From 1f64a1a33b2636ef2e863a0b577c8d8d50233580 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sun, 24 Feb 2013 02:41:03 -0500 Subject: [PATCH] Replace the whole Matrix implementation with something a little better. Add the Linear.System module. Get all doctests working again. Update .ghci/cabal files. --- .ghci | 4 +- numerical-analysis.cabal | 4 +- src/Linear/Matrix.hs | 294 +++++++++++++++++---------------------- src/Linear/System.hs | 37 ++--- src/Linear/Vector.hs | 49 ++----- 5 files changed, 153 insertions(+), 235 deletions(-) diff --git a/.ghci b/.ghci index 16cb969..9a6a4d4 100644 --- a/.ghci +++ b/.ghci @@ -7,6 +7,7 @@ src/Integration/Simpson.hs src/Integration/Trapezoid.hs src/Linear/Matrix.hs + src/Linear/System.hs src/Linear/Vector.hs src/Misc.hs src/Normed.hs @@ -17,8 +18,9 @@ import BigFloat import Integration.Simpson import Integration.Trapezoid -import Linear.Vector import Linear.Matrix +import Linear.System +import Linear.Vector import Misc import Normed import ODE.IVP diff --git a/numerical-analysis.cabal b/numerical-analysis.cabal index 0024786..e459791 100644 --- a/numerical-analysis.cabal +++ b/numerical-analysis.cabal @@ -18,8 +18,8 @@ data-files: makefile library exposed-modules: Integration.Simpson, Integration.Trapezoid, - Linear.Matrix, Linear.Vector, Misc, Normed, ODE.IVP, Roots.Simple, - Roots.Fast + Linear.Matrix, Linear.System, Linear.Vector, Misc, Normed, ODE.IVP, + Roots.Simple, Roots.Fast build-depends: base >= 3 && < 5, diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs index 5bb5519..7452007 100644 --- a/src/Linear/Matrix.hs +++ b/src/Linear/Matrix.hs @@ -6,26 +6,42 @@ {-# LANGUAGE TypeFamilies #-} {-# LANGUAGE RebindableSyntax #-} +-- | Boxed matrices; that is, boxed m-vectors of boxed n-vectors. We +-- assume that the underlying representation is +-- Data.Vector.Fixed.Boxed.Vec for simplicity. It was tried in +-- generality and failed. +-- module Linear.Matrix where import Data.List (intercalate) import Data.Vector.Fixed ( - Dim, N1, - Vector + N2, + N3, + N4, + N5, + S, + Z, + mk1, + mk2, + mk3, + mk4, + mk5 ) import qualified Data.Vector.Fixed as V ( and, fromList, length, map, + maximum, replicate, toList, zipWith ) -import Data.Vector.Fixed.Internal (Arity, arity, S) +import Data.Vector.Fixed.Boxed (Vec) +import Data.Vector.Fixed.Internal (Arity, arity) import Linear.Vector import Normed @@ -41,16 +57,14 @@ import qualified Algebra.ToRational as ToRational import qualified Algebra.Transcendental as Transcendental import qualified Prelude as P -data Mat v w a = (Vector v (w a), Vector w a) => Mat (v (w a)) -type Mat1 a = Mat D1 D1 a -type Mat2 a = Mat D2 D2 a -type Mat3 a = Mat D3 D3 a -type Mat4 a = Mat D4 D4 a +data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a)) +type Mat1 a = Mat N1 N1 a +type Mat2 a = Mat N2 N2 a +type Mat3 a = Mat N3 N3 a +type Mat4 a = Mat N4 N4 a +type Mat5 a = Mat N5 N5 a --- We can't just declare that all instances of Vector are instances of --- Eq unfortunately. We wind up with an overlapping instance for --- w (w a). -instance (Eq a, Vector v Bool, Vector w Bool) => Eq (Mat v w a) where +instance (Eq a) => Eq (Mat m n a) where -- | Compare a row at a