Source of Erros in Numerical Methods

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1/11/2010 1

Sources of Error

Major: All Engineering Majors Authors: Autar Kaw, Luke Snyder

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

Undergraduates

http://numericalmethods.eng.usf.edu/http://numericalmethods.eng.usf.edu/


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Sources of Error

http://numericalmethods.eng.usf.edu



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Two sources of numerical error

1) Round off error 2) Truncation error


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Round-off Error



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Round off Error

• Caused by representing a numberapproximately

333333.031 ≅

...4142.12 ≅


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Problems created by round off error

• 28 Americans were killed on February 25,1991 by an Iraqi Scud missile in Dhahran,Saudi Arabia.

• The patriot defense system failed to trackand intercept the Scud. Why?


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Problem with Patriot missile

• Clock cycle of 1/10 seconds wasrepresented in 24-bit fixed pointregister created an error of 9.5 x

10-8

seconds.• The battery was on for 100

consecutive hours, thus causingan inaccuracy of

1hr 3600s

100hr 0.1s

s109.5 8 ×××= −

s342.0=


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Problem (cont.)

• The shift calculated in the ranging systemof the missile was 687 meters.

• The target was considered to be out ofrange at a distance greater than 137meters.


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Effect of Carrying SignificantDigits in Calculations



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Find the contraction in thediameter

dT T D Dc

a

T

T

)(∫=∆ α

Ta=80 oF; Tc=-108 oF; D=12.363”

α = a 0+ a 1T + a 2T 2


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Thermal ExpansionCoefficient vs Temperature

T D D ∆=∆ α

T(oF) α (μ in/in/ oF)-340 2.45

-300 3.07

-220 4.08

-160 4.72

-80 5.43

0 6.00

40 6.2480 6.47


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Regressing Data in Excel(general format)

α = -1E-05T 2 + 0.0062T + 6.0234


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Observed and Predicted Values

T(oF) α (μ in/in/ oF)Given

α (μ in/in/ oF)Predicted

-340 2.45 2.76

-300 3.07 3.26-220 4.08 4.18

-160 4.72 4.78

-80 5.43 5.460 6.00 6.02

40 6.24 6.26

80 6.47 6.46

α = -1E-05T 2 + 0.0062T + 6.0234


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Regressing Data in Excel(scientific format)

α = -1.2360E-05T 2 + 6.2714E-03T + 6.0234


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-340 2.45 2.46

-300 3.07 3.03-220 4.08 4.05

-160 4.72 4.70

-80 5.43 5.440 6.00 6.02

40 6.24 6.25

80 6.47 6.45

α = -1.2360E-05T 2 + 6.2714E-03T + 6.0234


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-340 2.45 2.46 2.76

-300 3.07 3.03 3.26

-220 4.08 4.05 4.18

-160 4.72 4.70 4.78

-80 5.43 5.44 5.460 6.00 6.02 6.02

40 6.24 6.25 6.26

80 6.47 6.45 6.46

α = -1.2360E-05T 2 + 6.2714E-03T + 6.0234

α = -1E-05T2

+ 0.0062T + 6.0234


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THE END


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Truncation Error



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Truncation error

• Error caused by truncating orapproximating a mathematicalprocedure.


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Example of Truncation Error

Taking only a few terms of a Maclaurin series toapproximate

....................!3!2

132 ++++= x x xe x

xe

If only 3 terms are used,

++−=

!21

2 x xe Error Truncation x


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Another Example of TruncationError

Using a finite x∆ to approximate )( x f ′

x

x f x x f x f

∆

−∆+≈′ )()()(

P

Q

secant line

tangent line

Figure 1. Approximate derivative using finite Δ x


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Another Example of TruncationError

Using finite rectangles to approximate anintegral.

y = x 2

0

30

60

90

0 1.5 3 4.5 6 7.5 9 10.5 12

y

x


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Example 1 —Maclaurin series

Calculate the value of2.1e with an absolute

relative approximate error of less than 1%.

...................!32.1

!22.1

2.1132

2.1 ++++=e

n

1 1 __ ___

2 2.2 1.2 54.545

3 2.92 0.72 24.658

4 3.208 0.288 8.9776

5 3.2944 0.0864 2.6226

6 3.3151 0.020736 0.62550

a E %a∈2.1e

6 terms are required. How many are required to getat least 1 significant digit correct in your answer?


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Example 2 —Differentiation

Find )3( f ′ for 2)( x x f = using x

x f x x f x f ∆

−∆+≈′ )()()(

and 2.0=∆ x

2.0)3()2.03(

)3(' f f

f −+

=

2.0)3()2.3( f f −=

2.032.3 22 −=

2.0924.10 −=

2.024.1= 2.6=

The actual value is,2)(' x x f = 632)3(' =×= f

Truncation error is then, 2.02.66 −=−

Can you find the truncation error with 1.0=∆ x


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Example 3 — Integration

Use two rectangles of equal width toapproximate the area under the curve for

2

)( x x f =

over the interval ]9,3[

y = x 2

0

30

60

90

0 3 6 9 12

y

x

∫

9

3

2 dx x


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Integration example (cont.)

)69()()36()(6

2

3

29

3

2 −+−===∫ x x x xdx x

3)6(3)3( 22 +=13510827 =+=

Choosing a width of 3, we have

Actual value is given by

∫9

3

2

dx x

9

3

3

3 = x

234339 33

= −

=

Truncation error is then99135234 =−

Can you find the truncation error with 4 rectangles?


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Additional Resources

For all resources on this topic such as digital audiovisuallectures, primers, textbook chapters, multiple-choice tests,worksheets in MATLAB, MATHEMATICA, MathCad andMAPLE, blogs, related physical problems, please visit

http://numericalmethods.eng.usf.edu/topics/sources_of_err or.html

http://numericalmethods.eng.usf.edu/topics/sources_of_error.htmlhttp://numericalmethods.eng.usf.edu/topics/sources_of_error.htmlhttp://numericalmethods.eng.usf.edu/topics/sources_of_error.htmlhttp://numericalmethods.eng.usf.edu/topics/sources_of_error.htmlhttp://numericalmethods.eng.usf.edu/topics/gaussian_elimination.html


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Source of Erros in Numerical Methods