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