MAT 4725 Numerical Analysis Section 8.2 Orthogonal Polynomials and Least Squares Approximations (Part II)

Preview Inner Product Spaces Gram-Schmidt Process

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

Recall (Linear Algebra) General Inner Product Spaces

Inner Product

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

Norm, Distance,…

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.

Gram-Schmidt Process

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.

Simple Example

Specific Inner Product Space

Definition 8.1

Theorem 8.2 Idea

Definition

Theorem 8.3

Example 1

Definition (Skip it for the rest)

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

Modification of the Least Squares Approximation Recall from part I

Least Squares Approximation of Functions

Normal Equations

Modification of the Least Squares Approximation

Where are the Improvements?

Definition 8.5

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

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

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

Gram-Schmidt Process

Legendre Polynomials

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

Example 2

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