Copyright	(c) Henning Thielemann 2009 Mikael Johansson 2006
Maintainer	numericprelude@henning-thielemann.de
Stability	provisional
Portability	requires multi-parameter type classes
Safe Haskell	None
Language	Haskell98

MathObj.Matrix

Description

Routines and abstractions for Matrices and basic linear algebra over fields or rings.

We stick to simple Int indices. Although advanced indices would be nice e.g. for matrices with sub-matrices, this is not easily implemented since arrays do only support a lower and an upper bound but no additional parameters.

ToDo: - Matrix inverse, determinant (see htam:Matrix)

Synopsis

data T a
type Dimension = Int
format :: Show a => T a -> String
transpose :: T a -> T a
rows :: T a -> [[a]]
columns :: T a -> [[a]]
index :: T a -> Dimension -> Dimension -> a
fromRows :: Dimension -> Dimension -> [[a]] -> T a
fromColumns :: Dimension -> Dimension -> [[a]] -> T a
fromList :: Dimension -> Dimension -> [a] -> T a
dimension :: T a -> (Dimension, Dimension)
numRows :: T a -> Dimension
numColumns :: T a -> Dimension
zipWith :: (a -> b -> c) -> T a -> T b -> T c
zero :: C a => Dimension -> Dimension -> T a
one :: C a => Dimension -> T a
diagonal :: C a => [a] -> T a
scale :: C a => a -> T a -> T a
random :: (RandomGen g, Random a) => Dimension -> Dimension -> g -> (T a, g)
randomR :: (RandomGen g, Random a) => Dimension -> Dimension -> (a, a) -> g -> (T a, g)

Documentation

data T a Source #

A matrix is a twodimensional array, indexed by integers.

Instances

Instances details

Functor T Source #
Instance details Defined in MathObj.Matrix Methods fmap :: (a -> b) -> T a -> T b # (<$) :: a -> T b -> T a #
C T Source #
Instance details Defined in MathObj.Matrix Methods zero :: C a => T a Source # (<+>) :: C a => T a -> T a -> T a Source # (*>) :: C a => a -> T a -> T a Source #
C a b => C a (T b) Source #
Instance details Defined in MathObj.Matrix Methods (*>) :: a -> T b -> T b Source #
Eq a => Eq (T a) Source #
Instance details Defined in MathObj.Matrix Methods (==) :: T a -> T a -> Bool # (/=) :: T a -> T a -> Bool #
Ord a => Ord (T a) Source #
Instance details Defined in MathObj.Matrix Methods compare :: T a -> T a -> Ordering # (<) :: T a -> T a -> Bool # (<=) :: T a -> T a -> Bool # (>) :: T a -> T a -> Bool # (>=) :: T a -> T a -> Bool # max :: T a -> T a -> T a # min :: T a -> T a -> T a #
Read a => Read (T a) Source #
Instance details Defined in MathObj.Matrix Methods readsPrec :: Int -> ReadS (T a) # readList :: ReadS [T a] # readPrec :: ReadPrec (T a) # readListPrec :: ReadPrec [T a] #
Show a => Show (T a) Source #
Instance details Defined in MathObj.Matrix Methods showsPrec :: Int -> T a -> ShowS # show :: T a -> String # showList :: [T a] -> ShowS #
C a => C (T a) Source #	genIntMatrix /\ \a -> genSameMatrix a /\ \b -> Laws.commutative (+) a b genIntMatrix /\ \a -> genSameMatrix a /\ \b -> genSameMatrix b /\ \c -> Laws.associative (+) a b c
Instance details Defined in MathObj.Matrix Methods zero :: T a Source # (+) :: T a -> T a -> T a Source # (-) :: T a -> T a -> T a Source # negate :: T a -> T a Source #
C a => C (T a) Source #	genIntMatrix /\ \a -> Laws.leftIdentity () (Matrix.one (Matrix.numRows a)) a genIntMatrix /\ \a -> Laws.rightIdentity () (Matrix.one (Matrix.numColumns a)) a genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> Laws.homomorphism Matrix.transpose () (flip ()) a b genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> genFactorMatrix b /\ \c -> Laws.associative () a b c genIntMatrix /\ \b -> genSameMatrix b /\ \c -> genFactorMatrix b /\ \a -> Laws.leftDistributive () (+) a b c genIntMatrix /\ \a -> genFactorMatrix a /\ \b -> genSameMatrix b /\ \c -> Laws.rightDistributive () (+) a b c QC.choose (0,10) /\ \k -> genDimension /\ \n -> genMatrixFor n n /\ \a -> a^k == nest (fromInteger k) ((a::Matrix.T Integer)) (Matrix.one n)
Instance details Defined in MathObj.Matrix Methods (*) :: T a -> T a -> T a Source # one :: T a Source # fromInteger :: Integer -> T a Source # (^) :: T a -> Integer -> T a Source #

type Dimension = Int Source #

format :: Show a => T a -> String Source #

transpose :: T a -> T a Source #

Transposition of matrices is just transposition in the sense of Data.List.

genIntMatrix /\ \a -> Matrix.rows a == Matrix.columns (Matrix.transpose a)

genIntMatrix /\ \a -> Matrix.columns a == Matrix.rows (Matrix.transpose a)

genIntMatrix /\ \a -> genSameMatrix a /\ \b -> Laws.homomorphism Matrix.transpose (+) (+) a b

rows :: T a -> [[a]] Source #

columns :: T a -> [[a]] Source #

index :: T a -> Dimension -> Dimension -> a Source #

fromRows :: Dimension -> Dimension -> [[a]] -> T a Source #

genIntMatrix /\ \a -> a == uncurry Matrix.fromRows (Matrix.dimension a) (Matrix.rows a)

fromColumns :: Dimension -> Dimension -> [[a]] -> T a Source #

genIntMatrix /\ \a -> a == uncurry Matrix.fromColumns (Matrix.dimension a) (Matrix.columns a)

fromList :: Dimension -> Dimension -> [a] -> T a Source #

dimension :: T a -> (Dimension, Dimension) Source #

numRows :: T a -> Dimension Source #

numColumns :: T a -> Dimension Source #

zipWith :: (a -> b -> c) -> T a -> T b -> T c Source #

zero :: C a => Dimension -> Dimension -> T a Source #

genIntMatrix /\ \a -> Laws.identity (+) (uncurry Matrix.zero $ Matrix.dimension a) a

one :: C a => Dimension -> T a Source #

diagonal :: C a => [a] -> T a Source #

genDimension /\ \n -> Matrix.one n == Matrix.diagonal (replicate n Ring.one :: [Integer])

scale :: C a => a -> T a -> T a Source #

random :: (RandomGen g, Random a) => Dimension -> Dimension -> g -> (T a, g) Source #

randomR :: (RandomGen g, Random a) => Dimension -> Dimension -> (a, a) -> g -> (T a, g) Source #