summaryrefslogtreecommitdiff
path: root/packages/base/src/Internal/Modular.hs
blob: eb0c5a88070f6956273d37cfc1dc854954cdf9f7 (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
478
479
480
{-# LANGUAGE CPP #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE TypeFamilies  #-}
{-# LANGUAGE TypeOperators #-}

{-# OPTIONS_GHC -fno-warn-missing-signatures #-}
{-# OPTIONS_GHC -fno-warn-missing-methods #-}

{- |
Module      :  Internal.Modular
Copyright   :  (c) Alberto Ruiz 2015
License     :  BSD3
Stability   :  experimental

Proof of concept of statically checked modular arithmetic.

-}

module Internal.Modular(
    Mod, type (./.)
) where

import Internal.Vector
import Internal.Matrix hiding (size)
import Internal.Numeric
import Internal.Element
import Internal.Container
import Internal.Vectorized (prodI,sumI,prodL,sumL)
import Internal.LAPACK (multiplyI, multiplyL)
import Internal.Algorithms(luFact,LU(..))
import Internal.Util(Normed(..),Indexable(..),
                     gaussElim, gaussElim_1, gaussElim_2,
                     luST, luSolve', luPacked', magnit, invershur)
import Internal.ST(mutable)
#if MIN_VERSION_base(4,11,0)
import GHC.TypeLits hiding (Mod)
#else
import GHC.TypeLits
#endif
import Data.Proxy(Proxy)
import Foreign.ForeignPtr(castForeignPtr)
import Foreign.Storable
import Data.Ratio
import Data.Complex
import Control.DeepSeq ( NFData(..) )
#if MIN_VERSION_base(4,11,0)
import Prelude hiding ((<>))
#endif



-- | Wrapper with a phantom integer for statically checked modular arithmetic.
newtype Mod (n :: Nat) t = Mod {unMod:: t}
  deriving (Storable)

instance (NFData t) => NFData (Mod n t)
  where
    rnf (Mod x) = rnf x

infixr 5 ./.
type (./.) x n = Mod n x

instance (Integral t, Enum t, KnownNat m) => Enum (Mod m t)
  where
    toEnum = l0 (\m x -> fromIntegral $ x `mod` (fromIntegral m))
    fromEnum = fromIntegral . unMod

instance (Eq t, KnownNat m) => Eq (Mod m t)
  where
    a == b = (unMod a) == (unMod b)

instance (Ord t, KnownNat m) => Ord (Mod m t)
  where
    compare a b = compare (unMod a) (unMod b)

instance (Integral t, Real t, KnownNat m, Integral (Mod m t)) => Real (Mod m t)
  where
    toRational x = toInteger x % 1

instance (Integral t, KnownNat m, Num (Mod m t)) => Integral (Mod m t)
  where
    toInteger = toInteger . unMod
    quotRem a b = (Mod q, Mod r)
      where
         (q,r) = quotRem (unMod a) (unMod b)

-- | this instance is only valid for prime m
instance (Show (Mod m t), Num (Mod m t), Eq t, KnownNat m) => Fractional (Mod m t)
  where
    recip x
        | x*r == 1  = r
        | otherwise = error $ show x ++" does not have a multiplicative inverse mod "++show m'
      where
        r = x^(m'-2 :: Integer)
        m' = fromIntegral . natVal $ (undefined :: Proxy m)
    fromRational x = fromInteger (numerator x) / fromInteger (denominator x)

l2 :: forall m a b c. (Num c, KnownNat m) => (c -> a -> b -> c) -> Mod m a -> Mod m b -> Mod m c
l2 f (Mod u) (Mod v) = Mod (f m' u v)
  where
    m' = fromIntegral . natVal $ (undefined :: Proxy m)

l1 :: forall m a b . (Num b, KnownNat m) => (b -> a -> b) -> Mod m a -> Mod m b
l1 f (Mod u) = Mod (f m' u)
  where
    m' = fromIntegral . natVal $ (undefined :: Proxy m)

