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Exercises 1.35 - 1.36
| | (ns e1-35-36
(:use [util.util :only (tolerance close-enough? phi)])
(:use [clojure.contrib.test-is :only (deftest is run-tests)]))
1.35
Show that the golden ratio (section 1.2.2) is a fixed point of the
transformation \(x \leftarrow 1 + 1/x\), and use this fact to compute by
means of the fixed-point procedure.
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|
| | (defn fixed-point [f first-guess]
(let [trynext (fn [guess]
(let [nextval (f guess)]
(if (close-enough? guess nextval tolerance)
nextval
(recur nextval))))]
(trynext first-guess)))
|
| | (deftest test-close
(is (close-enough?
(fixed-point (fn [x] (+ 1 (/ 1 x))) 1.0)
phi
tolerance)))
1.36
Find a solution to $$x^x = 1000$$ by finding a fixed point of
$$x \rightarrow log(1000)/log(x)$$.
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With no damping: \(log(1000) / log(x)\)
| | (deftest test-fixed-point-no-damping
(is (close-enough?
(fixed-point (fn [x] (/ (Math/log 1000) (Math/log x))) 2)
4.5555
tolerance)))
With damping: \((log(1000) / log(x) + x) / 2\)
| | (deftest test-fixed-point-damping
(is (close-enough?
(fixed-point (fn [x] (/ (+ (/ (Math/log 1000) (Math/log x)) x) 2)) 2)
4.5555
tolerance)))
Compare the number of steps this takes with and without average damping.
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