-{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE NoMonomorphismRestriction #-}
+{-# LANGUAGE ScopedTypeVariables #-}
{-# 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,
- Vector
- )
+ (!),
+ generate,
+ mk1,
+ mk2,
+ mk3,
+ mk4,
+ mk5 )
import qualified Data.Vector.Fixed as V (
+ and,
fromList,
- length,
+ head,
+ ifoldl,
+ ifoldr,
+ imap,
map,
- toList
- )
-import Data.Vector.Fixed.Internal (arity)
+ maximum,
+ replicate,
+ reverse,
+ toList,
+ zipWith )
+import Data.Vector.Fixed.Cont ( Arity, arity )
+import Linear.Vector ( Vec, delete )
+import Naturals ( N1, N2, N3, N4, N5, N6, N7, N8, N9, N10, S, Z )
+import Normed ( Normed(..) )
+
+import NumericPrelude hiding ( (*), abs )
+import qualified NumericPrelude as NP ( (*) )
+import qualified Algebra.Absolute as Absolute ( C )
+import Algebra.Absolute ( abs )
+import qualified Algebra.Additive as Additive ( C )
+import qualified Algebra.Algebraic as Algebraic ( C )
+import Algebra.Algebraic ( root )
+import qualified Algebra.Field as Field ( C )
+import qualified Algebra.Ring as Ring ( C )
+import qualified Algebra.Module as Module ( C )
+import qualified Algebra.RealRing as RealRing ( C )
+import qualified Algebra.ToRational as ToRational ( C )
+import qualified Algebra.Transcendental as Transcendental ( C )
+import qualified Prelude as P ( map )
+
+-- | Our main matrix type.
+data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
+
+-- Type synonyms for n-by-n matrices.
+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
+
+-- * Type synonyms for 1-by-n row "vectors".
+
+-- | Type synonym for row vectors expressed as 1-by-n matrices.
+type Row n a = Mat N1 n a
+
+type Row1 a = Row N1 a
+type Row2 a = Row N2 a
+type Row3 a = Row N3 a
+type Row4 a = Row N4 a
+type Row5 a = Row N5 a
-import Linear.Vector
+-- * Type synonyms for n-by-1 column "vectors".
+
+-- | Type synonym for column vectors expressed as n-by-1 matrices.
+type Col n a = Mat n N1 a
+
+type Col1 a = Col N1 a
+type Col2 a = Col N2 a
+type Col3 a = Col N3 a
+type Col4 a = Col N4 a
+type Col5 a = Col N5 a
+type Col6 a = Col N6 a
+type Col7 a = Col N7 a
+type Col8 a = Col N8 a
+type Col9 a = Col N9 a
+type Col10 a = Col N10 a -- We need a big column for Gaussian quadrature.
+
+
+instance (Eq a) => Eq (Mat m n a) where
+ -- | Compare a row at a time.
+ --
+ -- Examples:
+ --
+ -- >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
+ -- >>> let m2 = fromList [[1,2],[3,4]] :: Mat2 Int
+ -- >>> let m3 = fromList [[5,6],[7,8]] :: Mat2 Int
+ -- >>> m1 == m2
+ -- True
+ -- >>> m1 == m3
+ -- False
+ --
+ (Mat rows1) == (Mat rows2) =
+ V.and $ V.zipWith comp rows1 rows2
+ where
+ -- Compare a row, one column at a time.
+ comp row1 row2 = V.and (V.zipWith (==) row1 row2)
+
+
+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
+ -- who insists his vector lengths be statically checked at
+ -- compile-time).
+ --
+ -- Examples:
+ --
+ -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+ -- >>> show m
+ -- ((1,2),(3,4))
+ --
+ show (Mat rows) =
+ "(" ++ (intercalate "," (V.toList row_strings)) ++ ")"
+ where
+ row_strings = V.map show_vector rows
+ show_vector v1 =
+ "(" ++ (intercalate "," element_strings) ++ ")"
+ where
+ v1l = V.toList v1
+ element_strings = P.map show v1l
-type Mat v w a = Vn v (Vn w a)
-type Mat2 a = Mat Vec2D Vec2D a
-type Mat3 a = Mat Vec3D Vec3D a
-type Mat4 a = Mat Vec4D Vec4D a
-- | Convert a matrix to a nested list.
