src/Linear/System.hs

   1 {-# LANGUAGE RebindableSyntax #-}
   2 {-# LANGUAGE ScopedTypeVariables #-}
   3 {-# LANGUAGE TypeFamilies #-}
   4
   5 module Linear.System (
   6   backward_substitute,
   7   forward_substitute,
   8   solve_positive_definite )
   9 where
  10
  11 import qualified Algebra.Algebraic as Algebraic ( C )
  12 import Data.Vector.Fixed ( Arity, S )
  13 import NumericPrelude hiding ( (*), abs )
  14 import qualified NumericPrelude as NP ( (*) )
  15 import qualified Algebra.Field as Field ( C )
  16
  17 import Linear.Matrix (
  18   Col,
  19   Mat(..),
  20   (!!!),
  21   cholesky,
  22   construct,
  23   diagonal,
  24   dot,
  25   ifoldl2,
  26   is_lower_triangular,
  27   is_upper_triangular,
  28   ncols,
  29   row,
  30   set_idx,
  31   zip2,
  32   transpose )
  33
  34
  35 -- | Solve the system m' * x = b', where m' is lower-triangular. Will
  36 --   probably crash if m' is non-singular. The result is the vector x.
  37 --
  38 --   Examples:
  39 --
  40 --   >>> import Linear.Matrix ( Mat2, Mat3, frobenius_norm, fromList )
  41 --   >>> import Linear.Matrix ( vec2d, vec3d )
  42 --   >>> import Naturals ( N7 )
  43 --
  44 --   >>> let identity = fromList [[1,0,0],[0,1,0],[0,0,1]] :: Mat3 Double
  45 --   >>> let b = vec3d (1, 2, 3::Double)
  46 --   >>> forward_substitute identity b
  47 --   ((1.0),(2.0),(3.0))
  48 --   >>> (forward_substitute identity b) == b
  49 --   True
  50 --
  51 --   >>> let m = fromList [[1,0],[1,1]] :: Mat2 Double
  52 --   >>> let b = vec2d (1, 1::Double)
  53 --   >>> forward_substitute m b
  54 --   ((1.0),(0.0))
  55 --
  56 --   >>> let m = fromList [[4,0],[0,2]] :: Mat2 Double
  57 --   >>> let b = vec2d (2, 1.5 :: Double)
  58 --   >>> forward_substitute m b
  59 --   ((0.5),(0.75))
  60 --
  61 --   >>> let f1 = [0.0418]
  62 --   >>> let f2 = [0.0805]
  63 --   >>> let f3 = [0.1007]
  64 --   >>> let f4 = [-0.0045]
  65 --   >>> let f5 = [-0.0332]
  66 --   >>> let f6 = [-0.0054]
  67 --   >>> let f7 = [-0.0267]
  68 --   >>> let big_F = fromList [f1,f2,f3,f4,f5,f6,f7] :: Col N7 Double
  69 --   >>> let k1 = [6, -3, 0, 0, 0, 0, 0] :: [Double]
  70 --   >>> let k2 = [-3, 10.5, -7.5, 0, 0, 0, 0] :: [Double]
  71 --   >>> let k3 = [0, -7.5, 12.5, 0, 0, 0, 0] :: [Double]
  72 --   >>> let k4 = [0, 0, 0, 6, 0, 0, 0] :: [Double]
  73 --   >>> let k5 = [0, 0, 0, 0, 6, 0, 0] :: [Double]
  74 --   >>> let k6 = [0, 0, 0, 0, 0, 6, 0] :: [Double]
  75 --   >>> let k7 = [0, 0, 0, 0, 0, 0, 15] :: [Double]
  76 --   >>> let big_K = fromList [k1,k2,k3,k4,k5,k6,k7] :: Mat N7 N7 Double
  77 --   >>> let r = cholesky big_K
  78 --   >>> let rt = transpose r
  79 --   >>> let e1 = [0.0170647785413895] :: [Double]
  80 --   >>> let e2 = [0.0338] :: [Double]
  81 --   >>> let e3 = [0.07408] :: [Double]
  82 --   >>> let e4 = [-0.00183711730708738] :: [Double]
  83 --   >>> let e5 = [-0.0135538432434003] :: [Double]
  84 --   >>> let e6 = [-0.00220454076850486] :: [Double]
  85 --   >>> let e7 = [-0.00689391035624920] :: [Double]
  86 --   >>> let expected = fromList [e1,e2,e3,e4,e5,e6,e7] :: Col N7 Double
  87 --   >>> let actual = forward_substitute rt big_F
  88 --   >>> frobenius_norm (actual - expected) < 1e-10
  89 --   True
  90 --
  91 forward_substitute :: forall a m. (Eq a, Field.C a, Arity m)
  92                    => Mat (S m) (S m) a
  93                    -> Col (S m) a
  94                    -> Col (S m) a
  95 forward_substitute matrix b
  96   | not (is_lower_triangular matrix) =
  97       error "forward substitution on non-lower-triangular matrix"
  98   | otherwise = ifoldl2 f zero pairs
  99       where
 100         -- Pairs (m_ii, b_i) needed at each step.
