1. Numerical Integration
Basic Numerical Integration
We want to find integration of functions of various forms of the equation known as the Newton Cotes integration formulas.
Trapezoidal Rule
Simpson’s Rule
1/3 Rule
3/8 Rule
Midpoint
Gaussian Quadrature

2. Basic Numerical Integration
Weighted sum of function values
f(x) x x0 x1
xn-1 xn

3. Numerical Integration
Idea is to do integral in small parts, like the way you first learned integration - a summation
Numerical methods just try to make it faster and more accurate

4. Numerical Integration
Newton-Cotes Closed Formulae -- Use both end points
Trapezoidal Rule : Linear
Simpson’s 1/3-Rule : Quadratic
Simpson’s 3/8-Rule : Cubic
Boole’s Rule : Fourth-order
Newton-Cotes Open Formulae -- Use only interior points
midpoint rule

5. Trapezoid Rule
Straight-line approximation
f(x)
L(x) x x0 x1

6. Trapezoid Rule
Lagrange interpolation

7. Trapezoid Rule
Integrate to obtain the rule

8. Example:Trapezoid Rule
Evaluate the integral
Exact solution
Trapezoidal Rule

9. Simpson’s 1/3-Rule Approximate the function by a parabola L(x) f(x) x

10. Simpson’s 1/3-Rule

11. Simpson’s 1/3-Rule
Integrate the Lagrange interpolation

12. Simpson’s 3/8-Rule Approximate by a cubic polynomial L(x) f(x) x x0 h

13. Simpson’s 3/8-Rule

14. Example: Simpson’s Rules
Evaluate the integral
Simpson’s 1/3-Rule
Simpson’s 3/8-Rule

15. Midpoint Rule
Newton-Cotes Open Formula
f(x) x a xm b

16. Two-point Newton-Cotes Open Formula
Approximate by a straight line
f(x) x x0 h x1 h x2 h x3

17. Three-point Newton-Cotes Open Formula
Approximate by a parabola
f(x) x x0 h x1 h x2 h x3 h x4

18. Better Numerical Integration
Composite integration
Composite Trapezoidal Rule
Composite Simpson’s Rule
Richardson Extrapolation
Romberg integration

19. Apply trapezoid rule to multiple segments over integration limits
Two segments
Three segments
Four segments
Many segments

20. Composite Trapezoid Rule
f(x) x x0 h x1 h x2 h x3 h x4

21. Composite Trapezoid Rule
Evaluate the integral

22. Composite Trapezoid Example

23. Composite Trapezoid Rule with Unequal Segments
Evaluate the integral
h1 = 2, h2 = 1, h3 = 0.5, h4 = 0.5

24. Composite Simpson’s Rule
Piecewise Quadratic approximations
f(x)
…... x x0 h x1 h x2 h x3 h x4
xn-2
xn-1 xn

25. Composite Simpson’s Rule
Multiple applications of Simpson’s rule

26. Composite Simpson’s Rule
Evaluate the integral
n = 2, h = 2
n = 4, h = 1

27. Composite Simpson’s Example

28. Composite Simpson’s Rule with Unequal Segments
Evaluate the integral
h1 = 1.5, h2 = 0.5

29. Richardson Extrapolation
Use trapezoidal rule as an example
subintervals: n = 2j = 1, 2, 4, 8, 16, ….

30. Richardson Extrapolation
For trapezoidal rule
kth level of extrapolation

31. Romberg Integration
Accelerated Trapezoid Rule

32. Romberg Integration
Accelerated Trapezoid Rule

33. Romberg Integration Example

34. Gaussian Quadratures Newton-Cotes Formulae Gaussian Quadratures
use evenly-spaced functional values
Gaussian Quadratures
select functional values at non-uniformly distributed points to achieve higher accuracy
change of variables so that the interval of integration is [-1,1]
Gauss-Legendre formulae

35. Gaussian Quadrature on [-1, 1]x1 x2 -1 1
Choose (c1, c2, x1, x2) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3

36. Gaussian Quadrature on [-1, 1]
Exact integral for f = x0, x1, x2, x3
Four equations for four unknowns

37. Gaussian Quadrature on [-1, 1]x1 x2 x3 -1 1
Choose (c1, c2, c3, x1, x2, x3) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3,x4, x5

38. Gaussian Quadrature on [-1, 1]

39. Gaussian Quadrature on [-1, 1]
Exact integral for f = x0, x1, x2, x3, x4, x5

40. Gaussian Quadrature on [a, b]
Coordinate transformation from [a,b] to [-1,1] a t1 t2 b

41. Example: Gaussian Quadrature
Evaluate
Coordinate transformation
Two-point formula

42. Example: Gaussian Quadrature
Three-point formula
Four-point formula