1 {-# LANGUAGE ExistentialQuantification #-}
2 {-# LANGUAGE FlexibleContexts #-}
3 {-# LANGUAGE FlexibleInstances #-}
4 {-# LANGUAGE MultiParamTypeClasses #-}
5 {-# LANGUAGE NoMonomorphismRestriction #-}
6 {-# LANGUAGE ScopedTypeVariables #-}
7 {-# LANGUAGE TypeFamilies #-}
8 {-# LANGUAGE RebindableSyntax #-}
10 -- | Boxed matrices; that is, boxed m-vectors of boxed n-vectors. We
11 -- assume that the underlying representation is
12 -- Data.Vector.Fixed.Boxed.Vec for simplicity. It was tried in
13 -- generality and failed.
18 import Data.List (intercalate)
20 import Data.Vector.Fixed (
36 import qualified Data.Vector.Fixed as V (
46 import Data.Vector.Fixed.Cont ( Arity, arity )
47 import Linear.Vector ( Vec, delete, element_sum )
48 import Normed ( Normed(..) )
50 import NumericPrelude hiding ( (*), abs )
51 import qualified NumericPrelude as NP ( (*) )
52 import qualified Algebra.Absolute as Absolute ( C )
53 import Algebra.Absolute ( abs )
54 import qualified Algebra.Additive as Additive ( C )
55 import qualified Algebra.Algebraic as Algebraic ( C )
56 import Algebra.Algebraic ( root )
57 import qualified Algebra.Ring as Ring ( C )
58 import qualified Algebra.Module as Module ( C )
59 import qualified Algebra.RealRing as RealRing ( C )
60 import qualified Algebra.ToRational as ToRational ( C )
61 import qualified Algebra.Transcendental as Transcendental ( C )
62 import qualified Prelude as P ( map )
64 data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
65 type Mat1 a = Mat N1 N1 a
66 type Mat2 a = Mat N2 N2 a
67 type Mat3 a = Mat N3 N3 a
68 type Mat4 a = Mat N4 N4 a
69 type Mat5 a = Mat N5 N5 a
71 instance (Eq a) => Eq (Mat m n a) where
72 -- | Compare a row at a time.
76 -- >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
77 -- >>> let m2 = fromList [[1,2],[3,4]] :: Mat2 Int
78 -- >>> let m3 = fromList [[5,6],[7,8]] :: Mat2 Int
84 (Mat rows1) == (Mat rows2) =
85 V.and $ V.zipWith comp rows1 rows2
87 -- Compare a row, one column at a time.
88 comp row1 row2 = V.and (V.zipWith (==) row1 row2)
91 instance (Show a) => Show (Mat m n a) where
92 -- | Display matrices and vectors as ordinary tuples. This is poor
93 -- practice, but these results are primarily displayed
94 -- interactively and convenience trumps correctness (said the guy
95 -- who insists his vector lengths be statically checked at
100 -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
105 "(" ++ (intercalate "," (V.toList row_strings)) ++ ")"
107 row_strings = V.map show_vector rows
109 "(" ++ (intercalate "," element_strings) ++ ")"
112 element_strings = P.map show v1l
115 -- | Convert a matrix to a nested list.
116 toList :: Mat m n a -> [[a]]
117 toList (Mat rows) = map V.toList (V.toList rows)
119 -- | Create a matrix from a nested list.
120 fromList :: (Arity m, Arity n) => [[a]] -> Mat m n a
121 fromList vs = Mat (V.fromList $ map V.fromList vs)
124 -- | Unsafe indexing.
125 (!!!) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> a
126 (!!!) m (i, j) = (row m i) ! j
129 (!!?) :: Mat m n a -> (Int, Int) -> Maybe a
130 (!!?) m@(Mat rows) (i, j)
131 | i < 0 || j < 0 = Nothing
132 | i > V.length rows = Nothing
133 | otherwise = if j > V.length (row m j)
135 else Just $ (row m j) ! j
138 -- | The number of rows in the matrix.
139 nrows :: forall m n a. (Arity m) => Mat m n a -> Int
140 nrows _ = arity (undefined :: m)
142 -- | The number of columns in the first row of the
143 -- matrix. Implementation stolen from Data.Vector.Fixed.length.
144 ncols :: forall m n a. (Arity n) => Mat m n a -> Int
145 ncols _ = arity (undefined :: n)
148 -- | Return the @i@th row of @m@. Unsafe.
149 row :: Mat m n a -> Int -> (Vec n a)
150 row (Mat rows) i = rows ! i
153 -- | Return the @j@th column of @m@. Unsafe.
154 column :: Mat m n a -> Int -> (Vec m a)
155 column (Mat rows) j =
156 V.map (element j) rows
163 -- | Transpose @m@; switch it's columns and its rows. This is a dirty
164 -- implementation.. it would be a little cleaner to use imap, but it
165 -- doesn't seem to work.
