{-# 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
+ (!),
+ N1,
+ N2,
+ N3,
+ N4,
+ N5,
+ S,
+ Z,
+ generate,
+ mk1,
+ mk2,
+ mk3,
+ mk4,
+ mk5
)
import qualified Data.Vector.Fixed as V (
- Fun(..),
- N1,
and,
- eq,
- foldl,
fromList,
+ head,
length,
map,
maximum,
toList,
zipWith
)
-import Data.Vector.Fixed.Internal (Arity, arity, S, Dim)
+import Data.Vector.Fixed.Boxed (Vec)
+import Data.Vector.Fixed.Cont (Arity, arity)
import Linear.Vector
import Normed
import qualified NumericPrelude as NP ((*))
import qualified Algebra.Algebraic as Algebraic
import Algebra.Algebraic (root)
-import qualified Algebra.Absolute as Absolute
import qualified Algebra.Additive as Additive
import qualified Algebra.Ring as Ring
-import Algebra.Absolute (abs)
-import qualified Algebra.Field as Field
import qualified Algebra.Module as Module
-import qualified Algebra.RealField as RealField
import qualified Algebra.RealRing as RealRing
import qualified Algebra.ToRational as ToRational
import qualified Algebra.Transcendental as Transcendental
import qualified Prelude as P
-data Mat v w a = (Vector v (w a), Vector w a) => Mat (v (w a))
-type Mat1 a = Mat D1 D1 a
-type Mat2 a = Mat D2 D2 a
-type Mat3 a = Mat D3 D3 a
-type Mat4 a = Mat D4 D4 a
+data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n 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
--- We can't just declare that all instances of Vector are instances of
--- Eq unfortunately. We wind up with an overlapping instance for
--- w (w a).
-instance (Eq a, Vector v Bool, Vector w Bool) => Eq (Mat v w a) where
+instance (Eq a) => Eq (Mat m n a) where
-- | Compare a row at a time.
--
-- Examples:
comp row1 row2 = V.and (V.zipWith (==) row1 row2)
-instance (Show a, Vector v String, Vector w String) => Show (Mat v w a) where
+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
element_strings = P.map show v1l
-
-- | Convert a matrix to a nested list.
-toList :: Mat v w a -> [[a]]
+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 (w a), Vector w a, Vector v a) => [[a]] -> Mat v w a
+fromList :: (Arity m, Arity n) => [[a]] -> Mat m n a
fromList vs = Mat (V.fromList $ map V.fromList vs)
-- | Unsafe indexing.
-(!!!) :: (Vector w a) => Mat v w a -> (Int, Int) -> a
+(!!!) :: (Arity m, Arity n) => Mat m n a -> (Int, Int) -> a
(!!!) m (i, j) = (row m i) ! j
-- | Safe indexing.
-(!!?) :: Mat v w a -> (Int, Int) -> Maybe a
+(!!?) :: Mat m n a -> (Int, Int) -> Maybe a
(!!?) m@(Mat rows) (i, j)
| i < 0 || j < 0 = Nothing
| i > V.length rows = Nothing
-- | The number of rows in the matrix.
-nrows :: Mat v w a -> Int
-nrows (Mat rows) = V.length rows
+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 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 :: Mat v w a -> Int -> w a
+row :: Mat m n a -> Int -> (Vec n a)
row (Mat rows) i = rows ! i
-- | Return the @j@th column of @m@. Unsafe.
-column :: (Vector v a) => Mat v w a -> Int -> v a
+column :: Mat m n a -> Int -> (Vec m a)
column (Mat rows) j =
V.map (element j) rows
where
element = flip (!)
+
+
-- | 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.
-- >>> transpose m
-- ((1,3),(2,4))
--
-transpose :: (Vector w (v a),
- Vector v a,
- Vector w a)
- => Mat v w a
- -> Mat w v a
+transpose :: (Arity m, Arity n) => Mat m n a -> Mat n m a
transpose m = Mat $ V.fromList column_list
where
column_list = [ column m i | i <- [0..(ncols m)-1] ]
-- >>> symmetric m2
-- False
--
-symmetric :: (Vector v (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 (w a),
- Vector w a)
- => (Int -> Int -> a)
- -> Mat v w a
-construct lambda = Mat 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
-- >>> (transpose (cholesky m1)) * (cholesky m1)
-- ((20.000000000000004,-1.0),(-1.0,20.0))
--
-cholesky :: forall a v w.
- (Algebraic.C a,
- Vector v (w a),
- Vector w a,
- Vector v a)
- => (Mat v w a)
- -> (Mat v w a)
+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
| 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.
+--
+-- 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 :: (Eq a, Ring.C a, Arity m, Arity n)
+ => Mat m n a -> Bool
+is_upper_triangular m =
+ and $ concat results
+ where
+ results = [[ test i j | i <- [0..(nrows m)-1]] | j <- [0..(ncols m)-1] ]
+
+ test :: Int -> Int -> Bool
+ test i j
+ | i <= j = True
+ | otherwise = m !!! (i,j) == 0
+
+
+-- | Returns True if the given matrix is lower-triangular, and False
+-- otherwise.
