import Data.Vector.Fixed (
(!),
- N1,
- N2,
- N3,
- N4,
- N5,
- S,
- Z,
generate,
+ mk0,
mk1,
mk2,
mk3,
mk5 )
import qualified Data.Vector.Fixed as V (
and,
+ foldl,
fromList,
head,
ifoldl,
+ ifoldr,
imap,
map,
- maximum,
replicate,
reverse,
toList,
zipWith )
import Data.Vector.Fixed.Cont ( Arity, arity )
-import Linear.Vector ( Vec, delete, element_sum )
+import Linear.Vector ( Vec, delete )
+import Naturals
import Normed ( Normed(..) )
-import NumericPrelude hiding ( (*), abs )
+-- We want the "max" that works on Ord, not the one that only works on
+-- Bool/Integer from the Lattice class!
+import NumericPrelude hiding ( (*), abs, max)
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 )
+import qualified Prelude as P ( map, max)
-- | 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 Mat0 a = Mat Z Z 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
+type Mat6 a = Mat N6 N6 a
+type Mat7 a = Mat N7 N7 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 synonyms for 1-by-n row "vectors".
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
+-- * 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 synonyms for n-by-1 column "vectors".
+type Col0 a = Col Z 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
-
--- We need a big column for Gaussian quadrature.
-type N10 = S (S (S (S (S N5))))
+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
+type Col11 a = Col N11 a
+type Col12 a = Col N12 a
+type Col13 a = Col N13 a
+type Col14 a = Col N14 a
+type Col15 a = Col N15 a
+type Col16 a = Col N16 a
+type Col17 a = Col N17 a
+type Col18 a = Col N18 a
+type Col19 a = Col N19 a
+type Col20 a = Col N20 a
+type Col21 a = Col N21 a
+type Col22 a = Col N22 a
+type Col23 a = Col N23 a
+type Col24 a = Col N24 a
+type Col25 a = Col N25 a
+type Col26 a = Col N26 a
+type Col27 a = Col N27 a
+type Col28 a = Col N28 a
+type Col29 a = Col N29 a
+type Col30 a = Col N30 a
+type Col31 a = Col N31 a
+type Col32 a = Col N32 a
instance (Eq a) => Eq (Mat m n a) where
-- >>> m1 == m3
-- False
--
- (Mat rows1) == (Mat rows2) =
- V.and $ V.zipWith comp rows1 rows2
+ (Mat rows_one) == (Mat rows_two) =
+ V.and $ V.zipWith comp rows_one rows_two
where
-- Compare a row, one column at a time.
comp row1 row2 = V.and (V.zipWith (==) row1 row2)
-- >>> is_upper_triangular r
-- True
--
--- >>> import Naturals ( N7 )
-- >>> 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]
-- >>> 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
+cholesky :: forall m a. (Algebraic.C a, Arity m)
+ => (Mat m m a) -> (Mat m m a)
+cholesky m = ifoldl2 f zero m
where
- r :: Int -> Int -> a
- 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) NP.* (r k j) | k <- [0..i-1]]))/(r i i)
- | otherwise = 0
+ f :: Int -> Int -> (Mat m m a) -> a -> (Mat m m a)
+ f i j cur_R _ = set_idx cur_R (i,j) (r cur_R i j)
+
+ r :: (Mat m m a) -> Int -> Int -> a
+ r cur_R i j
+ | i == j = sqrt(m !!! (i,j) - sum [(cur_R !!! (k,i))^2 | k <- [0..i-1]])
+ | i < j = (((m !!! (i,j))
+ - sum [(cur_R !!! (k,i)) NP.* (cur_R !!! (k,j))
+ | k <- [0..i-1]]))/(cur_R !!! (i,i))
+ | otherwise = 0
+
-- | Returns True if the given matrix is upper-triangular, and False
is_triangular m = is_upper_triangular m || is_lower_triangular m
--- | Return the (i,j)th minor of 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
--- >>> minor m 0 0 :: Mat2 Int
+-- >>> preminor m 0 0 :: Mat2 Int
-- ((5,6),(8,9))
--- >>> minor m 1 1 :: Mat2 Int
+-- >>> preminor m 1 1 :: Mat2 Int
-- ((1,3),(7,9))
--
-minor :: (m ~ S r,
- n ~ S t,
- Arity r,
- Arity t)
- => Mat m n a
+preminor :: (Arity m, Arity n)
+ => Mat (S m) (S n) a
-> Int
-> Int
- -> Mat r t a
