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Extra info for Concrete Mathematics: A Foundation for Computer Science (1st Edition)
A sum that depends on more than one index can be summed first on any one of its indices. 16) we saw earlier: 7 7 aj,k[P(j,k)] = x aj,k = 7 7 aj,k[P(j,k)]. 27) Multiple C’s are evaluated right to left (inside-out). 4 MULTIPLE SUMS 35 Who’s panicking? I think this rule is fairly obvious compared to some of the stuff in Chapter 1. The middle term of this law is a sum over two indices. On the left, tj tk stands for summing first on k, then on j. On the right, tk xi stands for summing first on j, then on k.
Kenneth Iverson introduced a wonderful idea in his programming language APL [161, page 111, and we’ll see that it greatly simplifies many of the things we want to do in this book. The idea is simply to enclose a true-or-false statement in brackets, and to sav that the result is 1 if the statement is true. 0 if the statement is false. For example, I [p prime] = 1, 0, if p is a prime number; if p is not a prime number. 4) in the form x ak [P(k)] . 5) k If P(k) is false, the term ok[P(k)] is zero, so we can safely include it among the terms being summed.
1 n = b,b,-, . 2 SUMS AND RECURRENCES 29 We started with a t in the recur- rence, and worked hard to get rid of it. But then after applying a summation factor, we came up with another t. Are sums good, or bad, or what? 10), is therefore C, = 2(n + 1) f 1. k=l k+l The sum that remains is very similar to a quantity that arises frequently in applications. 13) f;. k=l The letter H stands for “harmonic”; H, is a harmonic number, so called because the kth harmonic produced by a violin string is the fundamental tone produced by a string that is l/k times as long.