1. MATH 174: NUMERICAL ANALYSIS I
1st Sem AY
Math Division, IMSP, UPLB

2. Numerical Differentiation
You can use interpolation or curve fitting but beware of POLYNOMIAL WIGGLES!!!

3. Numerical Differentiation
REVIEW
Definition of derivative of f at x :
provided that the limit exists.

4. Numerical Differentiation
REVIEW
From Taylor’s Theorem: If f єC2 then
where 𝝃 is between x and x+h.
The following difference formulas can be derived using Taylor’s Theorem.

5. Numerical Differentiation1
Numerical Differentiation
FINITE DIFFERENCE FORMULAS (Discretization) Two-point forward difference formula (first-order method):

6. Numerical Differentiation1
Numerical Differentiation
FINITE DIFFERENCE FORMULAS Two-point forward difference formula (first-order method): The error is proportional to h. As h0, the error0. Notice that if we divide h by 2, then the error is also cut in half!!!

7. Numerical DifferentiationP O L A
Numerical Differentiation
Notice that formulae 1 is just the slope of the interpolating line from the considered points.
Forward Difference
Backward Difference
Centered Difference
(See Formula 2)

8. Numerical Differentiation
Forward Difference

9. Numerical Differentiation
Backward Difference

10. Numerical Differentiation
Centered Difference

11. Numerical Differentiation
Minimize h

12. Numerical Differentiation2
Numerical Differentiation
FINITE DIFFERENCE FORMULAS 3-point centered difference formula (second-order method): If f єC3 where

13. Numerical Differentiation2
Numerical Differentiation
FINITE DIFFERENCE FORMULAS
3-point centered difference formula (second-order method): (reduces to 2-point)
where
Notice that h0 faster than the first-order method.

14. Numerical DifferentiationS
Numerical Differentiation
FINITE DIFFERENCE FORMULAS Other Formulas: etc… etc… etc…

15. Numerical Differentiation
2nd D E R I V A T e
Numerical Differentiation
FINITE DIFFERENCE FORMULAS 3-point centered difference formula (second-order method): where

16. Numerical Differentiation
3rd D E R I V A T e
Numerical Differentiation
FINITE DIFFERENCE FORMULAS 3-point centered difference formula (second-order method):

17. Numerical Differentiation
4th D E R I V A T e
Numerical Differentiation
FINITE DIFFERENCE FORMULAS 3-point centered difference formula (second-order method):

18. Numerical DifferentiationX R C I S
Numerical Differentiation
Solve for f’(5) and f”(5), where f(x)=sin(x)
Solve for the (instantaneous) rate of change at x=3, given the following data: x y 1 23 2 34 3 21 4 25

19. Numerical Differentiation
Problem arises:
As h decreases, round-off error becomes a problem.
Finite difference formulas for numerical differentiation is ill-conditioned. We need to intelligently choose a value for h.

20. Numerical Differentiation
RICHARDSON EXTRAPOLATION Creating higher-order approximation from existing formula F(h)
Here, f’(x)=Q. The new formula will have order from O(hn) to at least O(hn+1).

21. Numerical Differentiation
RICHARDSON EXTRAPOLATION Example: Increase the order of the 3-point Centered Difference Formula: Hence,
n=2

22. Numerical Differentiation
RICHARDSON EXTRAPOLATION By substitution,

23. Numerical Differentiation
RICHARDSON EXTRAPOLATION Therefore, the new formula is The order of this formula is at least 3 (actually, it is of order 4). What do you think is the problem here?