Propagation of Errors
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Transcript of Propagation of Errors
04/19/23http://
numericalmethods.eng.usf.edu 1
Propagation of Errors
Major: All Engineering Majors
Authors: Autar Kaw, Matthew Emmons
http://numericalmethods.eng.usf.edu
Transforming Numerical Methods Education for STEM Undergraduates
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Propagation of Errors
In numerical methods, the calculations are not made with exact numbers. How do these inaccuracies propagate through the calculations?
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Example 1:Find the bounds for the propagation in adding two numbers. For example if one is calculating X +Y where
X = 1.5 ± 0.05Y = 3.4 ± 0.04
SolutionMaximum possible value of X = 1.55 and Y = 3.44
Maximum possible value of X + Y = 1.55 + 3.44 = 4.99
Minimum possible value of X = 1.45 and Y = 3.36.
Minimum possible value of X + Y = 1.45 + 3.36 = 4.81
Hence 4.81 ≤ X + Y ≤4.99.
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Propagation of Errors In Formulas
f nn XXXXX ,,.......,,, 1321
f
nn
nn
XX
fX
X
fX
X
fX
X
ff
11
22
11
.......
If is a function of several variables then the maximum possible value of the error in is
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Example 2:The strain in an axial member of a square cross-section is given by
Given
Find the maximum possible error in the measured strain.
Eh
F2
N9.072 Fmm1.04 hGPa5.170 E
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Example 2:
)1070()104(
72923
610286.64 286.64
EE
hh
FF
Solution
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Example 2:
EhF 2
1
Eh
F
h 3
2
22Eh
F
E
EEh
Fh
Eh
FF
EhE
2232
21
92923
933923
105.1)1070()104(
72
0001.0)1070()104(
7229.0
)1070()104(
1
3955.5
Thus
Hence)3955.5286.64(
Example 3:Subtraction of numbers that are nearly equal can create unwanted inaccuracies. Using the formula for error propagation, show that this is true.
SolutionLet
Then
So the relative change is
yxz
yy
zx
x
zz
yx )1()1(
yx
yx
yx
z
z
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Example 3:For example if
001.02 x001.0003.2 y
|003.22|
001.0001.0
z
z
= 0.6667
= 66.67%
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Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/propagation_of_errors.html