+++ /dev/null
-{-# LANGUAGE ScopedTypeVariables #-}
-{-# LANGUAGE FlexibleContexts #-}
-{-# LANGUAGE FlexibleInstances #-}
-{-# LANGUAGE MultiParamTypeClasses #-}
-{-# LANGUAGE TypeFamilies #-}
-
-module Matrix
-where
-
-import Vector
-import Data.Vector.Fixed (
- Dim,
- Vector
- )
-import qualified Data.Vector.Fixed as V (
- fromList,
- length,
- map,
- toList
- )
-import Data.Vector.Fixed.Internal (arity)
-
-type Mat v w a = Vn v (Vn w a)
-type Mat2 a = Mat Vec2D Vec2D a
-type Mat3 a = Mat Vec3D Vec3D a
-type Mat4 a = Mat Vec4D Vec4D a
-
--- | Convert a matrix to a nested list.
-toList :: (Vector v (Vn w a), Vector w a) => Mat v w a -> [[a]]
-toList m = map V.toList (V.toList m)
-
--- | Create a matrix from a nested list.
-fromList :: (Vector v (Vn w a), Vector w a) => [[a]] -> Mat v w a
-fromList vs = V.fromList $ map V.fromList vs
-
-
--- | Unsafe indexing.
-(!!!) :: (Vector v (Vn w a), Vector w a) => Mat v w a -> (Int, Int) -> a
-(!!!) m (i, j) = (row m i) ! j
-
--- | Safe indexing.
-(!!?) :: (Vector v (Vn w a), Vector w a) => Mat v w a
- -> (Int, Int)
- -> Maybe a
-(!!?) m (i, j)
- | i < 0 || j < 0 = Nothing
- | i > V.length m = Nothing
- | otherwise = if j > V.length (row m j)
- then Nothing
- else Just $ (row m j) ! j
-
-
--- | The number of rows in the matrix.
-nrows :: forall v w a. (Vector v (Vn w a), Vector w a) => Mat v w a -> Int
-nrows = V.length
-
--- | The number of columns in the first row of the
--- matrix. Implementation stolen from Data.Vector.Fixed.length.
-ncols :: forall v w a. (Vector v (Vn w a), Vector w a) => Mat v w a -> Int
-ncols _ = arity (undefined :: Dim w)
-
--- | Return the @i@th row of @m@. Unsafe.
-row :: (Vector v (Vn w a), Vector w a) => Mat v w a
- -> Int
- -> Vn w a
-row m i = m ! i
-
-
--- | Return the @j@th column of @m@. Unsafe.
-column :: (Vector v a, Vector v (Vn w a), Vector w a) => Mat v w a
- -> Int
- -> Vn v a
-column m j =
- V.map (element j) m
- 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.
---
--- TODO: Don't cheat with fromList.
---
--- Examples:
---
--- >>> let m = fromList [[1,2], [3,4]] :: Mat2 Int
--- >>> transpose m
--- ((1,3),(2,4))
---
-transpose :: (Vector v (Vn w a),
- Vector w (Vn v a),
- Vector v a,
- Vector w a)
- => Mat v w a
- -> Mat w v a
-transpose m = V.fromList column_list
- where
- column_list = [ column m i | i <- [0..(ncols m)-1] ]
-
--- | Is @m@ symmetric?
---
--- Examples:
---
--- >>> let m1 = fromList [[1,2], [2,1]] :: Mat2 Int
--- >>> symmetric m1
--- True
---
--- >>> let m2 = fromList [[1,2], [3,1]] :: Mat2 Int
--- >>> symmetric m2
--- False
---
-symmetric :: (Vector v (Vn w a),
- Vector w a,
- v ~ w,
- Vector w Bool,
- Eq a)
- => Mat v w a
- -> Bool
-symmetric m =
- m == (transpose m)
-
-
--- | Construct a new matrix from a function @lambda@. The function
--- @lambda@ should take two parameters i,j corresponding to the
--- 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 (Vn w a),
- Vector w a)
- => (Int -> Int -> a)
- -> Mat v w a
-construct lambda = rows
- 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] ]
-
--- | 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)) `mult` (cholesky m1)
--- ((20.000000000000004,-1.0),(-1.0,20.0))
---
-cholesky :: forall a v w.
- (RealFloat a,
- Vector v (Vn w a),
- Vector w a)
- => (Mat v w a)
- -> (Mat v w a)
-cholesky m = construct r
- 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)*(r k j) | k <- [0..i-1]]))/(r i i)
- | otherwise = 0
-
--- | Matrix multiplication. Our 'Num' instance doesn't define one, and
--- we need additional restrictions on the result type anyway.
---
--- Examples:
---
--- >>> let m1 = fromList [[1,2,3], [4,5,6]] :: Mat Vec2D Vec3D Int
--- >>> let m2 = fromList [[1,2],[3,4],[5,6]] :: Mat Vec3D Vec2D Int
--- >>> m1 `mult` m2
--- ((22,28),(49,64))
---
-mult :: (Num a,
- Vector v (Vn w a),
- Vector w a,
- Vector w (Vn z a),
- Vector z a,
- Vector v (Vn z a))
- => Mat v w a
- -> Mat w z a
- -> Mat v z a
-mult m1 m2 = construct lambda
- where
- lambda i j =
- sum [(m1 !!! (i,k)) * (m2 !!! (k,j)) | k <- [0..(ncols m1)-1] ]
projects / numerical-analysis.git / blobdiff