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 (
28 import qualified Data.Vector.Fixed as V (
41 import Data.Vector.Fixed.Cont ( Arity, arity )
42 import Linear.Vector ( Vec, delete )
44 import Normed ( Normed(..) )
46 import NumericPrelude hiding ( (*), abs )
47 import qualified NumericPrelude as NP ( (*) )
48 import qualified Algebra.Absolute as Absolute ( C )
49 import Algebra.Absolute ( abs )
50 import qualified Algebra.Additive as Additive ( C )
51 import qualified Algebra.Algebraic as Algebraic ( C )
52 import Algebra.Algebraic ( root )
53 import qualified Algebra.Field as Field ( C )
54 import qualified Algebra.Ring as Ring ( C )
55 import qualified Algebra.Module as Module ( C )
56 import qualified Algebra.RealRing as RealRing ( C )
57 import qualified Algebra.ToRational as ToRational ( C )
58 import qualified Algebra.Transcendental as Transcendental ( C )
59 import qualified Prelude as P ( map )
61 -- | Our main matrix type.
62 data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
64 -- Type synonyms for n-by-n matrices.
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
70 type Mat6 a = Mat N6 N6 a
71 type Mat7 a = Mat N7 N7 a
73 -- * Type synonyms for 1-by-n row "vectors".
75 -- | Type synonym for row vectors expressed as 1-by-n matrices.
76 type Row n a = Mat N1 n a
78 type Row1 a = Row N1 a
79 type Row2 a = Row N2 a
80 type Row3 a = Row N3 a
81 type Row4 a = Row N4 a
82 type Row5 a = Row N5 a
84 -- * Type synonyms for n-by-1 column "vectors".
86 -- | Type synonym for column vectors expressed as n-by-1 matrices.
87 type Col n a = Mat n N1 a
89 type Col1 a = Col N1 a
90 type Col2 a = Col N2 a
91 type Col3 a = Col N3 a
92 type Col4 a = Col N4 a
93 type Col5 a = Col N5 a
94 type Col6 a = Col N6 a
95 type Col7 a = Col N7 a
96 type Col8 a = Col N8 a
97 type Col9 a = Col N9 a
98 type Col10 a = Col N10 a
99 type Col11 a = Col N11 a
100 type Col12 a = Col N12 a
101 type Col13 a = Col N13 a
102 type Col14 a = Col N14 a
103 type Col15 a = Col N15 a
104 type Col16 a = Col N16 a
105 type Col17 a = Col N17 a
106 type Col18 a = Col N18 a
107 type Col19 a = Col N19 a
108 type Col20 a = Col N20 a
109 type Col21 a = Col N21 a
110 type Col22 a = Col N22 a
111 type Col23 a = Col N23 a
112 type Col24 a = Col N24 a
113 type Col25 a = Col N25 a
114 type Col26 a = Col N26 a
115 type Col27 a = Col N27 a
116 type Col28 a = Col N28 a
117 type Col29 a = Col N29 a
118 type Col30 a = Col N30 a
119 type Col31 a = Col N31 a
120 type Col32 a = Col N32 a
123 instance (Eq a) => Eq (Mat m n a) where
124 -- | Compare a row at a time.
128 -- >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
129 -- >>> let m2 = fromList [[1,2],[3,4]] :: Mat2 Int
130 -- >>> let m3 = fromList [[5,6],[7,8]] :: Mat2 Int
136 (Mat rows_one) == (Mat rows_two) =
137 V.and $ V.zipWith comp rows_one rows_two
139 -- Compare a row, one column at a time.
140 comp row1 row2 = V.and (V.zipWith (==) row1 row2)
143 instance (Show a) => Show (Mat m n a) where
144 -- | Display matrices and vectors as ordinary tuples. This is poor
145 -- practice, but these results are primarily displayed
146 -- interactively and convenience trumps correctness (said the guy
147 -- who insists his vector lengths be statically checked at
152 -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
157 "(" ++ (intercalate "," (V.toList row_strings)) ++ ")"
159 row_strings = V.map show_vector rows
161 "(" ++ (intercalate "," element_strings) ++ ")"
164 element_strings = P.map show v1l
167 -- | Convert a matrix to a nested list.
168 toList :: Mat m n a -> [[a]]
169 toList (Mat rows) = map V.toList (V.toList rows)
172 -- | Create a matrix from a nested list.
