Successfully reported this slideshow.
Upcoming SlideShare
×

# Applied numerical methods lec11

2,219 views

Published on

Numerical Differentiation

Published in: Engineering
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

### Applied numerical methods lec11

1. 1. 1 Numerical Differentiation
2. 2. 2 Taylor’s Theorem h=(xi+1- xi)
3. 3. High-Accuracy Differentiation Formulas          2ii1i i hOh 2 xf - h xfxf xf         ixx i dx xdf xf   one more sample xi+2 = xi+1 + h        hO h xfxf xf i1i i              hO h xfx2fxf xf 2 i1i2i i                  2 2 i1i2ii1i i hOh 2h xfxfxf - h xfxf xf     2 one extra sample xi+1 = xi + h first-order accurate second-order accurate Big O notation describes the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.
4. 4. Forward difference xi1 xi xi+1 x h
5. 5. Backward difference xi1 xi xi+1 x h
6. 6. Centered difference xi1 xi xi+1 x 2h
7. 7. First Derivatives • Forward difference • Backward difference • Central difference )x(f  i-2 i-1 i i+1 i+2 1i1i 1i1i 1i1i 1i1i 1ii 1ii 1ii 1ii i1i i1i i1i i1i xx yy xx )x(f)x(f )x(f xx yy xx )x(f)x(f )x(f xx yy xx )x(f)x(f )x(f                               x y
8. 8. 8 Forward Finite-Divided Difference Formulas second-order accurate forward difference formula for f’(x)
9. 9. 9 Backward Finite-Divided Difference Formulas
10. 10. 10 Centered Finite-Divided Difference Formulas
11. 11. 11 Example        hO h xfxf xf i1i i    
12. 12. 12
13. 13. 13 Example of High-Accuracy Differentiation
14. 14. 14 Miscellaneous problems #1
15. 15. 15
16. 16. 16 Miscellaneous problems #2
17. 17. 17
18. 18. 18 Miscellaneous problems #3
19. 19. 19
20. 20. 20
21. 21. Homework 1- 2- 3-
22. 22. 22 4- 5-