time. -- -- Examples: @@ -70,7 +84,7 @@ instance (Eq a, Vector v Bool, Vector w Bool) => Eq (Mat v w a) where comp row1 row2 = V.and (V.zipWith (==) row1 row2) -instance (Show a, Vector v String, Vector w String) => Show (Mat v w a) where +instance (Show a) => Show (Mat m n a) where -- | Display matrices and vectors as ordinary tuples. This is poor -- practice, but these results are primarily displayed -- interactively and convenience trumps correctness (said the guy @@ -94,22 +108,21 @@ instance (Show a, Vector v String, Vector w String) => Show (Mat v w a) where element_strings = P.map show v1l - -- | Convert a matrix to a nested list. -toList :: Mat v w a -> [[a]] +toList :: Mat m n a -> [[a]] toList (Mat rows) = map V.toList (V.toList rows) -- | Create a matrix from a nested list. -fromList :: (Vector v (w a), Vector w a, Vector v a) => [[a]] -> Mat v w a +fromList :: (Arity m, Arity n) => [[a]] -> Mat m n a fromList vs = Mat (V.fromList $ map V.fromList vs) -- | Unsafe indexing. -(!!!) :: (Vector w a) => Mat v w a -> (Int, Int) -> a +(!!!) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> a (!!!) m (i, j) = (row m i) ! j -- | Safe indexing. -(!!?) :: Mat v w a -> (Int, Int) -> Maybe a +(!!?) :: Mat m n a -> (Int, Int) -> Maybe a (!!?) m@(Mat rows) (i, j) | i < 0 || j < 0 = Nothing | i > V.length rows = Nothing @@ -119,27 +132,30 @@ fromList vs = Mat (V.fromList $ map V.fromList vs) -- | The number of rows in the matrix. -nrows :: Mat v w a -> Int -nrows (Mat rows) = V.length rows +nrows :: forall m n a. (Arity m) => Mat m n a -> Int +nrows _ = arity (undefined :: m) -- | The number of columns in the first row of the -- matrix. Implementation stolen from Data.Vector.Fixed.length. -ncols :: forall v w a. (Vector w a) => Mat v w a -> Int -ncols _ = (arity (undefined :: Dim w)) +ncols :: forall m n a. (Arity n) => Mat m n a -> Int +ncols _ = arity (undefined :: n) + -- | Return the @i@th row of @m@. Unsafe. -row :: Mat v w a -> Int -> w a +row :: Mat m n a -> Int -> (Vec n a) row (Mat rows) i = rows ! i -- | Return the @j@th column of @m@. Unsafe. -column :: (Vector v a) => Mat v w a -> Int -> v a +column :: Mat m n a -> Int -> (Vec m a) column (Mat rows) j = V.map (element j) rows where element = flip (!) + + -- | Transpose @m@; switch it's columns and its rows. This is a dirty -- implementation.. it would be a little cleaner to use imap, but it -- doesn't seem to work. @@ -152,11 +168,7 @@ column (Mat rows) j = -- >>> transpose m -- ((1,3),(2,4)) -- -transpose :: (Vector w (v a), - Vector v a, - Vector w a) - => Mat v w a - -> Mat w v a +transpose :: (Arity m, Arity n) => Mat m n a -> Mat n m a transpose m = Mat $ V.fromList column_list where column_list = [ column m i | i <- [0..