import Data.Vector.Fixed (
(!),
- N1,
- N2,
- N3,
- N4,
- N5,
- S,
- Z,
generate,
mk1,
mk2,
mk3,
mk4,
- mk5
- )
+ mk5 )
import qualified Data.Vector.Fixed as V (
and,
fromList,
head,
ifoldl,
- length,
+ 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 Normed ( Normed(..) )
import NumericPrelude hiding ( (*), 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 )
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 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 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 Col10 a = Col N10 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
toList :: Mat m n a -> [[a]]
toList (Mat rows) = map V.toList (V.toList rows)
+
-- | Create a matrix from a nested list.
fromList :: (Arity m, Arity n) => [[a]] -> Mat m n a
fromList vs = Mat (V.fromList $ map V.fromList vs)
--- | Unsafe indexing.
+-- | Unsafe indexing. Much faster than the safe indexing.
(!!!) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> a
-(!!!) m (i, j) = (row m i) ! j
+(!!!) (Mat rows) (i, j) = (rows ! i) ! j
+
-- | Safe indexing.
-(!!?) :: Mat m n a -> (Int, Int) -> Maybe a
-(!!?) m@(Mat rows) (i, j)
- | i < 0 || j < 0 = Nothing
- | i > V.length rows = 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 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 m n a. (Arity n) => Mat m n a -> Int
ncols _ = arity (undefined :: n)
--- | Return the @i@th row of @m@. Unsafe.
-row :: Mat m n a -> Int -> (Vec n a)
-row (Mat rows) i = rows ! 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 =
+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 :: Mat m n a -> Int -> (Vec m a)
-column (Mat rows) j =
- V.map (element j) rows
- where
- element = flip (!)
-
-
-- | 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 =
+column :: (Arity m, Arity n) => Mat m n a -> Int -> Col m a
+column m j =
construct lambda
where
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.
--
-- ((1,3),(2,4))
--
transpose :: (Arity m, Arity n) => Mat m n a -> Mat n m a
-transpose m = Mat $ V.fromList column_list
+transpose matrix =
+ construct lambda
where
- column_list = [ column m i | i <- [0..(ncols m)-1] ]
+ lambda i j = matrix !!! (j,i)
-- | Is @m@ symmetric?
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
-- values on the diagonal of @r@ positive.
-- 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
+--
+-- >>> 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
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] ]
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 (S m) a) where
+ -- | Generic p-norms for vectors in R^n that are represented as n-by-1
-- matrices.
--
-- Examples:
-- >>> 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 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
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, n ~ N1, m ~ S t, Arity t)
- => Mat m n a
- -> Mat m n 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 = unscalar $ ((transpose v1) * v2)
-- | The angle between @v1@ and @v2@ in Euclidean space.
--
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 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
+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))
-- | 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:
--
-- >>> 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.
row_function rowinit idx r = V.ifoldl (g idx) rowinit r
+-- | 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
+
projects / numerical-analysis.git / blobdiff