1 / 19

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter 23. Numerical Differentiation. Numerical Differentiation. Forward difference. Taylor series :. Numerical Differentiation. 2. Backward difference.

celeste-aguirre
Download Presentation

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter23 Numerical Differentiation

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

  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): xi-2= 0.0 f(0.0) = 1.2 xi-1= 0.25 f(0.0) = 1.103516 xi = 0.5 f(0.5) = 0.925 xi+1 = 0.75 f(0.75) = 0.63633 xi+2 = 1.0 f(1.0) = 0.2 Forward Difference

  18. FirstDerivatives - Example: • Backward Difference • Central Difference

  19. Summary True value: f`(0.5) = -0.9125 Basic formulas High-Accuracy formulas

More Related