Donald G. M. Anderson

Iterative Procedures for Nonlinear Integral Equations

The numerical solution of nonlinear integral equations involves the iterative soIutioon of finite systems of nonlinear algebraic or transcendental equations. Certain corwent i o n a l techniqucs for treating such systems are reviewed in the context of a particular class of n o n l i n e a r equations. A procedure is synthesized to offset some of the disadvantages of these t e c h n i q u e s in this context; however, the procedure is not restricted to this pt~rticular class of s y s t e m s of nonlinear equations.

Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix

Necessary and sufficient conditions for an optical system characterized by a Mueller matrix to be characterized by a Mueller–Jones, and thence by a Jones, matrix are considered, and the issue of measurement error is examined. It is shown that a Mueller matrix can be expressed as a linear combination of at most four trace orthonormal Mueller–Jones matrices, and an algorithm for the construction is given.

The triple centre approximation for charge exchange in atomic scattering theory

A triple centre approximation is proposed to describe proton-hydrogen charge changing collisions. Favourable agreement with experiment encourages further research into the applicability of three centre expansions.

Comments on "Anderson Acceleration, Mixing and Extrapolation"

The Extrapolation Algorithm is a technique devised in 1962 for accelerating the rate of convergence of slowly converging Picard iterations for fixed point problems. Versions to this technique are now called Anderson Acceleration in the applied mathematics community and Anderson Mixing in the physics and chemistry communities, and these are related to several other methods extant in the literature. We seek here to broaden and deepen the conceptual foundations for these methods, and to clarify their relationship to certain iterative methods for root-finding problems. For this purpose, the Extrapolation Algorithm will be reviewed in some detail, and selected papers from the existing literature will be discussed, both from conceptual and implementation perspectives.

Approximation of multi-centre integrals arising in proton-hydrogen charge exchange calculations

A lg-magnitude phase representation is used to effect the efficient approximation of multi-centre integrals through polynomial interpolation. When the multi-centre integrals are evaluated by numerical quadrature, computation costs can be reduced by an order of magnitude or more.