Scientific Computing Group Seminars - Detail View

Speaker: S Leon

Affiliation: University of Massachusetts, Dartmouth

Talk Title: Gram-Schmidt Orthogonalization: 100 Years and More

Invited by: David Gottlieb

Time: Sept. 12 2008 11 a.m.

Location: 182 George Street, Room 110

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). 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. The two algorithms produce the same results when carried out in exact arithmetic, however, the Gram version, now known as the modified Gram-Schmidt process (MGS), produces superior results when carried out in finite precision arithmetic. In actuality, Gram rediscovered a method that first appears in an 1812 treatise by P. S. Laplace. While the MGS algorithm has 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 we present a comprehensive survey of the research on Gram-Schmidt orthogonalization and its related QR factorization. Among the topics covered are: the early papers on orthogonality and least squares, loss of orthogonality, reorthogonalization, super orthogonality, iterative refinement, rank revealing factorizations, and applications to Krylov subspace methods. Joint work with Walter Gander, ETH Zurich, Ake Bjorck, Linkoping University, and Julien Langou, University of Colorado at Denver