{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE NoMonomorphismRestriction #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE RebindableSyntax #-}
N5,
S,
Z,
+ generate,
mk1,
mk2,
mk3,
toList,
zipWith
)
-import Data.Vector.Fixed.Boxed (Vec)
-import Data.Vector.Fixed.Internal.Arity (Arity, arity)
-import Linear.Vector
-import Normed
-
-import NumericPrelude hiding ((*), abs)
-import qualified NumericPrelude as NP ((*))
-import qualified Algebra.Algebraic as Algebraic
-import Algebra.Algebraic (root)
-import qualified Algebra.Additive as Additive
-import qualified Algebra.Ring as Ring
-import qualified Algebra.Module as Module
-import qualified Algebra.RealRing as RealRing
-import qualified Algebra.ToRational as ToRational
-import qualified Algebra.Transcendental as Transcendental
-import qualified Prelude as P
+import Data.Vector.Fixed.Cont ( Arity, arity )
+import Linear.Vector ( Vec, delete, element_sum )
+import Normed ( Normed(..) )
+
+import NumericPrelude hiding ( (*), abs )
+import qualified NumericPrelude as NP ( (*) )
+import qualified Algebra.Absolute as Absolute ( C )
+import Algebra.Absolute ( abs )
+import qualified Algebra.Additive as Additive ( C )
+import qualified Algebra.Algebraic as Algebraic ( C )
+import Algebra.Algebraic ( root )
+import qualified Algebra.Ring as Ring ( C )
+import qualified Algebra.Module as Module ( C )
+import qualified Algebra.RealRing as RealRing ( C )
+import qualified Algebra.ToRational as ToRational ( C )
+import qualified Algebra.Transcendental as Transcendental ( C )
+import qualified Prelude as P ( map )
data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
type Mat1 a = Mat N1 N1 a
-- 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
-- | Returns True if the given matrix is upper-triangular, and False
--- otherwise.
+-- otherwise. The parameter @epsilon@ lets the caller choose a
+-- tolerance.
--
-- Examples:
--
--- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+-- >>> let m = fromList [[1,1],[1e-12,1]] :: Mat2 Double
-- >>> is_upper_triangular m
-- False
---
--- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
--- >>> is_upper_triangular m
+-- >>> is_upper_triangular' 1e-10 m
-- True
--
-is_upper_triangular :: (Eq a, Ring.C a, Arity m, Arity n)
- => Mat m n a -> Bool
-is_upper_triangular m =
+-- TODO:
+--
+-- 1. Don't cheat with lists.
+--
+is_upper_triangular' :: (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
+ => a -- ^ The tolerance @epsilon@.
+ -> Mat m n a
+ -> Bool
+is_upper_triangular' epsilon 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
+ -- use "less than or equal to" so zero is a valid epsilon
+ | otherwise = abs (m !!! (i,j)) <= epsilon
+
+
+-- | Returns True if the given matrix is upper-triangular, and False
+-- otherwise. A specialized version of 'is_upper_triangular\'' with
+-- @epsilon = 0@.
+--
+-- 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
+--
+-- TODO:
+--
+-- 1. The Ord constraint is too strong here, Eq would suffice.
+--
+is_upper_triangular :: (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
+ => Mat m n a -> Bool
+is_upper_triangular = is_upper_triangular' 0
-- | Returns True if the given matrix is lower-triangular, and False
--- otherwise.
+-- otherwise. This is a specialized version of 'is_lower_triangular\''
+-- with @epsilon = 0@.
--
-- Examples:
--
-- >>> is_lower_triangular m
-- False
--
-is_lower_triangular :: (Eq a,
+is_lower_triangular :: (Ord a,
Ring.C a,
+ Absolute.C a,
Arity m,
Arity n)
=> Mat m n a
is_lower_triangular = is_upper_triangular . transpose
+-- | Returns True if the given matrix is lower-triangular, and False
+-- otherwise. The parameter @epsilon@ lets the caller choose a
+-- tolerance.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,1e-12],[1,1]] :: Mat2 Double
+-- >>> is_lower_triangular m
+-- False
+-- >>> is_lower_triangular' 1e-12 m
+-- True
+--
+is_lower_triangular' :: (Ord a,
+ Ring.C a,
+ Absolute.C a,
+ Arity m,
+ Arity n)
+ => a -- ^ The tolerance @epsilon@.
+ -> Mat m n a
+ -> Bool
+is_lower_triangular' epsilon = (is_upper_triangular' epsilon) . transpose
+
+
-- | Returns True if the given matrix is triangular, and False
-- otherwise.
--
-- >>> is_triangular m
-- False
--
-is_triangular :: (Eq a,
+is_triangular :: (Ord a,
Ring.C a,
+ Absolute.C a,
Arity m,
Arity n)
=> Mat m n 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,
+instance (Ord a,
Ring.C a,
+ Absolute.C a,
Arity n,
Determined (Mat (S n) (S n)) a)
=> Determined (Mat (S (S n)) (S (S n))) a where
ToRational.C a,
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.
+ -- | Generic p-norms for vectors in R^n that are represented as nx1
+ -- matrices.
--
-- Examples:
--
-- 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
fromRational' $ toRational $ V.maximum $ V.map V.maximum rows
-
+-- | Compute the Frobenius norm of a matrix. This essentially treats
+-- the matrix as one long vector containing all of its entries (in
+-- any order, it doesn't matter).
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1, 2, 3],[4,5,6],[7,8,9]] :: Mat3 Double
+-- >>> frobenius_norm m == sqrt 285
+-- True
+--
+-- >>> let m = fromList [[1, -1, 1],[-1,1,-1],[1,-1,1]] :: Mat3 Double
+-- >>> frobenius_norm m == 3
+-- True
+--
+frobenius_norm :: (Algebraic.C a, Ring.C a) => Mat m n a -> a
+frobenius_norm (Mat rows) =
+ sqrt $ element_sum $ V.map row_sum rows
+ where
+ -- | Square and add up the entries of a row.
+ row_sum = element_sum . V.map (^2)
-- Vector helpers. We want it to be easy to create low-dimension
-- Examples:
--
-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
--- >>> diagonal m
+-- >>> diagonal_part m
-- ((1,0,0),(0,5,0),(0,0,9))
--
-diagonal :: (Arity m, Ring.C a)
+diagonal_part :: (Arity m, Ring.C a)
=> Mat m m a
-> Mat m m a
-diagonal matrix =
+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