X-Git-Url: http://gitweb.michael.orlitzky.com/?p=numerical-analysis.git;a=blobdiff_plain;f=src%2FLinear%2FMatrix.hs;h=34920b4f025e7e2027588ab07c14d6772b679ab2;hp=7452007990e0e015c45610b03605cf6cfc642aad;hb=4e464a486bef07db44de9c3d3fae0c8094401b09;hpb=1f64a1a33b2636ef2e863a0b577c8d8d50233580

diff --git a/src/Linear/Matrix.hs b/src/Linear/Matrix.hs
index 7452007..34920b4 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,
@@ -33,37 +36,61 @@ import Data.Vector.Fixed (
 import qualified Data.Vector.Fixed as V (
   and,
   fromList,
+  head,
   length,
   map,
   maximum,
   replicate,
   toList,
-  zipWith
-  )
-import Data.Vector.Fixed.Boxed (Vec)
-import Data.Vector.Fixed.Internal (Arity, arity)
-import Linear.Vector
-import Normed
-
-import NumericPrelude hiding ((*), abs)
-import qualified NumericPrelude as NP ((*))
-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
-import qualified Algebra.ToRational as ToRational
-import qualified Algebra.Transcendental as Transcendental
-import qualified Prelude as P
-
+  zipWith )
+import Data.Vector.Fixed.Cont ( Arity, arity )
+import Linear.Vector ( Vec, delete, element_sum )
+import Normed ( Normed(..) )
+
+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 ( C )
+import qualified Algebra.Algebraic as Algebraic ( C )
+import Algebra.Algebraic ( root )
+import qualified Algebra.Ring as Ring ( C )
+import qualified Algebra.Module as Module ( C )
+import qualified Algebra.RealRing as RealRing ( C )
+import qualified Algebra.ToRational as ToRational ( C )
+import qualified Algebra.Transcendental as Transcendental ( C )
+import qualified Prelude as P ( map )
+
+-- | Our main matrix type.
 data Mat m n a = (Arity m, Arity n) => Mat (Vec m (Vec n a))
+
+-- Type synonyms for n-by-n matrices.
 type Mat1 a = Mat N1 N1 a
 type Mat2 a = Mat N2 N2 a
 type Mat3 a = Mat N3 N3 a
 type Mat4 a = Mat N4 N4 a
 type Mat5 a = Mat N5 N5 a
 
+-- | Type synonym for row vectors expressed as 1-by-n matrices.
+type Row n a = Mat N1 n a
+
+-- Type synonyms for 1-by-n row "vectors".
+type Row1 a = Row N1 a
+type Row2 a = Row N2 a
+type Row3 a = Row N3 a
+type Row4 a = Row N4 a
+type Row5 a = Row N5 a
+
+-- | Type synonym for column vectors expressed as n-by-1 matrices.
+type Col n a = Mat n N1 a
+
+-- Type synonyms for n-by-1 column "vectors".
+type Col1 a = Col N1 a
+type Col2 a = Col N2 a
+type Col3 a = Col N3 a
+type Col4 a = Col N4 a
+type Col5 a = Col N5 a
+
 instance (Eq a) => Eq (Mat m n a) where
   -- | Compare a row at a time.
   --
@@ -146,6 +173,14 @@ row :: Mat m n a -> Int -> (Vec n a)
 row (Mat rows) i = rows ! i
 
 
+-- | Return the @i@th row of @m@ as a matrix. Unsafe.
+row' :: (Arity m, Arity n) => Mat m n a -> Int -> Row n a
+row' m i =
+  construct lambda
+  where
+    lambda _ j = m !!! (i, j)
+
+
 -- | Return the @j@th column of @m@. Unsafe.
 column :: Mat m n a -> Int -> (Vec m a)
 column (Mat rows) j =
@@ -154,6 +189,12 @@ column (Mat rows) j =
     element = flip (!)
 