-toList :: (Vector v (Vn w a), Vector w a) => Mat v w a -> [[a]]
-toList m = map V.toList (V.toList m)
+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 (Vn w a), Vector w a) => [[a]] -> Mat v w a
-fromList vs = V.fromList $ map V.fromList vs
+fromList :: (Arity m, Arity n) => [[a]] -> Mat m n a
+fromList vs = Mat (V.fromList $ map V.fromList vs)
--- | Unsafe indexing.
-(!!!) :: (Vector v (Vn w a), Vector w a) => Mat v w a -> (Int, Int) -> a
-(!!!) m (i, j) = (row m i) ! j
+-- | Unsafe indexing. Much faster than the safe indexing.
+(!!!) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> a
+(!!!) (Mat rows) (i, j) = (rows ! i) ! j
+
-- | Safe indexing.
-(!!?) :: (Vector v (Vn w a), Vector w a) => Mat v w a
- -> (Int, Int)
- -> Maybe a
-(!!?) m (i, j)
- | i < 0 || j < 0 = Nothing
- | i > V.length m = Nothing
- | otherwise = if j > V.length (row m j)
- then Nothing
- else Just $ (row m j) ! j
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+-- >>> m !!? (-1,-1)
+-- Nothing
+-- >>> m !!? (-1,0)
+-- Nothing
+-- >>> m !!? (-1,1)
+-- Nothing
+-- >>> m !!? (0,-1)
+-- Nothing
+-- >>> m !!? (0,0)
+-- Just 1
+-- >>> m !!? (0,1)
+-- Just 2
+-- >>> m !!? (1,-1)
+-- Nothing
+-- >>> m !!? (1,0)
+-- Just 3
+-- >>> m !!? (1,1)
+-- Just 4
+-- >>> m !!? (2,-1)
+-- Nothing
+-- >>> m !!? (2,0)
+-- Nothing
+-- >>> m !!? (2,1)
+-- Nothing
+-- >>> m !!? (2,2)
+-- Nothing
+--
+(!!?) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> Maybe a
+(!!?) matrix idx =
+ ifoldl2 f Nothing matrix
+ where
+ f k l found cur = if (k,l) == idx then (Just cur) else found
-- | The number of rows in the matrix.
-nrows :: forall v w a. (Vector v (Vn w a), Vector w a) => Mat v w a -> Int
-nrows = V.length
+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 v (Vn 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 :: (Vector v (Vn w a), Vector w a) => Mat v w a
- -> Int
- -> Vn w a
-row m i = m ! i
+
+-- | Return the @i@th row of @m@ as a matrix. Unsafe.
+row :: (Arity m, Arity n) => Mat m n a -> Int -> Row n a
+row m i =
+ construct lambda
+ where
+ lambda _ j = m !!! (i, j)
--- | Return the @j@th column of @m@. Unsafe.
-column :: (Vector v a, Vector v (Vn w a), Vector w a) => Mat v w a
- -> Int
- -> Vn v a
+-- | Return the @j@th column of @m@ as a matrix. Unsafe.
+column :: (Arity m, Arity n) => Mat m n a -> Int -> Col m a
column m j =
- V.map (element j) m
+ construct lambda
where
- element = flip (!)
+ lambda i _ = m !!! (i, j)
-- | 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.
+-- implementation, but I don't see a better way.
--
-- TODO: Don't cheat with fromList.
--
-- >>> transpose m
-- ((1,3),(2,4))
--
-transpose :: (Vector v (Vn w a),
- Vector w (Vn v a),
- Vector v a,
- Vector w a)
- => Mat v w a
- -> Mat w v a
-transpose m = V.fromList column_list
+transpose :: (Arity m, Arity n) => Mat m n a -> Mat n m a
+transpose matrix =
+ construct lambda
where
- column_list = [ column m i | i <- [0..(ncols m)-1] ]
+ lambda i j = matrix !!! (j,i)
+
-- | Is @m@ symmetric?
--
-- >>> symmetric m2
-- False
--
-symmetric :: (Vector v (Vn 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)
-- entries in the matrix. The i,j entry of the resulting matrix will
-- have the value returned by lambda i j.
--
--- TODO: Don't cheat with fromList.
