1. NUMERICAL ERROR ENGR 351 Numerical Methods for Engineers
Southern Illinois University Carbondale
College of Engineering
Dr. L.R. Chevalier

2. Copyright© 1999 by Lizette R. Chevalier
Permission is granted to students at Southern Illinois University at Carbondale
to make one copy of this material for use in the class ENGR 351, Numerical
Methods for Engineers. No other permission is granted.
All other rights are reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise, without
the prior written permission of the copyright owner.

3. SIMPLE STATISTICS
Arithmetic mean
Standard deviation

4. Simple Statistics cont.
Variance, sy2
Coefficient of variation

5. Pseudo Code Review for First Programming Assignment
Raw Data
Code
Output Data

6. ASCII Acronym for American Standard Code for Information Interchanged
Pronounced ask-key
Also referred to as a text file
Generate in C++, Fortran, Basic, etc. editors
Generate in Notepad
Can generate in Word, but need to save as a text file
Executable programs are never stored in ASCII format

7. Pseudo Code Review for First Programming Assignment
Raw Data
Code
Output Data
These are in ASCII format, unless you are using an executable version of your code. In that case, only the data (input and output) are ASCII format.

8. Pseudo Code for Average10
77.8
39.2
56.7
98.2
88.7
86.2
76.3
82.5
93.6
78.2
Data file
input.dat
Read an ASCII file that
1. Inputs number of students n
2. Inputs scores y(n)

9. Pseudo Code for Average
DIMENSION Y
OPEN INPUT.DAT
READ N
DO I=1,N
READ Y(N)
CONTINUE
DO J=1,N
SUM=SUM+Y(J)
AVERAGE = SUM/N
PRINT “AV”

10. PROBLEM
Modify the pseudo-code to calculate the variance.

11. Approximation and Errors Significant Figures
1.845
43,500 ? confidence
4.35 x significant figures
4.350 x significant figures
x significant figures

12. 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

13. increasing accuracy
increasing precision

14. 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

15. Error definitions cont.
True relative percent error
But we may not know the true answer apriori

16. Error definitions cont.
May not know the true answer apriori
This leads us to develop an iterative approach of numerical methods

17. Error definitions cont.
Usually not concerned with sign, but with tolerance
Want to assure a result is correct to n significant figures

18. Example Consider a series expansion to estimate trigonometric
functions
Estimate sin p/ 2 to three significant figures

19. 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
To gain insight consider the mathematical
formulation that is used widely in numerical
methods - TAYLOR SERIES

20. 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
Zero order approximation
This is good if the function is a constant.

21. Taylor Series Expansion
First order approximation
slope multiplied by distance
Still a straight line but capable of predicting an increase or decrease - LINEAR

22. Taylor Series Expansion
Second order approximation - captures some of the curvature

23. Taylor Series Expansion

24. Taylor Series Expansion

25. Example
Use zero through fourth order Taylor series expansion to approximate f(1) given f(0) = 1.2 (i.e. h = 1)
Note:
f(1) = 0.2

26. Functions with infinite number of derivatives
f(x) = cos x
f '(x) = -sin x
f "x) = -cos x
f "'(x) = sin x
Evaluate the system where xi = p/4 and xi+1 = p /3
h = p /3 - p /4 = p /12

27. Functions with infinite number of derivatives
Zero order
f(p /3) = cos (p/4 ) = et = 41.4%
First order
f(p /3) = cos (p/4 ) - sin (p/4 )(p/12) et = 4.4%
Second order
f(p /3) = et = 0.45%
By n = 6 et = 2.4 x 10-6 %

28. Exam Question
How many significant figures are in the following numbers? A B
C x 10-3
D. 23,000,000
E. 2.3 x 107

29. TAYLOR SERIES PROBLEM
Use zero- through fourth-order Taylor series expansions to
predict f(4) for f(x) = ln x using a base point at x = 2.
Compute the percent relative error et for each approximation.