 101         pairs :: Col (S m) (a,a)
 102         pairs = zip2 (diagonal matrix) b
 103
 104         f :: Int -> Int -> Col (S m) a -> (a, a) -> Col (S m) a
 105         f i _ x (mii, bi) = set_idx x (i,0) newval
 106           where
 107             newval = (bi - (x `dot` (transpose $ row matrix i))) / mii
 108
 109
 110
 111 -- | Solve the system m*x = b, where m is upper-triangular. Will
 112 --   probably crash if m is non-singular. The result is the vector x.
 113 --
 114 --   Examples:
 115 --
 116 --   >>> import Linear.Matrix ( Mat3, fromList, vec3d )
 117 --
 118 --   >>> let identity = fromList [[1,0,0],[0,1,0],[0,0,1]] :: Mat3 Double
 119 --   >>> let b = vec3d (1, 2, 3::Double)
 120 --   >>> backward_substitute identity b
 121 --   ((1.0),(2.0),(3.0))
 122 --   >>> (backward_substitute identity b) == b
 123 --   True
 124 --
 125 --   >>> let m1 = fromList [[1,1,1], [0,1,1], [0,0,1]] :: Mat3 Double
 126 --   >>> let b = vec3d (1,1,1::Double)
 127 --   >>> backward_substitute m1 b
 128 --   ((0.0),(0.0),(1.0))
 129 --
 130 backward_substitute :: forall m a. (Eq a, Field.C a, Arity m)
 131                     => Mat (S m) (S m) a
 132                     -> Col (S m) a
 133                     -> Col (S m) a
 134 backward_substitute m' b'
 135   | not (is_upper_triangular m') =
 136       error "backward substitution on non-upper-triangular matrix"
 137   | otherwise = x'
 138     where
 139       x' = construct lambda
 140
 141       -- Convenient accessor for the elements of b'.
 142       b :: Int -> a
 143       b k = b' !!! (k, 0)
 144
 145       -- Convenient accessor for the elements of m'.
 146       m :: Int -> Int -> a
 147       m i j = m' !!! (i, j)
 148
 149       -- Convenient accessor for the elements of x'.
 150       x :: Int -> a
 151       x k = x' !!! (k, 0)
 152
 153       -- The second argument to lambda should always be zero here, so we
 154       -- ignore it.
 155       lambda :: Int -> Int -> a
 156       lambda k _
 157         | k == n = (b k) / (m k k)
 158         | otherwise = ((b k) - sum [ (m k j) NP.* (x j) |
 159                                     j <- [k+1..n] ]) / (m k k)
 160         where
 161           n = (ncols m') - 1
 162
 163
 164 -- | Solve the linear system m*x = b where m is positive definite.
 165 --
 166 --   Examples:
 167 --
 168 --   >>> import Linear.Matrix ( Col4, frobenius_norm, fromList )
 169 --   >>> import Naturals ( N7 )
 170 --
 171 --   >>> let f1 = [0.0418]
 172 --   >>> let f2 = [0.0805]
 173 --   >>> let f3 = [0.1007]
 174 --   >>> let f4 = [-0.0045]
 175 --   >>> let f5 = [-0.0332]
 176 --   >>> let f6 = [-0.0054]
 177 --   >>> let f7 = [-0.0267]
 178 --   >>> let big_F = fromList [f1,f2,f3,f4,f5,f6,f7] :: Col N7 Double
 179 --
 180 --   >>> let k1 = [6, -3, 0, 0, 0, 0, 0] :: [Double]
 181 --   >>> let k2 = [-3, 10.5, -7.5, 0, 0, 0, 0] :: [Double]
 182 --   >>> let k3 = [0, -7.5, 12.5, 0, 0, 0, 0] :: [Double]
 183 --   >>> let k4 = [0, 0, 0, 6, 0, 0, 0] :: [Double]
 184 --   >>> let k5 = [0, 0, 0, 0, 6, 0, 0] :: [Double]
 185 --   >>> let k6 = [0, 0, 0, 0, 0, 6, 0] :: [Double]
 186 --   >>> let k7 = [0, 0, 0, 0, 0, 0, 15] :: [Double]
 187 --   >>> let big_K = fromList [k1,k2,k3,k4,k5,k6,k7] :: Mat N7 N7 Double
 188 --
 189 --   >>> let e1 = [1871/75000] :: [Double]
 190 --   >>> let e2 = [899/25000] :: [Double]
 191 --   >>> let e3 = [463/15625] :: [Double]
 192 --   >>> let e4 = [-3/4000] :: [Double]
 193 --   >>> let e5 = [-83/15000] :: [Double]
 194 --   >>> let e6 = [-9/10000] :: [Double]
 195 --   >>> let e7 = [-89/50000] :: [Double]
 196 --   >>> let expected = fromList [e1,e2,e3,e4,e5,e6,e7] :: Col N7 Double
 197 --   >>> let actual = solve_positive_definite big_K big_F
 198 --   >>> frobenius_norm (actual - expected) < 1e-12
 199 --   True
 200 --
 201 solve_positive_definite :: (Arity m, Algebraic.C a, Eq a, Field.C a)
 202                         => Mat (S m) (S m) a
 203                         -> Col (S m) a
 204                         -> Col (S m) a
 205 solve_positive_definite m b = x
 206   where
 207     r = cholesky m
 208     -- Now, r^T*r*x = b. Let r*x = y, so the system looks like
 209     -- r^T * y = b. We can solve this for y.
 210     y = forward_substitute (transpose r) b
 211     -- Now solve r*x = y to find the value of x.
 212     x = backward_substitute r y