1. EE3561_Unit 6(c)AL-DHAIFALLAH14351 EE 3561 : Computational Methods Unit 6 Numerical Differentiation Dr. Mujahed AlDhaifallah ( Term 342)

2. EE3561_Unit 6(c)AL-DHAIFALLAH14352 Lecture 17 Numerical Differentiation  First order derivatives  High order derivatives  Richardson Extrapolation  Examples

3. EE3561_Unit 6(c)AL-DHAIFALLAH14353 Motivation How do you evaluate the derivative of a tabulated function. How do we determine the velocity and acceleration from tabulated measurements. Time (second) Displacemen t (meters) 030.1 548.2 1050.0 1540.2 Calculus is the mathematics of change. Because engineers must continuously deal with systems and processes that change, they always need to estimate the value of f '(x) for a given function f(x).. Standing in the heart of calculus are the mathematical concepts of differentiation and integration:

4. EE3561_Unit 6(c)AL-DHAIFALLAH14354 Recall Numerical differentiation and integration The derivative represents the rate of change of a dependent variable with respect to an independent variable. The difference approximation If x is allowed to approach zero, the difference becomes a derivative

5. EE3561_Unit 6(c)AL-DHAIFALLAH14355 Difference Formulas

6. EE3561_Unit 6(c)AL-DHAIFALLAH14356 Recall

7. EE3561_Unit 6(c)AL-DHAIFALLAH14357 Three formula

8. EE3561_Unit 6(c)AL-DHAIFALLAH14358 Forward Difference formula

9. EE3561_Unit 6(c)AL-DHAIFALLAH14359 Central Difference formula

10. EE3561_Unit 6(c)AL-DHAIFALLAH143510 The Three formula (revisited)

11. EE3561_Unit 6(c)AL-DHAIFALLAH143511 Higher Order Formulas

12. EE3561_Unit 6(c)AL-DHAIFALLAH143512 Other Higher Order Formulas

13. EE3561_Unit 6(c)AL-DHAIFALLAH143513 HIGH-ACCURACY DIFFERENTIATION FORMULAS The forward Taylor series expansion is: From this, we can write Substitute the second derivative approximation into the formula to yield: By collecting terms

14. EE3561_Unit 6(c)AL-DHAIFALLAH143514 HIGH-ACCURACY DIFFERENTIATION FORMULAS  Inclusion of the 2nd derivative term has improved the accuracy to O(h 2 ).  This is the forward divided difference formula for the first derivative.

15. EE3561_Unit 6(c)AL-DHAIFALLAH143515 Forward Formulas

16. EE3561_Unit 6(c)AL-DHAIFALLAH143516 Backward Formulas

17. EE3561_Unit 6(c)AL-DHAIFALLAH143517 Centered Formulas

18. EE3561_Unit 6(c)AL-DHAIFALLAH143518 Example Estimate f '(1) for f(x) = e x + x using the centered formula of O(h 4 ) with h = 0.25. Solution From Table 23.3:

19. EE3561_Unit 6(c)AL-DHAIFALLAH143519 substituting the values results in

20. EE3561_Unit 6(c)AL-DHAIFALLAH143520 Numerical Differentiation  Richardson Extrapolation  Examples

21. EE3561_Unit 6(c)AL-DHAIFALLAH143521 Richardson Extrapolation

22. EE3561_Unit 6(c)AL-DHAIFALLAH143522 Richardson Extrapolation

23. EE3561_Unit 6(c)AL-DHAIFALLAH143523 Richardson Extrapolation Table D(0,0)=Φ(h) D(1,0)=Φ(h/2)D(1,1) D(2,0)=Φ(h/4)D(2,1)D(2,2) D(3,0)=Φ(h/8)D(3,1)D(3,2)D(3,3)

24. EE3561_Unit 6(c)AL-DHAIFALLAH143524 Richardson Extrapolation Table

25. EE3561_Unit 6(c)AL-DHAIFALLAH143525 Example

26. EE3561_Unit 6(c)AL-DHAIFALLAH143526 Example First Column

27. EE3561_Unit 6(c)AL-DHAIFALLAH143527 Example Richardson Table

28. EE3561_Unit 6(c)AL-DHAIFALLAH143528 Example Richardson Table 1.0848 1.08991.09157 1.09111.09157 This is the best estimate of the derivative of the function All entries of the Richardson table are estimates of the derivative of the function. The first column are estimates using the central difference formula with different h.

29. EE3561_Unit 6(c)AL-DHAIFALLAH143529 Summary  Several formulas are available to determine first order, second order or higher order derivatives  Richardson Extrapolation provides high accuracy estimates of the first order derivative