1. Numerical Integration of Functions
Chapter 18
Numerical Integration of Functions

2. Numerical Integration
Tabulated data – the accuracy of the integral is limited by the number of data points
Continuous function – we can generate as many f(x) as required to attain the required accuracy
Richardson extrapolation and Romberg integration
Gauss Quadratures

3. Round-off errors may limit the precision of lower-order Newton-Cotes composite integration formula
Use Romberg Integration for efficient integration

4. Romberg Integration
More efficient methods to achieve better accuracy have been developed
Romberg integration - uses Richardson extrapolation
Idea behind Richardson extrapolation -improve the estimate by eliminating the leading term of truncation error at coarser grid levels

5. Richardson Extrapolation
The exact integral can be represented as
This is true for any h = (ba)/n
Use trapezoidal rule as an example

6. Composite Trapezoidal Rule
Evaluate the integral

7. Richardson Extrapolation
Truncation error for trapezoidal rule
Substitute into the exact integral
Which leads to

8. Richardson Extrapolation
Plugging back into I = I(h) + E(h)
If h2 = h1/2, then

9. Richardson Extrapolation
Combine two O(h2) estimates to get an O(h4) estimate
Can also combine two O(h4) estimates to get an O(h6) estimate
Combine two O(h6) estimates to get an O(h8) estimate
Im and Il are more and less accurate estimates, respectively

10. Romberg Integration General form is called Romberg Integration
j: level of accuracy - j+1 is more accurate (more segments)
k: level of integration - k=1 is the original trapezoidal rule estimate (O(h2)), k=2 is improved (O(h4)), k=3 corresponds to O(h6), etc.

11. Romberg Integration
Accelerated Trapezoidal Rule

12. Romberg Integration
Accelerated Trapezoid Rule

13. Accelerated trapezoidal Rule
Romberg Integration
Accelerated trapezoidal Rule

14. » intg = romberg(‘example1’,0,pi,0.00001,2)
» intg = Romberg(‘example1’,0,pi, ,3)
» intg = romberg(‘example1’,0,pi, ,4)
» intg = romberg(‘example1’,0,pi, ,6)

15. CVEN 302-501 Homework No. 12 Chapter 17
Problem 17.3 (25), 17.5 (25)– Hand Calculation
Chapter 18
Problem 18.2 (25), 18.4 (25)
Due on Monday, 11/17/2008 at the beginning of the period

16. Gauss Quadrature Assume a and b are limits of integration
Trapezoidal rule should give exact results for constant and linear functions

17. Trapezoidal rule gives exact solution for constant and linear functions

18. Gauss Quadrature
Now instead of trapezoidal, which has fixed end points (a,b), let them float
4 unknowns - x0 ,x1 ,c0 ,c1
4 equations - constant, linear (had before in trapezoidal rule), quadratic, cubic
Integrate from -1 to 1 to simplify math

19. Trapezoidal vs. Gauss-Quadrature
Exact for constant and linear functions
Exact for constant, linear, quadratic and cubic functions

20. Gauss Quadrature
The idea is that if we evaluate the function at certain points (non-uniformly distributed), and sum with certain weights, we will get accurate integral
Evaluation points and weights are tabulated

21. Gauss-Legendre Quadrature
Choose (c0 , c1 , x0 , x1) to yield highest possible accuracy

22. Gauss Quadratures Newton-Cotes Formulas
use evenly-spaced functional values
Gauss Quadratures (Gauss-Legendre formulas)
change of variables so that the interval of integration is [1,1]
select functional values at non-uniformly distributed points to achieve higher accuracy

23. Gauss Quadrature on [a, b]
To go to [1,1] from other limits [a,b] - use linear transformation
Change from a  x  b to 1  xd  1
Coordinate transformation

24. Gauss Quadrature on [a, b]
Coordinate transformation from [a,b] to [1,1] a t1 t2 b

25. Gauss Quadrature on [1, 1]x0 x1 -1 1
Choose (c0 , c1 , x0 , x1) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3

26. Gauss Quadrature on [1, 1]
Exact integral for f = x0, x1, x2, x3
Four equations for four unknowns

27. Gauss Quadrature on [1, 1]x0 x1 x2 -1 1
Choose (c0, c1, c2, x0, x1, x2) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3,x4, x5

28. Gauss Quadrature on [1, 1]
Exact integral for f = x0, x1, x2, x3, x4, x5

29. Example: Gauss Quadrature
Evaluate
Coordinate transformation
Two-point formula

30. Example: Gauss Quadrature
Three-point formula
Four-point formula

31. Gauss-Legendre Formulas

32. Gauss Quadrature

33. Gauss Quadrature k = 2 Exact Q = -4.9348
» I=Gauss_quad(‘example1’,0,pi,2);
t =
c =
tt =
0.5774
cd = 1
» I
I =
» Q=quad8(‘example1’,0,pi)
Q =
k = 2
Exact Q =

34. Gauss Quadrature k = 5 Exact Q = -4.9348
» I=Gauss_quad(‘example1’,0,pi,5);
t =
c =
tt =
0.5385
0.9062
cd =
0.2369
0.4786
0.5689
» I
I =
Gauss Quadrature k = 5
Exact Q =

35. Adaptive Quadrature
Composite Simpson’s 1/3 rule requires the use of equally spaced points
Use adaptive refinement in regions of relatively abrupt changes
Estimate truncation error between two levels of refinement
Automatically adjust the step size so that small steps are taken in regions of sharp variations while larger steps are used elsewhere
MATLAB functions: quad and quadl

36. MATLAB Integration Methods
trapz(x,y)
* Composite trapezoid rule
q = quad(‘func’,xmin,xmax)
* Adaptive Simpson’s rule (p 439),
more efficient for low accuracies or non- smooth functions
q =quadl(‘func’,xmin, xmax)
* Labatto quadrature – more efficient for high
accuracies and smooth functions (p439)