1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
|
{-# LANGUAGE FlexibleContexts, FlexibleInstances #-}
{-# LANGUAGE RecordWildCards #-}
module Numeric.LinearAlgebra.Util.CG(
cgSolve, cgSolve',
CGState(..), R, V
) where
import Data.Packed.Numeric
import Numeric.Sparse
import Numeric.Vector()
import Numeric.LinearAlgebra.Algorithms(linearSolveLS, relativeError, NormType(..))
import Control.Arrow((***))
{-
import Util.Misc(debug, debugMat)
(//) :: Show a => a -> String -> a
infix 0 // -- , ///
a // b = debug b id a
(///) :: V -> String -> V
infix 0 ///
v /// b = debugMat b 2 asRow v
-}
type R = Double
type V = Vector R
data CGState = CGState
{ cgp :: V -- ^ conjugate gradient
, cgr :: V -- ^ residual
, cgr2 :: R -- ^ squared norm of residual
, cgx :: V -- ^ current solution
, cgdx :: R -- ^ normalized size of correction
}
cg :: Bool -> (V -> V) -> (V -> V) -> CGState -> CGState
cg sym at a (CGState p r r2 x _) = CGState p' r' r'2 x' rdx
where
ap1 = a p
ap | sym = ap1
| otherwise = at ap1
pap | sym = p <·> ap1
| otherwise = norm2 ap1 ** 2
alpha = r2 / pap
dx = scale alpha p
x' = x + dx
r' = r - scale alpha ap
r'2 = r' <·> r'
beta = r'2 / r2
p' = r' + scale beta p
rdx = norm2 dx / max 1 (norm2 x)
conjugrad
:: Bool -> GMatrix -> V -> V -> R -> R -> [CGState]
conjugrad sym a b = solveG (tr a !#>) (a !#>) (cg sym) b
solveG
:: (V -> V) -> (V -> V)
-> ((V -> V) -> (V -> V) -> CGState -> CGState)
-> V
-> V
-> R -> R
-> [CGState]
solveG mat ma meth rawb x0' ϵb ϵx
= takeUntil ok . iterate (meth mat ma) $ CGState p0 r0 r20 x0 1
where
a = mat . ma
b = mat rawb
x0 = if x0' == 0 then konst 0 (dim b) else x0'
r0 = b - a x0
r20 = r0 <·> r0
p0 = r0
nb2 = b <·> b
ok CGState {..}
= cgr2 <nb2*ϵb**2
|| cgdx < ϵx
takeUntil :: (a -> Bool) -> [a] -> [a]
takeUntil q xs = a++ take 1 b
where
(a,b) = break q xs
cgSolve
:: Bool -- ^ is symmetric
-> GMatrix -- ^ coefficient matrix
-> Vector Double -- ^ right-hand side
-> Vector Double -- ^ solution
cgSolve sym a b = cgx $ last $ cgSolve' sym 1E-4 1E-3 n a b 0
where
n = max 10 (round $ sqrt (fromIntegral (dim b) :: Double))
cgSolve'
:: Bool -- ^ symmetric
-> R -- ^ relative tolerance for the residual (e.g. 1E-4)
-> R -- ^ relative tolerance for δx (e.g. 1E-3)
-> Int -- ^ maximum number of iterations
-> GMatrix -- ^ coefficient matrix
-> V -- ^ initial solution
-> V -- ^ right-hand side
-> [CGState] -- ^ solution
cgSolve' sym er es n a b x = take n $ conjugrad sym a b x er es
--------------------------------------------------------------------------------
instance Testable GMatrix
where
checkT _ = (ok,info)
where
sma = convo2 20 3
x1 = vect [1..20]
x2 = vect [1..40]
sm = mkSparse sma
dm = toDense sma
s1 = sm !#> x1
d1 = dm #> x1
s2 = tr sm !#> x2
d2 = tr dm #> x2
sdia = mkDiagR 40 20 (vect [1..10])
s3 = sdia !#> x1
s4 = tr sdia !#> x2
ddia = diagRect 0 (vect [1..10]) 40 20
d3 = ddia #> x1
d4 = tr ddia #> x2
v = testb 40
s5 = cgSolve False sm v
d5 = denseSolve dm v
info = do
print sm
disp (toDense sma)
print s1; print d1
print s2; print d2
print s3; print d3
print s4; print d4
print s5; print d5
print $ relativeError Infinity s5 d5
ok = s1==d1
&& s2==d2
&& s3==d3
&& s4==d4
&& relativeError Infinity s5 d5 < 1E-10
disp = putStr . dispf 2
vect = fromList :: [Double] -> Vector Double
convomat :: Int -> Int -> AssocMatrix
convomat n k = [ ((i,j `mod` n),1) | i<-[0..n-1], j <- [i..i+k-1]]
convo2 :: Int -> Int -> AssocMatrix
convo2 n k = m1 ++ m2
where
m1 = convomat n k
m2 = map (((+n) *** id) *** id) m1
testb n = vect $ take n $ cycle ([0..10]++[9,8..1])
denseSolve a = flatten . linearSolveLS a . asColumn
-- mkDiag v = mkDiagR (dim v) (dim v) v
|
