The Mathematica Journal


Relating Orthonormal Polynomials, Gram—Schmidt Orthonormalization, QR Factorization, Normal Equations and Vandermonde and Hilbert Matrices

Gottlob Gienger
Published May 17, 2017

This didactic synthesis compares three solution methods for polynomial approximation and systematically presents their common characteristics and their close interrelations:

1. Classical GramSchmidt orthonormalization and Fourier approximation in
2. Linear least-squares solution via QR factorization on an equally spaced grid in
3. Linear least-squares solution via the normal equations method in and on an equally
    spaced grid in

The first two methods are linear least-squares systems with Vandermonde matrices ; the normal equations contain matrices of Hilbert type . The solutions on equally spaced grids in converge to the solutions in All solution characteristics and their relations are illustrated by symbolic or numeric examples and graphs. Read More »

Zachary H. Levine, J. J. Curry
Published March 28, 2017

The derivation of the scattering force and the gradient force on a spherical particle due to an electromagnetic wave often invokes the ClausiusMossotti factor, based on an ad hoc physical model. In this article, we derive the expressions including the ClausiusMossotti factor directly from the fundamental equations of classical electromagnetism. Starting from an analytic expression for the force on a spherical particle in a vacuum using the Maxwell stress tensor, as well as the Mie solution for the response of dielectric particles to an electromagnetic plane wave, we derive the scattering and gradient forces. In both cases, the ClausiusMossotti factor arises rigorously from the derivation without any physical argumentation. The limits agree with expressions in the literature. Read More »