---
-- Examples:
--
-- >>> let lambda i j = i + j
-- >>> construct lambda :: Mat3 Int
-- ((0,1,2),(1,2,3),(2,3,4))
--
-construct :: forall v w a.
- (Vector v (Vn w a),
- Vector w a)
- => (Int -> Int -> a)
- -> Mat v w a
-construct lambda = rows
+construct :: forall m n a. (Arity m, Arity n)
+ => (Int -> Int -> a) -> Mat m n a
+construct lambda = Mat $ generate make_row
where
- -- The arity trick is used in Data.Vector.Fixed.length.
- imax = (arity (undefined :: Dim v)) - 1
- jmax = (arity (undefined :: Dim w)) - 1
- row' i = V.fromList [ lambda i j | j <- [0..jmax] ]
- rows = V.fromList [ row' i | i <- [0..imax] ]
+ make_row :: Int -> Vec n a
+ make_row i = generate (lambda i)
+
+
+-- | Create an identity matrix with the right dimensions.
+--
+-- Examples:
+--
+-- >>> identity_matrix :: Mat3 Int
+-- ((1,0,0),(0,1,0),(0,0,1))
+-- >>> identity_matrix :: Mat3 Double
+-- ((1.0,0.0,0.0),(0.0,1.0,0.0),(0.0,0.0,1.0))
+--
+identity_matrix :: (Arity m, Ring.C a) => Mat m m a
+identity_matrix =
+ construct (\i j -> if i == j then (fromInteger 1) else (fromInteger 0))
+
-- | Given a positive-definite matrix @m@, computes the
-- upper-triangular matrix @r@ with (transpose r)*r == m and all
-- Examples:
--
-- >>> let m1 = fromList [[20,-1], [-1,20]] :: Mat2 Double
--- >>> cholesky m1
--- ((4.47213595499958,-0.22360679774997896),(0.0,4.466542286825459))
--- >>> (transpose (cholesky m1)) `mult` (cholesky m1)
--- ((20.000000000000004,-1.0),(-1.0,20.0))
---
-cholesky :: forall a v w.
- (RealFloat a,
- Vector v (Vn w a),
- Vector w a)
- => (Mat v w a)
- -> (Mat v w a)
+-- >>> let r = cholesky m1
+-- >>> frobenius_norm ((transpose r)*r - m1) < 1e-10
+-- True
+-- >>> is_upper_triangular r
+-- True
+--
+-- >>> let k1 = [6, -3, 0, 0, 0, 0, 0] :: [Double]
+-- >>> let k2 = [-3, 10.5, -7.5, 0, 0, 0, 0] :: [Double]
+-- >>> let k3 = [0, -7.5, 12.5, 0, 0, 0, 0] :: [Double]
+-- >>> let k4 = [0, 0, 0, 6, 0, 0, 0] :: [Double]
+-- >>> let k5 = [0, 0, 0, 0, 6, 0, 0] :: [Double]
+-- >>> let k6 = [0, 0, 0, 0, 0, 6, 0] :: [Double]
+-- >>> let k7 = [0, 0, 0, 0, 0, 0, 15] :: [Double]
+-- >>> let big_K = fromList [k1,k2,k3,k4,k5,k6,k7] :: Mat N7 N7 Double
+--
+-- >>> let e1 = [2.449489742783178,0,0,0,0,0,0] :: [Double]
+-- >>> let e2 = [-1.224744871391589,3,0,0,0,0,0] :: [Double]
+-- >>> let e3 = [0,-5/2,5/2,0,0,0,0] :: [Double]
+-- >>> let e4 = [0,0,0,2.449489742783178,0,0,0] :: [Double]
+-- >>> let e5 = [0,0,0,0,2.449489742783178,0,0] :: [Double]
+-- >>> let e6 = [0,0,0,0,0,2.449489742783178,0] :: [Double]
+-- >>> let e7 = [0,0,0,0,0,0,3.872983346207417] :: [Double]
+-- >>> let expected = fromList [e1,e2,e3,e4,e5,e6,e7] :: Mat N7 N7 Double
+--
+-- >>> let r = cholesky big_K
+-- >>> frobenius_norm (r - (transpose expected)) < 1e-12
+-- True
+--
+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
- r i j | i == j = sqrt(m !!! (i,j) - sum [(r k i)**2 | k <- [0..i-1]])
+ r i j | i == j = sqrt(m !!! (i,j) - sum [(r k i)^2 | k <- [0..i-1]])
| i < j =
- (((m !!! (i,j)) - sum [(r k i)*(r k j) | k <- [0..i-1]]))/(r i i)
+ (((m !!! (i,j)) - sum [(r k i) NP.* (r k j) | k <- [0..i-1]]))/(r i i)
| otherwise = 0
--- | Matrix multiplication. Our 'Num' instance doesn't define one, and
--- we need additional restrictions on the result type anyway.
