tailieunhanh - Burden - Numerical Analysis 5e (PWS, 1993) Epside 1 Part 2

Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) epside 1 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 16 CHAPTER 1 B Mathematical Preliminaries EXAMPLE 2 a. If p X 101 and p - X IO1 the absolute error is and the relative error is X Ỉ0 1. b. Ifp - X 10 3 andp X 10 3 the absolute error is X 10-4 and the relative error is X 10-1. c. If p X 104 and p X 104 the absolute error is X 103 and the relative error is X 10-1. This example shows that the same relative error X 10-1 occurs for widely varying absolute errors. As a measure of accuracy the absolute error may be misleading and the relative error more meaningful. H E Returning to the machine representation of numbers we see that the floating-point representation yz y for the number y has the relative error y fi y y If k decimal digits and chopping are used for the machine representation of y d2 . dk dk i . X 10 then y - y 0. dk 1. X 10 - 0. 10 04 d2 . X 10 i dk 2 . X IQ- 10 0- 4 i 4-2 di . . X 10 . Since dỵ 0 the minimal value of the denominator is . The numerator is bounded above by 1. As a consequence y-fltỳ SJ_X 10- 10-w y In a similar manner a bound for the relative error when using fc-digit rounding arithmetic X 10-A i. SeeExercise21. Note that the bounds for the relative error using k- digit arithmetic are independent of the number being represented. This is due to the manner in which the machine numbers are distributed along the real line. Because of the exponential form of the characteristic the same number of decimal machine numbers are used to represent each of the intervals 1 1 10 and 10 100 . In fact within the limits of the machine the number of decimal machine numbers in 10 10 is constant for all integers n. In addition to inaccurate representation of numbers the arithmetic performed in a computer is not exact. The arithmetic generally involves manipulating binary digits by various shifting or logical operations. Since the actual mechanics of these operations are not .

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