Lecture001_Errors

# Lecture001_Errors - Round-off and Truncation Errors...

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Round-off and Truncation Errors Round-off and Truncation Errors 1 • Quantization error • Discretization error • Other errors

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Round-off and Truncation Errors Learning Objectives • Compare and contrast accuracy and precision • Compute error and relative error • Define round-off error and describe examples of typical computations in which it can cause problems • Define truncation error and describe examples of typical computations in which it can cause problems • Use Taylor Series to approximate a function • Use Taylor Series to estimate truncation error • Summarize the trade-off between round-off and truncation error 2
Round-off and Truncation Errors Accuracy • How closely a computed or measured value agrees with the true value Precision • How closely computed or measured values agree with each other 3

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Round-off and Truncation Errors Figure 4.1 An example from marksmanship illustrating the concepts of accuracy and precision: ( a ) inaccurate and imprecise, ( b ) accurate and imprecise, ( c ) inaccurate and precise, and ( d ) accurate and precise. 4 Accuracy and Precision
Round-off and Truncation Errors Error Definitions • True error ( E t ): the difference between the true value and the approximation. • Absolute error (| E t |): the absolute difference between the true value and the approximation. • True fractional relative error: the true error divided by the true value. • Relative error ( t ): the true fractional relative error expressed as a percentage. t true value - approximation true value 100%

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Round-off and Truncation Errors Error Definitions • The previous definitions of error relied on knowing a true value. If that is not the case, approximations can be made to the error. • The approximate percent relative error can be given as the approximate error divided by the approximation, expressed as a percentage - though this presents the challenge of finding the approximate error! • For iterative processes, the error can be approximated as the difference in values between sucessive iterations. a approximation error approximation 100%
Round-off and Truncation Errors Using Error Estimates • Often, when performing calculations, we may not be concerned with the sign of the error but are interested in whether the absolute value of the percent relative error is lower than a prespecified tolerance s . For such cases, the computation is repeated until | a |< s • This relationship is referred to as a stopping criterion .

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## This note was uploaded on 02/01/2011 for the course BME 113L taught by Professor Emelianov during the Spring '11 term at University of Texas.

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Lecture001_Errors - Round-off and Truncation Errors...

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