By Yung-Kai Lai, C.-C. Jay Kuo, Jin Li

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As already mentioned, in this case we only need to count the calls of step (i) for m = 1,2, ... ,7. m 1 2 3 4 5 6 7 number of calls of step (i) 2 3 3 4 4 3 1 40 Chapter 1. Computing TJ! 86. It has been shown that for large n, M TA (n) 121n 2 = --2-lnn = O(lnn). 7r In the remainder of this chapter we restrict our attention to complexity in the worst-case sense. 4 The Complexity of Some Algorithms. In this sections we compute the time complexitity TX (n) of several of the algorithms presented above.

In this case we can define arithmetic operations as above, except that we have to take account of roundoff, which means that the result intervals must be enlarged. Interval arithmetic has been heavily studied, and many of the methods in numerical analysis have been built into the theory. The main problem is to develop methods such that the intervals in the calculation do not grow too much. This requires a clever combination of conventional techniques with those from interval analysis. For details, see the extensive literature and the book of R.

18 Chapter 1. Computing Example. Suppose we want to compute the integrals In = 1 xn --dx o X +5 1 for n = 0,1,2,· .. ,20. It is easy to see that the numbers In satisfy the recursion In + 5In- 1 = 1 1 o xn + 5x n- 1 dx = x+5 11 0 x n- 1 dx 1 n = -. = Starting with the value 10 In ~, this recursion can theoretically be used to find all of the numbers In = ~ - 5In- 1 • But if we carry out this process, it turns out that after only a few steps we already have wrong results, and that after a few more steps we even get negative numbers.