X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=66b22f9f9d964e843e596678c0f7ebe34e8e9a22;hb=af25e6f32f6787a747be63093adc10915ad60068;hp=7452007990e0e015c45610b03605cf6cfc642aad;hpb=1f64a1a33b2636ef2e863a0b577c8d8d50233580;p=numerical-analysis.git

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 7452007..66b22f9 100644
--- a/src/Linear/Matrix.hs
+++ b/src/Linear/Matrix.hs
@@ -2,6 +2,7 @@
 {-# LANGUAGE FlexibleContexts #-}
 {-# LANGUAGE FlexibleInstances #-}
 {-# LANGUAGE MultiParamTypeClasses #-}
+{-# LANGUAGE NoMonomorphismRestriction #-}
 {-# LANGUAGE ScopedTypeVariables #-}
 {-# LANGUAGE TypeFamilies #-}
 {-# LANGUAGE RebindableSyntax #-}
@@ -17,6 +18,7 @@ where
 import Data.List (intercalate)
 
 import Data.Vector.Fixed (
+  (!),
   N1,
   N2,
   N3,
@@ -24,6 +26,7 @@ import Data.Vector.Fixed (
   N5,
   S,
   Z,
+  generate,
   mk1,
   mk2,
   mk3,
@@ -32,7 +35,9 @@ import Data.Vector.Fixed (
   )
 import qualified Data.Vector.Fixed as V (
   and,
+  foldl,
   fromList,
+  head,
   length,
   map,
   maximum,
@@ -41,7 +46,7 @@ import qualified Data.Vector.Fixed as V (
   zipWith
   )
 import Data.Vector.Fixed.Boxed (Vec)
-import Data.Vector.Fixed.Internal (Arity, arity)
+import Data.Vector.Fixed.Cont (Arity, arity)
 import Linear.Vector
 import Normed
 
@@ -196,8 +201,6 @@ symmetric m =
 --   entries in the matrix. The i,j entry of the resulting matrix will
 --   have the value returned by lambda i j.
 --
---   TODO: Don't cheat with fromList.
---
 --   Examples:
 --
 --   >>> let lambda i j = i + j
@@ -206,15 +209,25 @@ symmetric m =
 --
 construct :: forall m n a. (Arity m, Arity n)
           => (Int -> Int -> a) -> Mat m n a
-construct lambda = Mat rows
+construct lambda = Mat $ generate make_row
   where
-    -- The arity trick is used in Data.Vector.Fixed.length.
-    imax = (arity (undefined :: m)) - 1
-    jmax = (arity (undefined :: n)) - 1
-    row' i = V.fromList [ lambda i j | j <- [0..jmax] ]
-    rows = V.fromList [ row' i | i <- [0..imax] ]
+    make_row :: Int -> Vec n a
+    make_row i = generate (lambda i)
 
 
+-- | Create an identity matrix with the right dimensions.
+--
+--   Examples:
+--
+--   >>> identity_matrix :: Mat3 Int
+--   ((1,0,0),(0,1,0),(0,0,1))
+--   >>> identity_matrix :: Mat3 Double
+--   ((1.0,0.0,0.0),(0.0,1.0,0.0),(0.0,0.0,1.0))
+--
+identity_matrix :: (Arity m, Ring.C a) => Mat m m a
+identity_matrix =
+  construct (\i j -> if i == j then (fromInteger 1) else (fromInteger 0))
+
 -- | Given a positive-definite matrix @m@, computes the
 --   upper-triangular matrix @r@ with (transpose r)*r == m and all
 --   values on the diagonal of @r@ positive.
@@ -340,13 +353,21 @@ class (Eq a, Ring.C a) => Determined p a where
   determinant :: (p a) -> a
 
 instance (Eq a, Ring.C a) => Determined (Mat (S Z) (S Z)) a where
-  determinant m = m !!! (0,0)
-
-instance (Eq a, Ring.C a, Arity m) => Determined (Mat m m) a where
-  determinant _ = undefined
+  determinant (Mat rows) = (V.head . V.head) rows
 
-instance (Eq a, Ring.C a, Arity n)
+instance (Eq a,
+          Ring.C a,
+          Arity n,
+          Determined (Mat (S n) (S n)) a)
          => Determined (Mat (S (S n)) (S (S n))) a where
+  -- | The recursive definition with a special-case for triangular matrices.
+  --
+  --   Examples:
+  --
+  --   >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+  --   >>> determinant m
+  --   -1
+  --
   determinant m
     | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ]
     | otherwise = determinant_recursive
@@ -356,13 +377,12 @@ instance (Eq a, Ring.C a, Arity n)
           det_minor i j = determinant (minor m i j)
 
           determinant_recursive =
-            sum [ (-1)^(1+(toInteger j)) NP.* (m' 0 j) NP.* (det_minor 0 j)
+            sum [ (-1)^(toInteger j) NP.* (m' 0 j) NP.* (det_minor 0 j)
               | j <- [0..(ncols m)-1] ]
 
