import Data.List (intercalate)
import Data.Vector.Fixed (
+ (!),
N1,
N2,
N3,
N5,
S,
Z,
+ generate,
mk1,
mk2,
mk3,
import qualified Data.Vector.Fixed as V (
and,
fromList,
+ head,
length,
map,
maximum,
zipWith
)
import Data.Vector.Fixed.Boxed (Vec)
-import Data.Vector.Fixed.Internal (Arity, arity)
+import Data.Vector.Fixed.Cont (Arity, arity)
import Linear.Vector
import Normed
-- 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 :: forall m n a. (Arity m, Arity n)
=> (Int -> Int -> a) -> Mat m n a
-construct lambda = Mat rows
+construct lambda = Mat $ generate make_row
where
- -- The arity trick is used in Data.Vector.Fixed.length.
- imax = (arity (undefined :: m)) - 1
- jmax = (arity (undefined :: n)) - 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
-- values on the diagonal of @r@ positive.
determinant :: (p a) -> a
instance (Eq a, Ring.C a) => Determined (Mat (S Z) (S Z)) a where
- determinant m = m !!! (0,0)
+ determinant (Mat rows) = (V.head . V.head) rows
-instance (Eq a, Ring.C a, Arity m) => Determined (Mat m m) a where
- determinant _ = undefined
-
-instance (Eq a, Ring.C a, Arity n)
+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
det_minor i j = determinant (minor m i j)
determinant_recursive =
- sum [ (-1)^(1+(toInteger j)) NP.* (m' 0 j) NP.* (det_minor 0 j)
+ sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (det_minor 0 j)
| j <- [0..(ncols m)-1] ]
--- | Matrix multiplication. Our 'Num' instance doesn't define one, and
--- we need additional restrictions on the result type anyway.
+-- | Matrix multiplication.
--
-- Examples:
--
instance (Algebraic.C a,
ToRational.C a,
- Arity m,
- Arity n)
- => Normed (Mat (S m) (S n) 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
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