m371-f10-hw1

# m371-f10-hw1 - ξ when x = 0 5 and x = 0 1(4(Cancellation...

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Homework 1 Math 371, Fall 2010 Assigned: Thursday, September 9, 2010 Due: Thursday, September 16, 2010 (1) (Finite precision numbers) The ﬂoating point representation of a real number takes the form x = ± (0 .a 1 a 2 . . . a n ) β · β e , where a 1 ̸ = 0, M e M . Suppose that β = 2, n = 4, and M = 5. (a) Find the smallest and largest positive numbers that can be represented in this ﬂoating point system. Give your answers in decimal form. (b) Find the ﬂoating point number in this system that is closest to 2. (2) (Rounding arithmetic) Use three-digit, decimal rounding arithmetic (i.e., β = 10 and n = 3) to compute the following sums. Add the numbers by hand in the specified order. (a) 6 k =1 1 3 k (b) 6 k =1 1 3 7 k (3) (Taylor series) (a) Calculate the first three terms of the Taylor series for f ( x ) = ln x about x = 2. Express the coeﬃcients analytically (not as decimals). (b) The first-order Taylor expansion of f ( x ) = exp x about x = 0 with exact remainder is f ( x ) = 1 + x + f ′′ ( ξ ) 2! x 2 for some point ξ between zero and x . Find an analytic expression for ξ . Use this expression to determine the numerical value of
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Unformatted text preview: ξ when x = 0 . 5 and x = 0 . 1. (4) (Cancellation Errors, p52 #12) Near certain values of x , the following functions cannot be accurately computed using the given formula on account of arithmetic cancellations. Identify the values of x where cancellation occurs (e.g., near x = 0 or when x is large and positive). Propose a reformulation that removes the problem (e.g., using Taylor series, rationalization, trigonometric identities, etc.). (a) f ( x ) = 1 + cos x (b) f ( x ) = e − x + sin x − 1 (c) f ( x ) = ln x − ln(1 /x ) (d) f ( x ) = √ x 2 + 1 − √ x 2 + 4 (e) f ( x ) = 1 − 2 sin 2 x (f) f ( x ) = x − sin x (g) f ( x ) = ln x − 1 (5) (Rate of convergence of a sequence, p27 #1) Compute the following limits and determine the corresponding rates of convergence. (a) lim n →∞ n − 1 n 3 +2 (b) lim n →∞ ( √ n + 1 − √ n ) (c) lim n →∞ sin n n 1...
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