1. The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter 23 Numerical Differentiation

2. 1.Forward difference Taylor series :

3. Numerical Differentiation 2. Backward difference

4. Numerical Differentiation 3. Centered difference

5. High Accuracy Differentiation Formulas High-accuracy finite-difference formulas can be generated by including additional terms from the Taylor series expansion. An example: High-accuracy forward-difference formula for the first derivative.

6. Derivation: High-accuracy forward- difference formula for f`(x) (1) (2) Equ. 1 can be multiplied by 2 and subtracted from equ. 2: Solve: Second derivative forward finite divided difference

7. Derivation: High-accuracy forward-difference formula for f`(x) Taylor series expansion Solve for f’(x) High-accuracy forward-difference formula Substitute the forward- difference approx. of f”(x)

8. Derivation: High-accuracy forward-difference formula for f`(x) Similar improved versions can be developed for the backward and centered formulas as well as for the approximations of the higher derivatives.

9. Higher Order Forward Divided Difference

10. Higher Order Backward Divided Difference

11. Higher Order Central Divided Difference

12. First Derivatives - Example: Use forward,backward and centered difference approximations to estimate the first derivative of at x = 0.5 with h = 0.5 and 0.25 (exact sol. = -0.9125) Forward Difference Backward Difference

13. First Derivatives - Example: Central Difference

14. Forward Finite-divided differences

15. Backward finite-divided differences

16. Centered Finite-Divided Differences

17. First Derivatives - Example: Employing the high-accuracy formulas (h=0.25): x i-2 = 0.0 f(0.0) = 1.2 x i-1 = 0.25 f(0.0) = 1.103516 x i = 0.5 f(0.5) = 0.925 x i+1 = 0.75 f(0.75) = 0.63633 x i+2 = 1.0 f(1.0) = 0.2 Forward Difference

18. First Derivatives - Example: Backward Difference Central Difference

19. Summary h = 0.25Forward O(h 2 ) Backward O(h 2 ) Centered O(h 4 ) Estimate-0.859375-0.878125-0.9125 |t||t| 5.82%3.77%0% Basic formulas h = 0.25Forward O(h) Backward O(h) Centered O(h 2 ) Estimate-1.155-0.714-0.934 |t||t| 26.5%21.7%2.4% High- Accuracy formulas True value: f`(0.5) = -0.9125