+
+-- | Returns True if the given matrix is upper-triangular, and False
+-- otherwise. The parameter @epsilon@ lets the caller choose a
+-- tolerance.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,1],[1e-12,1]] :: Mat2 Double
+-- >>> is_upper_triangular m
+-- False
+-- >>> is_upper_triangular' 1e-10 m
+-- True
+--
+is_upper_triangular' :: forall m n a.
+ (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
+ => a -- ^ The tolerance @epsilon@.
+ -> Mat m n a
+ -> Bool
+is_upper_triangular' epsilon matrix =
+ ifoldl2 f True matrix
+ where
+ f :: Int -> Int -> Bool -> a -> Bool
+ f _ _ False _ = False
+ f i j True x
+ | i <= j = True
+ -- use "less than or equal to" so zero is a valid epsilon
+ | otherwise = abs x <= epsilon
+
+
+-- | Returns True if the given matrix is upper-triangular, and False
+-- otherwise. We don't delegate to the general
+-- 'is_upper_triangular'' here because it imposes additional
+-- typeclass constraints throughout the library.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+-- >>> is_upper_triangular m
+-- False
+--
+-- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
+-- >>> is_upper_triangular m
+-- True
+--
+is_upper_triangular :: forall m n a.
+ (Eq a, Ring.C a, Arity m, Arity n)
+ => Mat m n a -> Bool
+is_upper_triangular matrix =
+ ifoldl2 f True matrix
+ where
+ f :: Int -> Int -> Bool -> a -> Bool
+ f _ _ False _ = False
+ f i j True x
+ | i <= j = True
+ | otherwise = x == 0
+
+
+
+-- | Returns True if the given matrix is lower-triangular, and False
+-- otherwise.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+-- >>> is_lower_triangular m
+-- True
+--
+-- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
+-- >>> is_lower_triangular m
+-- False
+--
+is_lower_triangular :: (Eq a,
+ Ring.C a,
+ Arity m,
+ Arity n)
+ => Mat m n a
+ -> Bool
+is_lower_triangular = is_upper_triangular . transpose
+
+
+-- | Returns True if the given matrix is lower-triangular, and False
+-- otherwise. The parameter @epsilon@ lets the caller choose a
+-- tolerance.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,1e-12],[1,1]] :: Mat2 Double
+-- >>> is_lower_triangular m
+-- False
+-- >>> is_lower_triangular' 1e-12 m
+-- True
+--
+is_lower_triangular' :: (Ord a,
+ Ring.C a,
+ Absolute.C a,
+ Arity m,
+ Arity n)
+ => a -- ^ The tolerance @epsilon@.
+ -> Mat m n a
+ -> Bool
+is_lower_triangular' epsilon = (is_upper_triangular' epsilon) . transpose
+
+
+-- | Returns True if the given matrix is triangular, and False
+-- otherwise.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+-- >>> is_triangular m
+-- True
+--
+-- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
+-- >>> is_triangular m
+-- True
+--
+-- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+-- >>> is_triangular m
+-- False
+--
+is_triangular :: (Ord a,
+ Ring.C a,
+ Absolute.C a,
+ Arity m,
+ Arity n)
+ => Mat m n a
+ -> Bool
+is_triangular m = is_upper_triangular m || is_lower_triangular m
+
+
+-- | Delete the @i@th row and @j@th column from the matrix. The name
+-- \"preminor\" is made up, but is meant to signify that this is
+-- usually used in the computationof a minor. A minor is simply the
+-- determinant of a preminor in that case.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> preminor m 0 0 :: Mat2 Int
+-- ((5,6),(8,9))
+-- >>> preminor m 1 1 :: Mat2 Int
+-- ((1,3),(7,9))
+--
+preminor :: (Arity m, Arity n)
+ => Mat (S m) (S n) a
+ -> Int
+ -> Int
+ -> Mat m n a
+preminor (Mat rows) i j = m
+ where
+ rows' = delete rows i
+ m = Mat $ V.map ((flip delete) j) rows'
+
+
+-- | Compute the i,jth minor of a @matrix@.
