By VICTOR SHOUP
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10. Suppose that f and g are functions deﬁned on the integers k, k + 1, . , and that g is eventually positive. For n ≥ k, deﬁne F (n) := n n i=k f (i) and G(n) := i=k g(i). Show that if f = O(g) and G is eventually positive, then F = O(G). 11. Suppose that f and g are functions deﬁned on the integers k, k + 1, . , both of which are eventually positive. For n ≥ k, deﬁne F (n) := n n i=k f (i) and G(n) := i=k g(i). Show that if f ∼ g and G(n) → ∞ as n → ∞, then F ∼ G. The following two exercises are continuous variants of the previous two exercises.
Explain your ﬁndings. 20. This exercise is also for C /Java programmers. Suppose that values of type int are stored using a 32-bit 2’s complement representation, and that all basic arithmetic operations are computed correctly modulo 232 , even if an “overﬂow” happens to occur. Also assume that double precision ﬂoating point has 53 bits of precision, and that all basic arithmetic TEAM LinG 46 Computing with large integers operations give a result with a relative error of at most 2−53 . Also assume that conversion from type int to double is exact, and that conversion from double to int truncates the fractional part.
2) holds. 2) are precisely those integers that are congruent to 3 modulo 7, which we can list as follows: . . , −18, −11, −4, 3, 10, 17, 24, . . ✷ In the next section, we shall give a systematic treatment of the problem of solving linear congruences, such as the one appearing in the previous example. 1. Let x, y, n ∈ Z with n > 0 and x ≡ y (mod n). Also, let a0 , a1 , . . , ak be integers. Show that a0 + a1 x + · · · + ak xk ≡ a0 + a1 y + · · · + ak y k (mod n). 2. Let a, b, n, n ∈ Z with n > 0 and n | n.