Lecture 10 - CHAPTER 4 Round-Off and Truncation Errors...

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CHAPTER 4 Round-Off and Truncation Errors
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Sources of Errors Modeling error : Cd assumed to be constant; point mass, not flexible body Empirical measurements : g=9.81, Planck const Inexact input data : balloon data for weather prediction Blunder Round-off error : π 3.14159 not 3.14159265358979323846 ... Truncation error : e x 1 + x + x 2 + x 3 + x 4 for small x
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Numerical Error Truncation error : Method dependent Errors resulting from using an approximation rather than an exact procedure--- Taylor series expansion Round-off error : Machine dependent Resulting from using an approximate number to represent exact number: Errors resulting from not being able to represent true value e.g.: 0.1 =1/10 = 1/2 4 +1/2 5 +1/2 8 +1/2 9 +1/2 12 +1/2 13 +… = 0.000 1100 1100 1100 1100 1100 1100 1100 .... (in binary) Pattern repeats; never ending; the number is inexact . 2 3 ( ) ( ) ( ) ( ) ( ) . .. 2! 3! i i i i i h h f x h f x hf x f x f x ′′ ′′′ + = + + + + 3.141593, 2.718282 e π= = ??? Is such an “ inexactness ” or round-off error important???
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x=0.1=0.0625+0.03125+0.0390625+0.001953125+0.00048828125… n 1/2^n x= 0.1 1 0.5 2 0.25 0.125 0.0625 0.0375 5 0.03125 => 0.00625 6 0.015625 7 0.0078125 8 0.00390625 => 0.002343750 9 0.001953125 => 0.000390625 10 0.0009765625 11 0.00048828125 12 0.000244140625 => 0.0001464843750 13 0.0001220703125 => 0.0000244140625 14 0.00006103515625 15 0.000030517578125 16 0.000015258789063 17 0.000007629394531 18 0.000003814697266
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Lecture 10 - CHAPTER 4 Round-Off and Truncation Errors...

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