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

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

4. 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

5. 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

6. increasing accuracy increasing precision Accuracy and Precision

7. 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

8. Error definitions cont. True relative percent error

9. 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?

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

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

12. Example Consider a series expansion to estimate trigonometric functions Estimate sin(  / 2) to three significant figures. Calculate  t and  a. STRATEGY

13. Strategy Stop when  a ≤  s

14. 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

15. 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

16. 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

17. TAYLOR SERIES Zero order approximation This is good if the function is a constant.

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

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

20. Taylor Series Expansion

22. Example Use zero through fourth order Taylor series expansion to approximate f(1) given f(0) = 1.2 (i.e. h = 1). Calculate  t after each step. Note: f(1) = 0.2 STRATEGY

23. 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,  t Estimate the function using the first and second term Calculate the error,  t Progressively add terms

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