summaryrefslogtreecommitdiff
path: root/packages/base/src/Numeric/LinearAlgebra/Util.hs
blob: 6bb9d150085eef65b4932662e608d65b304fb7e9 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE ViewPatterns #-}


-----------------------------------------------------------------------------
{- |
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
    vector, matrix,
    disp,
    formatSparse,
    approxInt,
    dispDots,
    dispBlanks,
    formatShort,
    dispShort,
    zeros, ones,
    diagl,
    row,
    col,
    (&), (¦), (——), (#),
    (?), (¿),
    Indexable(..), size,
    Numeric,
    rand, randn,
    cross,
    norm,
    ℕ,ℤ,ℝ,ℂ,iC,
    Normed(..), norm_Frob, norm_nuclear,
    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,Normed)
--import qualified Numeric.LinearAlgebra.Algorithms as A
import Numeric.Matrix()
import Numeric.Vector()
import Numeric.LinearAlgebra.Random
import Numeric.LinearAlgebra.Util.Convolution
import Control.Monad(when)
import Text.Printf
import Data.List.Split(splitOn)
import Data.List(intercalate)

type ℝ = Double
type ℕ = Int
type ℤ = Int
type ℂ = Complex Double

-- | imaginary unit
iC :: ℂ
iC = 0:+1

{- | create a real vector

>>> vector [1..5]
fromList [1.0,2.0,3.0,4.0,5.0]

-}
vector :: [ℝ] -> Vector ℝ
vector = fromList

{- | create a real matrix

>>> matrix 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 ]

-}
matrix
  :: Int -- ^ columns
  -> [ℝ] -- ^ elements
  -> Matrix ℝ
matrix 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 = putStr . 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

class Normed a
  where
    norm_0   :: a -> ℝ
    norm_1   :: a -> ℝ
    norm_2   :: a -> ℝ
    norm_Inf :: a -> ℝ


instance Normed (Vector ℝ)
  where
    norm_0 v = sumElements (step (abs v - scalar (eps*normInf v)))
    norm_1 = pnorm PNorm1
    norm_2 = pnorm PNorm2
    norm_Inf = pnorm Infinity

instance Normed (Vector ℂ)
  where
    norm_0 v = sumElements (step (fst (fromComplex (abs v)) - scalar (eps*normInf v)))
    norm_1 = pnorm PNorm1
    norm_2 = pnorm PNorm2
    norm_Inf = pnorm Infinity

instance Normed (Matrix ℝ)
  where
    norm_0 = norm_0 . flatten
    norm_1 = pnorm PNorm1
    norm_2 = pnorm PNorm2
    norm_Inf = pnorm Infinity

instance Normed (Matrix ℂ)
  where
    norm_0 = norm_0 . flatten
    norm_1 = pnorm PNorm1
    norm_2 = pnorm PNorm2
    norm_Inf = pnorm Infinity


norm_Frob :: (Normed (Vector t), Element t) => Matrix t -> ℝ
norm_Frob = norm_2 . flatten

norm_nuclear :: Field t => Matrix t -> ℝ
norm_nuclear = sumElements . singularValues


-- | 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

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

infixl 0 ~!~
c ~!~ msg = when c (error msg)

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

formatSparse :: String -> String -> String -> Int -> Matrix Double -> String

formatSparse zeroI _zeroF sep _ (approxInt -> Just m) = format sep f m
  where
    f 0 = zeroI
    f x = printf "%.0f" x

formatSparse zeroI zeroF sep n m = format sep f m
  where
    f x | abs (x::Double) < 2*peps = zeroI++zeroF
        | abs (fromIntegral (round x::Int) - x) / abs x < 2*peps
            = printf ("%.0f."++replicate n ' ') x
        | otherwise = printf ("%."++show n++"f") x

approxInt m
    | norm_Inf (v - vi) < 2*peps * norm_Inf v = Just (reshape (cols m) vi)
    | otherwise = Nothing
  where
    v = flatten m
    vi = roundVector v

dispDots n = putStr . formatSparse "." (replicate n ' ') "  " n

dispBlanks n = putStr . formatSparse "" "" "  " n

formatShort sep fmt maxr maxc m = auxm4
  where
    (rm,cm) = size m
    (r1,r2,r3)
        | rm <= maxr = (rm,0,0)
        | otherwise  = (maxr-3,rm-maxr+1,2)
    (c1,c2,c3)
        | cm <= maxc = (cm,0,0)
        | otherwise  = (maxc-3,cm-maxc+1,2)
    [ [a,_,b]
     ,[_,_,_]
     ,[c,_,d]] = toBlocks [r1,r2,r3]
                          [c1,c2,c3] m
    auxm = fromBlocks [[a,b],[c,d]]
    auxm2
        | cm > maxc = format "|" fmt auxm
        | otherwise = format sep fmt auxm
    auxm3
        | cm > maxc = map (f . splitOn "|") (lines auxm2)
        | otherwise = (lines auxm2)
    f items = intercalate sep (take (maxc-3) items) ++ "  .. " ++
              intercalate sep (drop (maxc-3) items)
    auxm4
        | rm > maxr = unlines (take (maxr-3) auxm3 ++ vsep : drop (maxr-3) auxm3)
        | otherwise = unlines auxm3
    vsep = map g (head auxm3)
    g '.' = ':'
    g _ = ' '


dispShort :: Int -> Int -> Int -> Matrix Double -> IO ()
dispShort maxr maxc dec m =
    printf "%dx%d\n%s" (rows m) (cols m) (formatShort "  " fmt maxr maxc m)
  where
    fmt = printf ("%."++show dec ++"f")