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

2. Copyright © 2003 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.

4. Objectives To understand error terms Become familiar with notation and techniques used in this course

5. Approximation and Errors Significant Figures 4 significant figures 1.845 0.01845 0.0001845 43,500 ? confidence 4.35 x 10 4 3 significant figures 4.350 x 10 4 4 significant figures 4.3500 x 10 4 5 significant figures

6. 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

7. increasing accuracy increasing precision Accuracy and Precision

8. Error Definitions Numerical error - use of approximations to represent exact mathematical operations and quantities true value = approximation + error error,  t =true value - approximation subscript t represents the true error shortcoming....gives no sense of magnitude normalize by true value to get true relative error

9. Error definitions cont. True relative percent error

10. 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 relative error?

11. Error definitions cont. May not know the true answer apriori

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

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

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

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

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

17. 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

18. 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

19. TAYLOR SERIES Provides a means to predict a function value at one point in terms of the function value at and its derivatives at another point Zero order approximation

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

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

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

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

25. Taylor Series Expansion

27. 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

28. Solution n=0 f(1) = 1.2   = abs [(0.2 - 1.2)/0.2] x 100 = 500% n=1 f '(x) = -0.4x 3 - 0.45x 2 -x -0.25 f '(0) = -0.25 f(1) = 1.2 - 0.25h = 0.95  t =375%

29. n=2 f "=-1.2 x 2 - 0.9x -1 f "(0) = -1 f(1) = 0.45  t = 125% n=3 f "'=-2.4x - 0.9 f "'(0)=-0.9 f(1) = 0.3  t =50% Solution

30. n=4 f ""(0) = -2.4 f(1) = 0.2 EXACT Why does the fourth term give us an exact solution? The 5th derivative is zero In general, nth order polynomial, we get an exact solution with an nth order Taylor series Solution

32. Exam Question How many significant figures are in the following numbers? A. 3.215 B. 0.00083 C. 2.41 x 10 -3 D. 23,000,000 E. 2.3 x 10 7

33. 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  t for each approximation.

34. 1. Determine the step size h = 4 - 2 = 2 2. Determine the analytical solution f(4) = ln(4) = 1.3863 3. Determine the derivatives for f(2)