Why do we study numerical analysis? What are errors? #WikiCourses https://wikicourses.wikispaces.com/Num001+Numerical+Methods

1. Numerical Analysis - Introduction
Mohammad Tawfik
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Numerical Analysis
An Introduction

2. Numerical Analysis - Introduction
Mohammad Tawfik
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References
• Steven Chapra and Raymond Canale,
"Numerical Methods for Engineers," 5th
editions, ISBN 0-07-123140-4

3. Numerical Analysis - Introduction
Mohammad Tawfik
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Numerical Methods

4. Numerical Analysis - Introduction
Mohammad Tawfik
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Why Numerical?
• Ignorance
• Readily Available Packages
• Need to Develop New Techniques
• Good use of your computer!
• Re-Understand Mathematics

5. Numerical Analysis - Introduction
Mohammad Tawfik
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Mathematical Modeling &
Engineering Solutions

6. Numerical Analysis - Introduction
Mohammad Tawfik
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Mathematical Models

7. Numerical Analysis - Introduction
Mohammad Tawfik
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Example Problem
dt
dv
mmaF 
cvmgFFF UD 
cvmgvm
dt
dv
m  
m
cvmg
v

    mct
e
c
mg
tv /
1 


8. Numerical Analysis - Introduction
Mohammad Tawfik
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Solution

9. Numerical Analysis - Introduction
Mohammad Tawfik
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Approximate Solution
12
12
tt
vv
t
v
dt
dv






m
cvmg
tt
vv 



12
12
m
cvmg
tt
vv 1
12
12 




10. Numerical Analysis - Introduction
Mohammad Tawfik
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Approximate Solution
 
m
cvmg
ttvv 1
1212



11. Numerical Analysis - Introduction
Mohammad Tawfik
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Approximation and Round-Off
Errors

12. Numerical Analysis - Introduction
Mohammad Tawfik
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Errors!
• Round-off errors are due to the fact that
the computers present numbers in as a
finite number of bits and bytes!
• Truncation Errors are errors that emerge
from the approximation of the
mathematical model
• Model errors are due to the fact that the
mathematical model usually is an
approximation of the physical reality!

13. Numerical Analysis - Introduction
Mohammad Tawfik
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Significant Figures

14. Numerical Analysis - Introduction
Mohammad Tawfik
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Accuracy and Precision
• Accuracy how close
your solution is
• Precision is how close
your repetition of the
solution are!

15. Numerical Analysis - Introduction
Mohammad Tawfik
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Error Definition
• True Error
• True Relative Error
• Approximate Error
ionApproximatvalueTrueEt 
valueTrue
Et
t 
ionApproximatCurrent
ApproxPastionApproximatCurrent
a
.


16. Numerical Analysis - Introduction
Mohammad Tawfik
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Round-off Errors
210
10*210*510*3253 
21
10*510*225.0
4
1 

...10*310*310*310*3
...33333.0
3
1
4321




17. Numerical Analysis - Introduction
Mohammad Tawfik
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Computer Presentation of Numbers

18. Numerical Analysis - Introduction
Mohammad Tawfik
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When numbers are too small!

19. Numerical Analysis - Introduction
Mohammad Tawfik
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Truncation Errors

20. Numerical Analysis - Introduction
Mohammad Tawfik
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Taylor’s Expansion
          ...
!2
'''
2
0
0000 


xx
xfxxxfxfxf
         3
2
000
!2
''' hO
h
xfhxfxfxf 

21. Numerical Analysis - Introduction
Mohammad Tawfik
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Truncation Error!

22. Numerical Analysis - Introduction
Mohammad Tawfik
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Conclusion
• We need the numerical methods to solve
problems that do not have a solution or
have a complex mathematical solution
• Errors have different sources: Truncation,
round off, and model errors
• Error is evaluated using approximate
solutions