-minor (Mat rows) i j = m
+ -> 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],[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 (Mat rows) = (V.head . V.head) rows
+ determinant = unscalar
instance (Ord a,
Ring.C a,
where
m' i j = m !!! (i,j)
- det_minor i j = determinant (minor m i j)
-
determinant_recursive =
- sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (det_minor 0 j)
+ sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (minor m 0 j)
| j <- [0..(ncols m)-1] ]
--
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)
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 rows_one) + (Mat rows_two) =
+ Mat $ V.zipWith (V.zipWith (+)) rows_one rows_two
- (Mat rows1) - (Mat rows2) =
- Mat $ V.zipWith (V.zipWith (-)) rows1 rows2
+ (Mat rows_one) - (Mat rows_two) =
+ Mat $ V.zipWith (V.zipWith (-)) rows_one rows_two
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
x *> (Mat rows) = Mat $ V.map (V.map (NP.* x)) rows
-instance (Algebraic.C a,
+instance (Absolute.C a,
+ Algebraic.C a,
ToRational.C a,
Arity m)
- => Normed (Mat (S m) N1 a) where
- -- | Generic p-norms for vectors in R^n that are represented as nx1
+ => Normed (Col m a) where
+ -- | Generic p-norms for vectors in R^m that are represented as m-by-1
-- matrices.
--
-- Examples:
-- >>> norm_p 2 v1
-- 5.0
--
+ -- >>> let v1 = vec2d (-1,1) :: Col2 Double
+ -- >>> norm_p 1 v1 :: Double
+ -- 2.0
+ --
+ -- >>> let v1 = vec0d :: Col0 Double
+ -- >>> norm v1
+ -- 0.0
+ --
norm_p p (Mat rows) =
- (root p') $ sum [fromRational' (toRational x)^p' | x <- xs]
+ (root p') $ sum [fromRational' (toRational $ abs x)^p' | x <- xs]
where
p' = toInteger p
xs = concat $ V.toList $ V.map V.toList rows
-- 5
--
norm_infty (Mat rows) =
- fromRational' $ toRational $ V.maximum $ V.map V.maximum rows
+ fromRational' $ toRational
+ $ (V.foldl P.max 0) $ V.map (V.foldl P.max 0) rows
-- | Compute the Frobenius norm of a matrix. This essentially treats
-- >>> 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
-- >>> fixed_point g eps u0
-- ((1.0728549599342185),(1.0820591495686167))
--
+vec0d :: Col0 a
+vec0d = Mat mk0
+
vec1d :: (a) -> Col1 a
vec1d (x) = Mat (mk1 (mk1 x))
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))
-dot :: (RealRing.C a, m ~ S t, Arity t)
- => Col m a
- -> Col m a
+-- 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 = ((transpose v1) * v2) !!! (0, 0)
+v1 `dot` v2 = element_sum2 $ zipwith2 (NP.*) v1 v2
-- | The angle between @v1@ and @v2@ in Euclidean space.
-- 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 column matrices.
+
+-- | 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
--- >>> colzip m1 m2
+-- >>> zip2 m1 m2
-- (((1,1)),((1,2)),((1,3)))
--
-colzip :: Arity m => Col m a -> Col m a -> Col m (a,a)
-colzip c1 c2 =
+-- >>> 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 = (c1 !!! (i,j), c2 !!! (i,j))
+ lambda i j = (m1 !!! (i,j), m2 !!! (i,j), m3 !!! (i,j))
--- | Zip together two column matrices using the supplied function.
+-- | 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
--- >>> colzipwith (^) c1 c2
+-- >>> zipwith2 (^) c1 c2
-- ((1),(32),(729))
--
-colzipwith :: Arity m
- => (a -> a -> b)
- -> Col m a
- -> Col m a
- -> Col m b
-colzipwith f c1 c2 =
+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))
-- | Fold over the entire matrix passing the coordinates @i@ and @j@
--- (of the row/column) to the accumulation function.
+-- (of the row/column) to the accumulation function. The fold occurs
+-- from top-left to bottom-right.
--
-- Examples:
--
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@.
--
--
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