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{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE FunctionalDependencies #-}
{-# LANGUAGE UndecidableInstances #-}
-----------------------------------------------------------------------------
-- |
-- Module : Numeric.Container
-- Copyright : (c) Alberto Ruiz 2010-14
-- License : BSD3
-- Maintainer : Alberto Ruiz
-- Stability : provisional
--
-- Basic numeric operations on 'Vector' and 'Matrix', including conversion routines.
--
-- The 'Container' class is used to define optimized generic functions which work
-- on 'Vector' and 'Matrix' with real or complex elements.
--
-- Some of these functions are also available in the instances of the standard
-- numeric Haskell classes provided by "Numeric.LinearAlgebra".
--
-----------------------------------------------------------------------------
{-# OPTIONS_HADDOCK hide #-}
module Numeric.Container (
-- * Basic functions
module Data.Packed,
konst, build,
linspace,
diag, ident,
ctrans,
-- * Generic operations
Container,
add, mul, sub, divide, equal, scaleRecip, addConstant,
scalar, conj, scale, arctan2, cmap,
atIndex, minIndex, maxIndex, minElement, maxElement,
sumElements, prodElements,
step, cond, find, assoc, accum,
Transposable(..), Linear(..),
-- * Matrix product
Product(..), udot, dot, (◇),
Mul(..),
Contraction(..),(<.>),
optimiseMult,
mXm,mXv,vXm,LSDiv(..),
outer, kronecker,
-- * Random numbers
RandDist(..),
randomVector,
gaussianSample,
uniformSample,
-- * Element conversion
Convert(..),
Complexable(),
RealElement(),
RealOf, ComplexOf, SingleOf, DoubleOf,
IndexOf,
module Data.Complex,
-- * IO
module Data.Packed.IO,
-- * Misc
Testable(..)
) where
import Data.Packed hiding (stepD, stepF, condD, condF, conjugateC, conjugateQ)
import Data.Packed.Internal.Numeric
import Data.Complex
import Numeric.LinearAlgebra.Algorithms(Field,linearSolveSVD)
import Data.Monoid(Monoid(mconcat))
import Data.Packed.IO
import Numeric.LinearAlgebra.Random
------------------------------------------------------------------
{- | Creates a real vector containing a range of values:
>>> linspace 5 (-3,7::Double)
fromList [-3.0,-0.5,2.0,4.5,7.0]@
>>> linspace 5 (8,2+i) :: Vector (Complex Double)
fromList [8.0 :+ 0.0,6.5 :+ 0.25,5.0 :+ 0.5,3.5 :+ 0.75,2.0 :+ 1.0]
Logarithmic spacing can be defined as follows:
@logspace n (a,b) = 10 ** linspace n (a,b)@
-}
linspace :: (Container Vector e) => Int -> (e, e) -> Vector e
linspace 0 (a,b) = fromList[(a+b)/2]
linspace n (a,b) = addConstant a $ scale s $ fromList $ map fromIntegral [0 .. n-1]
where s = (b-a)/fromIntegral (n-1)
--------------------------------------------------------
{- | Matrix product, matrix - vector product, and dot product (equivalent to 'contraction')
(This operator can also be written using the unicode symbol ◇ (25c7).)
