1 Numerical Differentiation Jyun-Ming Chen

2 Contents Forward, Backward, Central Difference Richardson Extrapolation

3 Taylor’s Expansion

4 Forward Difference Formula for Geometrically h

5 Backward Difference Formula for Similarly h Geometrically

6 Central Difference Formula for –)–)

7 2h Geometrically

8 Example, Calculate f’(1) using FD, BD, CD

9 Example (cont) h=0.1 h=0.05 FD: BD: CD:

10 Example (cont) Remarks: –FD, BD, CD each involves 2 function calls, 1 subtraction, and 1 division: same computation time –CD is the most accurate (hence, the most recommended method) –However, sometimes, CD cannot be applied

11 Forward Difference Formula for –2  ）

12 Backward Difference Formula for –2  ）

13 Central Difference Formula for +)+) Similar remark on the selection of FD|BD|CD applies for f”(x)

14 More Accurate FD Formula for

15 Better accuracy can be achieved using this formula But, it involves more computations: –3 function calls, two +/–, one division Trade-off: –More computation is the price you paid for better accuracy Similar idea applies to more accurate BD formula More Accurate FD Formula (cont)

16 Richardson Extrapolation Idea: exact = computed+ error The truncation error is of the form: ch k –where c is some constant Use different h to estimate the truncation error Use extrapolation to get more accurate result

17 Example: CD for f’(x) Using Different h (h 1, h 2 ) ： c 1 and c 2 could be different Richardson Extrapolation (cont)

18 If Richardson Extrapolation (cont)

19 f(x) = x 3. Use CD with Richardson extrapolation to compute f’(1) Example Magic? Coincidence?

20 Revisit CD Formula for –)–) Change notation:

21 Error Analysis Assuming Eliminate a 1 to get better accuracy

22 Error Analysis (cont) 44 –) O(h4)O(h4)

23 Revisit Previous Example f(x) = x 3. Use CD with Richardson extrapolation to compute f’(1) a 2 involves f (5) (x), hence, the exact solution is no surprise.

24 Remark How much effort did we use to get this level of accuracy? –F(h): f(x+h), f(x-h); one –, one  –F(h/2): f(x+h/2), f(x-h/2); one –, one  –R.E.: two , one –

25 Summary (Textbook p.373)

26

27 אקסטרפולציית ריצ'רדסון לביטול איבר השגיאה המוביל