1. NUMERICAL DIFFERENTIATION Forward Difference Formula
The derivative of f (x) at x0 is:
An approximation to this is:
for small values of h.
Forward Difference Formula

2. Find an approximate value for
0.1
0.01
0.001
The exact value of

3. Lagrange Interpolating Polynomial
Assume that a function goes through three points:
Lagrange Interpolating Polynomial

6. If the points are equally spaced, i.e.,

7. Three-point formula:

8. If the points are equally spaced with x0 in the middle:

9. Another Three-point formula:

10. Alternate approach (Error estimate)
Take Taylor series expansion of f(x+h) about x:
(1)

11. Forward Difference Formula

12. (2)

13. (1)
(2)
2 X Eqn. (1) – Eqn. (2)

15. Three-point Formula

16. The Second Three-point Formula
Take Taylor series expansion of f(x+h) about x:
Take Taylor series expansion of f(x-h) about x:
Subtract one expression from another

18. Second Three-point Formula

19. Summary of Errors
Forward Difference Formula
Error term

20. Summary of Errors continued
First Three-point Formula
Error term

21. Summary of Errors continued
Second Three-point Formula
Error term

22. Example:
Find the approximate value of
with
1.9
2.0
2.1
2.2

23. Using the Forward Difference formula:

24. Using the 1st Three-point formula:

25. Using the 2nd Three-point formula:
The exact value of

26. Comparison of the results with h = 0.1
The exact value of is
Formula
Error
Forward Difference
1st Three-point
2nd Three-point

27. Second-order Derivative
Add these two equations.

29. NUMERICAL INTEGRATION
area under the curve f(x) between
In many cases a mathematical expression for f(x) is unknown and in some cases even if f(x) is known its complex form makes it difficult to perform the integration.

32. Area of the trapezoid
The length of the two parallel sides of the trapezoid are: f(a) and f(b)
The height is b-a

33. Simpson’s Rule:

40. Composite Numerical Integration

41. Riemann Sum
The area under the curve is subdivided into n subintervals. Each subinterval is treated as a rectangle. The area of all subintervals are added to determine the area under the curve.
There are several variations of Riemann sum as applied to composite integration.

42. In Left Riemann sum, the left-side sample of the function is used as the height of the individual rectangle.

43. In Right Riemann sum, the right-side sample of the function is used as the height of the individual rectangle.

44. In the Midpoint Rule, the sample at the middle of the subinterval is used as the height of the individual rectangle.

45. Composite Trapezoidal Rule:
Divide the interval into n subintervals and apply Trapezoidal Rule in each subinterval.
where

46. Find
by dividing the interval into 20 subintervals.

48. Composite Simpson’s Rule:
Divide the interval into n subintervals and apply Simpson’s Rule on each consecutive pair of subinterval. Note that n must be even.

49. where
Find
by dividing the interval into 20 subintervals.