(ncols m)-1] ] @@ -174,13 +186,7 @@ transpose m = Mat $ V.fromList column_list -- >>> symmetric m2 -- False -- -symmetric :: (Vector v (w a), - Vector w a, - v ~ w, - Vector w Bool, - Eq a) - => Mat v w a - -> Bool +symmetric :: (Eq a, Arity m) => Mat m m a -> Bool symmetric m = m == (transpose m) @@ -198,19 +204,17 @@ symmetric m = -- >>> construct lambda :: Mat3 Int -- ((0,1,2),(1,2,3),(2,3,4)) -- -construct :: forall v w a. - (Vector v (w a), - Vector w a) - => (Int -> Int -> a) - -> Mat v w a +construct :: forall m n a. (Arity m, Arity n) + => (Int -> Int -> a) -> Mat m n a construct lambda = Mat rows where -- The arity trick is used in Data.Vector.Fixed.length. - imax = (arity (undefined :: Dim v)) - 1 - jmax = (arity (undefined :: Dim w)) - 1 + imax = (arity (undefined :: m)) - 1 + jmax = (arity (undefined :: n)) - 1 row' i = V.fromList [ lambda i j | j <- [0..jmax] ] rows = V.fromList [ row' i | i <- [0..imax] ] + -- | Given a positive-definite matrix @m@, computes the -- upper-triangular matrix @r@ with (transpose r)*r == m and all -- values on the diagonal of @r@ positive. @@ -223,13 +227,8 @@ construct lambda = Mat rows -- >>> (transpose (cholesky m1)) * (cholesky m1) -- ((20.000000000000004,-1.0),(-1.0,20.0)) -- -cholesky :: forall a v w. - (Algebraic.C a, - Vector v (w a), - Vector w a, - Vector v a) - => (Mat v w a) - -> (Mat v w a) +cholesky :: forall m n a. (Algebraic.C a, Arity m, Arity n) + => (Mat m n a) -> (Mat m n a) cholesky m = construct r where r :: Int -> Int -> a @@ -252,7 +251,8 @@ cholesky m = construct r -- >>> is_upper_triangular m -- True -- -is_upper_triangular :: (Eq a, Ring.C a, Vector w a) => Mat v w a -> Bool +is_upper_triangular :: (Eq a, Ring.C a, Arity m, Arity n) + => Mat m n a -> Bool is_upper_triangular m = and $ concat results where @@ -279,10 +279,9 @@ is_upper_triangular m = -- is_lower_triangular :: (Eq a, Ring.C a, - Vector w a, - Vector w (v a), - Vector v a) - => Mat v w a + Arity m, + Arity n) + => Mat m n a -> Bool is_lower_triangular = is_upper_triangular . transpose @@ -306,10 +305,9 @@ is_lower_triangular = is_upper_triangular . transpose -- is_triangular :: (Eq a, Ring.C a, - Vector w a, - Vector w (v a), - Vector v a) - => Mat v w a + Arity m, + Arity n) + => Mat m n a -> Bool is_triangular m = is_upper_triangular m || is_lower_triangular m @@ -324,73 +322,60 @@ is_triangular m = is_upper_triangular m || is_lower_triangular m -- >>> minor m 1 1 :: Mat2 Int -- ((1,3),(7,9)) -- -minor :: (Dim v ~ S (Dim u), - Dim w ~ S (Dim z), - Vector z a, - Vector u (w a), - Vector u (z a)) - => Mat v w a +minor :: (m ~ S r, + n ~ S t, + Arity r, + Arity t) + => Mat m n a -> Int -> Int - -> Mat u z a + -> Mat r t a minor (Mat rows) i j = m where rows' = delete rows i m = Mat $ V.map ((flip delete) j) rows' -determinant :: (Eq a, - Ring.C a, - Vector w a, - Vector w (v a), - Vector v a, - Dim v ~ S r, - Dim w ~ S t) - => Mat v w a - -> a -determinant m - | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ] - | otherwise = undefined --determinant_recursive m - -{- -determinant_recursive :: forall v w a r c. - (Eq a, - Ring.C a, - Vector w a) - => Mat (v r) (w c) a - -> a -determinant_recursive m - | (ncols m) == 0 || (nrows m) == 0 = error "don't do that" - | (ncols m) == 1 && (nrows m) == 1 = m !!! (0,0) -- Base case - | otherwise = - sum [ (-1)^(1+(toInteger j)) NP.* (m' 1 j) NP.* (det_minor 1 j) - | j <- [0..