4.2.4 DISTRIBUTION OF FLOATING POINT NUMBERS 243 Pl(lons) (Best web design)
4.2.4 DISTRIBUTION OF FLOATING POINT NUMBERS 243 Pl(lons) = -&rl -1 + [lOrl -10 +. . . + [lo - rl -ion-l + [lO s] + 1 - 10%) = –&1+ 10 +. . . + 10n-l) + O(n) + Lions] -1 -10 -. *. -10 ) = -y g lo? -lon+l) + lonsl+ O(n)). (8) 102 ( As n + 00, P1(lOns) therefore approaches the limiting value 1 + (r -10)/9s. The above calculation for the case s 5 r can be modified so that it is valid for s > r if we replace [lOns] + 1 by [lO+]; when s 2 r, we therefore obtain the limiting value lO(r -1)/9s. [See J. F ranel, Naturforschende Gesellschaft, Vierteljahrsschrift 62 (Ziirich, 1917), 286-295.1 In other words, the sequence (PI(n)) has subsequences (Pl(lOns)) whose limit goes from (T - 1)/9 up to lO(r -1)/9r and down again to (T - 1)/9, as s goes from 1 to T to 10. We see that PI(n) has no limit as n -+ m; and the values of PI(n) for large n are not particularly good approximations to our conjectured limit logI T either! Since PI(~) doesn t approach a limit, we can try to use the same idea as (7) once again, to average out the anomalous behavior. In general, let &+1(4= ; c Pm(k). l