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1 | {-# LANGUAGE TypeFamilies #-} | ||
2 | {-# LANGUAGE FlexibleContexts #-} | ||
3 | {-# LANGUAGE FlexibleInstances #-} | ||
4 | {-# LANGUAGE MultiParamTypeClasses #-} | ||
5 | {-# LANGUAGE FunctionalDependencies #-} | ||
6 | {-# LANGUAGE UndecidableInstances #-} | ||
7 | |||
8 | ----------------------------------------------------------------------------- | ||
9 | -- | | ||
10 | -- Module : Numeric.Container | ||
11 | -- Copyright : (c) Alberto Ruiz 2010-14 | ||
12 | -- License : GPL | ||
13 | -- | ||
14 | -- Maintainer : Alberto Ruiz <aruiz@um.es> | ||
15 | -- Stability : provisional | ||
16 | -- Portability : portable | ||
17 | -- | ||
18 | -- Basic numeric operations on 'Vector' and 'Matrix', including conversion routines. | ||
19 | -- | ||
20 | -- The 'Container' class is used to define optimized generic functions which work | ||
21 | -- on 'Vector' and 'Matrix' with real or complex elements. | ||
22 | -- | ||
23 | -- Some of these functions are also available in the instances of the standard | ||
24 | -- numeric Haskell classes provided by "Numeric.LinearAlgebra". | ||
25 | -- | ||
26 | ----------------------------------------------------------------------------- | ||
27 | {-# OPTIONS_HADDOCK hide #-} | ||
28 | |||
29 | module Numeric.Container ( | ||
30 | -- * Basic functions | ||
31 | module Data.Packed, | ||
32 | konst, build, | ||
33 | linspace, | ||
34 | diag, ident, | ||
35 | ctrans, | ||
36 | -- * Generic operations | ||
37 | Container(..), | ||
38 | -- * Matrix product | ||
39 | Product(..), udot, dot, (◇), | ||
40 | Mul(..), | ||
41 | Contraction(..), | ||
42 | optimiseMult, | ||
43 | mXm,mXv,vXm,LSDiv(..), | ||
44 | outer, kronecker, | ||
45 | -- * Element conversion | ||
46 | Convert(..), | ||
47 | Complexable(), | ||
48 | RealElement(), | ||
49 | |||
50 | RealOf, ComplexOf, SingleOf, DoubleOf, | ||
51 | |||
52 | IndexOf, | ||
53 | module Data.Complex | ||
54 | ) where | ||
55 | |||
56 | import Data.Packed hiding (stepD, stepF, condD, condF, conjugateC, conjugateQ) | ||
57 | import Data.Packed.Numeric | ||
58 | import Data.Complex | ||
59 | import Numeric.LinearAlgebra.Algorithms(Field,linearSolveSVD) | ||
60 | import Data.Monoid(Monoid(mconcat)) | ||
61 | |||
62 | ------------------------------------------------------------------ | ||
63 | |||
64 | {- | Creates a real vector containing a range of values: | ||
65 | |||
66 | >>> linspace 5 (-3,7::Double) | ||
67 | fromList [-3.0,-0.5,2.0,4.5,7.0]@ | ||
68 | |||
69 | >>> linspace 5 (8,2+i) :: Vector (Complex Double) | ||
70 | fromList [8.0 :+ 0.0,6.5 :+ 0.25,5.0 :+ 0.5,3.5 :+ 0.75,2.0 :+ 1.0] | ||
71 | |||
72 | Logarithmic spacing can be defined as follows: | ||
73 | |||
74 | @logspace n (a,b) = 10 ** linspace n (a,b)@ | ||
75 | -} | ||
76 | linspace :: (Container Vector e) => Int -> (e, e) -> Vector e | ||
77 | linspace 0 (a,b) = fromList[(a+b)/2] | ||
78 | linspace n (a,b) = addConstant a $ scale s $ fromList $ map fromIntegral [0 .. n-1] | ||
79 | where s = (b-a)/fromIntegral (n-1) | ||
80 | |||
81 | -------------------------------------------------------- | ||
82 | |||
83 | class Contraction a b c | a b -> c | ||
84 | where | ||
85 | infixl 7 <.> | ||
86 | {- | Matrix product, matrix vector product, and dot product | ||
87 | |||
88 | Examples: | ||
89 | |||
90 | >>> let a = (3><4) [1..] :: Matrix Double | ||
91 | >>> let v = fromList [1,0,2,-1] :: Vector Double | ||
92 | >>> let u = fromList [1,2,3] :: Vector Double | ||
93 | |||
94 | >>> a | ||
95 | (3><4) | ||
96 | [ 1.0, 2.0, 3.0, 4.0 | ||
97 | , 5.0, 6.0, 7.0, 8.0 | ||
98 | , 9.0, 10.0, 11.0, 12.0 ] | ||
99 | |||
100 | matrix × matrix: | ||
101 | |||
102 | >>> disp 2 (a <.> trans a) | ||
103 | 3x3 | ||
104 | 30 70 110 | ||
105 | 70 174 278 | ||
106 | 110 278 446 | ||
107 | |||
108 | matrix × vector: | ||
109 | |||
110 | >>> a <.> v | ||
111 | fromList [3.0,11.0,19.0] | ||
112 | |||
113 | dot product: | ||
114 | |||
115 | >>> u <.> fromList[3,2,1::Double] | ||
116 | 10 | ||
117 | |||
118 | For complex vectors the first argument is conjugated: | ||
119 | |||
120 | >>> fromList [1,i] <.> fromList[2*i+1,3] | ||
121 | 1.0 :+ (-1.0) | ||
122 | |||
123 | >>> fromList [1,i,1-i] <.> complex a | ||
124 | fromList [10.0 :+ 4.0,12.0 :+ 4.0,14.0 :+ 4.0,16.0 :+ 4.0] | ||