167 -- TODO: Don't cheat with fromList.
171 -- >>> let m = fromList [[1,2], [3,4]] :: Mat2 Int
175 transpose :: (Arity m, Arity n) => Mat m n a -> Mat n m a
176 transpose m = Mat $ V.fromList column_list
178 column_list = [ column m i | i <- [0..(ncols m)-1] ]
181 -- | Is @m@ symmetric?
185 -- >>> let m1 = fromList [[1,2], [2,1]] :: Mat2 Int
189 -- >>> let m2 = fromList [[1,2], [3,1]] :: Mat2 Int
193 symmetric :: (Eq a, Arity m) => Mat m m a -> Bool
198 -- | Construct a new matrix from a function @lambda@. The function
199 -- @lambda@ should take two parameters i,j corresponding to the
200 -- entries in the matrix. The i,j entry of the resulting matrix will
201 -- have the value returned by lambda i j.
205 -- >>> let lambda i j = i + j
206 -- >>> construct lambda :: Mat3 Int
207 -- ((0,1,2),(1,2,3),(2,3,4))
209 construct :: forall m n a. (Arity m, Arity n)
210 => (Int -> Int -> a) -> Mat m n a
211 construct lambda = Mat $ generate make_row
213 make_row :: Int -> Vec n a
214 make_row i = generate (lambda i)
217 -- | Create an identity matrix with the right dimensions.
221 -- >>> identity_matrix :: Mat3 Int
222 -- ((1,0,0),(0,1,0),(0,0,1))
223 -- >>> identity_matrix :: Mat3 Double
224 -- ((1.0,0.0,0.0),(0.0,1.0,0.0),(0.0,0.0,1.0))
226 identity_matrix :: (Arity m, Ring.C a) => Mat m m a
228 construct (\i j -> if i == j then (fromInteger 1) else (fromInteger 0))
230 -- | Given a positive-definite matrix @m@, computes the
231 -- upper-triangular matrix @r@ with (transpose r)*r == m and all
232 -- values on the diagonal of @r@ positive.
236 -- >>> let m1 = fromList [[20,-1], [-1,20]] :: Mat2 Double
238 -- ((4.47213595499958,-0.22360679774997896),(0.0,4.466542286825459))
239 -- >>> (transpose (cholesky m1)) * (cholesky m1)
240 -- ((20.000000000000004,-1.0),(-1.0,20.0))
242 cholesky :: forall m n a. (Algebraic.C a, Arity m, Arity n)
243 => (Mat m n a) -> (Mat m n a)
244 cholesky m = construct r
247 r i j | i == j = sqrt(m !!! (i,j) - sum [(r k i)^2 | k <- [0..i-1]])
249 (((m !!! (i,j)) - sum [(r k i) NP.* (r k j) | k <- [0..i-1]]))/(r i i)
253 -- | Returns True if the given matrix is upper-triangular, and False
254 -- otherwise. The parameter @epsilon@ lets the caller choose a
259 -- >>> let m = fromList [[1,1],[1e-12,1]] :: Mat2 Double
260 -- >>> is_upper_triangular m
262 -- >>> is_upper_triangular' 1e-10 m
267 -- 1. Don't cheat with lists.
269 is_upper_triangular' :: (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
270 => a -- ^ The tolerance @epsilon@.
273 is_upper_triangular' epsilon m =
276 results = [[ test i j | i <- [0..(nrows m)-1]] | j <- [0..(ncols m)-1] ]
278 test :: Int -> Int -> Bool
281 -- use "less than or equal to" so zero is a valid epsilon
282 | otherwise = abs (m !!! (i,j)) <= epsilon
285 -- | Returns True if the given matrix is upper-triangular, and False
286 -- otherwise. A specialized version of 'is_upper_triangular\'' with
291 -- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
292 -- >>> is_upper_triangular m
295 -- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
296 -- >>> is_upper_triangular m
301 -- 1. The Ord constraint is too strong here, Eq would suffice.
303 is_upper_triangular :: (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
305 is_upper_triangular = is_upper_triangular' 0
308 -- | Returns True if the given matrix is lower-triangular, and False
309 -- otherwise. This is a specialized version of 'is_lower_triangular\''
310 -- with @epsilon = 0@.
314 -- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
315 -- >>> is_lower_triangular m
318 -- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
319 -- >>> is_lower_triangular m
322 is_lower_triangular :: (Ord a,
329 is_lower_triangular = is_upper_triangular . transpose
332 -- | Returns True if the given matrix is lower-triangular, and False
333 -- otherwise. The parameter @epsilon@ lets the caller choose a
338 -- >>> let m = fromList [[1,1e-12],[1,1]] :: Mat2 Double
339 -- >>> is_lower_triangular m
341 -- >>> is_lower_triangular' 1e-12 m
344 is_lower_triangular' :: (Ord a,
349 => a -- ^ The tolerance @epsilon@.
