Gram‐Schmidt orthogonalization: 100 years and more

In 1907, Erhard Schmidt published a paper in which he introduced an orthogonalization algorithm that has since become known as the classical Gram‐Schmidt process. Schmidt claimed that his procedure was essentially the same as an earlier one published by J. P. Gram in 1883. The Schmidt version was the first to become popular and widely used. An algorithm related to a modified version of the process appeared in an 1820 treatise by P. S. Laplace. Although related algorithms have been around for almost 200 years, it is the Schmidt paper that led to the popularization of orthogonalization techniques. The year 2007 marked the 100th anniversary of that paper. In celebration of that anniversary, we present a comprehensive survey of the research on Gram‐Schmidt orthogonalization and its related QR factorization. Its application for solving least squares problems and in Krylov subspace methods are also reviewed. Software and implementation aspects are also discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Measurement of the wavelength dependence of beam divergence for photonic crystal fiber

We report measurements of the wavelength dependence of beam divergence for single-mode photonic crystal fiber. These measurements confirm predictions of strongly wavelength-dependent beam divergence, consistent with the effective-index model for the photonic crystal cladding material.     

Matrix completion with ε-algorithm

Numerical Algorithms - We show in this paper how the convergence of an algorithm for matrix completion can be significantly improved by applying Wynn’s ε-algorithm. Straightforward...     

Méthodes de relaxation d'ondes (SWR) pour l'équation de la chaleur en dimension 1

We introduce Schwarz Waveform Relaxation algorithms (SWR) for the heat equation which have a much faster convergence rate than the classical one due to optimized transmission conditions between subdomains. We analyze the asymptotic dependence of the convergence rate with respect to the size of the overlap and the time step. To cite this article: M.J. Gander, L. Halpern, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation

Optimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behaviour. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2d case, which uses a different Robin condition for neighbouring subdomains at their common interface, and which we call two‐sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency an asymptotic convergence factor of 1 – O(h^1/4) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O(1/ω^γ) for γ?1, then the optimized asymptotic convergence factor is 1 – O(ω^(1–2γ)/8). We illustrate our analysis with 2d numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.     

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