Chapter 1 Mathematical Preliminaries

1.3 Algorithms and ConvergenceDefinition 1.8 Suppose {β n } ∞ n=1 is a sequence to converge to zero, and {α n } ∞ n=1 convergesto α. If exists a positive constant K such that|α n − α| ≤ K|β n |, for large n,then we say that {α n } ∞ n=1 converges to α with rate of convergence O(β n ). Writingα n = α + O(β n ). Usually, we use β n = 1 for some number p. We are generallynp interested in the largest value of p with α n = α + O(1/n p ).Definition 1.9 Suppose lim G(h) = 0 and lim F(h) = L. If exists a positive constanth→0 h→0K such that|F(h) − L| ≤ K|G(h)|, for sufficiently small h,then we write F(h) = L + O(G(h)). Usually, we use G(h) = h p for some number p. Weare generally interested in the largest value of p with F(h) = L + O(h p ).6

1.4 Homework chapter 11-1 (Ex. 1.1.6) Suppose f ∈ C[a,b] and f ′ (x) exists on (a,b). Show that if f ′ (x) ≠ 0for all x ∈ (a,b) then there can exist at most one p ∈ [a,b] with f(p) = 0.1-2 (Ex. 1.1.9) Find the second Taylor polynomial P 2 (x) for f(x) = e x cos x aboutx 0 = 0.a. Use P 2 (0.5) to approximate f(0.5). Find an upper bound for error |f(0.5) −P 2 (0.5)| using the error formula and compare it to the actual error.b. Find a bound for the error |f(x) − P 2 (x)| in using P 2 (x) to approximate f(x)on [0, 1].c. Approximate ∫ 10 f(x)dx using ∫ 10 P 2(x)dx.d. Find an upper bound for the error in (c) using ∫ 10 |R 2(x)|dx and compare thebound to the actual error.1-3 (Ex. 1.1.19) Let f(x) = e x and x 0 = 0. Find the n−th Taylor polynomial P n (x)for f(x) about x 0 . Find a value of n necessary for P n (x) to approximate f(x) towithin 10 −6 on [0, 0.5].1-4 (Ex. 1.2.4) Perform the following computations (i) exactly, (ii) using three-digitchopping arithmetic, and (iii) using three-digit rounding arithmetic. (iv) Computethe relative error in parts (ii) and (iii).a.c.45 + 1 3( 13 11)− 3 + 3 20b.d.45 × 1 3( 13 + 3 11)− 3201-5 (Ex. 1.2.24) Suppose that fl(y) is a k−digit rounding approximation to y. Showthat∣ ∣∣∣∣ y − fl(y)y ∣ ≤ 0.5 × 10−k+1 .1-6 (Ex. 1.3.7) Find the rates of convergence of the following functionssin hsin h − h cosh(i) lim = 1, (ii) lim= 0.h→0 h h→0 h7