--
-- Examples:
--
--- >>> let m1 = fromList [[1,2,3], [4,5,6]] :: Mat Vec2D Vec3D Int
--- >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat Vec3D Vec2D Int
--- >>> m1 `mult` m2
+-- >>> let m1 = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Double
+-- >>> minor m1 1 1
+-- -12.0
+--
+minor :: (Arity m, Determined (Mat m m) a)
+ => Mat (S m) (S m) a
+ -> Int
+ -> Int
+ -> a
+minor matrix i j = determinant (preminor matrix 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 = unscalar
+
+instance (Ord a,
+ Ring.C a,
+ Absolute.C a,
+ Arity n,
+ Determined (Mat (S n) (S n)) a)
+ => Determined (Mat (S (S n)) (S (S n))) a where
+ -- | The recursive definition with a special-case for triangular matrices.
+ --
+ -- Examples:
+ --
+ -- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+ -- >>> determinant m
+ -- -1
+ --
+ determinant m
+ | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ]
+ | otherwise = determinant_recursive
+ where
+ m' i j = m !!! (i,j)
+
+ determinant_recursive =
+ sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (minor m 0 j)
+ | j <- [0..(ncols m)-1] ]
+
+
+
+-- | Matrix multiplication.
+--
+-- Examples:
+--
+-- >>> 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))
--
-mult :: (Num a,
- Vector v (Vn w a),
- Vector w a,
- Vector w (Vn z a),
- Vector z a,
- Vector v (Vn z a))
- => Mat v w a
- -> Mat w z a
- -> Mat v z a
-mult m1 m2 = construct lambda
- where
- lambda i j =
- sum [(m1 !!! (i,k)) * (m2 !!! (k,j)) | k <- [0..(ncols m1)-1] ]
+infixl 7 *
+(*) :: (Ring.C a, Arity m, Arity n, Arity p)
+ => 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 = (transpose $ row m1 i) `dot` (column m2 j)
+
+
+
+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
+
+ (Mat rows1) - (Mat rows2) =
+ Mat $ V.zipWith (V.zipWith (-)) rows1 rows2
+
+ zero = Mat (V.replicate $ V.replicate (fromInteger 0))
+
+
+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
+
+
+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
+
+
+instance (Absolute.C a,
+ Algebraic.C a,
+ ToRational.C a,
+ Arity m)
+ => Normed (Col (S m) a) where
+ -- | Generic p-norms for vectors in R^n that are represented as n-by-1
+ -- matrices.
+ --
+ -- Examples:
+ --
+ -- >>> let v1 = vec2d (3,4)
+ -- >>> norm_p 1 v1
+ -- 7.0
+ -- >>> norm_p 2 v1
+ -- 5.0
+ --
+ -- >>> let v1 = vec2d (-1,1) :: Col2 Double
+ -- >>> norm_p 1 v1 :: Double
+ -- 2.0
+ --
+ norm_p p (Mat rows) =
+ (root p') $ sum [fromRational' (toRational $ abs x)^p' | x <- xs]
+ where
+ p' = toInteger p
+ xs = concat $ V.toList $ V.map V.toList rows
+
+ -- | The infinity norm.
+ --
+ -- Examples:
+ --
+ -- >>> let v1 = vec3d (1,5,2)
+ -- >>> norm_infty v1
+ -- 5
+ --
+ norm_infty (Mat rows) =
+ fromRational' $ toRational $ V.maximum $ V.map V.maximum rows
+
+
+-- | Compute the Frobenius norm of a matrix. This essentially treats
+-- the matrix as one long vector containing all of its entries (in
+-- any order, it doesn't matter).
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1, 2, 3],[4,5,6],[7,8,9]] :: Mat3 Double
+-- >>> frobenius_norm m == sqrt 285
+-- True
+--
+-- >>> let m = fromList [[1, -1, 1],[-1,1,-1],[1,-1,1]] :: Mat3 Double
+-- >>> frobenius_norm m == 3
+-- True
+--
+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
+ squares = map2 (^2) matrix
+
+
+-- Vector helpers. We want it to be easy to create low-dimension
+-- column vectors, which are nx1 matrices.
