80 lines
2.7 KiB
Haskell
80 lines
2.7 KiB
Haskell
import Data.List
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import Text.Printf (printf)
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eps = 1e-6 :: Double
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-- a matrix is represented as a list of columns
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mmult :: Num a => [[a]] -> [[a]] -> [[a]]
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nth :: Num a => [[a]] -> Int -> Int -> a
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mmult_num :: Num a => [[a]] -> a -> [[a]]
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madd :: Num a => [[a]] -> [[a]] -> [[a]]
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idMatrix :: Num a => Int -> Int -> [[a]]
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adjustWithE :: [[Double]] -> Int -> [[Double]]
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mmult a b = [ [ sum $ zipWith (*) ak bj | ak <- (transpose a) ] | bj <- b ]
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nth mA i j = (mA !! j) !! i
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mmult_num mA n = map (\c -> map (*n) c) mA
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madd mA mB = zipWith (\c1 c2 -> zipWith (+) c1 c2) mA mB
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idMatrix n m = [ [if (i==j) then 1 else 0 | i <- [1..n]] | j <- [1..m]]
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adjustWithE mA n = let lA = length mA in
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(idMatrix n (n - lA)) ++ (map (\c -> (take (n - lA) (repeat 0.0)) ++ c ) mA)
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-- auxiliary functions
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sqsum :: Floating a => [a] -> a
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norm :: Floating a => [a] -> a
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epsilonize :: [[Double]] -> [[Double]]
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sqsum a = foldl (\x y -> x + y*y) 0 a
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norm a = sqrt $! sqsum a
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epsilonize mA = map (\c -> map (\x -> if abs x <= eps then 0 else x) c) mA
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-- Householder transformation; householder A = (Q, R)
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uTransform :: [Double] -> [Double]
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hMatrix :: [Double] -> Int -> Int -> [[Double]]
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householder :: [[Double]] -> ([[Double]], [[Double]])
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-- householder_rec Q R A
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householder_rec :: [[Double]] -> [[Double]] -> Int -> ([[Double]], [[Double]])
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uTransform a = let t = (head a) + (signum (head a))*(norm a) in
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1 : map (\x -> x/t) (tail a)
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hMatrix a n i = let u = uTransform (drop i a) in
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madd
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(idMatrix (n-i) (n-i))
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(mmult_num
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(mmult [u] (transpose [u]))
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((/) (-2) (sqsum u)))
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householder_rec mQ mR 0 = (mQ, mR)
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householder_rec mQ mR n = let mSize = length mR in
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let mH = adjustWithE (hMatrix (mR!!(mSize - n)) mSize (mSize - n)) mSize in
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householder_rec (mmult mQ mH) (mmult mH mR) (n - 1)
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householder mA = let mSize = length mA in
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let (mQ, mR) = householder_rec (idMatrix mSize mSize) mA mSize in
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(epsilonize mQ, epsilonize mR)
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backSubstitution :: [[Double]] -> [Double] -> [Double] -> [Double]
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backSubstitution mR [] res = res
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backSubstitution mR@(hR:tR) q@(h:t) res =
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let x = (h / (head hR)) in
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backSubstitution
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(map tail tR)
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(tail (zipWith (-) q (map (*x) hR)))
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(x : res)
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showMatrix :: [[Double]] -> String
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showMatrix mA =
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concat $ intersperse "\n"
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(map (\x -> unwords $ printf "%10.4f" <$> (x::[Double])) (transpose mA))
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mY = [[12, 6, -4], [-51, 167, 24], [4, -68, -41]] :: [[Double]]
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q = [21, 245, 35] :: [Double]
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main = let (mQ, mR) = householder mY in
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putStrLn ("Q: \n" ++ showMatrix mQ) >>
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putStrLn ("R: \n" ++ showMatrix mR) >>
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putStrLn ("q: \n" ++ show q) >>
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putStrLn ("x: \n" ++ show (backSubstitution (reverse (map reverse mR)) (reverse q) []))
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