173 fromList :: (Arity m, Arity n) => [[a]] -> Mat m n a
174 fromList vs = Mat (V.fromList $ map V.fromList vs)
177 -- | Unsafe indexing. Much faster than the safe indexing.
178 (!!!) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> a
179 (!!!) (Mat rows) (i, j) = (rows ! i) ! j
186 -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
214 (!!?) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> Maybe a
216 ifoldl2 f Nothing matrix
218 f k l found cur = if (k,l) == idx then (Just cur) else found
221 -- | The number of rows in the matrix.
222 nrows :: forall m n a. (Arity m) => Mat m n a -> Int
223 nrows _ = arity (undefined :: m)
226 -- | The number of columns in the first row of the
227 -- matrix. Implementation stolen from Data.Vector.Fixed.length.
228 ncols :: forall m n a. (Arity n) => Mat m n a -> Int
229 ncols _ = arity (undefined :: n)
232 -- | Return the @i@th row of @m@ as a matrix. Unsafe.
233 row :: (Arity m, Arity n) => Mat m n a -> Int -> Row n a
237 lambda _ j = m !!! (i, j)
240 -- | Return the @j@th column of @m@ as a matrix. Unsafe.
241 column :: (Arity m, Arity n) => Mat m n a -> Int -> Col m a
245 lambda i _ = m !!! (i, j)
248 -- | Transpose @m@; switch it's columns and its rows. This is a dirty
249 -- implementation, but I don't see a better way.
251 -- TODO: Don't cheat with fromList.
255 -- >>> let m = fromList [[1,2], [3,4]] :: Mat2 Int
259 transpose :: (Arity m, Arity n) => Mat m n a -> Mat n m a
263 lambda i j = matrix !!! (j,i)
266 -- | Is @m@ symmetric?
270 -- >>> let m1 = fromList [[1,2], [2,1]] :: Mat2 Int
274 -- >>> let m2 = fromList [[1,2], [3,1]] :: Mat2 Int
278 symmetric :: (Eq a, Arity m) => Mat m m a -> Bool
283 -- | Construct a new matrix from a function @lambda@. The function
284 -- @lambda@ should take two parameters i,j corresponding to the
285 -- entries in the matrix. The i,j entry of the resulting matrix will
286 -- have the value returned by lambda i j.
290 -- >>> let lambda i j = i + j
291 -- >>> construct lambda :: Mat3 Int
292 -- ((0,1,2),(1,2,3),(2,3,4))
294 construct :: forall m n a. (Arity m, Arity n)
295 => (Int -> Int -> a) -> Mat m n a
296 construct lambda = Mat $ generate make_row
298 make_row :: Int -> Vec n a
299 make_row i = generate (lambda i)
302 -- | Create an identity matrix with the right dimensions.
306 -- >>> identity_matrix :: Mat3 Int
307 -- ((1,0,0),(0,1,0),(0,0,1))
308 -- >>> identity_matrix :: Mat3 Double
309 -- ((1.0,0.0,0.0),(0.0,1.0,0.0),(0.0,0.0,1.0))
311 identity_matrix :: (Arity m, Ring.C a) => Mat m m a
313 construct (\i j -> if i == j then (fromInteger 1) else (fromInteger 0))
316 -- | Given a positive-definite matrix @m@, computes the
317 -- upper-triangular matrix @r@ with (transpose r)*r == m and all
318 -- values on the diagonal of @r@ positive.
