4.2.2 ACCURACY OF FLOATING POINT ARITHMETIC 229 14. (Web hosting servers)
4.2.2 ACCURACY OF FLOATING POINT ARITHMETIC 229 14. [MZ7] Find a suitable E such that (U @ w) @ w Z=Z u @ (v @ w) (E), when unnormalized multiplication is being used. (This generalizes (39), since unnormalized multiplication is exactly the same as normalized multiplication when the input operands U, w, and w are normalized.) b 15. [A4Z4] (H. Bj6rk.) Does the computed midpoint of an interval always lie between the endpoints? (In other words, does u 5 21 imply that u 5 (U $ V) @ 2 5 v?) 16. [A&%] (a) What is (… ((x1 $ ~2) $ 2s) @ … $ 2,) when n = lo6 and xk = 1.1111111 for all k, using eight-digit floating decimal arithmetic? (b) What happens when Eq. (14) is used to calculate the standard deviation of these particular values xk? What happens when Eqs. (15) and (16) are used instead? (c) Prove that Sk 2 0 in (16), for all choices of x1, . . . , zk. 17. [%I Write a MIX subroutine, FCMP, that compares the floating point number u in location ACC with the floating point number v in register A, and that sets the comparison indicator to LESS, EQUAL, or GREATER, according as IL < U, u -V, or u > w (6); here E is stored in location EPSILON as a nonnegative fixed point quantity with the decimal point assumed at the left of the word. Assume normalized inputs. 18. [M40] In unnormalized arithmetic is there a suitable number E such that b 19. [MSO] (W. M. Kahan.) Consider the following procedure for floating point sum- mation of x1, . . . , xn: so = co = 0; yk = xk 8 ck-1, Sk = Sk-1 $ yk, ck = (Sk @ Sk-l) 8 yk, for 1 5 k 5 n. Let the relative errors in these operations be defined by the equations yk = (xk -ck-l)(l + vk), Sk = (Sk-1 + yk)(l + ok), ck = ((Sk -Sk-l)(l +-/k) -yk)(l + Sk), where Iqkl, lokl, lykl, 16kl 5 t. Prove that Sn = c l