New England Numerical Analysis Day 2009

University of Rhode Island
Department of Mathematics, Lippitt Hall
April 4, 2009, 9:00am-5:30pm
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Gram-Schmidt Orthogonalization: 100 Years and More

Steven J. Leon, UMass Dartmouth

(joint work with Ake Bjorck, Linkoping University, Walter Gander, ETH Zurich)

Abstract: In 1907 Erhard Schmidt published a paper where he introduced an orthogonalization algorithm that has since become known as the classical Gram-Schmidt process (CGS). In the paper, Schmidt claimed that his procedure was essentially the same as an earlier one published by J.~P.~Gram in 1883. In actuality, a modified version of the Gram-Schmidt algorithm (MGS) first appears in an 1812 treatise by P. S. Laplace. While versions of the algorithm have been around for almost 200 years, it is the Schmidt paper that led to the popularization of orthonormalization techniques. The year 2007 marked the 100th anniversary of that paper. In celebration of that anniversary our paper presents a comprehensive survey of the research on Gram-Schmidt orthogonalization and its related QR factorization. Topics covered include: the early history of orthogonalization and least squares, loss of orthogonality, reorthogonalization, super orthogonalization, rank revealing factorizations, and applications of Gram-Schmidt to Krylov subspace methods.

The talk will first focus on the early history and the works of Laplace, Cauchy, Bienayme, Gram, and Schmidt. We then discuss the work of Rutishauser, much of which is not that well known, and the important work of Ake Bjorck on the numerics of the Gram-Schmidt algorithm and least squares.



Organizers

Jim Baglama (URI), Tom Bella (URI), Li Wu (URI).

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