1. MAT 4725 Numerical Analysis Section 8.2 Orthogonal Polynomials and Least Squares Approximations (Part II) http://myhome.spu.edu/lauw

2. Preview Inner Product Spaces Gram-Schmidt Process

3. A Different Technique for Least Squares Approximation Computationally Efficient Once P n (x) is known, it is easy to determine P n+1 (x)

4. Recall (Linear Algebra) General Inner Product Spaces

5. Inner Product

6. Example 0 Let f,g  C[a,b]. Show that is an inner product on C[a,b]

7. Norm, Distance,…

8. Orthonormal Bases A basis S for an inner product space V is orthonormal if 1. For u,v  S, =0. 2. For u  S, u is a unit vector.

9. Gram-Schmidt Process

11. The component in v 2 that is “parallel” to w 1 is removed to get w 2. So w 1 is “perpendicular” to w 2.

12. Simple Example

13. Specific Inner Product Space

14. Definition 8.1

15. Theorem 8.2 Idea

16. Definition

17. Theorem 8.3

18. Example 1

19. Definition (Skip it for the rest)

20. Weight Functions to assign varying degree of importance to certain portion of the interval

21. Modification of the Least Squares Approximation Recall from part I

22. Least Squares Approximation of Functions

24. Normal Equations

25. Modification of the Least Squares Approximation

30. Where are the Improvements?

34. Definition 8.5

36. Theorem 8.6 a k are easier to solve a k are “reusable”

37. Theorem 8.6 a k are easier to solve a k are “reusable”

38. Where to find Orthogonal Poly.? the Gram-Schmidt Process

39. Gram-Schmidt Process

41. Legendre Polynomials

43. Example 2 Find the least squares approx. of f(x)=sin(  x) on [-1,1] by the Legendre Polynomials.

44. Example 2

50. Homework Download Homework