{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE ScopedTypeVariables #-}

-- | QR factorization via Givens rotations.
--
module Linear.QR (
  givens_rotator,
  qr )
where

import qualified Algebra.Ring as Ring ( C )
import qualified Algebra.Algebraic as Algebraic ( C )
import Data.Vector.Fixed ( ifoldl )
import Data.Vector.Fixed.Cont ( Arity )
import NumericPrelude hiding ( (*) )

import Linear.Matrix (
  Mat(..),
  (*),
  (!!!),
  construct,
  identity_matrix,
  transpose )


-- | Construct a givens rotation matrix that will operate on row @i@
--   and column @j@. This is done to create zeros in some column of
--   the target matrix. You must also supply that column's @i@th and
--   @j@th entries as arguments.
--
--   Examples (Watkins, p. 193):
--
--   >>> import Normed ( Normed(..) )
--   >>> import Linear.Vector ( Vec2, Vec3 )
--   >>> import Linear.Matrix ( Mat2, Mat3, fromList, frobenius_norm )
--   >>> import qualified Data.Vector.Fixed as V ( map )
--
--   >>> let m  = givens_rotator 0 1 1 1 :: Mat2 Double
--   >>> let m2 = fromList [[1, -1],[1, 1]] :: Mat2 Double
--   >>> m == (1 / (sqrt 2) :: Double) *> m2
--   True
--
--   >>> let m = fromList [[2,3],[5,7]] :: Mat2 Double
--   >>> let rot =  givens_rotator 0 1 2.0 5.0 :: Mat2 Double
--   >>> ((transpose rot) * m) !!! (1,0) < 1e-12
--   True
--   >>> let (Mat rows) = rot
--   >>> let (Mat cols) = transpose rot
--   >>> V.map norm rows :: Vec2 Double
--   fromList [1.0,1.0]
--   >>> V.map norm cols :: Vec2 Double
--   fromList [1.0,1.0]
--
--   >>> let m = fromList [[12,-51,4],[6,167,-68],[-4,24,-41]] :: Mat3 Double
--   >>> let rot = givens_rotator 1 2 6 (-4) :: Mat3 Double
--   >>> let ex_rot_r1 = [1,0,0] :: [Double]
--   >>> let ex_rot_r2 = [0,0.83205,-0.55470] :: [Double]
--   >>> let ex_rot_r3 = [0, 0.55470, 0.83205] :: [Double]
--   >>> let ex_rot = fromList [ex_rot_r1, ex_rot_r2, ex_rot_r3] :: Mat3 Double
--   >>> frobenius_norm ((transpose rot) - ex_rot) < 1e-4
--   True
--   >>> ((transpose rot) * m) !!! (2,0) == 0
--   True
--   >>> let (Mat rows) = rot
--   >>> let (Mat cols) = transpose rot
--   >>> V.map norm rows :: Vec3 Double
--   fromList [1.0,1.0,1.0]
--   >>> V.map norm cols :: Vec3 Double
--   fromList [1.0,1.0,1.0]
--
givens_rotator :: forall m a. (Eq a, Ring.C a, Algebraic.C a, Arity m)
               => Int -> Int -> a -> a -> Mat m m a
givens_rotator i j xi xj =
  construct f
  where
    xnorm = sqrt $ xi^2 + xj^2
    c = if xnorm == (fromInteger 0) then (fromInteger 1) else xi / xnorm
    s = if xnorm == (fromInteger 0) then (fromInteger 0) else xj / xnorm

    f :: Int -> Int -> a
    f y z
      | y == i && z == i = c
      | y == j && z == j = c
      | y == i && z == j = negate s
      | y == j && z == i = s
      | y == z = fromInteger 1
      | otherwise = fromInteger 0


-- | Compute the QR factorization of a matrix (using Givens
--   rotations). This is accomplished with two folds: the first one
--   traverses the columns of the matrix from left to right, and the
--   second traverses the entries of the column from top to
--   bottom.
--
--   The state that is passed through the fold is the current (q,r)
--   factorization. We keep the pair updated by multiplying @q@ and
--   @r@ by the new rotator (or its transpose).
--
--   We do not require that the diagonal elements of R are positive,
--   so our factorization is a little less unique than usual.
--
--   Examples:
--
--   >>> import Linear.Matrix
--
--   >>> let m = fromList [[1,2],[1,3]] :: Mat2 Double
--   >>> let (q,r) = qr m
--   >>> let c = (1 / (sqrt 2 :: Double))
--   >>> let ex_q = c *> (fromList [[1,-1],[1,1]] :: Mat2 Double)
--   >>> let ex_r = c *> (fromList [[2,5],[0,1]] :: Mat2 Double)
--   >>> frobenius_norm (q - ex_q) == 0
--   True
--   >>> frobenius_norm (r - ex_r) == 0
--   True
--   >>> let m' = q*r
--   >>> frobenius_norm (m - m') < 1e-10
--   True
--   >>> is_upper_triangular' 1e-10 r
--   True
--
--   >>> let m = fromList [[2,3],[5,7]] :: Mat2 Double
--   >>> let (q,r) = qr m
--   >>> frobenius_norm (m - (q*r)) < 1e-12
--   True
--   >>> is_upper_triangular' 1e-10 r
--   True
--
--   >>> let m = fromList [[12,-51,4],[6,167,-68],[-4,24,-41]] :: Mat3 Double
--   >>> let (q,r) = qr m
--   >>> frobenius_norm (m - (q*r)) < 1e-12
--   True
--   >>> is_upper_triangular' 1e-10 r
--   True
--
qr :: forall m n a. (Arity m, Arity n, Eq a, Algebraic.C a, Ring.C a, Show a)
   => Mat m n a -> (Mat m m a, Mat m n a)
qr matrix =
  ifoldl col_function initial_qr columns
  where
    Mat columns = transpose matrix
    initial_qr = (identity_matrix, matrix)

    -- | Process the column and spit out the current QR
    --   factorization. In the first column, we want to get rotators
    --   Q12, Q13, Q14,... In the second column, we want rotators Q23,
    --   Q24, Q25,...
    col_function (q,r) col_idx col =
      ifoldl (f col_idx) (q,r) col

    -- | Process the entries in a column, doing basically the same
    --   thing as col_dunction does. It updates the QR factorization,
    --   maybe, and returns the current one.
    f col_idx (q,r) idx _ -- ignore the current element
      | idx <= col_idx = (q,r)
--          trace ("---------------\nidx: " ++ (show idx) ++ ";\ncol_idx: " ++ (show col_idx) ++ "; leaving it alone") (q,r) -- leave it alone.
      | otherwise = (q*rotator, (transpose rotator)*r)
--          trace ("---------------\nidx: " ++ (show idx) ++ ";\ncol_idx: " ++ (show col_idx) ++ ";\nupdating Q and R;\nq: " ++ (show q) ++ ";\nr " ++ (show r) ++ ";\nnew q: " ++ (show $ q*rotator) ++ ";\nnew r: " ++ (show $ (transpose rotator)*r) ++ ";\ny: " ++ (show y) ++ ";\nr[i,j]: " ++ (show (r !!! (col_idx, col_idx))))
--          (q*rotator, (transpose rotator)*r)
          where
            y = r !!! (idx, col_idx)
            rotator :: Mat m m a
            rotator = givens_rotator col_idx idx (r !!! (col_idx, col_idx)) y