Exercise 1.16 (#11)

src/sicp/chapter_1/ex_1_16.clj

@@ -0,0 +1,24 @@
+(ns sicp.chapter-1.ex-1-16)
+
+; Exercise 1.16
+; Design a procedure that evolves an iterative exponentiation process
+; that uses successive squaring and uses a logarithmic number of steps, as does fast-expt.
+
+; Hint:
+; Using the observation that (b^(n/2))^2 = (b^2)^(n/2),
+; keep, along with the exponent n and the base b, an additional state variable a,
+; and define the state transformation in such a way that the product ab^n is unchanged from state to state.
+; At the beginning of the process a is taken to be 1, and the answer is given by the value of a at the end of the
+; process.
+; In general, the technique of defining an invariant quantity that remains unchanged from
+; state to state is a powerful way to think about the design of iterative algorithms.
+
+(defn fast-expt
+  [b n product]
+  (cond (= n 0) product
+        (even? n) (fast-expt (* b b) (/ n 2) product)
+        :else (fast-expt b (- n 1) (* product b))))
+
+(defn expt
+  [b n]
+  (fast-expt b n 1))

test/sicp/chapter_1/ex_1_16_test.clj

@@ -0,0 +1,7 @@
+(ns sicp.chapter-1.ex-1-16-test
+  (:require [clojure.test :refer :all]
+            [sicp.chapter-1.ex-1-16 :refer [expt]]))
+
+(deftest expt-test
+  (is (= 1 (expt 1 2)))
+  (is (= 3486784401 (expt 3 20))))