From: Michael Orlitzky <michael@orlitzky.com>
Date: Sun, 2 Feb 2014 17:12:09 +0000 (-0500)
Subject: Add the Linear.QR module and update the ghci/cabal files.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=e01a21693319ed1bf888d543caf817e1c803a88b;p=numerical-analysis.git

Add the Linear.QR module and update the ghci/cabal files.
---

diff --git a/.ghci b/.ghci
index b8019ba..942d73d 100644
--- a/.ghci
+++ b/.ghci
@@ -8,6 +8,7 @@
   src/Integration/Trapezoid.hs
   src/Linear/Iteration.hs
   src/Linear/Matrix.hs
+  src/Linear/QR.hs
   src/Linear/System.hs
   src/Linear/Vector.hs
   src/Misc.hs
@@ -22,6 +23,7 @@ import Integration.Simpson
 import Integration.Trapezoid
 import Linear.Iteration
 import Linear.Matrix
+import Linear.QR
 import Linear.System
 import Linear.Vector
 import Misc
diff --git a/numerical-analysis.cabal b/numerical-analysis.cabal
index 2bc7468..fe47ead 100644
--- a/numerical-analysis.cabal
+++ b/numerical-analysis.cabal
@@ -17,9 +17,19 @@ description:
 data-files: makefile
 
 library
-  exposed-modules: Integration.Simpson, Integration.Trapezoid,
-    Linear.Iteration, Linear.Matrix, Linear.System, Linear.Vector,
-    Misc, Normed, ODE.IVP, Polynomials.Orthogonal, Roots.Simple,
+  exposed-modules:
+    Integration.Simpson,
+    Integration.Trapezoid,
+    Linear.Iteration,
+    Linear.Matrix,
+    Linear.QR,
+    Linear.System,
+    Linear.Vector,
+    Misc,
+    Normed,
+    ODE.IVP,
+    Polynomials.Orthogonal,
+    Roots.Simple,
     Roots.Fast
 
   build-depends:
diff --git a/src/Linear/QR.hs b/src/Linear/QR.hs
new file mode 100644
index 0000000..58027bb
--- /dev/null
+++ b/src/Linear/QR.hs
@@ -0,0 +1,92 @@
+{-# LANGUAGE NoImplicitPrelude #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+
+-- | QR factorization via Givens rotations.
+--
+module Linear.QR (
+  givens_rotator,
+  qr )
+where
+
+import qualified Algebra.Ring as Ring ( C )
+import qualified Algebra.Algebraic as Algebraic ( C )
+import Data.Vector.Fixed ( ifoldl )
+import Data.Vector.Fixed.Cont ( Arity )
+import NumericPrelude hiding ( (*) )
+
+import Linear.Matrix (
+  Mat(..),
+  (*),
+  (!!!),
+  construct,
+  identity_matrix,
+  transpose )
+
+
+-- | Construct a givens rotation matrix that will operate on row @i@
+--   and column @j@. This is done to create zeros in some column of
+--   the target matrix. You must also supply that column's @i@th and
+--   @j@th entries as arguments.
+--
+--   Examples (Watkins, p. 193):
+--
+--   >>> import Linear.Matrix ( Mat2, fromList )
+--   >>> let m  = givens_rotator 0 1 1 1 :: Mat2 Double
+--   >>> let m2 = fromList [[1, -1],[1, 1]] :: Mat2 Double
+--   >>> m == (1 / (sqrt 2) :: Double) *> m2
+--   True
+--
+givens_rotator :: forall m a. (Ring.C a, Algebraic.C a, Arity m)
+               => Int -> Int -> a -> a -> Mat m m a
+givens_rotator i j xi xj =
+  construct f
+  where
+    xnorm = sqrt $ xi^2 + xj^2
+    c = xi / xnorm
+    s = xj / xnorm
+
+    f :: Int -> Int -> a
+    f y z
+      | y == i && z == i = c
+      | y == j && z == j = c
+      | y == i && z == j = negate s
+      | y == j && z == i = s
+      | y == z = fromInteger 1
+      | otherwise = fromInteger 0
+
+
+-- | Compute the QR factorization of a matrix (using Givens
+--   rotations). This is accomplished with two folds: the first one
+--   traverses the columns of the matrix from left to right, and the
+--   second traverses the entries of the column from top to
+--   bottom.
+--
+--   The state that is passed through the fold is the current (q,r)
+--   factorization. We keep the pair updated by multiplying @q@ and
+--   @r@ by the new rotator (or its transpose).
+--
+qr :: forall m n a. (Arity m, Arity n, Algebraic.C a, Ring.C a)
+   => Mat m n a -> (Mat m m a, Mat m n a)
+qr matrix =
+  ifoldl col_function initial_qr columns
+  where
+    Mat columns = transpose matrix
+    initial_qr = (identity_matrix, matrix)
+
+    -- | Process the column and spit out the current QR
+    --   factorization. In the first column, we want to get rotators
+    --   Q12, Q13, Q14,... In the second column, we want rotators Q23,
+    --   Q24, Q25,...
+    col_function (q,r) col_idx col =
+      ifoldl (f col_idx) (q,r) col
+
+    -- | Process the entries in a column, doing basically the same
+    --   thing as col_dunction does. It updates the QR factorization,
+    --   maybe, and returns the current one.
+    f col_idx (q,r) idx x
+      | idx <= col_idx = (q,r) -- leave it alone.
+      | otherwise =
+          (q*rotator, (transpose rotator)*r)
+          where
+            rotator :: Mat m m a
+            rotator = givens_rotator col_idx idx (r !!! (idx, col_idx)) x