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Chapter02 - CHAPTER 2 NUMERICAL ERRORS All numerical...

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2-1 CHAPTER 2 NUMERICAL ERRORS All numerical calculations are prone to numerical errors. This chapter examines two important sources of error: round-off error and truncation error . Round-off errors are related to the discrete representation of numbers in computers. Truncation errors are related to the numerical algorithms and formulae used in the calculations. I. Error definitions A. Absolute vs. Relative Numerical methods are all approximation methods. They give approximated values rather than true values. The absolute error is defined as: Value True Value ed Approximat Error Absolute = However, it is more convenient to use the relative error: Value ed Approximat Value True Value ed Approximat Value True Value True Value ed Approximat Error Relative = The second expression is used when the true value is unknown, which is typically the case. EE 3108 Semster B 2007/2008 S C Chan 2-2 Example: What is the relative error in using 0.33 to approximate 3 1 ? Solution: The relative error is: 01 . 0 3 1 3 1 33 . 0 = . We can also express the relative error in percentage as: % %= × . 1 100 01 0 . B. Accuracy vs. Precision When experiments or calculations are performed repeatedly, the concepts of accuracy and precision are important: Accuracy measures the proximity between the calculated values and the true value . Precision measures the consistency among repeated calculations . The concepts are illustrated in the following diagram.
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