1. Numerical differentiation
Recall finite differences from first week
Derived from Taylor series

2. Neglecting all tersms higher than first order
That’s the forward difference - also backwards and centered difference

3. Why is centered finite difference O(h2)?
Subtract second equation from first

4. We can combine Taylor series expansions in many different ways to get estimates of derivatives
Example: backwards second derivative, O(h2)
Start with

5. Multiply first equation by -5, second equation by 4 and add together+

6. Multiply third equation by -1 and add to above result+
Rearrange

7. Where did I get -5, 4,-1?
We multiply 1st equation by a, second by b, third by c

8. Now sum all equations and collect terms
Decide what derivatives we want to make disappear - want a second derivative only - eliminate first and third

9. Three unknowns - 2 equations - make an assumption
Let c=-1
Can solve by hand

10. If we have more derivatives to get rid of, use matrix methods - always one assumption

11. More Richardson extrapolation
Recall
Can do the same thing with derivatives

12. Use same approach as Romberg integration with halving the step size
Example: Formula for active lateral pressure coefficient Ka with internal angle of friction f and wall with slope b and flat top is
Use Richardson/Romberg approach to estimate
at b=10 degrees and f=15 degrees

13. Use O(h2) estimates to get O(h6) estimate

14. Now do Richardson/Romberg trick

15. Derivatives of unequally spaced data
Can use matrix approach with different amounts of h
Example: given values of f at x=(1,2,5.5,9) determine f’’ at 2

16. Let h=1, x=2 (values at 1,2,5.5,9)
Equations to get rid of f’ and f’’’ are
and assume a value for c

17. Let c=1, then a= , b=
then or

18. Derivatives of unequally spaced data
Another way is to take derivative of interpolating polynomial
Lagrange polynomial - second order in this case

19. Derivatives and integrals with errors in data
Errors in data points can cause problems
esp. with differentiation
Example: with and without noise
True derivative is 2x-6

20. Look at ratio of noise in y to noise in dy/dx
For differentiation, fit a smooth line to the data, then take derivative