+
+-- | Convenient constructor for 2D vectors.
+--
+-- Examples:
+--
+-- >>> import Roots.Simple
+-- >>> let fst m = m !!! (0,0)
+-- >>> let snd m = m !!! (1,0)
+-- >>> let h = 0.5 :: Double
+-- >>> 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) -> Col1 a
+vec1d (x) = Mat (mk1 (mk1 x))
+
+vec2d :: (a,a) -> Col2 a
+vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y))
+
+vec3d :: (a,a,a) -> Col3 a
+vec3d (x,y,z) = Mat (mk3 (mk1 x) (mk1 y) (mk1 z))
+
+vec4d :: (a,a,a,a) -> Col4 a
+vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z))
+
+vec5d :: (a,a,a,a,a) -> Col5 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 -> Mat1 a
+scalar x = Mat (mk1 (mk1 x))
+
+-- Get the scalar value out of a 1x1 matrix.
+unscalar :: Mat1 a -> a
+unscalar (Mat rows) = V.head $ V.head rows
+
+
+dot :: (Ring.C a, Arity m)
+ => Col (S m) a
+ -> Col (S m) a
+ -> a
+v1 `dot` v2 = element_sum2 $ zipwith2 (NP.*) v1 v2
+
+
+-- | The angle between @v1@ and @v2@ in Euclidean space.
+--
+-- Examples:
+--
+-- >>> let v1 = vec2d (1.0, 0.0)
+-- >>> let v2 = vec2d (0.0, 1.0)
+-- >>> angle v1 v2 == pi/2.0
+-- True
+--
+angle :: (Transcendental.C a,
+ RealRing.C a,
+ m ~ S t,
+ Arity t,
+ ToRational.C a)
+ => Col m a
+ -> Col m a
+ -> a
+angle v1 v2 =
+ acos theta
+ where
+ theta = (recip norms) NP.* (v1 `dot` v2)
+ norms = (norm v1) NP.* (norm v2)
+
+
+-- | Retrieve the diagonal elements of the given matrix as a \"column
+-- vector,\" i.e. a m-by-1 matrix. We require the matrix to be
+-- square to avoid ambiguity in the return type which would ideally
+-- have dimension min(m,n) supposing an m-by-n matrix.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> diagonal m
+-- ((1),(5),(9))
+--
+diagonal :: (Arity m) => Mat m m a -> Col m a
+diagonal matrix =
+ construct lambda
+ where
+ lambda i _ = matrix !!! (i,i)
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+-- containing only the on-diagonal entries of @matrix@. The
+-- off-diagonal entries are set to zero.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> diagonal_part m
+-- ((1,0,0),(0,5,0),(0,0,9))
+--
+diagonal_part :: (Arity m, Ring.C a)
+ => Mat m m a
+ -> Mat m m a
+diagonal_part matrix =
+ construct lambda
+ where
+ lambda i j = if i == j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+-- containing only the on-diagonal and below-diagonal entries of
+-- @matrix@. The above-diagonal entries are set to zero.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> lt_part m
+-- ((1,0,0),(4,5,0),(7,8,9))
+--
+lt_part :: (Arity m, Ring.C a)
+ => Mat m m a
+ -> Mat m m a
+lt_part matrix =
+ construct lambda
+ where
+ lambda i j = if i >= j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+-- containing only the below-diagonal entries of @matrix@. The on-
+-- and above-diagonal entries are set to zero.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> lt_part_strict m
+-- ((0,0,0),(4,0,0),(7,8,0))
+--
+lt_part_strict :: (Arity m, Ring.C a)
+ => Mat m m a
+ -> Mat m m a
+lt_part_strict matrix =
+ construct lambda
+ where
+ lambda i j = if i > j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+-- containing only the on-diagonal and above-diagonal entries of
+-- @matrix@. The below-diagonal entries are set to zero.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> ut_part m
+-- ((1,2,3),(0,5,6),(0,0,9))
+--
+ut_part :: (Arity m, Ring.C a)
+ => Mat m m a
+ -> Mat m m a
+ut_part = transpose . lt_part . transpose
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+-- containing only the above-diagonal entries of @matrix@. The on-
+-- and below-diagonal entries are set to zero.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> ut_part_strict m
+-- ((0,2,3),(0,0,6),(0,0,0))
+--
+ut_part_strict :: (Arity m, Ring.C a)
+ => Mat m m a
+ -> Mat m m a
+ut_part_strict = transpose . lt_part_strict . transpose
+
+
+-- | Compute the trace of a square matrix, the sum of the elements
+-- which lie on its diagonal. We require the matrix to be
+-- square to avoid ambiguity in the return type which would ideally
+-- have dimension min(m,n) supposing an m-by-n matrix.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> trace m
+-- 15
+--
+trace :: (Arity m, Ring.C a) => Mat m m a -> a
+trace = element_sum2 . diagonal
+
+
+
+-- | Zip together two matrices.
