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number theory. COMPLEXITY ANALYSIS The complexity analysis of a recursive algorithm depends upon the solubility of its recurrence relation. The general technique is to let T(n) be the number of steps required to carry out the algorithm on a problem of size n. The recursive part of the algorithm translates into a recurrence relation on T(n). Its solution is then the complexity function for the algorithm. The recursive factorial function runs in O(n) time. Let T(n) be the number of recursive calls made from the initial call f(n) to the function in Example 9.2 on page 165. Then T(0) = T(1) = 0, because if n < 2, no recursive calls are made. If n > 1, then the line
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(5.23)
return n*f(n-1);
where G m i n ( zis another minimum phase sequence. Because Hm,,(z) is minimum phase, lakI < I . Let H ( z ) be the ) causal nonminimum phase sequence that is formed by replacing the zero at z = a k with a zero at z = ] / a ; :
return n*f(n-1);
H ( z )=
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will return the correct value n! because (by the inductive hypothesis) the call f(n-1) will return (n 1)! and n! = n (n 1). Note that we are using the strong principle of mathematical induction here (also called the second principle of mathematical induction). In this version, the inductive hypothesis allows us to assume that all the preceding statements are true. In the weak (or first ) principle, we are allowed to assume that only the single preceding statement is true. But since these two principles are equivalent (i.e., they are both valid methods of proof), it is usually better to apply strong induction. The Euclidean algorithm is correct. We can use (strong) induction to prove this fact. (See page 322.) If m and n are equal, then that number is their greatest common divisor. So the function returns the correct value in that case because of the part
I - akz-'
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executes, making the recursive call f(n-1). Then the total number of recursive calls is 1 plus the number of calls that are made from f(n-1). That translates into the recurrence relation T(n) = 1 + T(n 1) The solution to this recurrence is T(n) = n 1, for n > 0 This conclusion is obtained in two stages: First we find the solution; then we use induction to prove that it is correct. The simplest technique for finding the solution to a recurrence relation is to make a table of values and look for a pattern. This recurrence relation says that each value of T(n) is 1 more than the previous value. So the solution f(n) = n 1 is pretty obvious.
Hminiz)
if (m == n) { return n; } If m and n are not equal, then the function returns either gcd(m,n-m) or gcd(m-n,n). To see that this too is the correct value, we need only realize that all three pairs (m,n), (m,n-m), and (m-n,n) will always have the same greatest common divisor. This fact is a theorem from
hm,.(n) = gmin(n) - akgmln(n - 1 ) h ( n ) = gmin(n - 1) - a,l~mln(n)
CHAP. 9]
which is greater than zero because \ak < 1. Therefore, 1
number theory. COMPLEXITY ANALYSIS The complexity analysis of a recursive algorithm depends upon the solubility of its recurrence relation. The general technique is to let T(n) be the number of steps required to carry out the algorithm on a problem of size n. The recursive part of the algorithm translates into a recurrence relation on T(n). Its solution is then the complexity function for the algorithm. The recursive factorial function runs in O(n) time. Let T(n) be the number of recursive calls made from the initial call f(n) to the function in Example 9.2 on page 165. Then T(0) = T(1) = 0, because if n < 2, no recursive calls are made. If n > 1, then the line
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