{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE NoMonomorphismRestriction #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE RebindableSyntax #-}
maximum,
replicate,
toList,
- zipWith
- )
-import Data.Vector.Fixed.Boxed (Vec)
-import Data.Vector.Fixed.Cont (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
-
+ zipWith )
+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 )
+
+-- | Our main matrix type.
data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
+
+-- Type synonyms for n-by-n matrices.
type Mat1 a = Mat N1 N1 a
type Mat2 a = Mat N2 N2 a
type Mat3 a = Mat N3 N3 a
type Mat4 a = Mat N4 N4 a
type Mat5 a = Mat N5 N5 a
+-- | 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 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
+
instance (Eq a) => Eq (Mat m n a) where
-- | Compare a row at a time.
--
row (Mat rows) i = rows ! i
+-- | Return the @i@th row of @m@ as a matrix. Unsafe.
+row' :: (Arity m, Arity n) => Mat m n a -> Int -> Row n a
+row' m i =
+ construct lambda
+ where
+ lambda _ j = m !!! (i, j)
+
+
-- | Return the @j@th column of @m@. Unsafe.
column :: Mat m n a -> Int -> (Vec m a)
column (Mat rows) j =
element = flip (!)
+-- | Return the @j@th column of @m@ as a matrix. Unsafe.
+column' :: (Arity m, Arity n) => Mat m n a -> Int -> Col m a
+column' m j =
+ construct lambda
+ where
+ lambda i _ = m !!! (i, j)
-- | Transpose @m@; switch it's columns and its rows. This is a dirty
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:
--
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
-- >>> fixed_point g eps u0
-- ((1.0728549599342185),(1.0820591495686167))
--
-vec1d :: (a) -> Mat N1 N1 a
+vec1d :: (a) -> Col1 a
vec1d (x) = Mat (mk1 (mk1 x))
-vec2d :: (a,a) -> Mat N2 N1 a
+vec2d :: (a,a) -> Col2 a
vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y))
-vec3d :: (a,a,a) -> Mat N3 N1 a
+vec3d :: (a,a,a) -> Col3 a
vec3d (x,y,z) = Mat (mk3 (mk1 x) (mk1 y) (mk1 z))
-vec4d :: (a,a,a,a) -> Mat N4 N1 a
+vec4d :: (a,a,a,a) -> Col4 a
vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z))
-vec5d :: (a,a,a,a,a) -> Mat N5 N1 a
+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 -> Mat N1 N1 a
+scalar :: a -> Mat1 a
scalar x = Mat (mk1 (mk1 x))
dot :: (RealRing.C a, n ~ N1, m ~ S t, Arity t)
norms = (norm v1) NP.* (norm v2)
+-- | Retrieve the diagonal elements of the given matrix as a \"column
+-- vector,\" i.e. a m-by-1 matrix. We require the matrix to be
+-- square to avoid ambiguity in the return type which would ideally
+-- have dimension min(m,n) supposing an m-by-n matrix.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> diagonal m
+-- ((1),(5),(9))
+--
+diagonal :: (Arity m) => Mat m m a -> Col m a
+diagonal matrix =
+ construct lambda
+ where
+ lambda i _ = matrix !!! (i,i)
+
-- | Given a square @matrix@, return a new matrix of the same size
-- containing only the on-diagonal entries of @matrix@. The
=> Mat m m a
-> Mat m m a
ut_part_strict = transpose . lt_part_strict . transpose
+
+
+-- | Compute the trace of a square matrix, the sum of the elements
+-- which lie on its diagonal. We require the matrix to be
+-- square to avoid ambiguity in the return type which would ideally
+-- have dimension min(m,n) supposing an m-by-n matrix.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> trace m
+-- 15
+--
+trace :: (Arity m, Ring.C a) => Mat m m a -> a
+trace matrix =
+ let (Mat rows) = diagonal matrix
+ in
+ element_sum $ V.map V.head rows
projects / numerical-analysis.git / blobdiff