{-# OPTIONS -fno-implicit-prelude -fglasgow-exts #-} {- | Copyright : (c) Mikael Johansson 2006 Maintainer : mik@math.uni-jena.de Stability : provisional Portability : requires multi-parameter type classes Routines and abstractions for Matrices and basic linear algebra over fields or rings. -} module MathObj.Matrix where import qualified Algebra.Module as Module import qualified Algebra.Vector as Vector import qualified Algebra.Ring as Ring import qualified Algebra.Additive as Additive import Algebra.Module((*>)) import Algebra.Ring((*), fromInteger, scalarProduct) import Algebra.Additive((+), (-), zero, subtract) import Data.Array (Array, listArray, elems, bounds, (!), ixmap, range) import qualified Data.List as List import Control.Monad (liftM2) import Control.Exception (assert) import NumericPrelude.List (outerProduct) import NumericPrelude(Integer) import PreludeBase hiding (zipWith) {- | A matrix is a twodimensional array of ring elements, indexed by integers. -} data {-(Ring.C a) =>-} T a = Cons (Array (Integer, Integer) a) deriving (Eq,Ord,Read) {- | Transposition of matrices is just transposition in the sense of Data.List. -} -- candidate for Utility twist :: (Integer,Integer) -> (Integer,Integer) twist (x,y) = (y,x) transpose :: T a -> T a transpose (Cons m) = let (lower,upper) = bounds m in Cons (ixmap (twist lower, twist upper) twist m) rows :: T a -> [[a]] rows (Cons m) = let ((lr,lc), (ur,uc)) = bounds m in outerProduct (curry(m!)) (range (lr,ur)) (range (lc,uc)) columns :: T a -> [[a]] columns (Cons m) = let ((lr,lc), (ur,uc)) = bounds m in outerProduct (curry(m!)) (range (lc,uc)) (range (lr,ur)) fromList :: Integer -> Integer -> [a] -> T a fromList m n xs = Cons (listArray ((1,1),(m,n)) xs) instance (Ring.C a, Show a) => Show (T a) where show m = List.unlines $ map (\r -> "(" ++ r ++ ")") $ map (unwords . map show) $ rows m dimension :: T a -> (Integer,Integer) dimension (Cons m) = uncurry subtract (bounds m) + (1,1) numRows :: T a -> Integer numRows = fst . dimension numColumns :: T a -> Integer numColumns = snd . dimension -- These implementations may benefit from a better exception than -- just assertions to validate dimensionalities instance (Additive.C a) => Additive.C (T a) where (+) = zipWith (+) (-) = zipWith (-) zero = zeroMatrix 1 1 zipWith :: (a -> b -> c) -> T a -> T b -> T c zipWith op mM@(Cons m) nM@(Cons n) = let d = dimension mM em = elems m en = elems n in assert (d == dimension nM) $ uncurry fromList d (List.zipWith op em en) zeroMatrix :: (Additive.C a) => Integer -> Integer -> T a zeroMatrix m n = fromList m n $ List.repeat zero -- List.replicate (fromInteger (m*n)) zero instance (Ring.C a) => Ring.C (T a) where mM * nM = assert (numRows mM == numColumns nM) $ fromList (numColumns mM) (numRows nM) (liftM2 scalarProduct (rows mM) (columns nM)) fromInteger n = fromList 1 1 [fromInteger n] instance Functor T where fmap f (Cons m) = Cons (fmap f m) instance Vector.C T where zero = zero (<+>) = (+) (*>) = Vector.functorScale instance Module.C a b => Module.C a (T b) where x *> m = fmap (x*>) m {- | What more do we need from our matrix class? We have addition, subtraction and multiplication, and thus composition of generic free-module-maps. We're going to want to solve linear equations with or without fields underneath, so we're going to want an implementation of the Gaussian algorithm as well as most probably Smith normal form. Determinants are cool, and these are to be calculated either with the Gaussian algorithm or some other goodish method. -} {- | We'll want generic linear equation solving, returning one solution, any solution really, or nothing. Basically, this is asking for the preimage of a given vector over the given map, so a_11 x_1 + .. + a_1n x_n = y_1 ... a_m1 x_1 + .. + a_mn a_n = y_m has really x_1,...,x_n as a preimage of the vector y_1,..,y_m under the map (a_ij), since obviously y_1,..,y_m = (a_ij) x_1,..,x_n So, generic linear equation solving boils down to the function -} preimage :: (Ring.C a) => T a -> T a -> Maybe (T a) preimage a y = assert (numRows a == numRows y && -- they match numColumns y == 1) -- and y is a column vector Nothing {- Cf. /usr/lib/hugs/demos/Matrix.hs -}