1. Today’s class Romberg integration Gauss quadrature Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

2. Numerical Integration
Multiple application of the Newton-Cotes Formulas will improve the accuracy of the approximation
As you increase the number of segments, you reduce the error
However, if you increase the number of segments too much, round-off errors begin to dominate and the error will start to increase
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

3. Numerical Integration
a=0, b = 0.8
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

4. Romberg integration Richardson’s extrapolation
Perform a numerical algorithm using multiple values of a parameter h and then extrapolate that result to the limit h=0
With numerical integration use two estimates of the integral to come up with a third more accurate approximation
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

5. Richardson’s Extrapolation
The estimated integral is a function of subinterval size. With smaller h, usually I(h), the integral estimate, is more accurate, which means the error caused by choosing this interval size is smaller.
If we have two integral esitmates, each one is obtained by using a different segment size, or equivalently different number of segments. We can use estimate the error for each integral estimate using our early analysis. If we are able to estimate this error, we can use this error to update the estimate so the new one is closer to the true integral. However, we cannot estimate this one because we don’t know this item related to second order derivative. So with two e
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

6. Richardson’s Extrapolation
We’ve used the error calculation to come up with a new estimate
It can also be shown that this new estimate is O(h4), whereas trapezoidal rule is only O(h2)
In 1978, two researchers show that this new esitmate is much more accuract with an error term in the forth order of the subinterval size.
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

7. Richardson’s Extrapolation
Special case where you always halve the interval - i.e. h2=h1/2
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

8. Richardson’s Extrapolation
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

9. Richardson’s Extrapolation
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

10. Romberg Integration Accelerated Trapezoid Rule k: level of integration+1
k: level of integration
j: level of accurancy
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

11. Romberg Integration Accelerated Trapezoid Rule Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

12. Romberg Integration Termination Criteria Trapezoidal method
9 iterations before hitting precision limit (n=256)
511 function evaluations
Romberg integration
3 iterations before hitting precision limit
15 function evaluations
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

13. Romberg Integration very good convergence properties
less susceptible to round-off error than Trapezoidal or Simpson’s rule
extra levels of extrapolation require very little computational work
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

14. Gauss Quadratures Newton-Cotes Formulas Gauss Quadratures
use evenly-spaced functional values
Gauss Quadratures
select functional values at non-uniformly distributed points to achieve higher accuracy
Gauss-Legendre formulas
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

15. Gauss Quadratures Trapezoidal Method Numerical Methods Prof. Jinbo Bi
Lecture 13
Prof. Jinbo Bi
CSE, UConn

16. Gauss Quadratures
Find interior points so that the trapezoidal area outside the curve is equal to the area below the curve and above the trapezoid
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

17. Gauss Quadratures Method of Undetermined Coefficients
Trapezoidal method should be exact for
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

18. Gauss Quadratures Two-Point Gauss Legendre Derivation
Find a solution to following equation over range [-1:1]
Using similar reasoning as before, solve for the four unknowns using the following equations
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

19. Gauss Quadratures
This formula gives an integral estimate that is third-order accurate
To be usable, the bounds of the definite integral have to be from -1 to 1
It is easy to convert by using a new variable xd
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

20. Gauss Quadratures 0.4dxd Numerical Methods Prof. Jinbo Bi Lecture 13
CSE, UConn

21. Gauss Quadratures Equivalent accuracy to Simpson’s 1/3 rule (O(n3))
Fewer function evaluations
Can be extended to higher-point versions
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

22. Gauss Quadratures Numerical Methods Prof. Jinbo Bi Lecture 13
CSE, UConn

23. Gauss Quadratures on [-1, 1]
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

24. Gauss Quadratures on [-1, 1]
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

25. Example: Gauss Quadratures
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

26. Gauss Quadratures High accuracy with few function evaluations
Error is proportional to the (2n+2)th derivative
Function must be known - not appropriate for tabular data
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

27. Improper integrals
How do you handle integrals where one of the bounds of the integral is ±∞?
Do a translation of the bounds into a proper integral
Works as long as a is -∞ and b is negative or a is positive and b is ∞.
The function f(x) must also asymptotically approach zero at least as fast as 1/x2
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

28. Improper integrals Example: Normal distribution
Split the integral at a point where the function starts to approach zero faster than 1/x2
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

29. Summary Integration Techniques Gaussian Quadrature Improper integrals
Trapezoidal Rule : Linear
Simpson’s 1/3-Rule : Quadratic
Simpson’s 3/8-Rule : Cubic
Improvement techniques
Multiple application or composite methods
Romberg integration
Gaussian Quadrature
Improper integrals
Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn

30. Next class Numerical Differentiation Read Chapter 23 Numerical Methods
Lecture 13
Prof. Jinbo Bi
CSE, UConn