352 is_lower_triangular' epsilon = (is_upper_triangular' epsilon) . transpose
355 -- | Returns True if the given matrix is triangular, and False
360 -- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
361 -- >>> is_triangular m
364 -- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
365 -- >>> is_triangular m
368 -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
369 -- >>> is_triangular m
372 is_triangular :: (Ord a,
379 is_triangular m = is_upper_triangular m || is_lower_triangular m
382 -- | Return the (i,j)th minor of m.
386 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
387 -- >>> minor m 0 0 :: Mat2 Int
389 -- >>> minor m 1 1 :: Mat2 Int
400 minor (Mat rows) i j = m
402 rows' = delete rows i
403 m = Mat $ V.map ((flip delete) j) rows'
406 class (Eq a, Ring.C a) => Determined p a where
407 determinant :: (p a) -> a
409 instance (Eq a, Ring.C a) => Determined (Mat (S Z) (S Z)) a where
410 determinant (Mat rows) = (V.head . V.head) rows
416 Determined (Mat (S n) (S n)) a)
417 => Determined (Mat (S (S n)) (S (S n))) a where
418 -- | The recursive definition with a special-case for triangular matrices.
422 -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
427 | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ]
428 | otherwise = determinant_recursive
432 det_minor i j = determinant (minor m i j)
434 determinant_recursive =
435 sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (det_minor 0 j)
436 | j <- [0..(ncols m)-1] ]
440 -- | Matrix multiplication.
444 -- >>> let m1 = fromList [[1,2,3], [4,5,6]] :: Mat N2 N3 Int
445 -- >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat N3 N2 Int
450 (*) :: (Ring.C a, Arity m, Arity n, Arity p)
454 (*) m1 m2 = construct lambda
457 sum [(m1 !!! (i,k)) NP.* (m2 !!! (k,j)) | k <- [0..(ncols m1)-1] ]
461 instance (Ring.C a, Arity m, Arity n) => Additive.C (Mat m n a) where
463 (Mat rows1) + (Mat rows2) =
464 Mat $ V.zipWith (V.zipWith (+)) rows1 rows2
466 (Mat rows1) - (Mat rows2) =
467 Mat $ V.zipWith (V.zipWith (-)) rows1 rows2
469 zero = Mat (V.replicate $ V.replicate (fromInteger 0))
472 instance (Ring.C a, Arity m, Arity n, m ~ n) => Ring.C (Mat m n a) where
473 -- The first * is ring multiplication, the second is matrix
478 instance (Ring.C a, Arity m, Arity n) => Module.C a (Mat m n a) where
479 -- We can multiply a matrix by a scalar of the same type as its
481 x *> (Mat rows) = Mat $ V.map (V.map (NP.* x)) rows
484 instance (Algebraic.C a,
487 => Normed (Mat (S m) N1 a) where
488 -- | Generic p-norms for vectors in R^n that are represented as nx1
493 -- >>> let v1 = vec2d (3,4)
499 norm_p p (Mat rows) =
500 (root p') $ sum [fromRational' (toRational x)^p' | x <- xs]
503 xs = concat $ V.toList $ V.map V.toList rows
505 -- | The infinity norm.
509 -- >>> let v1 = vec3d (1,5,2)
513 norm_infty (Mat rows) =
514 fromRational' $ toRational $ V.maximum $ V.map V.maximum rows
517 -- | Compute the Frobenius norm of a matrix. This essentially treats
518 -- the matrix as one long vector containing all of its entries (in
519 -- any order, it doesn't matter).
523 -- >>> let m = fromList [[1, 2, 3],[4,5,6],[7,8,9]] :: Mat3 Double
524 -- >>> frobenius_norm m == sqrt 285
527 -- >>> let m = fromList [[1, -1, 1],[-1,1,-1],[1,-1,1]] :: Mat3 Double
528 -- >>> frobenius_norm m == 3
531 frobenius_norm :: (Algebraic.C a, Ring.C a) => Mat m n a -> a
532 frobenius_norm (Mat rows) =
533 sqrt $ element_sum $ V.map row_sum rows
535 -- | Square and add up the entries of a row.
536 row_sum = element_sum . V.map (^2)
539 -- Vector helpers. We want it to be easy to create low-dimension
540 -- column vectors, which are nx1 matrices.
542 -- | Convenient constructor for 2D vectors.