322 -- >>> let m1 = fromList [[20,-1], [-1,20]] :: Mat2 Double
323 -- >>> let r = cholesky m1
324 -- >>> frobenius_norm ((transpose r)*r - m1) < 1e-10
326 -- >>> is_upper_triangular r
329 -- >>> let k1 = [6, -3, 0, 0, 0, 0, 0] :: [Double]
330 -- >>> let k2 = [-3, 10.5, -7.5, 0, 0, 0, 0] :: [Double]
331 -- >>> let k3 = [0, -7.5, 12.5, 0, 0, 0, 0] :: [Double]
332 -- >>> let k4 = [0, 0, 0, 6, 0, 0, 0] :: [Double]
333 -- >>> let k5 = [0, 0, 0, 0, 6, 0, 0] :: [Double]
334 -- >>> let k6 = [0, 0, 0, 0, 0, 6, 0] :: [Double]
335 -- >>> let k7 = [0, 0, 0, 0, 0, 0, 15] :: [Double]
336 -- >>> let big_K = fromList [k1,k2,k3,k4,k5,k6,k7] :: Mat N7 N7 Double
338 -- >>> let e1 = [2.449489742783178,0,0,0,0,0,0] :: [Double]
339 -- >>> let e2 = [-1.224744871391589,3,0,0,0,0,0] :: [Double]
340 -- >>> let e3 = [0,-5/2,5/2,0,0,0,0] :: [Double]
341 -- >>> let e4 = [0,0,0,2.449489742783178,0,0,0] :: [Double]
342 -- >>> let e5 = [0,0,0,0,2.449489742783178,0,0] :: [Double]
343 -- >>> let e6 = [0,0,0,0,0,2.449489742783178,0] :: [Double]
344 -- >>> let e7 = [0,0,0,0,0,0,3.872983346207417] :: [Double]
345 -- >>> let expected = fromList [e1,e2,e3,e4,e5,e6,e7] :: Mat N7 N7 Double
347 -- >>> let r = cholesky big_K
348 -- >>> frobenius_norm (r - (transpose expected)) < 1e-12
351 cholesky :: forall m n a. (Algebraic.C a, Arity m, Arity n)
352 => (Mat m n a) -> (Mat m n a)
353 cholesky m = construct r
356 r i j | i == j = sqrt(m !!! (i,j) - sum [(r k i)^2 | k <- [0..i-1]])
358 (((m !!! (i,j)) - sum [(r k i) NP.* (r k j) | k <- [0..i-1]]))/(r i i)
362 -- | Returns True if the given matrix is upper-triangular, and False
363 -- otherwise. The parameter @epsilon@ lets the caller choose a
368 -- >>> let m = fromList [[1,1],[1e-12,1]] :: Mat2 Double
369 -- >>> is_upper_triangular m
371 -- >>> is_upper_triangular' 1e-10 m
374 is_upper_triangular' :: forall m n a.
375 (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
376 => a -- ^ The tolerance @epsilon@.
379 is_upper_triangular' epsilon matrix =
380 ifoldl2 f True matrix
382 f :: Int -> Int -> Bool -> a -> Bool
383 f _ _ False _ = False
386 -- use "less than or equal to" so zero is a valid epsilon
387 | otherwise = abs x <= epsilon
390 -- | Returns True if the given matrix is upper-triangular, and False
391 -- otherwise. We don't delegate to the general
392 -- 'is_upper_triangular'' here because it imposes additional
393 -- typeclass constraints throughout the library.
397 -- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
398 -- >>> is_upper_triangular m
401 -- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
402 -- >>> is_upper_triangular m
405 is_upper_triangular :: forall m n a.
406 (Eq a, Ring.C a, Arity m, Arity n)
408 is_upper_triangular matrix =
409 ifoldl2 f True matrix
411 f :: Int -> Int -> Bool -> a -> Bool
412 f _ _ False _ = False
419 -- | Returns True if the given matrix is lower-triangular, and False
424 -- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
425 -- >>> is_lower_triangular m
428 -- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
429 -- >>> is_lower_triangular m
432 is_lower_triangular :: (Eq a,
438 is_lower_triangular = is_upper_triangular . transpose
441 -- | Returns True if the given matrix is lower-triangular, and False
442 -- otherwise. The parameter @epsilon@ lets the caller choose a
447 -- >>> let m = fromList [[1,1e-12],[1,1]] :: Mat2 Double
448 -- >>> is_lower_triangular m
450 -- >>> is_lower_triangular' 1e-12 m
453 is_lower_triangular' :: (Ord a,
458 => a -- ^ The tolerance @epsilon@.
461 is_lower_triangular' epsilon = (is_upper_triangular' epsilon) . transpose
464 -- | Returns True if the given matrix is triangular, and False
469 -- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
470 -- >>> is_triangular m
473 -- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
474 -- >>> is_triangular m
477 -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
478 -- >>> is_triangular m
481 is_triangular :: (Ord a,
488 is_triangular m = is_upper_triangular m || is_lower_triangular m
491 -- | Delete the @i@th row and @j@th column from the matrix. The name
492 -- \"preminor\" is made up, but is meant to signify that this is
493 -- usually used in the computationof a minor. A minor is simply the
494 -- determinant of a preminor in that case.