l0 :: forall m a b . (Num b, KnownNat m) => (b -> a -> b) -> a -> Mod m b
l0 f u = Mod (f m' u)
  where
    m' = fromIntegral . natVal $ (undefined :: Proxy m)


instance Show t => Show (Mod n t)
  where
    show = show . unMod

instance (Integral t, KnownNat n) => Num (Mod n t)
  where
    (+) = l2 (\m a b -> (a + b) `mod` (fromIntegral m))
    (*) = l2 (\m a b -> (a * b) `mod` (fromIntegral m))
    (-) = l2 (\m a b -> (a - b) `mod` (fromIntegral m))
    abs = l1 (const abs)
    signum = l1 (const signum)
    fromInteger = l0 (\m x -> fromInteger x `mod` (fromIntegral m))


instance KnownNat m => Element (Mod m I)
  where
    constantD x n = i2f (constantD (unMod x) n)
    extractR ord m mi is mj js = i2fM <$> extractR ord (f2iM m) mi is mj js
    setRect i j m x = setRect i j (f2iM m) (f2iM x)
    sortI = sortI . f2i
    sortV = i2f . sortV . f2i
    compareV u v = compareV (f2i u) (f2i v)
    selectV c l e g = i2f (selectV c (f2i l) (f2i e) (f2i g))
    remapM i j m = i2fM (remap i j (f2iM m))
    rowOp c a i1 i2 j1 j2 x = rowOpAux (c_rowOpMI m') c (unMod a) i1 i2 j1 j2 (f2iM x)
      where
        m' = fromIntegral . natVal $ (undefined :: Proxy m)
    gemm u a b c = gemmg (c_gemmMI m') (f2i u) (f2iM a) (f2iM b) (f2iM c)
      where
        m' = fromIntegral . natVal $ (undefined :: Proxy m)

instance KnownNat m => Element (Mod m Z)
  where
    constantD x n = i2f (constantD (unMod x) n)
    extractR ord m mi is mj js = i2fM <$> extractR ord (f2iM m) mi is mj js
    setRect i j m x = setRect i j (f2iM m) (f2iM x)
    sortI = sortI . f2i
    sortV = i2f . sortV . f2i
    compareV u v = compareV (f2i u) (f2i v)
    selectV c l e g = i2f (selectV c (f2i l) (f2i e) (f2i g))
    remapM i j m = i2fM (remap i j (f2iM m))
    rowOp c a i1 i2 j1 j2 x = rowOpAux (c_rowOpML m') c (unMod a) i1 i2 j1 j2 (f2iM x)
      where
        m' = fromIntegral . natVal $ (undefined :: Proxy m)
    gemm u a b c = gemmg (c_gemmML m') (f2i u) (f2iM a) (f2iM b) (f2iM c)
      where
        m' = fromIntegral . natVal $ (undefined :: Proxy m)


instance KnownNat m => CTrans (Mod m I)
instance KnownNat m => CTrans (Mod m Z)


instance KnownNat m => Container Vector (Mod m I)
  where
    conj' = id
    size' = dim
    scale' s x = vmod (scale (unMod s) (f2i x))
    addConstant c x = vmod (addConstant (unMod c) (f2i x))
    add' a b = vmod (add' (f2i a) (f2i b))
    sub a b = vmod (sub (f2i a) (f2i b))
    mul a b = vmod (mul (f2i a) (f2i b))
    equal u v = equal (f2i u) (f2i v)
    scalar' x = fromList [x]
    konst' x = i2f . konst (unMod x)
    build' n f = build n (fromIntegral . f)
    cmap' = mapVector
    atIndex' x k = fromIntegral (atIndex (f2i x) k)
    minIndex'     = minIndex . f2i
    maxIndex'     = maxIndex . f2i
    minElement'   = Mod . minElement . f2i
    maxElement'   = Mod . maxElement . f2i
    sumElements'  = fromIntegral . sumI m' . f2i
      where
        m' = fromIntegral . natVal $ (undefined :: Proxy m)
    prodElements' = fromIntegral . prodI m' . f2i
      where
        m' = fromIntegral . natVal $ (undefined :: Proxy m)
    step'         = i2f . step . f2i
    find' = findV
    assoc' = assocV
    accum' = accumV
    ccompare' a b = ccompare (f2i a) (f2i b)
    cselect' c l e g = i2f $ cselect c (f2i l) (f2i e) (f2i g)
    scaleRecip s x = scale' s (cmap recip x)
    divide x y = mul x (cmap recip y)
    arctan2' = undefined
    cmod' m = vmod . cmod' (unMod m) . f2i
    fromInt' = vmod
    toInt'   = f2i
    fromZ'   = vmod . fromZ'
    toZ'     = toZ' . f2i

