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Exercises 1.37 - 1.39
| | (ns e1-37-38-39
(:use [util.util :only (tolerance close-enough? const phi e square)])
(:use [clojure.contrib.generic.math-functions :only (tan)])
(:use [clojure.contrib.test-is :only (deftest is run-tests)]))
1.37
| |
|
| | (defn cont-frac [n d k]
(letfn [(iter [ind]
(if (> ind k)
0
(/ (n ind) (+ (d ind) (iter (inc ind))))))]
(iter 1)))
cont-frac-iter follows the same pattern as all of the recursive -> iterating
process transformations:
- add an accumulating parameter to the physically recursive function
- return it instead of zero in the end of the computation
| | (defn cont-frac-iter [n d k]
(let [iter (fn [ind result]
(if (> ind k)
result
(recur (inc ind) (/ (n ind) (+ (d ind) result)))))]
(iter 1 0)))
Both of the tests should result in the same value \(\approx 1/phi\).
| | (deftest test-cont-frac
(close-enough?
(cont-frac (const 1.0) (const 1.0) 100)
(/ 1 phi)
0.001)
(close-enough?
(cont-frac-iter (const 1.0) (const 1.0) 100)
(/ 1 phi)
0.001))
1.38
| |
Result should be \(\approx e - 2\).
| | (deftest test-cont-frac-non-iter
(close-enough?
(cont-frac (const 1.0)
;; Series = [1, 2, 1, 1, 4, 1, 1, 6, ..]
;; If we index the series beginning with 1, we can determine the value of the
;; corresponding element of the series using the index:
;; `if` (mod (index - 2) 3) == 0 `=>` series[index] == ((index - 2) * 2) + 2 `else` series[index] == 1
(fn [ind]
(let [base-ind (- ind 2)]
(if (= 0 (rem base-ind 3))
(+ 2 (* 2 (/ base-ind 3)))
1)))
100)
(- e 2)
0.001))
1.39
| |
|
| | (defn tan-cf [x k]
(cont-frac (fn [ind]
(if (= ind 1) x (- (square x))))
(fn [ind] (dec (* ind 2)))
k))
|
| | (deftest test-tan-cf
(close-enough?
(tan-cf 20 100)
(tan 20)
0.001))
| | |