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aimz1 - AIMS Exercise Set 1 Peter J Olver 1 Determine the...

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AIMS Exercise Set # 1 Peter J. Olver 1. Determine the form of the single precision floating point arithmetic used in the computers at AIMS. What is the largest number that can be accurately represented? What is the smallest positive number n 1 ? The second smallest positive number n 2 ? Which is larger: the gap between n 1 and 0 or the gap betweeen n 1 and n 2 ? Discuss. 2. Determine the value of each of the following quantities using 4 digit rounding and four digit chopping arithmetic. Find the abolute and relative errors of your approximation. ( a ) π + e - cos 22 , ( b ) e π - π e log 10 11 . 3. ( a ) To how many significant decimal digits do the numbers 10002 and 10001 agree? ( b ) Subtract the two numbers. How many significant decimal digits are lost in the computation? ( c ) How might you rearrange the computation to obtain a more accurate answer. 4. ( a ) Verify that f ( x ) = 1 - sin x and g ( x ) = cos 2 x 1 + sin x are identical functions. ( b ) Which function should be used for computations when x is near 1 2 π ? Why? ( c ) Which function should be used for computations when x is near 3 2 π ? Why? 5. Horner’s Method ( a ) Suppose x is a real number and n a positive integer. How many multiplications are
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