenation of real vectors
infixl 3 &
(&) :: Vector Double -> Vector Double -> Vector Double
a & b = vjoin [a,b]
{- | horizontal concatenation of real matrices
(unicode 0x00a6, broken bar)
>>> ident 3 ¦ konst 7 (3,4)
(3><7)
[ 1.0, 0.0, 0.0, 7.0, 7.0, 7.0, 7.0
, 0.0, 1.0, 0.0, 7.0, 7.0, 7.0, 7.0
, 0.0, 0.0, 1.0, 7.0, 7.0, 7.0, 7.0 ]
-}
infixl 3 ¦
(¦) :: Matrix Double -> Matrix Double -> Matrix Double
a ¦ b = fromBlocks [[a,b]]
-- | vertical concatenation of real matrices
--
-- (unicode 0x2014, em dash)
(——) :: Matrix Double -> Matrix Double -> Matrix Double
infixl 2 ——
a —— b = fromBlocks [[a],[b]]
(#) :: Matrix Double -> Matrix Double -> Matrix Double
infixl 2 #
a # b = fromBlocks [[a],[b]]
-- | create a single row real matrix from a list
row :: [Double] -> Matrix Double
row = asRow . fromList
-- | create a single column real matrix from a list
col :: [Double] -> Matrix Double
col = asColumn . fromList
{- | extract rows
>>> (20><4) [1..] ? [2,1,1]
(3><4)
[ 9.0, 10.0, 11.0, 12.0
, 5.0, 6.0, 7.0, 8.0
, 5.0, 6.0, 7.0, 8.0 ]
-}
infixl 9 ?
(?) :: Element t => Matrix t -> [Int] -> Matrix t
(?) = flip extractRows
{- | extract columns
(unicode 0x00bf, inverted question mark, Alt-Gr ?)
>>> (3><4) [1..] ¿ [3,0]
(3><2)
[ 4.0, 1.0
, 8.0, 5.0
, 12.0, 9.0 ]
-}
infixl 9 ¿
(¿) :: Element t => Matrix t -> [Int] -> Matrix t
(¿)= flip extractColumns
cross :: Vector Double -> Vector Double -> Vector Double
-- ^ cross product (for three-element real vectors)
cross x y | dim x == 3 && dim y == 3 = fromList [z1,z2,z3]
| otherwise = error $ "cross ("++show x++") ("++show y++")"
where
[x1,x2,x3] = toList x
[y1,y2,y3] = toList y
z1 = x2*y3-x3*y2
z2 = x3*y1-x1*y3
z3 = x1*y2-x2*y1
norm :: Vector Double -> Double
-- ^ 2-norm of real vector
norm = pnorm PNorm2
norm_2 :: Normed Vector t => Vector t -> RealOf t
norm_2 = pnorm PNorm2
norm_1 :: Normed Vector t => Vector t -> RealOf t
norm_1 = pnorm PNorm1
norm_Inf :: Normed Vector t => Vector t -> RealOf t
norm_Inf = pnorm Infinity
norm_0 :: Vector ℝ -> ℝ
norm_0 v = sumElements (step (abs v - scalar (eps*mx)))
where
mx = norm_Inf v
norm_Frob :: Normed Matrix t => Matrix t -> RealOf t
norm_Frob = pnorm Frobenius
norm_nuclear :: Field t => Matrix t -> ℝ
norm_nuclear = sumElements . singularValues
mnorm_2 :: Normed Matrix t => Matrix t -> RealOf t
mnorm_2 = pnorm PNorm2
mnorm_1 :: Normed Matrix t => Matrix t -> RealOf t
mnorm_1 = pnorm PNorm1
mnorm_Inf :: Normed Matrix t => Matrix t -> RealOf t
mnorm_Inf = pnorm Infinity
mnorm_0 :: Matrix ℝ -> ℝ
mnorm_0 = norm_0 . flatten
-- | Obtains a vector in the same direction with 2-norm=1
unitary :: Vector Double -> Vector Double
unitary v = v / scalar (norm v)
-- | trans . inv
mt :: Matrix Double -> Matrix Double
mt = trans . inv
--------------------------------------------------------------------------------
{- |
>>> size $ fromList[1..10::Double]
10
>>> size $ (2><5)[1..10::Double]
(2,5)
-}
size :: Container c t => c t -> IndexOf c
size = size'
{- |
>>> vect [1..10] ! 3
4.0
>>> mat 5 [1..15] ! 1
fromList [6.0,7.0,8.0,9.0,10.0]
>>> mat 5 [1..15] ! 1 ! 3
9.0
-}
class Indexable c t | c -> t , t -> c
where
infixl 9 !
