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Then line 12 executes, this time passing 1 to n, 'C' to x, 'A' to y, and 'B' to z. In that call, n = 1 again, so line 8 executes, printing the third line of output:
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Move top disk from peg C to peg B.
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That call returns to where the second recursive call had left off at line 12. Since that is the last executable statement in the method, it also returns, back to where the first recursive call had left off at line 10. So it proceeds to line 11 with n = 3, x = 'A', y = 'B', and z = 'C'. That prints the fourth line of output:
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Move top disk from peg A to peg C.
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Then line 12 executes, passing 2 to n, 'B' to x, 'A' to y, and 'C' to z. That call, HanoiTowers(2, 'B', 'A', 'C'), recursively moves the stack of two disks from peg B to to peg C via peg A and generates the last three lines of output:
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Move top disk from peg B to peg A. Move top disk from peg B to peg C. Move top disk from peg A to peg C.
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Since the previous four moves had already transferred the largest disk from peg A to peg C, this completes the task.
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MUTUAL RECURSION When a function calls itself, it is called direct recursion. Another form of recursion is when a function calls other functions that call other functions that eventually call the original function. This is called indirect recursion. Its most common form is when two functions call each other. This is called mutual recursion. (See Figure 9.6.)
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Figure 9.6 Types of recursion
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EXAMPLE 9.16 The Sine and Cosine Functions Computed by Mutual Recursion
The sine and cosine functions from trigonometry can be defined in several different ways, and there are several different algorithms for computing their values. The simplest (although not the most efficient) is via mutual recursion. It is based upon the identities: sin2 = 2sin cos cos2 = 1 2(sin )2 and the two Taylor polynomials: sin x x x 3/6
RECURSION
[CHAP. 9
cos x 1 x 2/2 which are close approximations for small values of x.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
public class TestMutualRecursion { public static void main(String[] args) { String fmt1 = "%18s%18s%18s%n"; String fmt2 = "%18.13f%18.13f%18.13f%n"; System.out.printf(fmt1, "s(x) ", "Math.sin(x) ", "error "); for (double x = 0.0; x < 1.0; x += 0.1) { System.out.printf(fmt2, s(x), Math.sin(x), Math.sin(x) - s(x)); } System.out.printf(fmt1, "c(x) ", "Math.cos(x) ", "error "); for (double x = 0.0; x < 1.0; x += 0.1) { System.out.printf(fmt2, c(x), Math.cos(x), c(x) - Math.cos(x)); } } public static double s(double x) { if (-0.005 < x && x < 0.005) { return x - x*x*x/6; // basis } return 2*s(x/2)*c(x/2); // recursion } public static double c(double x) { if (-0.005 < x && x < 0.005) { return 1.0 - x*x/2; // basis } return 1 - 2*s(x/2)*s(x/2); // recursion } }
The output is:
Math.sin(x) error 0.0000000000000 0.0000000000000 0.0998334166468 0.0000000000005 0.1986693307951 0.0000000000009 0.2955202066613 0.0000000000069 0.3894183423087 0.0000000000018 0.4794255386042 0.0000000000051 0.5646424733950 0.0000000000120 0.6442176872377 0.0000000000015 0.7173560908995 0.0000000000027 0.7833269096275 0.0000000000043 0.8414709848079 0.0000000000063 Math.cos(x) error 1.0000000000000 0.0000000000000 0.9950041652780 0.0000000000000 0.9800665778412 0.0000000000002 0.9553364891256 0.0000000000021 0.9210609940029 0.0000000000007 0.8775825618904 0.0000000000028 0.8253356149097 0.0000000000082 0.7648421872845 0.0000000000013 0.6967067093472 0.0000000000027 0.6216099682707 0.0000000000054 0.5403023058681 0.0000000000098 This works because on each recursive call x is divided by 2, and eventually it reaches the basis criterion (-0.005 < x && x < 0.005), which stops the recursion. s(x) 0.0000000000000 0.0998334166464 0.1986693307941 0.2955202066544 0.3894183423069 0.4794255385991 0.5646424733831 0.6442176872362 0.7173560908969 0.7833269096232 0.8414709848016 c(x) 1.0000000000000 0.9950041652781 0.9800665778414 0.9553364891277 0.9210609940036 0.8775825618932 0.8253356149179 0.7648421872857 0.6967067093499 0.6216099682760 0.5403023058779
CHAP. 9]
RECURSION
Review Questions
9.1 9.2 9.3 9.4 9.5
A recursive function must have two parts: its basis and its recursive part. Explain what each of these is and why it is essential to recursion. How many recursive calls will the call f(10) to the recursive factorial function (Example 9.2 on page 165) generate How many recursive calls will the call fib(6) to the recursive Fibonacci function (Example 9.6 on page 167) generate What are the advantages and disadvantages of implementing a recursive solution instead of an iterative solution What is the difference between direct recursion and indirect recursion
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