derivative.clj

(ns euler.utils.derivative
  (:use clojure.contrib.math))

(defn take-with-epsilon
  "Take from a list of numbers xs until two numbers are within the epsilon eps.
   Check a maximum of `limit` times. Returns the last number of the satisfying
   pair or nil of no match is found."
  [xs eps limit]

  (let [next (rest xs)]
    (loop [prev-x (first xs) x (first next) xs (rest next) i 0]
     (if (or (= i limit) (nil? x))
       nil
       (let [scaled-eps (if (or (zero? prev-x) (zero? x))
                          eps
                          (* eps (abs x)))]
         (if (<= (abs (- x prev-x)) scaled-eps)
           x
           (recur x (first xs) (rest xs) (inc i))))))))

(defn n-derivative [f x]
  (let [halve #(/ % 2)
        h-slope #(/ (- (f (+ x %)) (f (- x %))) (* 2 %))
        hs (iterate halve 2)]
    (take-with-epsilon (map h-slope hs) 0.00001 50)))