2. Chapter 7 Numerical Differentiation and Integration

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

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

5. Basic Issues in Integration
What does an integral
represent?
= AREA
= VOLUME

6. Basic definition of an integral::= =
sum of Height x Width

7. Objective:
Evaluate I = without
doing calculation analytically.
When would we want to do this?

8. 1. Integrand is too complicated to integrate analytically.
2. Integrand is not precisely defined by an equation,i.e., we are given a set of data (xi,ƒ(xi)), i=1,...,n.
All methods are applicable to integrands that are functions.
Some are applicable to tabulated values.

9. Key concepts:
Integration is a summing process. Thus virtually all numerical approximations can be represented by
I = =

10. where:
x = weights
xi = sampling points
Et = truncation error
2. Closed & Open forms:
Closed forms include the end
points a & b in xi.
Open forms do not.

11. NUMERICAL INTEGRATION
Consider the definite integral

12. where f (x) is known either explicitly or is given as a table of values corresponding to some values of x, whether equispaced or not. Integration of such functions can be carried out using numerical techniques.

13. Of course, we assume that the function to be integrated is smooth and Riemann integrable in the interval of integration. In the following section, we shall develop Newton-Cotes

14. formulae based on
interpolation which form the
basis for trapezoidal rule and
Simpson’s rule of numerical
integration.

15. NEWTON-COTES INTERGRATION FORMULAE
In this method, as in the case of numerical differentiation, we shall approximate the given tabulated function, by a polynomial Pn(x) and then integrate this polynomial.

16. Suppose, we are given the data (xi, yi), i = 0(1)n, at equispaced points with spacing h = xi+1 – xi, we can represent the polynomial by any standard interpolation polynomial. Suppose, we use Lagrangian approximation, then we have

17. with associated error given by

18. where
and

19. Then, we obtain an equivalent integration formula to the definite integral in the form

20. where ck are the weighting coefficients given by

21. which are also called Cotes numbers
which are also called Cotes numbers. Let the equispaced nodes are defined by

22. so that xk – x1 = ( k – 1)h etc. Now, we shall change the variable x to p such that, x = x0 + ph, then we can rewrite equations.

24. as

25. and

26. or

27. Also, noting that dx = h dp. The limits of the integral in Equation
change from 0 to n and
equation reduces to

28. The error in approximating the integral can be obtained from

29. Where x0 < ξ < xn. For illustration, consider the cases for n = 1, 2; For which we get

30. and

31. Thus, the integration formula is found to be

32. This equation represents the Trapezoidal rule in the interval [x0, x1] with error term. Geometrically, it represents an area between the curve y = f (x), the x-axis and the ordinates erected
at x = x0 ( = a) and x = x1
as shown in the figure.

33. yn-1 y3 y2 y1 y0 yn
xn = b
xn-1 x3 x2 x1
x0 = a X O Y
(x2, y2)
(x1, y1)
(x0, y0)
y = f(x)

34. This area is approximated by the trapezium formed by replacing the curve with its secant line drawn between the end points (x0, y0) and (x1, y1).

35. For n =2, We have

37. and the error term is given by

38. Thus, for n = 2, the integration takes the form

39. This is known as Simpson’s 1/3 rule
This is known as Simpson’s 1/3 rule. Geometrically, this equation represents the area between the curve y = f (x), the x-axis and the ordinates at x = x0 and x2 after replacing the arc of the curve between (x0, y0) and (x2, y2) by an arc of a quadratic polynomial as in the figure

40. xn = b
xn-1 x3 x2 x1
x0 = a X O Y
(x2, y2)
(x0, y0) y2 y1 y0
y = f(x)

41. Thus Simpson’s 1/3 rule is based on fitting three points with a quadratic.
Similarly, for n = 3, the integration is found to be

43. This is known as Simpson’s 3/8 rule, which is based on fitting four points by a cubic. Still higher order Newton-Cotes integration formulae can be derived for large values of n.

44. But for all practical purposes,
Simpson’s 1/3 rule is found to
be sufficiently accurate.