498 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
499 -- >>> preminor m 0 0 :: Mat2 Int
501 -- >>> preminor m 1 1 :: Mat2 Int
504 preminor :: (Arity m, Arity n)
509 preminor (Mat rows) i j = m
511 rows' = delete rows i
512 m = Mat $ V.map ((flip delete) j) rows'
515 -- | Compute the i,jth minor of a @matrix@.
519 -- >>> let m1 = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Double
523 minor :: (Arity m, Determined (Mat m m) a)
528 minor matrix i j = determinant (preminor matrix i j)
530 class (Eq a, Ring.C a) => Determined p a where
531 determinant :: (p a) -> a
533 instance (Eq a, Ring.C a) => Determined (Mat (S Z) (S Z)) a where
534 determinant = unscalar
540 Determined (Mat (S n) (S n)) a)
541 => Determined (Mat (S (S n)) (S (S n))) a where
542 -- | The recursive definition with a special-case for triangular matrices.
546 -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
551 | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ]
552 | otherwise = determinant_recursive
556 determinant_recursive =
557 sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (minor m 0 j)
558 | j <- [0..(ncols m)-1] ]
562 -- | Matrix multiplication.
566 -- >>> let m1 = fromList [[1,2,3], [4,5,6]] :: Mat N2 N3 Int
567 -- >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat N3 N2 Int
572 (*) :: (Ring.C a, Arity m, Arity n, Arity p)
576 (*) m1 m2 = construct lambda
578 lambda i j = (transpose $ row m1 i) `dot` (column m2 j)
582 instance (Ring.C a, Arity m, Arity n) => Additive.C (Mat m n a) where
584 (Mat rows_one) + (Mat rows_two) =
585 Mat $ V.zipWith (V.zipWith (+)) rows_one rows_two
587 (Mat rows_one) - (Mat rows_two) =
588 Mat $ V.zipWith (V.zipWith (-)) rows_one rows_two
590 zero = Mat (V.replicate $ V.replicate (fromInteger 0))
593 instance (Ring.C a, Arity m, Arity n, m ~ n) => Ring.C (Mat (S m) (S n) a) where
594 -- The first * is ring multiplication, the second is matrix
599 instance (Ring.C a, Arity m, Arity n) => Module.C a (Mat m n a) where
600 -- We can multiply a matrix by a scalar of the same type as its
602 x *> (Mat rows) = Mat $ V.map (V.map (NP.* x)) rows
605 instance (Absolute.C a,
609 => Normed (Col (S m) a) where
610 -- | Generic p-norms for vectors in R^n that are represented as n-by-1
615 -- >>> let v1 = vec2d (3,4)
621 -- >>> let v1 = vec2d (-1,1) :: Col2 Double
622 -- >>> norm_p 1 v1 :: Double
625 norm_p p (Mat rows) =
626 (root p') $ sum [fromRational' (toRational $ abs x)^p' | x <- xs]
629 xs = concat $ V.toList $ V.map V.toList rows
631 -- | The infinity norm.
635 -- >>> let v1 = vec3d (1,5,2)
639 norm_infty (Mat rows) =
640 fromRational' $ toRational $ V.maximum $ V.map V.maximum rows
643 -- | Compute the Frobenius norm of a matrix. This essentially treats
644 -- the matrix as one long vector containing all of its entries (in
645 -- any order, it doesn't matter).
649 -- >>> let m = fromList [[1, 2, 3],[4,5,6],[7,8,9]] :: Mat3 Double
650 -- >>> frobenius_norm m == sqrt 285
653 -- >>> let m = fromList [[1, -1, 1],[-1,1,-1],[1,-1,1]] :: Mat3 Double
654 -- >>> frobenius_norm m == 3
657 frobenius_norm :: (Arity m, Arity n, Algebraic.C a, Ring.C a)
660 frobenius_norm matrix =
661 sqrt $ element_sum2 $ squares
663 squares = map2 (^2) matrix
666 -- Vector helpers. We want it to be easy to create low-dimension
667 -- column vectors, which are nx1 matrices.
669 -- | Convenient constructor for 2D vectors.