(ncols m)-1] ] - where - m' i j = m !!! (i,j) - - det_minor :: Int -> Int -> a - det_minor i j = determinant (minor m i j) --} +class (Eq a, Ring.C a) => Determined p a where + determinant :: (p a) -> a + +instance (Eq a, Ring.C a) => Determined (Mat (S Z) (S Z)) a where + determinant m = m !!! (0,0) + +instance (Eq a, Ring.C a, Arity m) => Determined (Mat m m) a where + determinant _ = undefined + +instance (Eq a, Ring.C a, Arity n) + => Determined (Mat (S (S n)) (S (S n))) a where + determinant m + | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ] + | otherwise = determinant_recursive + where + m' i j = m !!! (i,j) + + det_minor i j = determinant (minor m i j) + + determinant_recursive = + sum [ (-1)^(1+(toInteger j)) NP.* (m' 0 j) NP.* (det_minor 0 j) + | j <- [0..(ncols m)-1] ] + + -- | Matrix multiplication. Our 'Num' instance doesn't define one, and -- we need additional restrictions on the result type anyway. -- -- Examples: -- --- >>> let m1 = fromList [[1,2,3], [4,5,6]] :: Mat D2 D3 Int --- >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat D3 D2 Int +-- >>> let m1 = fromList [[1,2,3], [4,5,6]] :: Mat N2 N3 Int +-- >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat N3 N2 Int -- >>> m1 * m2 -- ((22,28),(49,64)) -- infixl 7 * -(*) :: (Ring.C a, - Vector v a, - Vector w a, - Vector z a, - Vector v (z a)) - => Mat v w a - -> Mat w z a - -> Mat v z a +(*) :: (Ring.C a, Arity m, Arity n, Arity p) + => Mat m n a + -> Mat n p a + -> Mat m p a (*) m1 m2 = construct lambda where lambda i j = @@ -398,10 +383,7 @@ infixl 7 * -instance (Ring.C a, - Vector v (w a), - Vector w a) - => Additive.C (Mat v w a) where +instance (Ring.C a, Arity m, Arity n) => Additive.C (Mat m n a) where (Mat rows1) + (Mat rows2) = Mat $ V.zipWith (V.zipWith (+)) rows1 rows2 @@ -412,20 +394,13 @@ instance (Ring.C a, zero = Mat (V.replicate $ V.replicate (fromInteger 0)) -instance (Ring.C a, - Vector v (w a), - Vector w a, - v ~ w) - => Ring.C (Mat v w a) where +instance (Ring.C a, Arity m, Arity n, m ~ n) => Ring.C (Mat m n a) where -- The first * is ring multiplication, the second is matrix -- multiplication. m1 * m2 = m1 * m2 -instance (Ring.C a, - Vector v (w a), - Vector w a) - => Module.C a (Mat v w a) where +instance (Ring.C a, Arity m, Arity n) => Module.C a (Mat m n a) where -- We can multiply a matrix by a scalar of the same type as its -- elements. x *> (Mat rows) = Mat $ V.map (V.map (NP.* x)) rows @@ -433,11 +408,9 @@ instance (Ring.C a, instance (Algebraic.C a, ToRational.C a, - Vector v (w a), - Vector w a, - Vector v a, - Vector v [a]) - => Normed (Mat v w a) where + Arity m, + Arity n) + => Normed (Mat (S m) (S n) a) where -- | Generic p-norms. The usual norm in R^n is (norm_p 2). We treat -- all matrices as big vectors. -- @@ -455,9 +428,7 @@ instance (Algebraic.C a, p' = toInteger p xs = concat $ V.toList $ V.map V.toList rows - -- | The infinity norm. We don't use V.maximum here because it - -- relies on a type constraint that the vector be non-empty and I - -- don't know how to pattern match it away. + -- | The infinity norm. -- -- Examples: -- @@ -465,11 +436,8 @@ instance (Algebraic.C a, -- >>> norm_infty v1 -- 5 -- - norm_infty m@(Mat rows) - | nrows m == 0 || ncols m == 0 = 0 - | otherwise = - fromRational' $ toRational $ - P.maximum $ V.toList $ V.map (P.maximum . V.toList) rows + norm_infty (Mat rows) = + fromRational' $ toRational $ V.maximum $ V.map V.maximum rows @@ -483,41 +451,40 @@ instance (Algebraic.C a, -- Examples: -- -- >>> import Roots.Simple +-- >>> let fst m = m !!! (0,0) +-- >>> let snd m = m !!! (1,0) -- >>> let h = 0.5 :: Double --- >>> let g1 (Mat (D2 (D1 x) (D1 y))) = 1.0 + h NP.