+--
+-- TODO: don't cheat with construct (map V.zips instead).
+--
+-- Examples:
+--
+-- >>> let m1 = fromList [[1],[1],[1]] :: Col3 Int
+-- >>> let m2 = fromList [[1],[2],[3]] :: Col3 Int
+-- >>> zip2 m1 m2
+-- (((1,1)),((1,2)),((1,3)))
+--
+-- >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
+-- >>> let m2 = fromList [[1,1],[1,1]] :: Mat2 Int
+-- >>> zip2 m1 m2
+-- (((1,1),(2,1)),((3,1),(4,1)))
+--
+zip2 :: (Arity m, Arity n) => Mat m n a -> Mat m n b -> Mat m n (a,b)
+zip2 m1 m2 =
+ construct lambda
+ where
+ lambda i j = (m1 !!! (i,j), m2 !!! (i,j))
+
+
+-- | Zip together three matrices.
+--
+-- TODO: don't cheat with construct (map V.zips instead).
+--
+-- Examples:
+--
+-- >>> let m1 = fromList [[1],[1],[1]] :: Col3 Int
+-- >>> let m2 = fromList [[1],[2],[3]] :: Col3 Int
+-- >>> let m3 = fromList [[4],[5],[6]] :: Col3 Int
+-- >>> zip2three m1 m2 m3
+-- (((1,1,4)),((1,2,5)),((1,3,6)))
+--
+-- >>> let m1 = fromList [[1,2],[3,4]] :: Mat2 Int
+-- >>> let m2 = fromList [[1,1],[1,1]] :: Mat2 Int
+-- >>> let m3 = fromList [[8,2],[6,3]] :: Mat2 Int
+-- >>> zip2three m1 m2 m3
+-- (((1,1,8),(2,1,2)),((3,1,6),(4,1,3)))
+--
+zip2three :: (Arity m, Arity n)
+ => Mat m n a
+ -> Mat m n a
+ -> Mat m n a
+ -> Mat m n (a,a,a)
+zip2three m1 m2 m3 =
+ construct lambda
+ where
+ lambda i j = (m1 !!! (i,j), m2 !!! (i,j), m3 !!! (i,j))
+
+
+-- | Zip together two matrices using the supplied function.
+--
+-- Examples:
+--
+-- >>> let c1 = fromList [[1],[2],[3]] :: Col3 Integer
+-- >>> let c2 = fromList [[4],[5],[6]] :: Col3 Integer
+-- >>> zipwith2 (^) c1 c2
+-- ((1),(32),(729))
+--
+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
+ lambda i j = f (c1 !!! (i,j)) (c2 !!! (i,j))
+
+
+-- | Map a function over a matrix of any dimensions.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+-- >>> map2 (^2) m
+-- ((1,4),(9,16))
+--
+map2 :: (a -> b) -> Mat m n a -> Mat m n b
+map2 f (Mat rows) =
+ Mat $ V.map g rows
+ where
+ g = V.map f
+
+
+-- | Fold over the entire matrix passing the coordinates @i@ and @j@
+-- (of the row/column) to the accumulation function. The fold occurs
+-- from top-left to bottom-right.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> ifoldl2 (\i j cur _ -> cur + i + j) 0 m
+-- 18
+--
+ifoldl2 :: forall a b m n.
