From: Michael Orlitzky Date: Wed, 19 Feb 2014 01:25:05 +0000 (-0500) Subject: Generalize the signature of zipwith2. X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=6892f38cf4dab275a91fbe06bfb8efe229518336;p=numerical-analysis.git Generalize the signature of zipwith2. Implement the dot product using zipwith2. Implement matrix multiplication using the dot product. Implement the frobenius_norm with zipwith2. --- diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs index 8266557..8006bde 100644 --- a/src/Linear/Matrix.hs +++ b/src/Linear/Matrix.hs @@ -39,7 +39,7 @@ import qualified Data.Vector.Fixed as V ( toList, zipWith ) import Data.Vector.Fixed.Cont ( Arity, arity ) -import Linear.Vector ( Vec, delete, element_sum ) +import Linear.Vector ( Vec, delete ) import Naturals ( N1, N2, N3, N4, N5, N6, N7, N8, N9, N10, S, Z ) import Normed ( Normed(..) ) @@ -546,13 +546,12 @@ instance (Ord a, -- infixl 7 * (*) :: (Ring.C a, Arity m, Arity n, Arity p) - => Mat m n a - -> Mat n p a - -> Mat m p a + => Mat (S m) (S n) a + -> Mat (S n) (S p) a + -> Mat (S m) (S p) a (*) m1 m2 = construct lambda where - lambda i j = - sum [(m1 !!! (i,k)) NP.* (m2 !!! (k,j)) | k <- [0..(ncols m1)-1] ] + lambda i j = (transpose $ row m1 i) `dot` (column m2 j) @@ -567,7 +566,7 @@ instance (Ring.C a, Arity m, Arity n) => Additive.C (Mat m n a) where zero = Mat (V.replicate $ V.replicate (fromInteger 0)) -instance (Ring.C a, Arity m, Arity n, m ~ n) => Ring.C (Mat m n a) where +instance (Ring.C a, Arity m, Arity n, m ~ n) => Ring.C (Mat (S m) (S n) a) where -- The first * is ring multiplication, the second is matrix -- multiplication. m1 * m2 = m1 * m2 @@ -631,12 +630,13 @@ instance (Absolute.C a, -- >>> frobenius_norm m == 3 -- True -- -frobenius_norm :: (Algebraic.C a, Ring.C a) => Mat m n a -> a -frobenius_norm (Mat rows) = - sqrt $ element_sum $ V.map row_sum rows +frobenius_norm :: (Arity m, Arity n, Algebraic.C a, Ring.C a) + => Mat m n a + -> a +frobenius_norm matrix = + sqrt $ element_sum2 $ squares where - -- | Square and add up the entries of a row. - row_sum = element_sum . V.map (^2) + squares = map2 (^2) matrix -- Vector helpers. We want it to be easy to create low-dimension @@ -688,7 +688,7 @@ dot :: (Ring.C a, Arity m) => Col (S m) a -> Col (S m) a -> a -v1 `dot` v2 = unscalar $ ((transpose v1) * v2) +v1 `dot` v2 = element_sum2 $ zipwith2 (NP.*) v1 v2 -- | The angle between @v1@ and @v2@ in Euclidean space. @@ -834,10 +834,8 @@ ut_part_strict = transpose . lt_part_strict . transpose -- 15 -- trace :: (Arity m, Ring.C a) => Mat m m a -> a -trace matrix = - let (Mat rows) = diagonal matrix - in - element_sum $ V.map V.head rows +trace = element_sum2 . diagonal + -- | Zip together two matrices. @@ -901,11 +899,11 @@ zip2three m1 m2 m3 = -- >>> zipwith2 (^) c1 c2 -- ((1),(32),(729)) -- -zipwith2 :: Arity m - => (a -> a -> b) - -> Col m a - -> Col m a - -> Col m b +zipwith2 :: (Arity m, Arity n) + => (a -> b -> c) + -> Mat m n a + -> Mat m n b + -> Mat m n c zipwith2 f c1 c2 = construct lambda where @@ -956,6 +954,18 @@ ifoldl2 f initial (Mat rows) = row_function rowinit idx r = V.ifoldl (g idx) rowinit r +-- | Left fold over the entries of a matrix (top-left to bottom-right). +-- +foldl2 :: forall a b m n. + (b -> a -> b) + -> b + -> Mat m n a + -> b +foldl2 f initial matrix = + -- Use the index fold but ignore the index arguments. + let g _ _ = f in ifoldl2 g initial matrix + + -- | Fold over the entire matrix passing the coordinates @i@ and @j@ -- (of the row/column) to the accumulation function. The fold occurs -- from bottom-right to top-left. @@ -1081,3 +1091,35 @@ inverse matrix = where lambda i j = cofactor matrix i j + + +-- | Retrieve the rows of a matrix as a column matrix. If the given +-- matrix is m-by-n, the result would be an m-by-1 column whose +-- entries are 1-by-n row matrices. +-- +-- Examples: +-- +-- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int +-- >>> (rows2 m) !!! (0,0) +-- ((1,2)) +-- >>> (rows2 m) !!! (1,0) +-- ((3,4)) +-- +rows2 :: (Arity m, Arity n) + => Mat m n a + -> Col m (Row n a) +rows2 (Mat rows) = + Mat $ V.map (mk1. Mat . mk1) rows + + + +-- | Sum the elements of a matrix. +-- +-- Examples: +-- +-- >>> let m = fromList [[1,-1],[3,4]] :: Mat2 Int +-- >>> element_sum2 m +-- 7 +-- +element_sum2 :: (Arity m, Arity n, Additive.C a) => Mat m n a -> a +element_sum2 = foldl2 (+) zero