--
-- Examples:
--
--- >>> let m1 = fromList [[1,2,3], [4,5,6]] :: Mat D2 D3 Int
--- >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat D3 D2 Int
+-- >>> 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 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 :: (Eq a,
+ Ring.C a,
+ Arity m,
+ Arity n)
+ => Mat m n a
+ -> Bool
+is_triangular m = is_upper_triangular m || is_lower_triangular m
+
+
+-- | Return the (i,j)th minor of m.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> minor m 0 0 :: Mat2 Int
+-- ((5,6),(8,9))
+-- >>> minor m 1 1 :: Mat2 Int
+-- ((1,3),(7,9))
+--
+minor :: (m ~ S r,
+ n ~ S t,
+ Arity r,
+ Arity t)
+ => Mat m n a
+ -> Int
+ -> Int
+ -> Mat r t a
+minor (Mat rows) i j = m
+ where
+ rows' = delete rows i
+ m = Mat $ V.map ((flip delete) j) rows'
+
+
+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
+
+instance (Eq a,
+ Ring.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)
+
+ det_minor i j = determinant (minor m i j)
+
+ determinant_recursive =
+ sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (det_minor 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))
--
infixl 7 *
-(*) :: (Ring.C a,
- Vector v a,
- Vector w a,
- Vector z a,
- Vector v (z a))
- => Mat v w a
- -> Mat w z a
- -> Mat v z a
+(*) :: (Ring.C a, Arity m, Arity n, Arity p)
+ => Mat m n a
+ -> Mat n p a
+ -> Mat m p a
(*) m1 m2 = construct lambda
where
lambda i j =
-instance (Ring.C a,
- Vector v (w a),
- Vector w a)
- => Additive.C (Mat v w a) where
+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
zero = Mat (V.replicate $ V.replicate (fromInteger 0))
-instance (Ring.C a,
- Vector v (w a),
- Vector w a,
- v ~ w)
- => Ring.C (Mat v w a) where
+instance (Ring.C a, Arity m, Arity n, m ~ n) => Ring.C (Mat m n a) where
-- The first * is ring multiplication, the second is matrix
-- multiplication.
- m1 * m2 = m1 * m1
+ m1 * m2 = m1 * m2
-instance (Ring.C a,
- Vector v (w a),
- Vector w a)
- => Module.C a (Mat v w a) where
+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 (Algebraic.C a,
ToRational.C a,
- Vector v (w a),
- Vector w a,
- Vector v a,
- Vector v [a])
- => Normed (Mat v w a) where
+ Arity m)
+ => Normed (Mat (S m) N1 a) where
-- | Generic p-norms. The usual norm in R^n is (norm_p 2). We treat
-- all matrices as big vectors.
--
-- 5.0
--
norm_p p (Mat rows) =
- (root p') $ sum [(fromRational' $ toRational x)^p' | x <- xs]
+ (root p') $ sum [fromRational' (toRational x)^p' | x <- xs]
where
p' = toInteger p
xs = concat $ V.toList $ V.map V.toList rows
- -- | The infinity norm. We don't use V.maximum here because it
- -- relies on a type constraint that the vector be non-empty and I
- -- don't know how to pattern match it away.
+ -- | The infinity norm.
--
-- Examples:
--
-- >>> norm_infty v1
-- 5
--
- norm_infty m@(Mat rows)
- | nrows m == 0 || ncols m == 0 = 0
- | otherwise =
- fromRational' $ toRational $
- P.maximum $ V.toList $ V.map (P.maximum . V.toList) rows
+ norm_infty (Mat rows) =
+ fromRational' $ toRational $ V.maximum $ V.map V.maximum rows
-- Examples:
--
-- >>> import Roots.Simple
+-- >>> let fst m = m !!! (0,0)
+-- >>> let snd m = m !!! (1,0)
-- >>> let h = 0.5 :: Double
--- >>> let g1 (Mat (D2 (D1 x) (D1 y))) = 1.0 + h*exp(-(x^2))/(1.0 + y^2)
--- >>> let g2 (Mat (D2 (D1 x) (D1 y))) = 0.5 + h*atan(x^2 + y^2)
+-- >>> 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)
+-- ((1.0728549599342185),(1.0820591495686167))
--
-vec2d :: (a,a) -> Mat D2 D1 a
-vec2d (x,y) = Mat (D2 (D1 x) (D1 y))
+vec1d :: (a) -> Mat N1 N1 a
+vec1d (x) = Mat (mk1 (mk1 x))
+
+vec2d :: (a,a) -> Mat N2 N1 a
+vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y))
-vec3d :: (a,a,a) -> Mat D3 D1 a
-vec3d (x,y,z) = Mat (D3 (D1 x) (D1 y) (D1 z))
+vec3d :: (a,a,a) -> Mat N3 N1 a
+vec3d (x,y,z) = Mat (mk3 (mk1 x) (mk1 y) (mk1 z))
-vec4d :: (a,a,a,a) -> Mat D4 D1 a
-vec4d (w,x,y,z) = Mat (D4 (D1 w) (D1 x) (D1 y) (D1 z))
+vec4d :: (a,a,a,a) -> Mat N4 N1 a
+vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z))
+
+vec5d :: (a,a,a,a,a) -> Mat N5 N1 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 -> Mat D1 D1 a
-scalar x = Mat (D1 (D1 x))
-
-dot :: (RealRing.C a,
- Dim w ~ V.N1,
- Dim v ~ S n,
- Vector v a,
- Vector w a,
- Vector w (v a),
- Vector w (w a))
- => Mat v w a
- -> Mat v w a
+scalar :: a -> Mat N1 N1 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
-> a
v1 `dot` v2 = ((transpose v1) * v2) !!! (0, 0)
--
angle :: (Transcendental.C a,
RealRing.C a,
- Dim w ~ V.N1,
- Dim v ~ S n,
- Vector w (w a),
- Vector v [a],
- Vector v a,
- Vector w a,
- Vector v (w a),
- Vector w (v a),
+ n ~ N1,
+ m ~ S t,
+ Arity t,
ToRational.C a)
- => Mat v w a
- -> Mat v w a
+ => Mat m n a
+ -> Mat m n a
-> a
angle v1 v2 =
acos theta
where
theta = (recip norms) NP.* (v1 `dot` v2)
norms = (norm v1) NP.* (norm v2)
+
+
+
+-- | 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
projects / numerical-analysis.git / blobdiff