Consider the problem of evaluating the integrals Yn = dx x+10 for n = 1, 2,... Analytically we notice that Also, 1 Yn + 10yn-1 = L x+10x-1 x+10 1 dx = xn-1 dx n Yo = S 1 1 dx = ln(11) - In(10). x+10 This gives us the following algorithm for computing Yo, Y1,...: 1. Compute yo ln(11) In(10). = 2. For n = 1, 2, ..., evaluate yn = 1 - 10yn-1. (a) Show that yn < 1 for all n and yn decreases monotonically to 0 as n . (A picture would suffice for this!) (b) Program the algorithm in a computer and find yn for n = 1,2, Why does this happen? 30. What happens? (c) Derive an algorithm for computing the values of these integrals based on evaluating the value of Yn-1 from the value of Yn. (d) Suppose you want to calculate the values of yo, Y1, YN for some number N with an absolute error of less than for some > 0. Since the naive algorithm above produces large errors (even though it is exact in theory), you will use the algorithm in (c). But you need a starting value. Show that there exists M = N such that if you take YM O and use the algorithm in (c), then, if calculations are performed with infinite precision, the absolute errors in the calculations of Yo, Y1, ..., YN will be less than . = 3 (e) Explain why rounding errors in the computer do not produce excessive errors using this algorithm. (f) Use your algorithm (and the computer) to find the value of 30 to an accuracy of 105. Explain how you chose M in this case.