 
 
--- | Matrix multiplication. Our 'Num' instance doesn't define one, and
---   we need additional restrictions on the result type anyway.
+-- | Matrix multiplication.
 --
 --   Examples:
 --
@@ -408,9 +428,8 @@ instance (Ring.C a, Arity m, Arity n) => Module.C a (Mat m n a) where
 
 instance (Algebraic.C a,
           ToRational.C a,
-          Arity m,
-          Arity n)
-         => Normed (Mat (S m) (S n) a) where
+          Arity m)
+         => Normed (Mat (S m) N1 a) where
   -- | Generic p-norms. The usual norm in R^n is (norm_p 2). We treat
   --   all matrices as big vectors.
   --
@@ -423,7 +442,7 @@ instance (Algebraic.C a,
   --   5.0
   --
   norm_p p (Mat rows) =
-    (root p') $ sum [(fromRational' $ toRational x)^p' | x <- xs]
+    (root p') $ sum [fromRational' (toRational x)^p' | x <- xs]
     where
       p' = toInteger p
       xs = concat $ V.toList $ V.map V.toList rows
@@ -440,7 +459,31 @@ instance (Algebraic.C a,
     fromRational' $ toRational $ V.maximum $ V.map V.maximum rows
 
 
+-- | Compute the Frobenius norm of a matrix. This essentially treats
+--   the matrix as one long vector containing all of its entries (in
+--   any order, it doesn't matter).
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1, 2, 3],[4,5,6],[7,8,9]] :: Mat3 Double
+--   >>> frobenius_norm m == sqrt 285
+--   True
+--
+--   >>> let m = fromList [[1, -1, 1],[-1,1,-1],[1,-1,1]] :: Mat3 Double
+--   >>> frobenius_norm m == 3
+--   True
+--
+frobenius_norm :: (Algebraic.C a, Ring.C a) => Mat m n a -> a
+frobenius_norm (Mat rows) =
+  sqrt $ vsum $ V.map row_sum rows
+  where
+    -- | The \"sum\" function defined in fixed-vector requires a 'Num'
+    --   constraint whereas we want to use the classes from
+    --   numeric-prelude.
+    vsum = V.foldl (+) (fromInteger 0)
 
+    -- | Square and add up the entries of a row.
+    row_sum = vsum . V.map (^2)
 
 
 -- Vector helpers. We want it to be easy to create low-dimension
@@ -512,3 +555,93 @@ angle v1 v2 =
   where
    theta = (recip norms) NP.* (v1 `dot` v2)
    norms = (norm v1) NP.* (norm v2)
+
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+--   containing only the on-diagonal entries of @matrix@. The
+--   off-diagonal entries are set to zero.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> diagonal_part m
+--   ((1,0,0),(0,5,0),(0,0,9))
+--
+diagonal_part :: (Arity m, Ring.C a)
+         => Mat m m a
+         -> Mat m m a
+diagonal_part matrix =
+  construct lambda
+  where
+    lambda i j = if i == j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+--   containing only the on-diagonal and below-diagonal entries of
+--   @matrix@. The above-diagonal entries are set to zero.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> lt_part m
+--   ((1,0,0),(4,5,0),(7,8,9))
+--
+lt_part :: (Arity m, Ring.C a)
+        => Mat m m a
+        -> Mat m m a
+lt_part matrix =
+  construct lambda
+  where
+    lambda i j = if i >= j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+--   containing only the below-diagonal entries of @matrix@. The on-
+--   and above-diagonal entries are set to zero.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> lt_part_strict m
+--   ((0,0,0),(4,0,0),(7,8,0))
+--
+lt_part_strict :: (Arity m, Ring.C a)
+        => Mat m m a
+        -> Mat m m a
+lt_part_strict matrix =
+  construct lambda
+  where
+    lambda i j = if i > j then matrix !!! (i,j) else 0
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+--   containing only the on-diagonal and above-diagonal entries of
+--   @matrix@. The below-diagonal entries are set to zero.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> ut_part m
+--   ((1,2,3),(0,5,6),(0,0,9))
+--
+ut_part :: (Arity m, Ring.C a)
+        => Mat m m a
+        -> Mat m m a
+ut_part = transpose . lt_part . transpose
+
+
+-- | Given a square @matrix@, return a new matrix of the same size
+--   containing only the above-diagonal entries of @matrix@. The on-
+--   and below-diagonal entries are set to zero.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> ut_part_strict m
+--   ((0,2,3),(0,0,6),(0,0,0))
+--
+ut_part_strict :: (Arity m, Ring.C a)
+        => Mat m m a
+        -> Mat m m a
+ut_part_strict = transpose . lt_part_strict . transpose