-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Algorithms using the Repa array library.
--   
--   NOTE: You must use the GHC head branch > 6.13.20100309 to get
--   decent performance. Reusable algorithms using the Repa array library.
@package repa-algorithms
@version 1.1.0.0


-- | Algorithms operating on matrices.
--   
--   These functions should give performance comparable with nested loop C
--   implementations, but not block-based, cache friendly, SIMD using,
--   vendor optimised implementions. If you care deeply about runtime
--   performance then you may be better off using a binding to LAPACK, such
--   as hvector.
module Data.Array.Repa.Algorithms.Matrix

-- | Matrix-matrix multiply.
multiplyMM :: Array DIM2 Double -> Array DIM2 Double -> Array DIM2 Double


-- | Strict complex doubles.
module Data.Array.Repa.Algorithms.Complex

-- | Strict complex doubles.
type Complex = Double :*: Double

-- | Take the magnitude of a complex number.
mag :: Complex -> Double

-- | Take the argument (phase) of a complex number, in the range [-pi ..
--   pi].
arg :: Complex -> Double

-- | Strict pair
data (:*:) a b :: * -> * -> *
(:*:) :: !a -> !b -> :*: a b
instance Fractional Complex
instance Num Complex


-- | Calculation of roots of unity for the forward and inverse DFT/FFT.
module Data.Array.Repa.Algorithms.DFT.Roots

-- | Calculate roots of unity for the forward transform.
calcRootsOfUnity :: (Shape sh) => (sh :. Int) -> Array (sh :. Int) Complex

-- | Calculate roots of unity for the inverse transform.
calcInverseRootsOfUnity :: (Shape sh) => (sh :. Int) -> Array (sh :. Int) Complex


-- | Compute the Discrete Fourier Transform (DFT) along the low order
--   dimension of an array.
--   
--   This uses the naive algorithm and takes O(n^2) time. However, you can
--   transform an array with an arbitray extent, unlike with FFT which
--   requires each dimension to be a power of two.
--   
--   The <a>dft</a> and <a>idft</a> functions also compute the roots of
--   unity needed. If you need to transform several arrays with the same
--   extent then it is faster to compute the roots once using
--   <a>calcRootsOfUnity</a> or <a>calcInverseRootsOfUnity</a>, then call
--   <a>dftWithRoots</a> directly.
--   
--   You can also compute single values of the transform using
--   <a>dftWithRootsSingle</a>.
module Data.Array.Repa.Algorithms.DFT

-- | Compute the DFT along the low order dimension of an array.
dft :: (Shape sh) => Array (sh :. Int) Complex -> Array (sh :. Int) Complex

-- | Compute the inverse DFT along the low order dimension of an array.
idft :: (Shape sh) => Array (sh :. Int) Complex -> Array (sh :. Int) Complex

-- | Generic function for computation of forward or inverse DFT. This
--   function is also useful if you transform many arrays with the same
--   extent, and don't want to recompute the roots for each one. The extent
--   of the given roots must match that of the input array, else
--   <a>error</a>.
dftWithRoots :: (Shape sh) => Array (sh :. Int) Complex -> Array (sh :. Int) Complex -> Array (sh :. Int) Complex

-- | Compute a single value of the DFT. The extent of the given roots must
--   match that of the input array, else <a>error</a>.
dftWithRootsSingle :: (Shape sh) => Array (sh :. Int) Complex -> Array (sh :. Int) Complex -> (sh :. Int) -> Complex


-- | Applying these transforms to the input of a DFT causes the output to
--   be centered so that the zero frequency is in the middle.
module Data.Array.Repa.Algorithms.DFT.Center

-- | Apply the centering transform to a vector.
centerVector :: Array DIM1 Complex -> Array DIM1 Complex

-- | Apply the centering transform to a matrix.
centerMatrix :: Array DIM2 Complex -> Array DIM2 Complex


-- | Fast computation of Discrete Fourier Transforms using the
--   Cooley-Tuckey algorithm.
--   
--   Time complexity is O(n log n) in the size of the input.
--   
--   Input dimensions must be powers of two, else <a>error</a>.
--   
--   The <a>fft</a> and <a>ifft</a> functions (and friends) also compute
--   the roots of unity needed. If you need to transform several arrays
--   with the same extent then it is faster to compute the roots once using
--   <a>calcRootsOfUnity</a> or <a>calcInverseRootsOfUnity</a>, then call
--   <a>fftWithRoots</a> directly.
--   
--   The inverse transforms provided also perform post-scaling so that
--   <a>ifft</a> is the true inverse of <a>fft</a>. If you don't want that
--   then call <a>fftWithRoots</a> directly.
--   
--   The functions <a>fft2d</a> and <a>fft3d</a> require their inputs to be
--   squares (and cubes) respectively. This allows them to reuse the same
--   roots-of-unity when transforming along each axis. If you need to
--   transform rectanglular arrays then call <a>fftWithRoots</a> directly.
module Data.Array.Repa.Algorithms.FFT

-- | Compute the DFT along the low order dimension of an array.
fft :: (Shape sh) => Array (sh :. Int) Complex -> Array (sh :. Int) Complex

-- | Compute the inverse DFT along the low order dimension of an array.
ifft :: (Shape sh) => Array (sh :. Int) Complex -> Array (sh :. Int) Complex

-- | Compute the DFT of a square matrix. If the matrix is not square then
--   <a>error</a>.
fft2d :: Array DIM2 Complex -> Array DIM2 Complex

-- | Compute the inverse DFT of a square matrix.
ifft2d :: Array DIM2 Complex -> Array DIM2 Complex

-- | Compute the DFT of a 3d cube. If the array is not a cube then
--   <a>error</a>.
fft3d :: Array DIM3 Complex -> Array DIM3 Complex

-- | Compute the inverse DFT of a 3d cube. If the array is not a cube then
--   <a>error</a>.
ifft3d :: Array DIM3 Complex -> Array DIM3 Complex

-- | Generic function for computation of forward or inverse Discrete
--   Fourier Transforms. Computation is along the low order dimension of
--   the array.
fftWithRoots :: (Shape sh) => Array (sh :. Int) Complex -> Array (sh :. Int) Complex -> Array (sh :. Int) Complex