1. Today’s class Multiple Variable Linear Regression
Polynomial Interpolation
Lagrange Interpolation
Newton Interpolation
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

2. Mid-term Exam 2 Average = 83, Standard deviation = 27
Max = 100, Min = 60
Numerical Methods Fall 2010
Lecture 21
Prof. Lei Wang
ECE, UConn

3. Multiple Variable Linear Least Square Approximation
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

4. Multiple Linear Regression
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

5. General Linear Least Squares
Simple linear, polynomial, and multiple linear regressions are special cases of the general linear least squares model
Linear in ai, but zi may be highly nonlinear
Examples:
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

6. General Linear Least Squares
General equation in matrix form
Where
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

7. General Linear Least Squares
As usual, take partial derivatives to minimize the square errors Sr
This leads to the normal equations
Solve this for {a}
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

8. Interpolation & Extrapolation
data to be found are within the range of observed data.
Extrapolation
data to be found are beyond the range of observation data. (may not be reliable)
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

9. Interpolation
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

10. Polynomial Interpolation
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

11. Linear Interpolation Errors are larger when the interval is larger.
Smaller interval provides a better estimate.
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

12. Linear Interpolation
Quadratic interpolation gives a better estimate than linear interpolation when the change in a function is smooth and slowly.
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

13. Direct Method
Solve the system of equations for the coefficients. It is time consuming and the coefficient matrix may be ill-conditioned.
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

14. Newton (divided difference) Interpolation polynomials
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

15. Newton (divided difference) Interpolation polynomials
x2-x0
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

16. General Scheme for Divided Difference Coefficients
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

17. General Scheme for Divided Difference Coefficients
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

18. Example Estimate ln2 with data points at (1,0) and (4,1.386294)
Linear interpolation
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

19. Example
Estimate ln2 with data points at (1,0), (4, ), and (5, )
Quadratic interpolation
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

20. Example
Estimate ln2 with data points at (1,0), (4, ), (5, ), (6, )
Cubic interpolation
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

21. Newton’s Interpolating Polynomial
Error
Similar to Taylor series remainder
Proportional to n+1th finite divided difference
Also proportional to the distance of the data points from x
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

22. Lagrange Interpolation Polynomials
Nth order polynomial: interpolate n+1 points
The Lagrangian coefficient, Li(x) will be 1 at x=xi and 0 at all other data points
Thus, fn(xi)=f(xi) for all i, meaning that the Lagrangian polynomial passes through all the data points as expected.
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

23. Lagrange Interpolating Polynomials
Linear polynomial: interpolate 2 points
2nd order polynomial: interpolate 3 points
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

24. Lagrange Interpolating Polynomials
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

25. Example
Estimate ln2 with data points at (1,0), (4, ), (5, )
2nd order Lagrangian interpolation
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

26. Lagrange & Newton Interpolation
The forms of the Lagrangian and Newton polynomials look different, but they are actually equivalent if the data points are the same
The Lagrangian coefficients are easier to calculate since the divided difference computation is not required
However, if you are adding new data points, the Newton method is more efficient since the earlier coefficients remain the same
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn

27. Next class Spline Interpolation Fourier Approximation HW8 due Dec 8
Numerical Methods
Lecture 20
Prof. Jinbo Bi
CSE, UConn