mk2,
mk3,
mk4,
- mk5
- )
+ mk5 )
import qualified Data.Vector.Fixed as V (
and,
fromList,
map,
maximum,
replicate,
+ reverse,
toList,
zipWith )
import Data.Vector.Fixed.Cont ( Arity, arity )
-- Examples:
--
-- >>> let m1 = fromList [[20,-1], [-1,20]] :: Mat2 Double
--- >>> cholesky m1
--- ((4.47213595499958,-0.22360679774997896),(0.0,4.466542286825459))
--- >>> (transpose (cholesky m1)) * (cholesky m1)
--- ((20.000000000000004,-1.0),(-1.0,20.0))
+-- >>> let r = cholesky m1
+-- >>> frobenius_norm ((transpose r)*r - m1) < 1e-10
+-- True
+-- >>> 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]
+-- >>> 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)
-- >>> is_upper_triangular' 1e-10 m
-- True
--
--- TODO:
---
--- 1. Don't cheat with lists.
---
-is_upper_triangular' :: (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
+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 m =
- and $ concat results
+is_upper_triangular' epsilon matrix =
+ ifoldl2 f True matrix
where
- results = [[ test i j | i <- [0..(nrows m)-1]] | j <- [0..(ncols m)-1] ]
-
- test :: Int -> Int -> Bool
- test i j
+ 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 (m !!! (i,j)) <= epsilon
+ | otherwise = abs x <= epsilon
-- | Returns True if the given matrix is upper-triangular, and False
--- otherwise. A specialized version of 'is_upper_triangular\'' with
--- @epsilon = 0@.
+-- otherwise. We don't delegate to the general
+-- 'is_upper_triangular'' here because it imposes additional
+-- typeclass constraints throughout the library.
--
-- Examples:
--
-- >>> is_upper_triangular m
-- True
--
--- TODO:
---
--- 1. The Ord constraint is too strong here, Eq would suffice.
---
-is_upper_triangular :: (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
+is_upper_triangular :: forall m n a.
+ (Eq a, Ring.C a, Arity m, Arity n)
=> Mat m n a -> Bool
-is_upper_triangular = is_upper_triangular' 0
+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. This is a specialized version of 'is_lower_triangular\''
--- with @epsilon = 0@.
+-- otherwise.
--
-- Examples:
--
-- >>> is_lower_triangular m
-- False
--
-is_lower_triangular :: (Ord a,
+is_lower_triangular :: (Eq a,
Ring.C a,
- Absolute.C a,
Arity m,
Arity n)
=> Mat m n a
scalar :: a -> Mat1 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
+dot :: (RealRing.C a, m ~ S t, Arity t)
+ => Col m a
+ -> Col m a
-> a
v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0)
--
angle :: (Transcendental.C a,
RealRing.C a,
- n ~ N1,
m ~ S t,
Arity t,
ToRational.C a)
- => Mat m n a
- -> Mat m n a
+ => Col m a
+ -> Col m a
-> a
angle v1 v2 =
acos theta
element_sum $ V.map V.head rows
--- | 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 a -> Mat m n (a,a)
+zip2 m1 m2 =
construct lambda
where
- lambda i j = (c1 !!! (i,j), c2 !!! (i,j))
+ lambda i j = (m1 !!! (i,j), m2 !!! (i,j))
--- | Zip together two column matrices using the supplied function.
+-- | 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
--- >>> colzipwith (^) c1 c2
+-- >>> zipwith2 (^) c1 c2
-- ((1),(32),(729))
--
-colzipwith :: Arity m
+zipwith2 :: Arity m
=> (a -> a -> b)
-> Col m a
-> Col m a
-> Col m b
-colzipwith f c1 c2 =
+zipwith2 f c1 c2 =
construct lambda
where
lambda i j = f (c1 !!! (i,j)) (c2 !!! (i,j))
-- Examples:
--
-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
--- >>> ifoldl2 (\i j cur _ -> cur + i + j) 0 m
+-- >>> ifoldl2 (\i j cur _ -> cur + i + j) 0 m
-- 18
--
ifoldl2 :: forall a b m n.
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
+
+
projects / numerical-analysis.git / blobdiff