Add tolerant versions of is_{upper,lower}_triangular.

[numerical-analysis.git] / src / Linear / Matrix.hs

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 69a74b50eeacf7ea944c790d3858dd0cba12c266..c6f4a83c71af1c5231113ec555fa79f2dcfcc038 100644 (file)
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -48,11 +48,13 @@ import Data.Vector.Fixed.Cont (Arity, arity)
 import Linear.Vector
 import Normed
 
-import NumericPrelude hiding ((*), abs)
-import qualified NumericPrelude as NP ((*))
+import NumericPrelude hiding ( (*), abs )
+import qualified NumericPrelude as NP ( (*) )
+import qualified Algebra.Absolute as Absolute ( C )
+import Algebra.Absolute ( abs )
+import qualified Algebra.Additive as Additive
 import qualified Algebra.Algebraic as Algebraic
 import Algebra.Algebraic (root)
-import qualified Algebra.Additive as Additive
 import qualified Algebra.Ring as Ring
 import qualified Algebra.Module as Module
 import qualified Algebra.RealRing as RealRing
@@ -250,21 +252,26 @@ cholesky m = construct r
 
 
 -- | Returns True if the given matrix is upper-triangular, and False
---   otherwise.
+--   otherwise. The parameter @epsilon@ lets the caller choose a
+--   tolerance.
 --
 --   Examples:
 --
---   >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+--   >>> let m = fromList [[1,1],[1e-12,1]] :: Mat2 Double
 --   >>> is_upper_triangular m
 --   False
---
---   >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
---   >>> is_upper_triangular m
+--   >>> is_upper_triangular' 1e-10 m
 --   True
 --
-is_upper_triangular :: (Eq a, Ring.C a, Arity m, Arity n)
-                    => Mat m n a -> Bool
-is_upper_triangular m =
+--   TODO:
+--
+--     1. Don't cheat with lists.
+--
+is_upper_triangular' :: (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
+                    => a -- ^ The tolerance @epsilon@.
+                    -> Mat m n a
+                    -> Bool
+is_upper_triangular' epsilon m =
   and $ concat results
   where
     results = [[ test i j | i <- [0..(nrows m)-1]] | j <- [0..(ncols m)-1] ]
@@ -272,11 +279,36 @@ is_upper_triangular m =
     test :: Int -> Int -> Bool
     test i j
       | i <= j = True
-      | otherwise = m !!! (i,j) == 0
+      -- use "less than or equal to" so zero is a valid epsilon
+      | otherwise = abs (m !!! (i,j)) <= epsilon
+
+
+-- | Returns True if the given matrix is upper-triangular, and False
+--   otherwise. A specialized version of 'is_upper_triangular\'' with
+--   @epsilon = 0@.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+--   >>> is_upper_triangular m
+--   False
+--
+--   >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
+--   >>> is_upper_triangular m
+--   True
+--
+--   TODO:
+--
+--     1. The Ord constraint is too strong here, Eq would suffice.
+--
+is_upper_triangular :: (Ord a, Ring.C a, Absolute.C a, Arity m, Arity n)
+                    => Mat m n a -> Bool
+is_upper_triangular = is_upper_triangular' 0
 
 
 -- | Returns True if the given matrix is lower-triangular, and False
---   otherwise.
+--   otherwise. This is a specialized version of 'is_lower_triangular\''
+--   with @epsilon = 0@.
 --
 --   Examples:
 --
@@ -288,8 +320,9 @@ is_upper_triangular m =
 --   >>> is_lower_triangular m
 --   False
 --
-is_lower_triangular :: (Eq a,
+is_lower_triangular :: (Ord a,
                         Ring.C a,
+                        Absolute.C a,
                         Arity m,
                         Arity n)
                     => Mat m n a
@@ -297,6 +330,29 @@ is_lower_triangular :: (Eq a,
 is_lower_triangular = is_upper_triangular . transpose
 
 
+-- | Returns True if the given matrix is lower-triangular, and False
+--   otherwise. The parameter @epsilon@ lets the caller choose a
+--   tolerance.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,1e-12],[1,1]] :: Mat2 Double
+--   >>> is_lower_triangular m
+--   False
+--   >>> is_lower_triangular' 1e-12 m
+--   True
+--
+is_lower_triangular' :: (Ord a,
+                         Ring.C a,
+                         Absolute.C a,
+                         Arity m,
+                         Arity n)
+                    => a -- ^ The tolerance @epsilon@.
+                    -> Mat m n a
+                    -> Bool
+is_lower_triangular' epsilon = (is_upper_triangular' epsilon) . transpose
+
+
 -- | Returns True if the given matrix is triangular, and False
 --   otherwise.
 --
@@ -314,8 +370,9 @@ is_lower_triangular = is_upper_triangular . transpose
 --   >>> is_triangular m
 --   False
 --
-is_triangular :: (Eq a,
+is_triangular :: (Ord a,
                   Ring.C a,
+                  Absolute.C a,
                   Arity m,
                   Arity n)
               => Mat m n a
@@ -353,8 +410,9 @@ class (Eq a, Ring.C a) => Determined p a where
 instance (Eq a, Ring.C a) => Determined (Mat (S Z) (S Z)) a where
   determinant (Mat rows) = (V.head . V.head) rows
 
-instance (Eq a,
+instance (Ord a,
           Ring.C a,
+          Absolute.C a,
           Arity n,
           Determined (Mat (S n) (S n)) a)
          => Determined (Mat (S (S n)) (S (S n))) a where