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By using recursion the need for a looping statement has been removed. To determine if x is contained within A(l:n) this function is initially invoked as ans SEARCH(l ). (See the exercises for an improvement to SEARCH). 20 Introducdon Removing Recursion We are in a mild dilemma. Some design techniques are inherently recursive and so recursion is a natural way to describe algorithms obtained from these techniques. Also, it is often easier to prove a recursive algorithm correct than it is to prove the corresponding iterative algorithm correct.
Write a recursive program which computes the binomial coefficient BINOM = BINOM(n - 1,m) + BINOM(n - 1,m - 1) and BINOM(n,O) = BINOM(n,n) = 1. (n,m) using the recursive definition BINOM(n,m) 21. ). 22. Prove that 1 + 2 + 3 + · · · + n = n(n + 1)/2. ) 23. 1 by adding the values for the following columns: log log n, n 2 log n, n 3 log n, and nn. 24. 1 by adding rows for the following values of n: 64, 128, 256, 512 and 1024. Use approximation whereever it is necessary. 25. 7, the frequency of execution of every statement is fixed by n except "k - i".
Computer science and its relation to mathematics", by D. E. Knuth, American Math Monthly, April, 1974. Other important textbooks on the theory and complexity of algorithms are The Design and Analysis of Computer Algorithms, by A. V. E. Hopcroft, J. D. Ullman, Addison-Wesley, Reading, Mass. (1974). The computational complexity of algebraic and numeric problems by A. Borodin and I. Munro, American Elsevier, New York, 1975. " Elements of Combinatorial Computing, by M. B. Wells, Pergamon Press, Oxford (1971).