673 -- >>> import Roots.Simple
674 -- >>> let fst m = m !!! (0,0)
675 -- >>> let snd m = m !!! (1,0)
676 -- >>> let h = 0.5 :: Double
677 -- >>> let g1 m = 1.0 + h NP.* exp(-((fst m)^2))/(1.0 + (snd m)^2)
678 -- >>> let g2 m = 0.5 + h NP.* atan((fst m)^2 + (snd m)^2)
679 -- >>> let g u = vec2d ((g1 u), (g2 u))
680 -- >>> let u0 = vec2d (1.0, 1.0)
681 -- >>> let eps = 1/(10^9)
682 -- >>> fixed_point g eps u0
683 -- ((1.0728549599342185),(1.0820591495686167))
685 vec1d :: (a) -> Col1 a
686 vec1d (x) = Mat (mk1 (mk1 x))
688 vec2d :: (a,a) -> Col2 a
689 vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y))
691 vec3d :: (a,a,a) -> Col3 a
692 vec3d (x,y,z) = Mat (mk3 (mk1 x) (mk1 y) (mk1 z))
694 vec4d :: (a,a,a,a) -> Col4 a
695 vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z))
697 vec5d :: (a,a,a,a,a) -> Col5 a
698 vec5d (v,w,x,y,z) = Mat (mk5 (mk1 v) (mk1 w) (mk1 x) (mk1 y) (mk1 z))
701 -- Since we commandeered multiplication, we need to create 1x1
702 -- matrices in order to multiply things.
703 scalar :: a -> Mat1 a
704 scalar x = Mat (mk1 (mk1 x))
706 -- Get the scalar value out of a 1x1 matrix.
707 unscalar :: Mat1 a -> a
708 unscalar (Mat rows) = V.head $ V.head rows
711 dot :: (Ring.C a, Arity m)
715 v1 `dot` v2 = element_sum2 $ zipwith2 (NP.*) v1 v2
718 -- | The angle between @v1@ and @v2@ in Euclidean space.
722 -- >>> let v1 = vec2d (1.0, 0.0)
723 -- >>> let v2 = vec2d (0.0, 1.0)
724 -- >>> angle v1 v2 == pi/2.0
727 angle :: (Transcendental.C a,
738 theta = (recip norms) NP.* (v1 `dot` v2)
739 norms = (norm v1) NP.* (norm v2)
742 -- | Retrieve the diagonal elements of the given matrix as a \"column
743 -- vector,\" i.e. a m-by-1 matrix. We require the matrix to be
744 -- square to avoid ambiguity in the return type which would ideally
745 -- have dimension min(m,n) supposing an m-by-n matrix.
749 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
753 diagonal :: (Arity m) => Mat m m a -> Col m a
757 lambda i _ = matrix !!! (i,i)
760 -- | Given a square @matrix@, return a new matrix of the same size
761 -- containing only the on-diagonal entries of @matrix@. The
762 -- off-diagonal entries are set to zero.
766 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
767 -- >>> diagonal_part m
768 -- ((1,0,0),(0,5,0),(0,0,9))
770 diagonal_part :: (Arity m, Ring.C a)
773 diagonal_part matrix =
776 lambda i j = if i == j then matrix !!! (i,j) else 0
779 -- | Given a square @matrix@, return a new matrix of the same size
780 -- containing only the on-diagonal and below-diagonal entries of
781 -- @matrix@. The above-diagonal entries are set to zero.
785 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
787 -- ((1,0,0),(4,5,0),(7,8,9))
789 lt_part :: (Arity m, Ring.C a)
795 lambda i j = if i >= j then matrix !!! (i,j) else 0
798 -- | Given a square @matrix@, return a new matrix of the same size
799 -- containing only the below-diagonal entries of @matrix@. The on-
800 -- and above-diagonal entries are set to zero.
804 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
805 -- >>> lt_part_strict m
806 -- ((0,0,0),(4,0,0),(7,8,0))
808 lt_part_strict :: (Arity m, Ring.C a)
811 lt_part_strict matrix =
814 lambda i j = if i > j then matrix !!! (i,j) else 0
817 -- | Given a square @matrix@, return a new matrix of the same size
818 -- containing only the on-diagonal and above-diagonal entries of
819 -- @matrix@. The below-diagonal entries are set to zero.