instance KnownNat m => Container Vector (Mod m Z)
  where
    conj' = id
    size' = dim
    scale' s x = vmod (scale (unMod s) (f2i x))
    addConstant c x = vmod (addConstant (unMod c) (f2i x))
    add' a b = vmod (add' (f2i a) (f2i b))
    sub a b = vmod (sub (f2i a) (f2i b))
    mul a b = vmod (mul (f2i a) (f2i b))
    equal u v = equal (f2i u) (f2i v)
    scalar' x = fromList [x]
    konst' x = i2f . konst (unMod x)
    build' n f = build n (fromIntegral . f)
    cmap' = mapVector
    atIndex' x k = fromIntegral (atIndex (f2i x) k)
    minIndex'     = minIndex . f2i
    maxIndex'     = maxIndex . f2i
    minElement'   = Mod . minElement . f2i
    maxElement'   = Mod . maxElement . f2i
    sumElements'  = fromIntegral . sumL m' . f2i
      where
        m' = fromIntegral . natVal $ (undefined :: Proxy m)
    prodElements' = fromIntegral . prodL m' . f2i
      where
        m' = fromIntegral . natVal $ (undefined :: Proxy m)
    step'         = i2f . step . f2i
    find' = findV
    assoc' = assocV
    accum' = accumV
    ccompare' a b = ccompare (f2i a) (f2i b)
    cselect' c l e g = i2f $ cselect c (f2i l) (f2i e) (f2i g)
    scaleRecip s x = scale' s (cmap recip x)
    divide x y = mul x (cmap recip y)
    arctan2' = undefined
    cmod' m = vmod . cmod' (unMod m) . f2i
    fromInt' = vmod . fromInt'
    toInt'   = toInt . f2i
    fromZ'   = vmod
    toZ'     = f2i


instance (Storable t, Indexable (Vector t) t) => Indexable (Vector (Mod m t)) (Mod m t)
  where
    (!) = (@>)

type instance RealOf (Mod n I) = I
type instance RealOf (Mod n Z) = Z

instance KnownNat m => Product (Mod m I) where
    norm2      = undefined
    absSum     = undefined
    norm1      = undefined
    normInf    = undefined
    multiply   = lift2m (multiplyI m')
      where
        m' = fromIntegral . natVal $ (undefined :: Proxy m)

instance KnownNat m => Product (Mod m Z) where
    norm2      = undefined
    absSum     = undefined
    norm1      = undefined
    normInf    = undefined
    multiply   = lift2m (multiplyL m')
      where
        m' = fromIntegral . natVal $ (undefined :: Proxy m)

instance KnownNat m => Normed (Vector (Mod m I))
  where
    norm_0 = norm_0 . toInt
    norm_1 = norm_1 . toInt
    norm_2 = norm_2 . toInt
    norm_Inf = norm_Inf . toInt

instance KnownNat m => Normed (Vector (Mod m Z))
  where
    norm_0 = norm_0 . toZ
    norm_1 = norm_1 . toZ
    norm_2 = norm_2 . toZ
    norm_Inf = norm_Inf . toZ


instance KnownNat m => Numeric (Mod m I)
instance KnownNat m => Numeric (Mod m Z)

