From: Michael Orlitzky <michael@orlitzky.com>
Date: Tue, 4 Feb 2014 17:16:52 +0000 (-0500)
Subject: Add the eigenvectors_symmetric function and begin writing tests for it.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=8ba752ee2e6a9e4f80843c21b52e2d534fde01c0;p=numerical-analysis.git

Add the eigenvectors_symmetric function and begin writing tests for it.
---

diff --git a/src/Linear/QR.hs b/src/Linear/QR.hs
index 79ca5f1..f9c5e30 100644
--- a/src/Linear/QR.hs
+++ b/src/Linear/QR.hs
@@ -5,23 +5,27 @@
 --
 module Linear.QR (
   eigenvalues,
+  eigenvectors_symmetric,
   givens_rotator,
   qr )
 where
 
 import qualified Algebra.Ring as Ring ( C )
 import qualified Algebra.Algebraic as Algebraic ( C )
+import Control.Arrow ( first )
 import Data.Vector.Fixed ( N1, S, ifoldl )
 import Data.Vector.Fixed.Cont ( Arity )
 import NumericPrelude hiding ( (*) )
 
 import Linear.Matrix (
+  Col,
   Mat(..),
   (*),
   (!!!),
   construct,
   diagonal,
   identity_matrix,
+  symmetric,
   transpose )
 
 
@@ -193,7 +197,7 @@ qr matrix =
 eigenvalues :: forall m a. (Arity m, Algebraic.C a, Eq a)
              => Int
              -> Mat (S m) (S m) a
-             -> Mat (S m) N1 a
+             -> Col (S m) a
 eigenvalues iterations matrix =
   diagonal (ut_approximation iterations)
   where
@@ -201,3 +205,64 @@ eigenvalues iterations matrix =
     ut_approximation 0 = matrix
     ut_approximation k = rk*qk where (qk,rk) = qr (ut_approximation (k-1))
 
+
+-- | Compute the eigenvalues and eigenvectors of a symmetric matrix
+--   using an iterative QR algorithm. This is similar to what we do in
+--   'eigenvalues' except we also return the product of all \"Q\"
+--   matrices that we have generated. This turns out to me the matrix
+--   of eigenvectors when the original matrix is symmetric. For
+--   references see Goluv and Van Loan, \"Matrix Computations\", or
+--   \"Calculation of Gauss Quadrature Rules\" by Golub and Welsch.
+--
+--   Examples:
+--
+--   >>> import Linear.Matrix ( Col2, Col3, Mat2, Mat3 )
+--   >>> import Linear.Matrix ( frobenius_norm, fromList, identity_matrix )
+--
+--   >>> let m = identity_matrix :: Mat3 Double
+--   >>> let (vals, vecs) = eigenvectors_symmetric 100 m
+--   >>> let expected_vals = fromList [[1],[1],[1]] :: Col3 Double
+--   >>> let expected_vecs = m
+--   >>> vals == expected_vals
+--   True
+--   >>> vecs == expected_vecs
+--   True
+--
+--   >>> let m = fromList [[3,2,4],[2,0,2],[4,2,3]] :: Mat3 Double
+--   >>> let (vals, vecs) = eigenvectors_symmetric 1000 m
+--   >>> let expected_vals = fromList [[8],[-1],[-1]] :: Col3 Double
+--   >>> let v0' = vec3d (2, 1, 2) :: Col3 Double
+--   >>> let v0  = (1 / (norm v0') :: Double) *> v0'
+--   >>> let v1' = vec3d (1, -2, 0) :: Col3 Double
+--   >>> let v1  = (1 / (norm v1') :: Double) *> v1'
+--   >>> let v2' = vec3d (4, 2, 5) :: Col3 Double
+--   >>> let v2  = (1 / (norm v2') :: Double) *> v2'
+--   >>> frobenius_norm (vals - expected_vals)
+--
+eigenvectors_symmetric :: forall m a. (Arity m, Algebraic.C a, Eq a)
+                       => Int
+                       -> Mat (S m) (S m) a
+                       -> (Col (S m) a, Mat (S m) (S m) a)
+eigenvectors_symmetric iterations matrix
+  | iterations <  0 = error "negative iterations requested"
+  | iterations == 0 = (diagonal matrix, identity_matrix)
+  | not $ symmetric matrix = error "argument is not symmetric"
+  | otherwise =
+      (values, vectors)
+      where
+        -- | We think of \"T\" as an approximation to an
+        -- upper-triangular matrix from which we get our
+        -- eigenvalues. The matrix \"P\" is the product of all
+        -- previous \"Q\"s and its columns approximate the
+        -- eigenvectors.
+        tp_pair :: Int -> (Mat (S m) (S m) a, Mat (S m) (S m) a)
+        tp_pair 0 = (matrix, identity_matrix)
+        tp_pair k = (tk,pk)
+                    where
+                      (t_prev, p_prev) = tp_pair (k-1)
+                      (qk,rk) = qr t_prev
+                      tk = rk*qk
+                      pk = p_prev*qk
+
+
+        (values, vectors) = (first diagonal) (tp_pair iterations)