823 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
825 -- ((1,2,3),(0,5,6),(0,0,9))
827 ut_part :: (Arity m, Ring.C a)
830 ut_part = transpose . lt_part . transpose
833 -- | Given a square @matrix@, return a new matrix of the same size
834 -- containing only the above-diagonal entries of @matrix@. The on-
835 -- and below-diagonal entries are set to zero.
839 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
840 -- >>> ut_part_strict m
841 -- ((0,2,3),(0,0,6),(0,0,0))
843 ut_part_strict :: (Arity m, Ring.C a)
846 ut_part_strict = transpose . lt_part_strict . transpose
849 -- | Compute the trace of a square matrix, the sum of the elements
850 -- which lie on its diagonal. We require the matrix to be
851 -- square to avoid ambiguity in the return type which would ideally
852 -- have dimension min(m,n) supposing an m-by-n matrix.
856 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
860 trace :: (Arity m, Ring.C a) => Mat m m a -> a
861 trace = element_sum2 . diagonal
865 -- | Zip together two matrices.
867 -- TODO: don't cheat with construct (map V.zips instead).
871 -- >>> let m1 = fromList [[1],[1],[1]] :: Col3 Int
872 -- >>> let m2 = fromList [[1],[2],[3]] :: Col3 Int
874 -- (((1,1)),((1,2)),((1,3)))
876 -- >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
877 -- >>> let m2 = fromList [[1,1],[1,1]] :: Mat2 Int
879 -- (((1,1),(2,1)),((3,1),(4,1)))
881 zip2 :: (Arity m, Arity n) => Mat m n a -> Mat m n b -> Mat m n (a,b)
885 lambda i j = (m1 !!! (i,j), m2 !!! (i,j))
888 -- | Zip together three matrices.
890 -- TODO: don't cheat with construct (map V.zips instead).
894 -- >>> let m1 = fromList [[1],[1],[1]] :: Col3 Int
895 -- >>> let m2 = fromList [[1],[2],[3]] :: Col3 Int
896 -- >>> let m3 = fromList [[4],[5],[6]] :: Col3 Int
897 -- >>> zip2three m1 m2 m3
898 -- (((1,1,4)),((1,2,5)),((1,3,6)))
900 -- >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
901 -- >>> let m2 = fromList [[1,1],[1,1]] :: Mat2 Int
902 -- >>> let m3 = fromList [[8,2],[6,3]] :: Mat2 Int
903 -- >>> zip2three m1 m2 m3
904 -- (((1,1,8),(2,1,2)),((3,1,6),(4,1,3)))
906 zip2three :: (Arity m, Arity n)
914 lambda i j = (m1 !!! (i,j), m2 !!! (i,j), m3 !!! (i,j))
917 -- | Zip together two matrices using the supplied function.
921 -- >>> let c1 = fromList [[1],[2],[3]] :: Col3 Integer
922 -- >>> let c2 = fromList [[4],[5],[6]] :: Col3 Integer
923 -- >>> zipwith2 (^) c1 c2
926 zipwith2 :: (Arity m, Arity n)
934 lambda i j = f (c1 !!! (i,j)) (c2 !!! (i,j))
937 -- | Map a function over a matrix of any dimensions.
941 -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
945 map2 :: (a -> b) -> Mat m n a -> Mat m n b
952 -- | Fold over the entire matrix passing the coordinates @i@ and @j@
953 -- (of the row/column) to the accumulation function. The fold occurs
954 -- from top-left to bottom-right.
958 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
959 -- >>> ifoldl2 (\i j cur _ -> cur + i + j) 0 m
962 ifoldl2 :: forall a b m n.
963 (Int -> Int -> b -> a -> b)
967 ifoldl2 f initial (Mat rows) =
968 V.ifoldl row_function initial rows
970 -- | The order that we need this in (so that @g idx@ makes sense)
971 -- is a little funny. So that we don't need to pass weird
972 -- functions into ifoldl2, we swap the second and third
973 -- arguments of @f@ calling the result @g@.
974 g :: Int -> b -> Int -> a -> b
977 row_function :: b -> Int -> Vec n a -> b
978 row_function rowinit idx r = V.ifoldl (g idx) rowinit r
981 -- | Left fold over the entries of a matrix (top-left to bottom-right).
983 foldl2 :: forall a b m n.
988 foldl2 f initial matrix =
989 -- Use the index fold but ignore the index arguments.
