From: Michael Orlitzky <michael@orlitzky.com>
Date: Sun, 24 Feb 2013 00:05:10 +0000 (-0500)
Subject: Add the Linear.System module.
X-Git-Url: http://gitweb.michael.orlitzky.com/?p=numerical-analysis.git;a=commitdiff_plain;h=7304f41e81fe97d40afe18b8215fb00a58702502

Add the Linear.System module.
---

diff --git a/src/Linear/System.hs b/src/Linear/System.hs
new file mode 100644
index 0000000..e0fdf1c
--- /dev/null
+++ b/src/Linear/System.hs
@@ -0,0 +1,104 @@
+{-# LANGUAGE RebindableSyntax #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+{-# LANGUAGE TypeFamilies #-}
+
+module Linear.System
+where
+
+import Data.Vector.Fixed (Dim, N1, Vector)
+
+import Linear.Matrix
+
+import NumericPrelude hiding ((*), abs)
+import qualified NumericPrelude as NP ((*))
+import qualified Algebra.Field as Field
+
+import Debug.Trace (trace, traceShow)
+
+-- | Solve the system m' * x = b', where m' is upper-triangular. Will
+--   probably crash if m' is non-singular. The result is the vector x.
+--
+--   Examples:
+--
+--   >>> let identity = fromList [[1,0,0],[0,1,0],[0,0,1]] :: Mat3 Double
+--   >>> let b = vec3d (1,2,3)
+--   >>> forward_substitute identity b
+--   ((1.0),(2.0),(3.0))
+--   >>> (forward_substitute identity b) == b
+--   True
+--
+--   >>> let m = fromList [[1,0],[1,1]] :: Mat2 Double
+--   >>> let b = vec2d (1,1)
+--   >>> forward_substitute m b
+--   ((1.0),(0.0))
+--
+forward_substitute :: forall a v w z.
+                      (Show a, Field.C a,
+                       Vector z a,
+                       Vector w (z a),
+                       Vector w a,
+                       Dim z ~ N1,
+                       v ~ w)
+                   => Mat v w a
+                   -> Mat w z a
+                   -> Mat w z a
+forward_substitute m' b' = x'
+  where
+    x' = construct lambda
+
+    -- Convenient accessor for the elements of b'.
+    b :: Int -> a
+    b k = b' !!! (k, 0)
+
+    -- Convenient accessor for the elements of m'.
+    m :: Int -> Int -> a
+    m i j = m' !!! (i, j)
+
+    -- Convenient accessor for the elements of x'.
+    x :: Int -> a
+    x k = x' !!! (k, 0)
+
+    -- The second argument to lambda should always be zero here, so we
+    -- ignore it.
+    lambda :: Int -> Int -> a
+    lambda 0 _ = (b 0) / (m 0 0)
+    lambda k _ = ((b k) - sum [ (m k j) NP.* (x j) |
+                                  j <- [0..k-1] ]) / (m k k)
+
+
+-- | Solve the system m*x = b, where m is lower-triangular. Will
+--   probably crash if m is non-singular. The result is the vector x.
+--
+--   Examples:
+--
+--   >>> let identity = fromList [[1,0,0],[0,1,0],[0,0,1]] :: Mat3 Double
+--   >>> let b = vec3d (1,2,3)
+--   >>> backward_substitute identity b
+--   ((1.0),(2.0),(3.0))
+--   >>> (backward_substitute identity b) == b
+--   True
+--
+backward_substitute :: (Show a, Field.C a,
+                        Vector z a,
+                        Vector v (w a),
+                        Vector w (z a),
+                        Vector w a,
+                        Dim z ~ N1,
+                        v ~ w)
+                    => Mat v w a
+                    -> Mat w z a
+                    -> Mat w z a
+backward_substitute m b =
+  forward_substitute (transpose m) b
+
+
+-- | Solve the linear system m*x = b where m is positive definite.
+{-
+solve_positive_definite :: Mat v w a -> Mat w z a
+solve_positive_definite m b = x
+  where
+    r = cholesky m
+    -- First we solve r^T * y == b for y. Then let y=r*x
+    rx = forward_substitute (transpose r) b
+    -- Now solve r*x == b.
+-}