125 | |||
126 | -} | ||
127 | (<.>) :: a -> b -> c | ||
128 | |||
129 | |||
130 | instance (Product t, Container Vector t) => Contraction (Vector t) (Vector t) t where | ||
131 | u <.> v = conj u `udot` v | ||
132 | |||
133 | instance Product t => Contraction (Matrix t) (Vector t) (Vector t) where | ||
134 | (<.>) = mXv | ||
135 | |||
136 | instance (Container Vector t, Product t) => Contraction (Vector t) (Matrix t) (Vector t) where | ||
137 | (<.>) v m = (conj v) `vXm` m | ||
138 | |||
139 | instance Product t => Contraction (Matrix t) (Matrix t) (Matrix t) where | ||
140 | (<.>) = mXm | ||
141 | |||
142 | |||
143 | -------------------------------------------------------------------------------- | ||
144 | |||
145 | class Mul a b c | a b -> c where | ||
146 | infixl 7 <> | ||
147 | -- | Matrix-matrix, matrix-vector, and vector-matrix products. | ||
148 | (<>) :: Product t => a t -> b t -> c t | ||
149 | |||
150 | instance Mul Matrix Matrix Matrix where | ||
151 | (<>) = mXm | ||
152 | |||
153 | instance Mul Matrix Vector Vector where | ||
154 | (<>) m v = flatten $ m <> asColumn v | ||
155 | |||
156 | instance Mul Vector Matrix Vector where | ||
157 | (<>) v m = flatten $ asRow v <> m | ||
158 | |||
159 | -------------------------------------------------------------------------------- | ||
160 | |||
161 | class LSDiv c where | ||
162 | infixl 7 <\> | ||
163 | -- | least squares solution of a linear system, similar to the \\ operator of Matlab\/Octave (based on linearSolveSVD) | ||
164 | (<\>) :: Field t => Matrix t -> c t -> c t | ||
165 | |||
166 | instance LSDiv Vector where | ||
167 | m <\> v = flatten (linearSolveSVD m (reshape 1 v)) | ||
168 | |||
169 | instance LSDiv Matrix where | ||
170 | (<\>) = linearSolveSVD | ||
171 | |||
172 | -------------------------------------------------------------------------------- | ||
173 | |||
174 | class Konst e d c | d -> c, c -> d | ||
175 | where | ||
176 | -- | | ||
177 | -- >>> konst 7 3 :: Vector Float | ||
178 | -- fromList [7.0,7.0,7.0] | ||
179 | -- | ||
180 | -- >>> konst i (3::Int,4::Int) | ||
181 | -- (3><4) | ||
182 | -- [ 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0 | ||
183 | -- , 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0 | ||
184 | -- , 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0, 0.0 :+ 1.0 ] | ||
185 | -- | ||
186 | konst :: e -> d -> c e | ||
187 | |||
188 | instance Container Vector e => Konst e Int Vector | ||
189 | where | ||
190 | konst = konst' | ||
191 | |||
192 | instance Container Vector e => Konst e (Int,Int) Matrix | ||
193 | where | ||
194 | konst = konst' | ||
195 | |||
196 | -------------------------------------------------------------------------------- | ||
197 | |||
198 | class Build d f c e | d -> c, c -> d, f -> e, f -> d, f -> c, c e -> f, d e -> f | ||
199 | where | ||
200 | -- | | ||
201 | -- >>> build 5 (**2) :: Vector Double | ||
202 | -- fromList [0.0,1.0,4.0,9.0,16.0] | ||
203 | -- | ||
204 | -- Hilbert matrix of order N: | ||
205 | -- | ||
206 | -- >>> let hilb n = build (n,n) (\i j -> 1/(i+j+1)) :: Matrix Double | ||
207 | -- >>> putStr . dispf 2 $ hilb 3 | ||
208 | -- 3x3 | ||
209 | -- 1.00 0.50 0.33 | ||
210 | -- 0.50 0.33 0.25 | ||
211 | -- 0.33 0.25 0.20 | ||
212 | -- | ||
213 | build :: d -> f -> c e | ||
214 | |||
215 | instance Container Vector e => Build Int (e -> e) Vector e | ||
216 | where | ||
217 | build = build' | ||
218 | |||
219 | instance Container Matrix e => Build (Int,Int) (e -> e -> e) Matrix e | ||
220 | where | ||
221 | build = build' | ||
222 | |||
223 | -------------------------------------------------------------------------------- | ||
224 | |||
225 | {- | alternative operator for '(\<.\>)' | ||
226 | |||
227 | x25c7, white diamond | ||
228 | |||
229 | -} | ||
230 | (◇) :: Contraction a b c => a -> b -> c | ||
231 | infixl 7 ◇ | ||
232 | (◇) = (<.>) | ||
233 | |||
234 | -- | dot product: @cdot u v = 'udot' ('conj' u) v@ | ||
235 | dot :: (Container Vector t, Product t) => Vector t -> Vector t -> t | ||
236 | dot u v = udot (conj u) v | ||
237 | |||
238 | -------------------------------------------------------------------------------- | ||
239 | |||
240 | optimiseMult :: Monoid (Matrix t) => [Matrix t] -> Matrix t | ||
241 | optimiseMult = mconcat | ||
242 | |||