Examples:
>>> let a = (3><4) [1..] :: Matrix Double
>>> let v = fromList [1,0,2,-1] :: Vector Double
>>> let u = fromList [1,2,3] :: Vector Double
>>> a
(3><4)
[ 1.0, 2.0, 3.0, 4.0
, 5.0, 6.0, 7.0, 8.0
, 9.0, 10.0, 11.0, 12.0 ]
matrix × matrix:
>>> disp 2 (a <.> trans a)
3x3
30 70 110
70 174 278
110 278 446
matrix × vector:
>>> a <.> v
fromList [3.0,11.0,19.0]
dot product:
>>> u <.> fromList[3,2,1::Double]
10
For complex vectors the first argument is conjugated:
>>> fromList [1,i] <.> fromList[2*i+1,3]
1.0 :+ (-1.0)
>>> fromList [1,i,1-i] <.> complex a
fromList [10.0 :+ 4.0,12.0 :+ 4.0,14.0 :+ 4.0,16.0 :+ 4.0]
-}
infixl 7 <.>
(<.>) :: Contraction a b c => a -> b -> c
(<.>) = contraction
class Contraction a b c | a b -> c
where
-- | Matrix product, matrix - vector product, and dot product
contraction :: a -> b -> c
instance (Product t, Container Vector t) => Contraction (Vector t) (Vector t) t where
u `contraction` v = conj u `udot` v
instance Product t => Contraction (Matrix t) (Vector t) (Vector t) where
contraction = mXv
instance (Container Vector t, Product t) => Contraction (Vector t) (Matrix t) (Vector t) where
contraction v m = (conj v) `vXm` m
instance Product t => Contraction (Matrix t) (Matrix t) (Matrix t) where
contraction = mXm
--------------------------------------------------------------------------------
class Mul a b c | a b -> c where
infixl 7 <>
-- | Matrix-matrix, matrix-vector, and vector-matrix products.
(<>) :: Product t => a t -> b t -> c t
instance Mul Matrix Matrix Matrix where
(<>) = mXm
instance Mul Matrix Vector Vector where
(<>) m v = flatten $ m <> asColumn v
instance Mul Vector Matrix Vector where
(<>) v m = flatten $ asRow v <> m
--------------------------------------------------------------------------------
class LSDiv c where
infixl 7 <\>
-- | least squares solution of a linear system, similar to the \\ operator of Matlab\/Octave (based on linearSolveSVD)
(<\>) :: Field t => Matrix t -> c t -> c t
instance LSDiv Vector where
m <\> v = flatten (linearSolveSVD m (reshape 1 v))
instance LSDiv Matrix where
(<\>) = linearSolveSVD
--------------------------------------------------------------------------------
class Konst e d c | d -> c, c -> d
where
-- |
-- >>> konst 7 3 :: Vector Float
-- fromList [7.0,7.0,7.0]
--
-- >>> konst i (3::Int,4::Int)
-- (3><4)
-- [ 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0
-- , 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0
-- , 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0 ]
--
konst :: e -> d -> c e
instance Container Vector e => Konst e Int Vector
where
konst = konst'
instance Container Vector e => Konst e (Int,Int) Matrix
where
konst = konst'
--------------------------------------------------------------------------------
class Build d f c e | d -> c, c -> d, f -> e, f -> d, f -> c, c e -> f, d e -> f
where
-- |
-- >>> build 5 (**2) :: Vector Double
-- fromList [0.0,1.0,4.0,9.0,16.0]
--
-- Hilbert matrix of order N:
--
-- >>> let hilb n = build (n,n) (\i j -> 1/(i+j+1)) :: Matrix Double
-- >>> putStr . dispf 2 $ hilb 3
-- 3x3
-- 1.00 0.50 0.33
-- 0.50 0.33 0.25
-- 0.33 0.25 0.20
--
build :: d -> f -> c e
instance Container Vector e => Build Int (e -> e) Vector e
where
build = build'
instance Container Matrix e => Build (Int,Int) (e -> e -> e) Matrix e
where
build = build'
--------------------------------------------------------------------------------
-- | alternative unicode symbol (25c7) for 'contraction'
(◇) :: Contraction a b c => a -> b -> c
infixl 7 ◇
(◇) = contraction
-- | dot product: @cdot u v = 'udot' ('conj' u) v@
dot :: (Container Vector t, Product t) => Vector t -> Vector t -> t
dot u v = udot (conj u) v
--------------------------------------------------------------------------------
optimiseMult :: Monoid (Matrix t) => [Matrix t] -> Matrix t
optimiseMult = mconcat
|