5. Matrix Norms, Eigenvalues, Eigenvectors, & Condition Numbers (Part 2)
The problem is that (speaking a bit loosely) each successive x' increasingly lies in
the space spanned by the lower powers of x and thus incrementally spans only a
rapidly shrinking portion of the y(x) function space. The solution is to construct an
orthonormal basis using the Gram-Schmidt process. Let:
= x° = 1 and proceed successively:
(x¹, x¹) - (x¹, Po(x))
- - - p1 p1
(a) Construct explicitly the first three elements of this orthonormal polynomial basis:
Po(x), P1 (x), and P2(x).
(b) What is the moment form of the least squares problem in the p basis and, in
particular, what are the corresponding coefficients Aij? If possible, interpret your
results as an explicit formula for the c:-
(c) Given the form taken by the Gram matrix in (b), discuss (with as few calculations
as possible) how your answers to 4(c)-(f) would differ in the p basis VS. xi basis.
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