 
+-- | Return the @j@th column of @m@ as a matrix. Unsafe.
+column' :: (Arity m, Arity n) => Mat m n a -> Int -> Col m a
+column' m j =
+  construct lambda
+  where
+    lambda i _ = m !!! (i, j)
 
 
 -- | Transpose @m@; switch it's columns and its rows. This is a dirty
@@ -196,8 +237,6 @@ 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
@@ -206,14 +245,24 @@ 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
@@ -239,21 +288,26 @@ 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] ]
@@ -261,11 +315,36 @@ 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:
 --
@@ -277,8 +356,9 @@ 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
@@ -286,6 +366,29 @@ 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.
 --
@@ -303,8 +406,9 @@ 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
@@ -340,13 +444,22 @@ 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 (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
+  -- | 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 +469,12 @@ 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:
 --
@@ -408,11 +520,10 @@ 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
-  -- | Generic p-norms. The usual norm in R^n is (norm_p 2). We treat
-  --   all matrices as big vectors.
+          Arity m)
+         => Normed (Mat (S m) N1 a) where
+  -- | Generic p-norms for vectors in R^n that are represented as nx1
+  --   matrices.
   --
   --   Examples:
   --
@@ -423,7 +534,7 @@ 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 +551,26 @@ 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 $ element_sum $ V.map row_sum rows
+  where
+    -- | Square and add up the entries of a row.
+    row_sum = element_sum . V.map (^2)
 
 
 -- Vector helpers. We want it to be easy to create low-dimension
@@ -462,24 +592,24 @@ instance (Algebraic.C a,
 --   >>> fixed_point g eps u0
 --   ((1.0728549599342185),(1.0820591495686167))
 --
-vec1d :: (a) -> Mat N1 N1 a
+vec1d :: (a) -> Col1 a
 vec1d (x) = Mat (mk1 (mk1 x))
 
-vec2d :: (a,a) -> Mat N2 N1 a
+vec2d :: (a,a) -> Col2 a
 vec2d (x,y) = Mat (mk2 (mk1 x) (mk1 y))
 
-vec3d :: (a,a,a) -> Mat N3 N1 a
+vec3d :: (a,a,a) -> Col3 a
 vec3d (x,y,z) = Mat (mk3 (mk1 x) (mk1 y) (mk1 z))
 
-vec4d :: (a,a,a,a) -> Mat N4 N1 a
+vec4d :: (a,a,a,a) -> Col4 a
 vec4d (w,x,y,z) = Mat (mk4 (mk1 w) (mk1 x) (mk1 y) (mk1 z))
 
-vec5d :: (a,a,a,a,a) -> Mat N5 N1 a
+vec5d :: (a,a,a,a,a) -> Col5 a
 vec5d (v,w,x,y,z) = Mat (mk5 (mk1 v) (mk1 w) (mk1 x) (mk1 y) (mk1 z))
 
 -- Since we commandeered multiplication, we need to create 1x1
 -- matrices in order to multiply things.
-scalar :: a -> Mat N1 N1 a
+scalar :: a -> Mat1 a
 scalar x = Mat (mk1 (mk1 x))
 
 dot :: (RealRing.C a, n ~ N1, m ~ S t, Arity t)
@@ -512,3 +642,128 @@ angle v1 v2 =
   where
    theta = (recip norms) NP.* (v1 `dot` v2)
    norms = (norm v1) NP.* (norm v2)
+
+
+-- | Retrieve the diagonal elements of the given matrix as a \"column
+--   vector,\" i.e. a m-by-1 matrix. We require the matrix to be
+--   square to avoid ambiguity in the return type which would ideally
+--   have dimension min(m,n) supposing an m-by-n matrix.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> diagonal m
+--   ((1),(5),(9))
+--
+diagonal :: (Arity m) => Mat m m a -> Col m a
+diagonal matrix =
+  construct lambda
+  where
+    lambda i _ = matrix !!! (i,i)
+
+
+-- | 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
+
+
+-- | Compute the trace of a square matrix, the sum of the elements
+--   which lie on its diagonal. We require the matrix to be
+--   square to avoid ambiguity in the return type which would ideally
+--   have dimension min(m,n) supposing an m-by-n matrix.
+--
+--   Examples:
+--
+--   >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+--   >>> trace m
+--   15
+--
+trace :: (Arity m, Ring.C a) => Mat m m a -> a
+trace matrix =
+  let (Mat rows) = diagonal matrix
+  in
+    element_sum $ V.map V.head rows