1. Numerical Analysis
Lecture 27

2. Chapter 7 Numerical Differentiation and Integration

3. INTRODUCTION
DIFFERENTIATION USING DIFFERENCE OPERATORS
DIFFERENTIATION USING INTERPOLATION
RICHARDSON’S EXTRAPOLATION METHOD
NUMERICAL INTEGRATION

4. NEWTON-COTES INTEGRATION FORMULAE
THE TRAPEZOIDAL RULE (COMPOSITE FORM )
SIMPSON’S RULES (COMPOSITE FORM )
ROMBERG’S INTEGRATION
DOUBLE INTEGRATION

5. DIFFERENTIATION USING DIFFERENCE OPERATORS

6. Introduction Consider a function of single variable y = f (x). If the
function is known and simple,
we can easily obtain its
derivative (s) or can evaluate
its definite integral

7. However, if we do not know the function as such or the function is complicated and is given in a tabular form at a set of points x0,x1,…,xn, we use only numerical methods for differentiation or integration of the given function.

8. We shall discuss numerical approximation to derivatives of functions of two or more variables in the lectures to follow when we shall talk about partial differential equations.

9. In what follows, we
shall derive and illustrate
various formulae for
numerical differentiation of a
function of a single variable
based on finite difference
operators and interpolation.

10. Subsequently, we shall develop Newton-Cotes formulae and related trapezoidal rule and Simpson’s rule for numerical integration of a function.

11. DIFFERENTIATION USING DIFFERENCE OPERATORS

12. We assume that the function
y = f (x) is given for the values
of the independent variable
x = x0 + ph, for p = 0, 1, 2, …
and so on.
To find the derivatives of
such a tabular function, we
proceed as follows

13. Case I:
Using forward difference operator ∆ and combining equations
and

14. Remember the Differential operator, D is known to represents the property

15. This would mean that in terms of ∆ :

16. Therefore,

17. In other words,

18. Also, we can easily verify

20. Case II:
Using backward difference operator , we have
On expansion, we have

21. we can also verify that

22. Hence,

23. and

24. The formulae for Dy0and
D2y0 are useful to calculate the first and second derivatives at the beginning of the table of values in terms of forward differences; while formulae for y’n and y’’n

25. are used to compute the first
and second derivatives near
the end points of the table, in
terms of backward differences.
Similar formulae can also be
derived for computing higher
order derivatives.

26. Case III: Using central difference operator δ and following the definitions of differential operator D, central difference operator δ and the shift operator E, we have

28. Therefore, we have
But,

30. Using the last two equations we get
That is,

32. Therefore,
Also

33. Hence

34. For calculating the second
derivative at an interior tabular
point, we use the equation

35. while for computing the first
derivative at an interior tabular point, we in general use another convenient form for D, which is derived as follows. Multiply the right hand side of

36. which is unity and noting the Binomial expansionby
which is unity and noting the
Binomial expansion

38. We get

39. On simplification, we obtain
Therefore the equation can also be written in another useful form as

41. The last two equations for y” and y’ respectively are known as Stirling’s formulae for computing the derivatives of a tabular function. Similar formulae can be derived for computing higher order derivatives of a tabular function.

42. The equation for y’ can also be
written as

43. In order to illustrate the use of formulae derived so far, for computing the derivatives of a tabulated function, we shall consider some examples in the next lecture.

44. Numerical Analysis
Lecture 27