2-1 CHAPTER 2 NUMERICAL ERRORS All numerical calculations are prone to numerical errors. This chapter examines two important sources of error: round-off errorand 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. RelativeNumerical methods are all approximation methods. They give approximated values rather than true values. The absolute error is defined as: ValueTrueValueedApproximatErrorAbsolute−=However, it is more convenient to use the relative error: ValueedApproximatValueTrueValueedApproximatValueTrueValueTrueValueedApproximatErrorRelative−≈−=The second expression is used when the true value is unknown, which is typically the case. EE 3108 Semster B 2007/2008 S C Chan2-2 Example: What is the relative error in using 0.33 to approximate31?Solution:The relative error is: 01.0313133.0−=−. We can also express the relative error in percentage as: %%=×.1100010−−. B. Accuracy vs. PrecisionWhen experiments or calculations are performed repeatedly, the concepts of accuracy and precision are important: •Accuracymeasures the proximity between the calculated values and thetrue value. •Precisionmeasures the consistency among repeated calculations. The concepts are illustrated in the following diagram.
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