In order to represent a nonnegative real number X in a computer with finiteprecision

Chapter 4, Problem 5

(choose chapter or problem)

In order to represent a nonnegative real number X in a computer with finiteprecision, the number is either rounded to obtain Xr, or chopped (or trun-cated) to obtain Xc [STER 1974]. The representation errors in the two casesare bounded by12 Yr = X Xr 12 and0 Yc = X Xc < 1(measured in the units of the last digit). It is common to assume that Yr and Ycare uniformly distributed over their respective ranges. Now assume two indepen-dent numbers X1 and X2 are being added. Compute the pdf, the mean, and thevariance of the cumulative error in the sum X1 + X2 in cases of both roundingand chopping. Next assume that n mutually independent real numbers are tobe added, each subject to rounding or chopping. What are the mean and thevariance of the cumulative error in the two cases? Compare the mean with theworst-case errors. For n = 4, 9, 16, 25, 36, 49, 100, estimate the probability thatthe computed sum will differ from the sum of the original numbers by more than0.5. [Hint: Use the central-limit theorem.]