Approximations and Round-Off Errors

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

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

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

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. 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.

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. 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. 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. 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 ex = 1, add terms one at a time to estimate e0.5 • Compute approximate error estimate a so that it falls below a pre-specified error criterion s, up to three significant figures.

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. Results

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