propagation of errors

04/19/23http://

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Propagation of Errors

Major: All Engineering Majors

Authors: Autar Kaw, Matthew Emmons

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Numerical Methods 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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