MATH 640 - Numerical Analysis
Test 3 Topics
Chapter 8: Approximation Theory
•Be able to explain the concepts behind, and compare and contrast Discrete Least Squares
Approximation(Section 8.1) and (Continuous) Least Squares Approximation(Section
8.2).
•Be able to list the error function possibilities discussed in class and the interpretation of each
(especially visually).
•Be able to set up both types least square approximations discussed (discrete and continuous).
•Know what it means for a set of functions {φ0, φ1,· · · , φn}to be linearly independent, or
orthogonal (or orthonormal) with respect to the weight function w(x).
•Know why we would want to bother dealing with writing our least squares approximating
polynomials in terms of these orthogonal polynomials, {φ0, φ1,· · · , φn}, rather than using the
first method discussed in Section 8.2.
•Be able to use the Gram Schmidt process (if provided with the formulas) to generate a set
of orthogonal polynomials with respect to a given weight function.
•Know what we mean by the set of trigonometric polynomials of degree less than or equal
to n,Tn.
•Be able to define and calculate the trigonometric polynomial (Fourier polynomial) of
fover [−π, π], Sn(x), which is also known as the continuous least squares trigonometric
approximation to fby the set of trigonometric polynomials of degree less than or equal to
nover [−π, π]. (And, if we let napproach infinity, then we call this the Fourier series for
f.)
•Know how to calculate the discrete least squares trigonometric approximation,Sn(x),
for some data over the interval [−π, π].
•Know and be able to derive (in full analytical detail) the normal equations associated
with any of the least squares approximation techniques (standard polynomials, orthogonal
polynomials, trigonometric polynomials, discrete, continuous, etc.).