546 -- >>> import Roots.Simple
547 -- >>> let fst m = m !!! (0,0)
548 -- >>> let snd m = m !!! (1,0)
549 -- >>> let h = 0.5 :: Double
550 -- >>> let g1 m = 1.0 + h NP.* exp(-((fst m)^2))/(1.0 + (snd m)^2)
551 -- >>> let g2 m = 0.5 + h NP.* atan((fst m)^2 + (snd m)^2)
552 -- >>> let g u = vec2d ((g1 u), (g2 u))
553 -- >>> let u0 = vec2d (1.0, 1.0)
554 -- >>> let eps = 1/(10^9)
555 -- >>> fixed_point g eps u0
556 -- ((1.0728549599342185),(1.0820591495686167))
558 vec1d :: (a) -> Mat N1 N1 a
559 vec1d (x) = Mat (mk1 (mk1 x))
561 vec2d :: (a,a) -> Mat N2 N1 a
562 vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y))
564 vec3d :: (a,a,a) -> Mat N3 N1 a
565 vec3d (x,y,z) = Mat (mk3 (mk1 x) (mk1 y) (mk1 z))
567 vec4d :: (a,a,a,a) -> Mat N4 N1 a
568 vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z))
570 vec5d :: (a,a,a,a,a) -> Mat N5 N1 a
571 vec5d (v,w,x,y,z) = Mat (mk5 (mk1 v) (mk1 w) (mk1 x) (mk1 y) (mk1 z))
573 -- Since we commandeered multiplication, we need to create 1x1
574 -- matrices in order to multiply things.
575 scalar :: a -> Mat N1 N1 a
576 scalar x = Mat (mk1 (mk1 x))
578 dot :: (RealRing.C a, n ~ N1, m ~ S t, Arity t)
582 v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0)
585 -- | The angle between @v1@ and @v2@ in Euclidean space.
589 -- >>> let v1 = vec2d (1.0, 0.0)
590 -- >>> let v2 = vec2d (0.0, 1.0)
591 -- >>> angle v1 v2 == pi/2.0
594 angle :: (Transcendental.C a,
606 theta = (recip norms) NP.* (v1 `dot` v2)
607 norms = (norm v1) NP.* (norm v2)
610 -- | Retrieve the diagonal elements of the given matrix as a \"column
611 -- vector,\" i.e. a m-by-1 matrix. We require the matrix to be
612 -- square to avoid ambiguity in the return type which would ideally
613 -- have dimension min(m,n) supposing an m-by-n matrix.
617 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
621 diagonal :: (Arity m) => Mat m m a -> Mat m N1 a
625 lambda i _ = matrix !!! (i,i)
628 -- | Given a square @matrix@, return a new matrix of the same size
629 -- containing only the on-diagonal entries of @matrix@. The
630 -- off-diagonal entries are set to zero.
634 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
635 -- >>> diagonal_part m
636 -- ((1,0,0),(0,5,0),(0,0,9))
638 diagonal_part :: (Arity m, Ring.C a)
641 diagonal_part matrix =
644 lambda i j = if i == j then matrix !!! (i,j) else 0
647 -- | Given a square @matrix@, return a new matrix of the same size
648 -- containing only the on-diagonal and below-diagonal entries of
649 -- @matrix@. The above-diagonal entries are set to zero.
653 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
655 -- ((1,0,0),(4,5,0),(7,8,9))
657 lt_part :: (Arity m, Ring.C a)
663 lambda i j = if i >= j then matrix !!! (i,j) else 0
666 -- | Given a square @matrix@, return a new matrix of the same size
667 -- containing only the below-diagonal entries of @matrix@. The on-
668 -- and above-diagonal entries are set to zero.
672 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
673 -- >>> lt_part_strict m
674 -- ((0,0,0),(4,0,0),(7,8,0))
676 lt_part_strict :: (Arity m, Ring.C a)
679 lt_part_strict matrix =
682 lambda i j = if i > j then matrix !!! (i,j) else 0
685 -- | Given a square @matrix@, return a new matrix of the same size
686 -- containing only the on-diagonal and above-diagonal entries of
687 -- @matrix@. The below-diagonal entries are set to zero.
691 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
693 -- ((1,2,3),(0,5,6),(0,0,9))
695 ut_part :: (Arity m, Ring.C a)
698 ut_part = transpose . lt_part . transpose
701 -- | Given a square @matrix@, return a new matrix of the same size
702 -- containing only the above-diagonal entries of @matrix@. The on-
703 -- and below-diagonal entries are set to zero.
707 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
708 -- >>> ut_part_strict m
709 -- ((0,2,3),(0,0,6),(0,0,0))
711 ut_part_strict :: (Arity m, Ring.C a)
714 ut_part_strict = transpose . lt_part_strict . transpose
717 -- | Compute the trace of a square matrix, the sum of the elements
718 -- which lie on its diagonal. We require the matrix to be
719 -- square to avoid ambiguity in the return type which would ideally
720 -- have dimension min(m,n) supposing an m-by-n matrix.
724 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
728 trace :: (Arity m, Ring.C a) => Mat m m a -> a
730 let (Mat rows) = diagonal matrix
732 element_sum $ V.map V.head rows