* exp(-(x^2))/(1.0 + y^2) --- >>> let g2 (Mat (D2 (D1 x) (D1 y))) = 0.5 + h NP.* atan(x^2 + y^2) +-- >>> let g1 m = 1.0 + h NP.* exp(-((fst m)^2))/(1.0 + (snd m)^2) +-- >>> let g2 m = 0.5 + h NP.* atan((fst m)^2 + (snd m)^2) -- >>> let g u = vec2d ((g1 u), (g2 u)) -- >>> let u0 = vec2d (1.0, 1.0) -- >>> let eps = 1/(10^9) -- >>> fixed_point g eps u0 -- ((1.0728549599342185),(1.0820591495686167)) -- -vec1d :: (a) -> Mat D1 D1 a -vec1d (x) = Mat (D1 (D1 x)) +vec1d :: (a) -> Mat N1 N1 a +vec1d (x) = Mat (mk1 (mk1 x)) -vec2d :: (a,a) -> Mat D2 D1 a -vec2d (x,y) = Mat (D2 (D1 x) (D1 y)) +vec2d :: (a,a) -> Mat N2 N1 a +vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y)) -vec3d :: (a,a,a) -> Mat D3 D1 a -vec3d (x,y,z) = Mat (D3 (D1 x) (D1 y) (D1 z)) +vec3d :: (a,a,a) -> Mat N3 N1 a +vec3d (x,y,z) = Mat (mk3 (mk1 x) (mk1 y) (mk1 z)) -vec4d :: (a,a,a,a) -> Mat D4 D1 a -vec4d (w,x,y,z) = Mat (D4 (D1 w) (D1 x) (D1 y) (D1 z)) +vec4d :: (a,a,a,a) -> Mat N4 N1 a +vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z)) + +vec5d :: (a,a,a,a,a) -> Mat N5 N1 a +vec5d (v,w,x,y,z) = Mat (mk5 (mk1 v) (mk1 w) (mk1 x) (mk1 y) (mk1 z)) -- Since we commandeered multiplication, we need to create 1x1 -- matrices in order to multiply things. -scalar :: a -> Mat D1 D1 a -scalar x = Mat (D1 (D1 x)) - -dot :: (RealRing.C a, - Dim w ~ N1, - Dim v ~ S n, - Vector v a, - Vector w a, - Vector w (v a), - Vector w (w a)) - => Mat v w a - -> Mat v w a +scalar :: a -> Mat N1 N1 a +scalar x = Mat (mk1 (mk1 x)) + +dot :: (RealRing.C a, n ~ N1, m ~ S t, Arity t) + => Mat m n a + -> Mat m n a -> a v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0) @@ -533,17 +500,12 @@ v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0) -- angle :: (Transcendental.C a, RealRing.C a, - Dim w ~ N1, - Dim v ~ S n, - Vector w (w a), - Vector v [a], - Vector v a, - Vector w a, - Vector v (w a), - Vector w (v a), + n ~ N1, + m ~ S t, + Arity t, ToRational.C a) - => Mat v w a - -> Mat v w a + => Mat m n a + -> Mat m n a -> a angle v1 v2 = acos theta diff --git a/src/Linear/System.hs b/src/Linear/System.hs index e0fdf1c..58d8d78 100644 --- a/src/Linear/System.hs +++ b/src/Linear/System.hs @@ -5,7 +5,7 @@ module Linear.System where -import Data.Vector.Fixed (Dim, N1, Vector) +import Data.Vector.Fixed (Arity, N1) import Linear.Matrix @@ -13,7 +13,6 @@ import NumericPrelude hiding ((*), abs) import qualified NumericPrelude as NP ((*)) import qualified Algebra.Field as Field -import Debug.Trace (trace, traceShow) -- | Solve the system m' * x = b', where m' is upper-triangular. Will -- probably crash if m' is non-singular. The result is the vector x. @@ -21,27 +20,21 @@ import Debug.Trace (trace, traceShow) -- Examples: -- -- >>> let identity = fromList [[1,0,0],[0,1,0],[0,0,1]] :: Mat3 Double --- >>> let b = vec3d (1,2,3) +-- >>> let b = vec3d (1, 2, 3::Double) -- >>> forward_substitute identity b -- ((1.0),(2.0),(3.0)) -- >>> (forward_substitute identity b) == b -- True -- -- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Double --- >>> let b = vec2d (1,1) +-- >>> let b = vec2d (1, 1::Double) -- >>> forward_substitute m b -- ((1.0),(0.0)) -- -forward_substitute :: forall a v w z. - (Show a, Field.C a, - Vector z a, - Vector w (z a), - Vector w a, - Dim z ~ N1, - v ~ w) - => Mat v w a - -> Mat w z a - -> Mat w z a +forward_substitute :: forall a m. (Field.C a, Arity m) + => Mat m m a + -> Mat m N1 a + -> Mat m N1 a forward_substitute m' b' = x' where x' = construct lambda @@ -72,22 +65,16 @@ forward_substitute m' b' = x' -- Examples: -- -- >>> let identity = fromList [[1,0,0],[0,1,0],[0,0,1]] :: Mat3 Double --- >>> let b = vec3d (1,2,3) +-- >>> let b = vec3d (1, 2, 3::Double) -- >>> backward_substitute identity b -- ((1.0),(2.0),(3.0)) -- >>> (backward_substitute identity b) == b -- True -- -backward_substitute :: (Show a, Field.C a, - Vector z a, - Vector v (w a), - Vector w (z a), - Vector w a, - Dim z ~ N1, - v ~ w) - => Mat v w a - -> Mat w z a - -> Mat w z a +backward_substitute :: (Field.C a, Arity m) + => Mat m m a + -> Mat m N1 a + -> Mat m N1 a backward_substitute m b = forward_substitute (transpose m) b diff --git a/src/Linear/Vector.hs b/src/Linear/Vector.hs index ec6ebcb..aa1568b 100644 --- a/src/Linear/Vector.hs +++ b/src/Linear/Vector.hs @@ -9,17 +9,12 @@ where import Data.Vector.Fixed ( Dim, - Fun(..), N1, - N2, - N3, N4, N5, S, Vector(..), - construct, fromList, - inspect, toList, ) import qualified Data.Vector.Fixed as V ( @@ -31,43 +26,13 @@ type Vec1 = Vec N1 type Vec4 = Vec N4 type Vec5 = Vec N5 --- * Low-dimension vector wrappers. --- --- These wrappers are instances of 'Vector', so they inherit all of --- the userful instances defined above. But, they use fixed --- constructors, so you can pattern match out the individual --- components. - -data D1 a = D1 a deriving (Show, Eq) -type instance Dim D1 = N1 -instance Vector D1 a where - inspect (D1 x) (Fun f) = f x - construct = Fun D1 - -data D2 a = D2 a a deriving (Show, Eq) -type instance Dim D2 = N2 -instance Vector D2 a where - inspect (D2 x y) (Fun f) = f x y - construct = Fun D2 - -data D3 a = D3 a a a deriving (Show, Eq) -type instance Dim D3 = N3 -instance Vector D3 a where - inspect (D3 x y z) (Fun f) = f x y z - construct = Fun D3 - -data D4 a = D4 a a a a deriving (Show, Eq) -type instance Dim D4 = N4 -instance Vector D4 a where - inspect (D4 w x y z) (Fun f) = f w x y z - construct = Fun D4 - -- | Unsafe indexing. -- -- Examples: -- --- >>> let v1 = D2 1 2 +-- >>> import Data.Vector.Fixed (mk2) +-- >>> let v1 = mk2 1 2 :: Vec2 Int -- >>> v1 ! 1 -- 2 -- @@ -78,7 +43,8 @@ instance Vector D4 a where -- -- Examples: -- --- >>> let v1 = D3 1 2 3 +-- >>> import Data.Vector.Fixed (mk3) +-- >>> let v1 = mk3 1 2 3 :: Vec3 Int -- >>> v1 !? 2 -- Just 3 -- >>> v1 !? 3 @@ -94,9 +60,10 @@ instance Vector D4 a where -- -- Examples: -- --- >>> let b = D3 1 2 3 --- >>> delete b 1 :: D2 Int --- D2 1 3 +-- >>> import Data.Vector.Fixed (mk3) +-- >>> let b = mk3 1 2 3 :: Vec3 Int +-- >>> delete b 1 :: Vec2 Int +-- fromList [1,3] -- delete :: (Vector v a, Vector w a, Dim v ~ S (Dim w)) => v a -> Int -> w a delete v1 idx = -- 2.44.2