Understanding Orthogonal Polynomials for Least Squares Approximations
Dive into the concept of orthogonal polynomials and their role in least squares approximations. Explore how to apply least squares regression to fit data points and functions, with a focus on minimizing errors for accurate approximations.
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Presentation Transcript
MAT 4725 Numerical Analysis Section 8.2 Orthogonal Polynomials and Least Squares Approximations Part I http://myhome.spu.edu/lauw
We have looked at how to... Local approximation What about....
Review Least Squares Approximation (Least Squares Regression)
Linear Least Squares Approx. Given data points ??,?? want to fit a straight line to the data. ?=1,2,..,?. We
Linear Least Squares Approx. = + y a 1 ix a 0 = + y a a x 0 1 ( , ) ix y i
Normal Equations m m + = ma a x y 0 1 i i = = 1 1 i i m m m + = 2 a x a x x y 0 1 i i i i = = = 1 1 1 i i i Solve for and a a 0 1
General Case Given data points ??,?? want to approximate the data with ?=1,2,..,?. We n = = + + + n k ( ) P x a a x a x a x 0 1 n n k = 0 k Find such that a k m 2 = ( ) E y P x i n i = 1 i is minimized
Least Squares Approximation Apply the Least Squares Approximation to functions (instead of data sets) Think of a function as a data set such that each point of the function is one datum.
Least Squares Approximation of Functions Given approximate ( ) by [ , ], C a b f x f ( ) f x n = k ( ) n P x a x ( ) nP x k = 0 k a b
Least Squares Approximation of Functions Find such that a k b ( ) f x 2 = ( ) f x ( ) E P x dx n a ( ) nP x is minimized a b n = k ( ) P x a x n k = 0 k
Normal Equations = b b n + = k j j ( ) f x x dx , 0,1, , a x dx j n k = 0 k a a Solve for a k
Example 1 Find the least squares approx. of ?(?) = sin(??) on [0,1] by ?2(?). = b b n + = k j j ( ) f x x dx , 0,1, , a x dx j n k = 0 k a a
Example 1 = ( ) f x sin( ) x 2( ) P x
Disadvantages Solving (? + 1) (? + 1) linear system Computing the approx. ??(?) does not help the computation of ??+1(?).
Homework Download Homework Read the Definitions again I do not have a solution in SageMath. I am counting on using your program for the future solution key!
Classwork Finish at least the first problem of your HW.