+ (Int -> Int -> b -> a -> b)
+ -> b
+ -> Mat m n a
+ -> b
+ifoldl2 f initial (Mat rows) =
+ V.ifoldl row_function initial rows
+ where
+ -- | The order that we need this in (so that @g idx@ makes sense)
+ -- is a little funny. So that we don't need to pass weird
+ -- functions into ifoldl2, we swap the second and third
+ -- arguments of @f@ calling the result @g@.
+ g :: Int -> b -> Int -> a -> b
+ g w x y = f w y x
+
+ row_function :: b -> Int -> Vec n a -> b
+ 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.
+--
+-- The order of the arguments in the supplied function are different
+-- from those in V.ifoldr; we keep them similar to ifoldl2.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> ifoldr2 (\i j cur _ -> cur + i + j) 0 m
+-- 18
+--
+ifoldr2 :: forall a b m n.
+ (Int -> Int -> b -> a -> b)
+ -> b
+ -> Mat m n a
+ -> b
+ifoldr2 f initial (Mat rows) =
+ V.ifoldr row_function initial rows
+ where
+ -- | Swap the order of arguments in @f@ so that it agrees with the
+ -- @f@ passed to ifoldl2.
+ g :: Int -> Int -> a -> b -> b
+ g w x y z = f w x z y
+
+ row_function :: Int -> Vec n a -> b -> b
+ row_function idx r rowinit = V.ifoldr (g idx) rowinit r
+
+
+-- | Map a function over a matrix of any dimensions, passing the
+-- coordinates @i@ and @j@ to the function @f@.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+-- >>> imap2 (\i j _ -> i+j) m
+-- ((0,1),(1,2))
+--
+imap2 :: (Int -> Int -> a -> b) -> Mat m n a -> Mat m n b
+imap2 f (Mat rows) =
+ Mat $ V.imap g rows
+ where
+ g i = V.imap (f i)
+
+
+-- | Reverse the order of elements in a matrix.
+--
+-- Examples:
+--
+-- >>> let m1 = fromList [[1,2,3]] :: Row3 Int
+-- >>> reverse2 m1
+-- ((3,2,1))
+--
+-- >>> let m1 = vec3d (1,2,3 :: Int)
+-- >>> reverse2 m1
+-- ((3),(2),(1))
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> reverse2 m
+-- ((9,8,7),(6,5,4),(3,2,1))
+--
+reverse2 :: (Arity m, Arity n) => Mat m n a -> Mat m n a
+reverse2 (Mat rows) = Mat $ V.reverse $ V.map V.reverse rows
+
+
+-- | Unsafely set the (i,j) element of the given matrix.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> set_idx m (1,1) 17
+-- ((1,2,3),(4,17,6),(7,8,9))
+--
+set_idx :: forall m n a.
+ (Arity m, Arity n)
+ => Mat m n a
+ -> (Int, Int)
+ -> a
+ -> Mat m n a
+set_idx matrix (i,j) newval =
+ imap2 updater matrix
+ where
+ updater :: Int -> Int -> a -> a
+ updater k l existing =
+ if k == i && l == j
+ then newval
+ else existing
+
+
+-- | Compute the i,jth cofactor of the given @matrix@. This simply
+-- premultiplues the i,jth minor by (-1)^(i+j).
+cofactor :: (Arity m, Determined (Mat m m) a)
+ => Mat (S m) (S m) a
+ -> Int
+ -> Int
+ -> a
+cofactor matrix i j =
+ (-1)^(toInteger i + toInteger j) NP.* (minor matrix i j)
+
+
+-- | Compute the inverse of a matrix using cofactor expansion
+-- (generalized Cramer's rule).
+--
+-- Examples:
+--
+-- >>> let m1 = fromList [[37,22],[17,54]] :: Mat2 Double
+-- >>> let e1 = [54/1624, -22/1624] :: [Double]
+-- >>> let e2 = [-17/1624, 37/1624] :: [Double]
+-- >>> let expected = fromList [e1, e2] :: Mat2 Double
+-- >>> let actual = inverse m1
+-- >>> frobenius_norm (actual - expected) < 1e-12
+-- True
+--
+inverse :: (Arity m,
+ Determined (Mat (S m) (S m)) a,
+ Determined (Mat m m) a,
+ Field.C a)
+ => Mat (S m) (S m) a
+ -> Mat (S m) (S m) a
+inverse matrix =
+ (1 / (determinant matrix)) *> (transpose $ construct lambda)
+ 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