i2f :: Storable t => Vector t -> Vector (Mod n t)
i2f v = unsafeFromForeignPtr (castForeignPtr fp) (i) (n)
    where (fp,i,n) = unsafeToForeignPtr v

f2i :: Storable t => Vector (Mod n t) -> Vector t
f2i v = unsafeFromForeignPtr (castForeignPtr fp) (i) (n)
    where (fp,i,n) = unsafeToForeignPtr v

f2iM :: (Element t, Element (Mod n t)) => Matrix (Mod n t) -> Matrix t
f2iM m = m { xdat = f2i (xdat m) }

i2fM :: (Element t, Element (Mod n t)) => Matrix t -> Matrix (Mod n t)
i2fM m = m { xdat = i2f (xdat m) }

vmod :: forall m t. (KnownNat m, Storable t, Integral t, Numeric t) => Vector t -> Vector (Mod m t)
vmod = i2f . cmod' m'
  where
    m' = fromIntegral . natVal $ (undefined :: Proxy m)

lift1 f a   = vmod (f (f2i a))
lift2 f a b = vmod (f (f2i a) (f2i b))

lift2m f a b = liftMatrix vmod (f (f2iM a) (f2iM b))

instance KnownNat m => Num (Vector (Mod m I))
  where
    (+) = lift2 (+)
    (*) = lift2 (*)
    (-) = lift2 (-)
    abs = lift1 abs
    signum = lift1 signum
    negate = lift1 negate
    fromInteger x = fromInt (fromInteger x)

instance KnownNat m => Num (Vector (Mod m Z))
  where
    (+) = lift2 (+)
    (*) = lift2 (*)
    (-) = lift2 (-)
    abs = lift1 abs
    signum = lift1 signum
    negate = lift1 negate
    fromInteger x = fromZ (fromInteger x)

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

instance (KnownNat m) => Testable (Matrix (Mod m I))
  where
    checkT _ = test

test = (ok, info)
  where
    v = fromList [3,-5,75] :: Vector (Mod 11 I)
    m = (3><3) [1..]   :: Matrix (Mod 11 I)

    a = (3><3) [1,2 , 3
               ,4,5 , 6
               ,0,10,-3] :: Matrix I

    b = (3><2) [0..] :: Matrix I

    am = fromInt a :: Matrix (Mod 13 I)
    bm = fromInt b :: Matrix (Mod 13 I)
    ad = fromInt a :: Matrix Double
    bd = fromInt b :: Matrix Double

    g = (3><3) (repeat (40000)) :: Matrix I
    gm = fromInt g :: Matrix (Mod 100000 I)

    lg = (3><3) (repeat (3*10^(9::Int))) :: Matrix Z
    lgm = fromZ lg :: Matrix (Mod 10000000000 Z)

    gen  n = diagRect 1 (konst 5 n) n n :: Numeric t => Matrix t
    
    rgen n = gen n :: Matrix R
    cgen n = complex (rgen n) + fliprl (complex (rgen n)) * scalar (0:+1) :: Matrix C
    sgen n = single (cgen n)
    
    checkGen x = norm_Inf $ flatten $ invg x <> x - ident (rows x)
    
    invg t = gaussElim t (ident (rows t))

    checkLU okf t = norm_Inf $ flatten (l <> u <> p - t)
      where
        (l,u,p,_) = luFact (LU x' p')
          where
            (x',p') = mutable (luST okf) t

    checkSolve aa = norm_Inf $ flatten (aa <> x - bb)
       where
         bb = flipud aa
         x = luSolve' (luPacked' aa) bb

    tmm = diagRect 1 (fromList [2..6]) 5 5 :: Matrix (Mod 19 I)

    info = do
        print v
        print m
        print (tr m)
        print $ v+v
        print $ m+m
        print $ m <> m
        print $ m #> v

