fromList,
head,
ifoldl,
+ ifoldr,
imap,
map,
maximum,
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
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] ]
scalar :: a -> Mat1 a
scalar x = Mat (mk1 (mk1 x))
-dot :: (RealRing.C a, m ~ S t, Arity t)
- => Col m a
- -> Col m 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.
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))
-- | 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:
--
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@.
--
--
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