(!) :: c -> Int -> t
instance Indexable (Vector Double) Double
where
(!) = (@>)
instance Indexable (Vector Float) Float
where
(!) = (@>)
instance Indexable (Vector (Complex Double)) (Complex Double)
where
(!) = (@>)
instance Indexable (Vector (Complex Float)) (Complex Float)
where
(!) = (@>)
instance Element t => Indexable (Matrix t) (Vector t)
where
m!j = subVector (j*c) c (flatten m)
where
c = cols m
--------------------------------------------------------------------------------
-- | Matrix of pairwise squared distances of row vectors
-- (using the matrix product trick in blog.smola.org)
pairwiseD2 :: Matrix Double -> Matrix Double -> Matrix Double
pairwiseD2 x y | ok = x2 `outer` oy + ox `outer` y2 - 2* x <> trans y
| otherwise = error $ "pairwiseD2 with different number of columns: "
++ show (size x) ++ ", " ++ show (size y)
where
ox = one (rows x)
oy = one (rows y)
oc = one (cols x)
one k = konst 1 k
x2 = x * x <> oc
y2 = y * y <> oc
ok = cols x == cols y
--------------------------------------------------------------------------------
-- | outer products of rows
rowOuters :: Matrix Double -> Matrix Double -> Matrix Double
rowOuters a b = a' * b'
where
a' = kronecker a (ones 1 (cols b))
b' = kronecker (ones 1 (cols a)) b
--------------------------------------------------------------------------------
-- | solution of overconstrained homogeneous linear system
null1 :: Matrix Double -> Vector Double
null1 = last . toColumns . snd . rightSV
-- | solution of overconstrained homogeneous symmetric linear system
null1sym :: Matrix Double -> Vector Double
null1sym = last . toColumns . snd . eigSH'
--------------------------------------------------------------------------------
vec :: Element t => Matrix t -> Vector t
-- ^ stacking of columns
vec = flatten . trans
vech :: Element t => Matrix t -> Vector t
-- ^ half-vectorization (of the lower triangular part)
vech m = vjoin . zipWith f [0..] . toColumns $ m
where
f k v = subVector k (dim v - k) v
dup :: (Num t, Num (Vector t), Element t) => Int -> Matrix t
-- ^ duplication matrix (@'dup' k \<> 'vech' m == 'vec' m@, for symmetric m of 'dim' k)
dup k = trans $ fromRows $ map f es
where
rs = zip [0..] (toRows (ident (k^(2::Int))))
es = [(i,j) | j <- [0..k-1], i <- [0..k-1], i>=j ]
f (i,j) | i == j = g (k*j + i)
| otherwise = g (k*j + i) + g (k*i + j)
g j = v
where
Just v = lookup j rs
vtrans :: Element t => Int -> Matrix t -> Matrix t
-- ^ generalized \"vector\" transposition: @'vtrans' 1 == 'trans'@, and @'vtrans' ('rows' m) m == 'asColumn' ('vec' m)@
vtrans p m | r == 0 = fromBlocks . map (map asColumn . takesV (replicate q p)) . toColumns $ m
| otherwise = error $ "vtrans " ++ show p ++ " of matrix with " ++ show (rows m) ++ " rows"
where
(q,r) = divMod (rows m) p
|