import Data.Vector.Fixed (
(!),
- N1,
- N2,
- N3,
- N4,
- N5,
- S,
- Z,
generate,
mk1,
mk2,
fromList,
head,
ifoldl,
+ ifoldr,
imap,
map,
maximum,
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
-- >>> 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]
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
-- >>> 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 :: (Arity m, Arity n) => Mat m n a -> Mat m n b -> Mat m n (a,b)
zip2 m1 m2 =
construct lambda
where
-- | 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@.
--
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