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A chapter from a book on numerical methods that discusses the importance of understanding errors in the effective use of these methods. The chapter covers the concept of error, approximations, and estimates of errors in numerical methods. It also discusses the identification, quantification, and minimization of errors. The document emphasizes the importance of error in engineering and the need to settle for approximations or estimates of errors in the absence of analytical solutions. The chapter provides general information on the quantification of error.
Typology: Exercises
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Dr. Mohammed Ali AlMahamdy
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Because so many of the methods in this book are straightforward in description and application, it would be very tempting at this point for us to proceed directly to the main body of the text and teach you how to use these techniques. However, understanding the concept of error is so important to the effective use of numerical methods that we have chosen to devote the next two chapters to this topic. The importance of error was introduced in our discussion of the falling parachutist in Chap. 1. Recall that we determined the velocity of a falling parachutist by both ana- lytical and numerical methods. Although the numerical technique yielded estimates that were close to the exact analytical solution, there was a discrepancy, or error, because the numerical method involved an approximation. Actually, we were fortunate in that case because the availability of an analytical solution allowed us to compute the error exactly. For many applied engineering problems, we cannot obtain analytical solutions. Therefore, we cannot compute exactly the errors associated with our numerical methods. In these cases, we must settle for approximations or estimates of the errors. Such errors are characteristic of most of the techniques described in this book. This statement might at irst seem contrary to what one normally conceives of as sound engineering. Students and practicing engineers constantly strive to limit errors in their work. When taking examinations or doing homework problems, you are penalized, not rewarded, for your errors. In professional practice, errors can be costly and sometimes catastrophic. If a structure or device fails, lives can be lost. Although perfection is a laudable goal, it is rarely, if ever, attained. For example, despite the fact that the model developed from Newton’s second law is an excellent approximation, it would never in practice exactly predict the parachutist’s fall. A variety of factors such as winds and slight variations in air resistance would result in deviations from the prediction. If these deviations are systematically high or low, then we might need to develop a new model. However, if they are randomly distributed and tightly grouped around the prediction, then the deviations might be considered negligible and the model deemed adequate. Numerical approximations also introduce similar discrepancies into the analysis. Again, the question is: How much the next error is present in our calculations and is it tolerable? This chapter and Chap. 4 cover basic topics related to the identiication, quan- tiication, and minimization of these errors. In this chapter, general information con- cerned with the quantiication of error is reviewed in the irst sections. This is
3.1 SIGNIFICANT FIGURES 57 one person might say 48.8, whereas another might say 48.9 km/h. Therefore, because of the limits of this instrument, only the irst two digits can be used with conidence. Estimates of the third digit (or higher) must be viewed as approximations. It would be ludicrous to claim, on the basis of this speedometer, that the automobile is traveling at 48.8642138 km/h. In contrast, the odometer provides up to six certain digits. From Fig. 3.1, we can conclude that the car has traveled slightly less than 87,324.5 km during its lifetime. In this case, the seventh digit (and higher) is uncertain. The concept of a signiicant igure, or digit, has been developed to formally designate the reliability of a numerical value. The signii cant digits of a number are those that can be used with conidence. They correspond to the number of certain digits plus one esti- mated digit. For example, the speedometer and the odometer in Fig. 3.1 yield readings of three and seven signiicant igures, respectively. For the speedometer, the two certain digits are 48. It is conventional to set the estimated digit at one-half of the smallest scale division on the measurement device. Thus the speedometer reading would consist of the three signiicant igures: 48.5. In a similar fashion, the odometer would yield a seven- signii cant-igure reading of 87,324.45. Although it is usually a straightforward procedure to ascertain the signiicant igures of a number, some cases can lead to confusion. For example, zeros are not always sig- niicant igures because they may be necessary just to locate a decimal point. The num- bers 0.00001845, 0.0001845, and 0.001845 all have four signiicant igures. Similarly, when trailing zeros are used in large numbers, it is not clear how many, if any, of the zeros are signiicant. For example, at face value the number 45,300 may have three, four, or ive signiicant digits, depending on whether the zeros are known with conidence. Such uncertainty can be resolved by using scientiic notation, where 4.53 3 104 , 4.530 3 104 , 4.5300 3 104 designate that the number is known to three, four, and ive signiicant igures, respectively. The concept of signiicant igures has two important implications for our study of numerical methods:
1. As introduced in the falling parachutist problem, numerical methods yield approxi- mate results. We must, therefore, develop criteria to specify how conident we are in our approximate result. One way to do this is in terms of signiicant igures. For example, we might decide that our approximation is acceptable if it is correct to four signiicant i gures. 2. Although quantities such as p, e , or 17 represent speciic quantities, they cannot be expressed exactly by a limited number of digits. For example, p 5 3.1415 92653589793238462643 p ad ini nitum. Because computers retain only a inite number of signiicant igures, such numbers can never be represented exactly. The omission of the remaining signii cant igures is called round-off error_._ Both round-off error and the use of signiicant igures to express our conidence in a numerical result will be explored in detail in subsequent sections. In addition, the concept of signiicant igures will have relevance to our dei nition of accuracy and preci- sion in the next section.
58 APPROXIMATIONS AND ROUND-OFF ERRORS
The errors associated with both calculations and measurements can be characterized with regard to their accuracy and precision. Accuracy refers to how closely a computed or measured value agrees with the true value. Precision refers to how closely individual computed or measured values agree with each other. These concepts can be illustrated graphically using an analogy from target practice. The bullet holes on each target in Fig. 3.2 can be thought of as the predictions of a nu- merical technique, whereas the bull’s-eye represents the truth. Inaccuracy (also called bias ) is deined as systematic deviation from the truth. Thus, although the shots in Fig. 3.2 c are more tightly grouped than those in Fig. 3.2 a, the two cases are equally biased because they are both centered on the upper left quadrant of the target. Imprecision (also called uncertainty ), on the other hand, refers to the magnitude of the scatter. Therefore, although Fig. 3.2 b and d are equally accurate (that is, centered on the bull’s-eye), the latter is more precise because the shots are tightly grouped. Numerical methods should be suficiently accurate or unbiased to meet the require- ments of a particular engineering problem. They also should be precise enough for adequate ( c ) ( a ) ( d ) ( b ) Increasing accuracy Increasing precision FIGURE 3. An example from marksmanship illustrating the concepts of accuracy and precision. ( a ) Inaccurate and imprecise; ( b ) accurate and imprecise; ( c ) inaccurate and precise; ( d ) accurate and precise.