990 let g _ _ = f in ifoldl2 g initial matrix
993 -- | Fold over the entire matrix passing the coordinates @i@ and @j@
994 -- (of the row/column) to the accumulation function. The fold occurs
995 -- from bottom-right to top-left.
997 -- The order of the arguments in the supplied function are different
998 -- from those in V.ifoldr; we keep them similar to ifoldl2.
1002 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
1003 -- >>> ifoldr2 (\i j cur _ -> cur + i + j) 0 m
1006 ifoldr2 :: forall a b m n.
1007 (Int -> Int -> b -> a -> b)
1011 ifoldr2 f initial (Mat rows) =
1012 V.ifoldr row_function initial rows
1014 -- | Swap the order of arguments in @f@ so that it agrees with the
1015 -- @f@ passed to ifoldl2.
1016 g :: Int -> Int -> a -> b -> b
1017 g w x y z = f w x z y
1019 row_function :: Int -> Vec n a -> b -> b
1020 row_function idx r rowinit = V.ifoldr (g idx) rowinit r
1023 -- | Map a function over a matrix of any dimensions, passing the
1024 -- coordinates @i@ and @j@ to the function @f@.
1028 -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
1029 -- >>> imap2 (\i j _ -> i+j) m
1032 imap2 :: (Int -> Int -> a -> b) -> Mat m n a -> Mat m n b
1033 imap2 f (Mat rows) =
1039 -- | Reverse the order of elements in a matrix.
1043 -- >>> let m1 = fromList [[1,2,3]] :: Row3 Int
1047 -- >>> let m1 = vec3d (1,2,3 :: Int)
1051 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
1053 -- ((9,8,7),(6,5,4),(3,2,1))
1055 reverse2 :: (Arity m, Arity n) => Mat m n a -> Mat m n a
1056 reverse2 (Mat rows) = Mat $ V.reverse $ V.map V.reverse rows
1059 -- | Unsafely set the (i,j) element of the given matrix.
1063 -- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
1064 -- >>> set_idx m (1,1) 17
1065 -- ((1,2,3),(4,17,6),(7,8,9))
1067 set_idx :: forall m n a.
1073 set_idx matrix (i,j) newval =
1074 imap2 updater matrix
1076 updater :: Int -> Int -> a -> a
1077 updater k l existing =
1083 -- | Compute the i,jth cofactor of the given @matrix@. This simply
1084 -- premultiplues the i,jth minor by (-1)^(i+j).
1085 cofactor :: (Arity m, Determined (Mat m m) a)
1086 => Mat (S m) (S m) a
1090 cofactor matrix i j =
1091 (-1)^(toInteger i + toInteger j) NP.* (minor matrix i j)
1094 -- | Compute the inverse of a matrix using cofactor expansion
1095 -- (generalized Cramer's rule).
1099 -- >>> let m1 = fromList [[37,22],[17,54]] :: Mat2 Double
1100 -- >>> let e1 = [54/1624, -22/1624] :: [Double]
1101 -- >>> let e2 = [-17/1624, 37/1624] :: [Double]
1102 -- >>> let expected = fromList [e1, e2] :: Mat2 Double
1103 -- >>> let actual = inverse m1
1104 -- >>> frobenius_norm (actual - expected) < 1e-12
1107 inverse :: (Arity m,
1108 Determined (Mat (S m) (S m)) a,
1109 Determined (Mat m m) a,
1111 => Mat (S m) (S m) a
1112 -> Mat (S m) (S m) a
1114 (1 / (determinant matrix)) *> (transpose $ construct lambda)
1116 lambda i j = cofactor matrix i j
1120 -- | Retrieve the rows of a matrix as a column matrix. If the given
1121 -- matrix is m-by-n, the result would be an m-by-1 column whose
1122 -- entries are 1-by-n row matrices.
1126 -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
1127 -- >>> (rows2 m) !!! (0,0)
1129 -- >>> (rows2 m) !!! (1,0)
1132 rows2 :: (Arity m, Arity n)
1136 Mat $ V.map (mk1. Mat . mk1) rows
1140 -- | Sum the elements of a matrix.
1144 -- >>> let m = fromList [[1,-1],[3,4]] :: Mat2 Int
1145 -- >>> element_sum2 m
1148 element_sum2 :: (Arity m, Arity n, Additive.C a) => Mat m n a -> a
1149 element_sum2 = foldl2 (+) zero