New function: Linear.Matrix.identity_matrix.

[numerical-analysis.git] / src / Roots / Simple.hs

diff --git a/src/Roots/Simple.hs b/src/Roots/Simple.hs
index d3c10cd8a3dfb95c001aeb5ce244b1bc7f14f7ed..03b39aeb643de47ed6546693c26346851b80b0fc 100644 (file)
--- a/src/Roots/Simple.hs
+++ b/src/Roots/Simple.hs
@@ -18,7 +18,6 @@ import Normed
 import qualified Roots.Fast as F
 
 import NumericPrelude hiding (abs)
-import qualified Algebra.Absolute as Absolute
 import Algebra.Absolute (abs)
 import qualified Algebra.Additive as Additive
 import qualified Algebra.Algebraic as Algebraic
@@ -57,7 +56,7 @@ has_root f a b epsilon =
 
 
 -- | We are given a function @f@ and an interval [a,b]. The bisection
---   method checks finds a root by splitting [a,b] in half repeatedly.
+--   method finds a root by splitting [a,b] in half repeatedly.
 --
 --   If one is found within some prescribed tolerance @epsilon@, it is
 --   returned. Otherwise, the interval [a,b] is split into two
@@ -69,8 +68,12 @@ has_root f a b epsilon =
 --
 --   Examples:
 --
---   >>> bisect cos 1 2 0.001
---   Just 1.5712890625
+--   >>> let actual = 1.5707963267948966
+--   >>> let Just root = bisect cos 1 2 0.001
+--   >>> root
+--   1.5712890625
+--   >>> abs (root - actual) < 0.001
+--   True
 --
 --   >>> bisect sin (-1) 1 0.001
 --   Just 0.0
@@ -85,11 +88,45 @@ bisect f a b epsilon =
   F.bisect f a b epsilon Nothing Nothing
 
 
+-- | We are given a function @f@ and an interval [a,b]. The trisection
+--   method finds a root by splitting [a,b] into three
+--   subintervals repeatedly.
+--
+--   If one is found within some prescribed tolerance @epsilon@, it is
+--   returned. Otherwise, the interval [a,b] is split into two
+--   subintervals [a,c] and [c,b] of equal length which are then both
+--   checked via the same process.
+--
+--   Returns 'Just' the value x for which f(x) == 0 if one is found,
+--   or Nothing if one of the preconditions is violated.
+--
+--   Examples:
+--
+--   >>> let actual = 1.5707963267948966
+--   >>> let Just root = trisect cos 1 2 0.001
+--   >>> root
+--   1.5713305898491083
+--   >>> abs (root - actual) < 0.001
+--   True
+--
+--   >>> trisect sin (-1) 1 0.001
+--   Just 0.0
+--
+trisect :: (RealField.C a, RealRing.C b)
+       => (a -> b) -- ^ The function @f@ whose root we seek
+       -> a       -- ^ The \"left\" endpoint of the interval, @a@
+       -> a       -- ^ The \"right\" endpoint of the interval, @b@
+       -> a       -- ^ The tolerance, @epsilon@
+       -> Maybe a
+trisect f a b epsilon =
+  F.trisect f a b epsilon Nothing Nothing
+
+
 -- | Find a fixed point of the function @f@ with the search starting
 --   at x0. We delegate to the version that returns the number of
 --   iterations and simply discard the number of iterations.
 --
-fixed_point :: (Normed a, Algebraic.C a, Algebraic.C b, RealField.C b)
+fixed_point :: (Normed a, Additive.C a, Algebraic.C b, RealField.C b)
             => (a -> a) -- ^ The function @f@ to iterate.
             -> b       -- ^ The tolerance, @epsilon@.
             -> a       -- ^ The initial value @x0@.
@@ -103,7 +140,7 @@ fixed_point f epsilon x0 =
 --   @epsilon@. We delegate to the version that returns the number of
 --   iterations and simply discard the fixed point.
 fixed_point_iteration_count :: (Normed a,
-                                Algebraic.C a,
+                                Additive.C a,
                                 RealField.C b,
                                 Algebraic.C b)
                             => (a -> a) -- ^ The function @f@ to iterate.
@@ -158,8 +195,8 @@ newton_iterations :: (Field.C a)
                     -> (a -> a) -- ^ The derivative of @f@
                     -> a       -- ^ Initial guess, x-naught
                     -> [a]
-newton_iterations f f' x0 =
-  iterate next x0
+newton_iterations f f' =
+  iterate next
   where
   next xn =
     xn - ( (f xn) / (f' xn) )
@@ -243,8 +280,8 @@ secant_iterations :: (Field.C a)
                     -> a       -- ^ Initial guess, x-naught
                     -> a       -- ^ Second initial guess, x-one
                     -> [a]
-secant_iterations f x0 x1 =
-  iterate2 g x0 x1
+secant_iterations f =
+  iterate2 g
   where
   g prev2 prev1 =
     let x_change = prev1 - prev2