1. 1 Approximations and Round-Off Errors Lecture Notes Dr. Rakhmad Arief Siregar Universiti Malaysia Perlis Applied Numerical Method for Engineers Chapter 3

2. 2 Background The numerical technique yielded estimates that were close to the exact analytical solution, as discussed in Chapter 1. There was a discrepancy, or error because numerical method involved an approximation. In many engineering applications, we cannot obtain analytical solutions, therefore we need to settle for the approximations or estimates of errors.

3. 3 Significant Figures Velocity = ? 49 or 48 km/h?

4. 4 Significant Figures or 48.9 km/h? or 48.864 km/h?

5. 5 Significant Figures The concept of significant figure, or digit, has been developed to formally designate the reliability of a numerical value. The significant digits of a number are those that can use with confidence

6. 6 Accuracy and Precision The errors associated with both calculations and measurements can be characterized with regard to their accuracy and precision. Accuracy refers to how closely a computed or measured value agrees with true value. Precision refers to how closely individual computed or measured value agree with each other.

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8. 8 Error definitions Numerical errors arise from the use of approximations to represent exact mathematical operations and quantities. True error: The true percent relative error:

9. 9 Ex. 3.1 Calculation of errors Suppose that you have the task of measuring the lengths of a bridge and a rivet and come up with 9999 and 9 cm, respectively. If the true values are 10,000 and 10 cm, respectively, compute (a) the true error and (b) the percent relative error for each case. Solution (a) The true error for bridge: The true error for rivet: Solution (b) The percent relative error for bridge: The true error for rivet:

10. 10 Real-world applications We will obviously NOT know the true value. For these situations, an alternative is to normalize the error using the best available estimate the true value. The approximate percent relative error: In the case of Iterative Methods The number of significant figures

11. 11 Ex. 3.2 Error estimates for iterative methods In mathematics, functions can often be represented by infinite series. For example, the exponential function can be computed using Thus, more terms are added in sequence, he approximation become a better and better estimate of the true value of e x = 1, add terms one at a time to estimate e 0.5 Compute approximate error estimate  a so that it falls below a pre-specified error criterion  s, up to three significant figures.

12. 12 Ex. 3.2 Error estimates for iterative methods Solution: Up to three significant figures: A true percent relative error (Eq. 3.3) An approximate estimate of error

13. 13 Results

14. 14 Round-Off Error Round-off errors originate from the fact that computers retain only a fixed number of significant figures during calculation. Number such as , e, or  7 cannot be expressed by a fixed number of significant figures The discrepancy introduced by this omission of significant figures is called round-off error