-- | The sequence x_{n} of values obtained by applying Newton's method
-- on the function @f@ and initial guess @x0@.
+--
+-- Examples:
+--
+-- Atkinson, p. 60.
+-- >>> let f x = x^6 - x - 1
+-- >>> let f' x = 6*x^5 - 1
+-- >>> tail $ take 4 $ newton_iterations f f' 2
+-- [1.6806282722513088,1.4307389882390624,1.2549709561094362]
+--
newton_iterations :: (Fractional a, Ord a)
=> (a -> a) -- ^ The function @f@ whose root we seek
-> (a -> a) -- ^ The derivative of @f@
+-- | Use Newton's method to find a root of @f@ near the initial guess
+-- @x0@. If your guess is bad, this will recurse forever!
+--
+-- Examples:
+--
+-- Atkinson, p. 60.
+--
+-- >>> let f x = x^6 - x - 1
+-- >>> let f' x = 6*x^5 - 1
+-- >>> let Just root = newtons_method f f' (1/1000000) 2
+-- >>> root
+-- 1.1347241385002211
+-- >>> abs (f root) < 1/100000
+-- True
+--
+-- >>> import Data.Number.BigFloat
+-- >>> let eps = 1/(10^20) :: BigFloat Prec50
+-- >>> let Just root = newtons_method f f' eps 2
+-- >>> root
+-- 1.13472413840151949260544605450647284028100785303643e0
+-- >>> abs (f root) < eps
+-- True
+--
newtons_method :: (Fractional a, Ord a)
=> (a -> a) -- ^ The function @f@ whose root we seek
-> (a -> a) -- ^ The derivative of @f@
= find (\x -> abs (f x) < epsilon) x_n
where
x_n = newton_iterations f f' x0
+
+
+
+-- | Takes a function @f@ of two arguments and repeatedly applies @f@
+-- to the previous two values. Returns a list containing all
+-- generated values, f(x0, x1), f(x1, x2), f(x2, x3)...
+--
+-- Examples:
+--
+-- >>> let fibs = iterate2 (+) 0 1
+-- >>> take 15 fibs
+-- [0,1,1,2,3,5,8,13,21,34,55,89,144,233,377]
+--
+iterate2 :: (a -> a -> a) -- ^ The function @f@
+ -> a -- ^ The initial value @x0@
+ -> a -- ^ The second value, @x1@
+ -> [a] -- ^ The result list, [x0, x1, ...]
+iterate2 f x0 x1 =
+ x0 : x1 : (go x0 x1)
+ where
+ go prev2 prev1 =
+ let next = f prev2 prev1 in
+ next : go prev1 next
+
+-- | The sequence x_{n} of values obtained by applying the secant
+-- method on the function @f@ and initial guesses @x0@, @x1@.
+--
+-- The recursion more or less implements a two-parameter 'iterate',
+-- although one list is passed to the next iteration (as opposed to
+-- one function argument, with iterate). At each step, we peel the
+-- first two elements off the list and then compute/append elements
+-- three, four... onto the end of the list.
+--
+-- Examples:
+--
+-- Atkinson, p. 67.
+-- >>> let f x = x^6 - x - 1
+-- >>> take 4 $ secant_iterations f 2 1
+-- [2.0,1.0,1.0161290322580645,1.190577768676638]
+--
+secant_iterations :: (Fractional a, Ord a)
+ => (a -> a) -- ^ The function @f@ whose root we seek
+ -> a -- ^ Initial guess, x-naught
+ -> a -- ^ Second initial guess, x-one
+ -> [a]
+secant_iterations f x0 x1 =
+ iterate2 g x0 x1
+ where
+ g prev2 prev1 =
+ let x_change = prev1 - prev2
+ y_change = (f prev1) - (f prev2)
+ in
+ (prev1 - (f prev1 * (x_change / y_change)))
+
+
+-- | Use the secant method to find a root of @f@ near the initial guesses
+-- @x0@ and @x1@. If your guesses are bad, this will recurse forever!
+--
+-- Examples:
+--
+-- Atkinson, p. 67.
+-- >>> let f x = x^6 - x - 1
+-- >>> let Just root = secant_method f (1/10^9) 2 1
+-- >>> root
+-- 1.1347241384015196
+-- >>> abs (f root) < (1/10^9)
+-- True
+--
+secant_method :: (Fractional a, Ord a)
+ => (a -> a) -- ^ The function @f@ whose root we seek
+ -> a -- ^ The tolerance epsilon
+ -> a -- ^ Initial guess, x-naught
+ -> a -- ^ Second initial guess, x-one
+ -> Maybe a
+secant_method f epsilon x0 x1
+ = find (\x -> abs (f x) < epsilon) x_n
+ where
+ x_n = secant_iterations f x0 x1
+
+
+
+fixed_point_iterations :: (a -> a) -- ^ The function @f@ to iterate.
+ -> a -- ^ The initial value @x0@.
+ -> [a] -- ^ The resulting sequence of x_{n}.
+fixed_point_iterations f x0 =
+ iterate f x0
+
+
+-- | Find a fixed point of the function @f@ with the search starting
+-- at x0. This will find the first element in the chain f(x0),
+-- f(f(x0)),... such that the magnitude of the difference between it
+-- and the next element is less than epsilon.
+--
+fixed_point :: (Num a, Ord a)
+ => (a -> a) -- ^ The function @f@ to iterate.
+ -> a -- ^ The tolerance, @epsilon@.
+ -> a -- ^ The initial value @x0@.
+ -> a -- ^ The fixed point.
+fixed_point f epsilon x0 =
+ fst winning_pair
+ where
+ xn = fixed_point_iterations f x0
+ xn_plus_one = tail $ fixed_point_iterations f x0
+
+ abs_diff v w =
+ abs (v - w)
+
+ -- The nth entry in this list is the absolute value of x_{n} -
+ -- x_{n+1}.
+ differences = zipWith abs_diff xn xn_plus_one
+
+ -- A list of pairs, (xn, |x_{n} - x_{n+1}|).
+ pairs = zip xn differences
+
+ -- The pair (xn, |x_{n} - x_{n+1}|) with
+ -- |x_{n} - x_{n+1}| < epsilon. The pattern match on 'Just' is
+ -- "safe" since the list is infinite. We'll succeed or loop
+ -- forever.
+ Just winning_pair = find (\(_, diff) -> diff < epsilon) pairs
projects / numerical-analysis.git / blobdiff