Post on 02-Jan-2016
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NUMERICAL ERRORStudent Notes
ENGR 351 Numerical Methods for EngineersSouthern Illinois University CarbondaleCollege of EngineeringDr. L.R. Chevalier
Objectives
• To understand error terms• Become familiar with notation and
techniques used in this course
Approximation and ErrorsSignificant Figures• 4 significant figures
• 1.845• 0.01845• 0.0001845
• 43,500 ? confidence• 4.35 x 104 3 significant figures• 4.350 x 104 4 significant figures• 4.3500 x 104 5 significant figures
Accuracy and Precision
• Accuracy - how closely a computed or measured value agrees with the true value
• Precision - how closely individual computed or measured values agree with each other• number of significant figures• spread in repeated measurements or
computations
increasing accuracy
incr
easi
ng p
reci
sion
Accuracy and Precision
Error Definitions
• Numerical error - use of approximations to represent exact mathematical operations and quantities
• true value = approximation + error• error, et=true value - approximation• subscript t represents the true error• shortcoming....gives no sense of magnitude• normalize by true value to get true relative
error
Error definitions cont.
valuetrue
valueestimatedvaluetrue
valuetrue
errortruet
100
• True relative percent error
Example
Consider a problem where the true answer is 7.91712. If you report the value as 7.92, answer the following questions.
1. How many significant figures did you use?2. What is the true error?3. What is the true relative percent error?
Error definitions cont.• May not know the true answer
apriori• This leads us to develop an iterative approach to numerical methods
100.
..
100
approxpresentapproxpreviousapproxpresent
ionapproximaterroreapproximat
a
Error definitions cont.
• Usually not concerned with sign, but with tolerance
• Want to assure a result is correct to n significant figures
%105.0 2 ns
sa
Example
Consider a series expansion to estimate trigonometric functions
xxxx
xx .....!7!5!3
sin753
Estimate sin(p/ 2) to three significant figures. Calculate et and ea. STRATEGY
Strategy
Terms Results t % a %
12345
Stop when ea ≤ es
Error Definitions cont.
• Round off error - originate from the fact that computers retain only a fixed number of significant figures
• Truncation errors - errors that result from using an approximation in place of an exact mathematical procedure
Error Definitions cont.
• Round off error - originate from the fact that computers retain only a fixed number of significant figures
• Truncation errors - errors that result from using an approximation in place of an exact mathematical procedureTo gain insight consider the mathematical
formulation that is used widely in numerical methods - TAYLOR SERIES
TAYLOR SERIES
• Provides a means to predict a function value at one point in terms of the function value at and its derivative at another point
TAYLOR SERIES
Zero order approximation
ii xfxf 1
This is good if the function is a constant.
Taylor Series Expansion
First order approximation
slope multiplied by distance
Still a straight line but capable of predicting an increase or decrease - LINEAR
iiiii xxxfxfxf 11 '
Taylor Series Expansion
Second order approximation - captures some of the curvature
2111 !2
''' ii
iiiiii xx
xfxxxfxfxf
Taylor Series Expansion
ii
nni
n
iiiii
xxsizestephwhere
Rhn
xf
hxf
hxf
hxfxfxf
1
......
321
!
!3
'''
!2
'''
Taylor Series Expansion
1
11
1
......
321
!1
!
!3'''
!2''
'
iin
n
n
ii
nni
n
iiiii
xxhn
fR
xxsizestephwhere
Rhn
xf
hxf
hxf
hxfxfxf
Example
Use zero through fourth order Taylor series expansion to approximate f(1) given f(0) = 1.2 (i.e. h = 1). Calculate et after each step.
f x 01 015 0 5 0 25 1 24 3 2. . . . .x x x x
Note:f(1) = 0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.2 0.4 0.6 0.8 1
x
f(x)
STRATEGY
Strategy• Estimate the function using only the first term
• Use x = 0 to estimate f(1), which is the y-value when x = 1
• Calculate error, et
• Estimate the function using the first and second term
• Calculate the error, et
• Progressively add terms
Objectives
• To understand error terms• Become familiar with notation and
techniques used in this course