        print $ am <> gaussElim am bm - bm
        print $ ad <> gaussElim ad bd - bd

        print g
        print $ g <> g
        print gm
        print $ gm <> gm

        print lg
        print $ lg <> lg
        print lgm
        print $ lgm <> lgm
        
        putStrLn "checkGen"
        print (checkGen (gen 5 :: Matrix R))
        print (checkGen (gen 5 :: Matrix Float))
        print (checkGen (cgen 5 :: Matrix C))
        print (checkGen (sgen 5 :: Matrix (Complex Float)))
        print (invg (gen 5) :: Matrix (Mod 7 I))
        print (invg (gen 5) :: Matrix (Mod 7 Z))
        
        print $ mutable (luST (const True)) (gen 5 :: Matrix R)
        print $ mutable (luST (const True)) (gen 5 :: Matrix (Mod 11 Z))

        putStrLn "checkLU"
        print $ checkLU (magnit 0) (gen 5 :: Matrix R)
        print $ checkLU (magnit 0) (gen 5 :: Matrix Float)
        print $ checkLU (magnit 0) (cgen 5 :: Matrix C)
        print $ checkLU (magnit 0) (sgen 5 :: Matrix (Complex Float))
        print $ checkLU (magnit 0) (gen 5 :: Matrix (Mod 7 I))
        print $ checkLU (magnit 0) (gen 5 :: Matrix (Mod 7 Z))

        putStrLn "checkSolve"
        print $ checkSolve (gen 5 :: Matrix R)
        print $ checkSolve (gen 5 :: Matrix Float)
        print $ checkSolve (cgen 5 :: Matrix C)
        print $ checkSolve (sgen 5 :: Matrix (Complex Float))
        print $ checkSolve (gen 5 :: Matrix (Mod 7 I))
        print $ checkSolve (gen 5 :: Matrix (Mod 7 Z))
        
        putStrLn "luSolve'"
        print $ luSolve' (luPacked' tmm) (ident (rows tmm))
        print $ invershur tmm


    ok = and
      [ toInt (m #> v) == cmod 11 (toInt m #> toInt v )
      , am <> gaussElim_1 am bm == bm
      , am <> gaussElim_2 am bm == bm
      , am <> gaussElim   am bm == bm
      , (checkGen (gen 5 :: Matrix R)) < 1E-15
      , (checkGen (gen 5 :: Matrix Float)) < 2E-7
      , (checkGen (cgen 5 :: Matrix C)) < 1E-15
      , (checkGen (sgen 5 :: Matrix (Complex Float))) < 3E-7
      , (checkGen (gen 5 :: Matrix (Mod 7 I))) == 0
      , (checkGen (gen 5 :: Matrix (Mod 7 Z))) == 0
      , (checkLU (magnit 1E-10) (gen 5 :: Matrix R)) < 2E-15
      , (checkLU (magnit 1E-5) (gen 5 :: Matrix Float)) < 1E-6
      , (checkLU (magnit 1E-10) (cgen 5 :: Matrix C)) < 5E-15
      , (checkLU (magnit 1E-5) (sgen 5 :: Matrix (Complex Float))) < 1E-6
      , (checkLU (magnit 0) (gen 5 :: Matrix (Mod 7 I))) == 0
      , (checkLU (magnit 0) (gen 5 :: Matrix (Mod 7 Z))) == 0
      , checkSolve (gen 5 :: Matrix R) < 2E-15
      , checkSolve (gen 5 :: Matrix Float) < 1E-6
      , checkSolve (cgen 5 :: Matrix C) < 4E-15
      , checkSolve (sgen 5 :: Matrix (Complex Float)) < 1E-6
      , checkSolve (gen 5 :: Matrix (Mod 7 I)) == 0
      , checkSolve (gen 5 :: Matrix (Mod 7 Z)) == 0
      , prodElements (konst (9:: Mod 10 I) (12::Int)) == product (replicate 12 (9:: Mod 10 I))
      , gm <> gm == konst 0 (3,3)
      , lgm <> lgm == konst 0 (3,3)
      , invershur tmm == luSolve' (luPacked' tmm) (ident (rows tmm))
      , luSolve' (luPacked' (tr $ ident 5 :: Matrix (I ./. 2))) (ident 5) == ident 5
      ]