From: Michael Orlitzky <michael@orlitzky.com>
Date: Wed, 5 Feb 2014 01:33:31 +0000 (-0500)
Subject: Add the Integration.Gaussian.nodes_and_weights function.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=907977c61c9d817e1270541ab481586a986cc21c;p=numerical-analysis.git

Add the Integration.Gaussian.nodes_and_weights function.
---

diff --git a/src/Integration/Gaussian.hs b/src/Integration/Gaussian.hs
index 3169b20..5d683f6 100644
--- a/src/Integration/Gaussian.hs
+++ b/src/Integration/Gaussian.hs
@@ -1,19 +1,28 @@
+{-# LANGUAGE FlexibleContexts #-}
 {-# LANGUAGE NoImplicitPrelude #-}
 {-# LANGUAGE ScopedTypeVariables #-}
 
 -- | Gaussian quadrature over [-1, 1] using the Legendre polynomials.
 --
 module Integration.Gaussian (
-  jacobi_matrix )
+  jacobi_matrix,
+  nodes_and_weights )
 where
 
 import Algebra.Absolute ( abs )
 import qualified Algebra.Algebraic as Algebraic ( C )
 import qualified Algebra.Ring as Ring ( C )
-import Data.Vector.Fixed ( Arity )
+import Data.Vector.Fixed ( Arity, S )
 import NumericPrelude hiding ( abs )
 
-import Linear.Matrix ( Mat(..), construct )
+import Linear.Matrix (
+  Col,
+  Mat(..),
+  construct,
+  identity_matrix,
+  matmap,
+  row',
+  transpose )
 import Linear.QR ( eigenvectors_symmetric )
 
 
@@ -37,7 +46,7 @@ import Linear.QR ( eigenvectors_symmetric )
 --   True
 --
 jacobi_matrix :: forall m a. (Arity m, Algebraic.C a, Ring.C a)
-              => Mat m m a
+              => Mat (S m) (S m) a
 jacobi_matrix =
   construct lambda
   where
@@ -57,3 +66,54 @@ jacobi_matrix =
           in  sqrt $ (coeff_a (k-1)) * (coeff_c k)
 
       | otherwise = fromInteger 0
+
+
+
+-- | Compute the nodes/weights for Gaussian quadrature over [-1,1]
+--   using the Legendre polynomials. The test cases can all be found
+--   on e.g. Wikipedia.
+--
+--   Examples:
+--
+--   >>> import Linear.Matrix ( Col2, frobenius_norm, fromList )
+--
+--   >>> let (ns,ws) = nodes_and_weights 1000 :: (Col2 Double, Col2 Double)
+--   >>> let c = 1/(sqrt 3)
+--   >>> let expected_ns = fromList [[-c],[c]] :: Col2 Double
+--   >>> let expected_ws = fromList [[1],[1]] :: Col2 Double
+--   >>> frobenius_norm (ns - expected_ns) < 1e-12
+--   True
+--   >>> frobenius_norm (ws - expected_ws) < 1e-12
+--   True
+--
+nodes_and_weights :: forall m a.
+                     (Arity m, Eq a, Algebraic.C a)
+                  => Int
+                  -> (Col (S m) a, Col (S m) a)
+nodes_and_weights iterations =
+  (nodes, weights)
+  where
+    jac = jacobi_matrix
+
+    -- A shift is needed to make sure the QR algorithm
+    -- converges. Since the roots (and thus eigenvalues) of orthogonal
+    -- polynomials are real and distinct, there does exist a shift
+    -- that will make it work. We guess lambda=1 makes it work.
+
+    shifted_jac = jac - identity_matrix
+    (shifted_nodes, vecs) = eigenvectors_symmetric iterations shifted_jac
+
+    ones :: Col (S m) a
+    ones = construct (\_ _ -> fromInteger 1)
+
+    nodes :: Col (S m) a
+    nodes = shifted_nodes + ones -- unshift the nodes
+
+    -- Get the first component of each column.
+    first_components = row' vecs 0
+
+    -- Square it and multiply by 2; see the Golub-Welsch paper for
+    -- this magic.
+    weights_row = matmap (\x -> (fromInteger 2)*x^2) first_components
+
+    weights = transpose $ weights_row