New THE MATHEMATICA JOURNAL article:
Polynomial L^2 Approximation
Relating Orthonormal Polynomials, GramSchmidt Orthonormalization, QR Factorization, Normal Equations and Vandermonde and Hilbert Matrices
by GOTTLOB GIENGER
ABSTRACT: This didactic synthesis compares three solution methods for polynomial L^2 approximation and systematically presents their common characteristics and their close interrelations:
- Classical GramSchmidt orthonormalization and Fourier approximation in L^2(a,b)
- Linear least-squares solution via QR factorization on an equally
spaced grid in [a,b] - Linear least-squares solution via the normal equations method in L^2(a,b)
and on an equally spaced grid in [a,b]
The first two methods are linear least-squares systems with Vandermonde matrices V ; the normal equations contain matrices of Hilbert type H=V^TV . The solutions on equally spaced grids in [a,b] converge to the solutions in L^2(a,b). All solution characteristics and their relations are illustrated by symbolic or numeric examples and graphs.
