import Data.Vector.Fixed (
(!),
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 Naturals ( N1, N2, N3, N4, N5, N6, N7, N8, N9, N10, S, Z )
+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.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 column vectors expressed as n-by-1 matrices.
type Col n a = Mat n N1 a
+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 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.
+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)
-- >>> 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
--
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
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
+ => Normed (Col m a) where
+ -- | Generic p-norms for vectors in R^m that are represented as m-by-1
-- matrices.
--
-- Examples:
-- >>> 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 $ abs x)^p' | x <- xs]
where
-- 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))
=> 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.
-- 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.
-- >>> zip2 m1 m2
-- (((1,1),(2,1)),((3,1),(4,1)))
--
-zip2 :: (Arity m, Arity n) => Mat m n a -> Mat m n a -> Mat m n (a,a)
+zip2 :: (Arity m, Arity n) => Mat m n a -> Mat m n b -> Mat m n (a,b)
zip2 m1 m2 =
construct lambda
where
-- >>> 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
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.
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
projects / numerical-analysis.git / blobdiff