X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FFEM%2FR1.hs;h=7b89ffabdf41513197e77d93f6952fd09dc3706d;hb=84bf90406341348a41c7cc57c3d365238690a930;hp=3e6869557d66f5601c769ff3863448a666729d60;hpb=7ac2f5fdd7db6c998949af5efdb45501dfb051da;p=numerical-analysis.git

diff --git a/src/FEM/R1.hs b/src/FEM/R1.hs
index 3e68695..7b89ffa 100644
--- a/src/FEM/R1.hs
+++ b/src/FEM/R1.hs
@@ -28,13 +28,14 @@
 module FEM.R1
 where
 
+import Algebra.Absolute ( abs )
 import qualified Algebra.Algebraic as Algebraic ( C )
 import qualified Algebra.Field as Field ( C )
 import qualified Algebra.RealField as RealField ( C )
 import qualified Algebra.ToRational as ToRational ( C )
 import Data.Vector.Fixed ( Arity, S )
-import NumericPrelude
-import qualified Prelude as P
+import NumericPrelude hiding ( abs )
+import Prelude ()
 
 import Integration.Gaussian ( gaussian )
 import Linear.Matrix (
@@ -43,6 +44,7 @@ import Linear.Matrix (
   Row,
   (!!!),
   construct,
+  dot,
   element_sum2,
   fromList,
   ifoldl2,
@@ -55,7 +57,7 @@ import Linear.Matrix (
   zip2,
   zipwith2 )
 import Linear.System ( solve_positive_definite )
-import Piecewise ( Piecewise(..), from_intervals )
+import Piecewise ( Piecewise(..), evaluate', from_intervals )
 import Polynomials.Orthogonal ( legendre )
 
 -- | Dirichlet boundary conditions. Since u(a)=u(b)=0 are fixed,
@@ -188,11 +190,11 @@ affine (x1,x2) x = (fromInteger 2)*(x - x1)/(x2 - x1) - (fromInteger 1)
 --   >>> phi 1
 --   7.0
 --
-affine_inv :: Field.C a => (a,a) -> (a -> a)
+affine_inv :: forall a. Field.C a => (a,a) -> (a -> a)
 affine_inv (x1,x2) x =
   x*(x2 - x1)/two + (x1 + x2)/two
   where
-    two = fromInteger 2
+    two = fromInteger 2 :: a
 
 
 -- * Load vector
@@ -219,9 +221,9 @@ big_N k x
   | otherwise =
       coeff * ( legendre k x - legendre (k-2) x )
       where
-        two = fromInteger 2
-        four = fromInteger 4
-        coeff = one / (sqrt (four*(fromInteger k) - two)) :: a
+        two = fromInteger 2 :: a
+        four = fromInteger 4 :: a
+        coeff = one / (sqrt (four*(fromInteger k) - two))
 
 
 -- | A matrix containing 'big_N' functions indexed by their
@@ -309,7 +311,7 @@ big_F pde params =
     accum i j prev_F this_N =
       prev_F + this_F
       where
-        two = fromInteger 2
+        two = fromInteger 2 :: a
         (x1,x2) = (mesh params) !!! (i,0)
         q = affine_inv (x1,x2)
         integrand x = ((f pde) (q x)) * (this_N x)
@@ -345,8 +347,8 @@ big_N' k x
   | k == 1 = one / (fromInteger 2)
   | otherwise = coeff * ( legendre k x )
       where
-        two = fromInteger 2
-        coeff = sqrt ((two*(fromInteger k) + one) / two) :: a
+        two = fromInteger 2 :: a
+        coeff = sqrt ((two*(fromInteger k) + one) / two)
 
 
 -- | The matrix of (N_i' * N_j') functions used in the integrand of
@@ -376,7 +378,7 @@ big_K_elem pde params _ k cur_K _ =
     accum i j prev_K these_N's =
       prev_K + this_K
       where
-        two = fromInteger 2
+        two = fromInteger 2 :: a
         (x1,x2) = (mesh params) !!! (k,0)
         q = affine_inv (x1,x2)
         integrand x = ((big_A pde) (q x)) * (these_N's x)
@@ -445,7 +447,7 @@ big_M_elem pde params _ k cur_M _ =
     accum i j prev_M these_Ns =
       prev_M + this_M
       where
-        two = fromInteger 2
+        two = fromInteger 2 :: a
         (x1,x2) = (mesh params) !!! (k,0)
         q = affine_inv (x1,x2)
         integrand x = ((c pde) (q x)) * (these_Ns x)
@@ -528,7 +530,7 @@ coefficients pde params =
 
 solution :: forall m n l a.
             (Arity m, Arity n, Arity l,
-            Algebraic.C a, Eq a, RealField.C a, ToRational.C a, Show a)
+            Algebraic.C a, Eq a, RealField.C a, ToRational.C a)
          => PDE a
          -> Params m n (S l) a
          -> Piecewise a
@@ -579,3 +581,41 @@ solution pde params =
           where
             xi = (affine interval) x
             combine ci ni = ci*(ni xi)
+
+
+energy_fem :: (Arity m, Arity n, Arity l,
+                Algebraic.C a, Eq a, RealField.C a, ToRational.C a)
+           => PDE a
+           -> Params m n (S l) a
+           -> a
+energy_fem pde params =
+  (coefficients pde params) `dot` (big_F pde params)
+
+
+relative_error :: forall m n l a.
+                  (Arity m, Arity n, Arity l,
+                   Algebraic.C a, Eq a, RealField.C a, ToRational.C a)
+                => PDE a
+                -> Params m n (S l) a
+                -> a -- ^ The energy norm of the true solution @u@
+                -> a
+relative_error pde params energy_true =
+  cent * sqrt(energy_true - (energy_fem pde params)/energy_true)
+  where
+    cent = fromInteger 100 :: a
+
+
+
+relative_error_pointwise :: forall m n l a.
+                            (Arity m, Arity n, Arity l,
+                            Algebraic.C a, Eq a, RealField.C a, ToRational.C a)
+                         => PDE a
+                         -> Params m n (S l) a
+                         -> (a -> a) -- ^ The true solution @u@
+                         -> a -- ^ The point @x@ at which to compute the error.
+                         -> a
+relative_error_pointwise pde params u x =
+  cent * ( abs $ (u x) - u_fem ) / ( abs $ u x )
+  where
+    u_fem = evaluate' (solution pde params) x
+    cent = fromInteger 100 :: a