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A History of Gram-Schmidt Orthogonalization

Steven J Leon
UMass Dartmouth


It has been more than a hundred years since the appearance of the landmark 1907 paper by Erhard Schmidt where he introduced a method for finding an orthonormal basis for the span of a set of linearly independent vectors. This method has since become known as the classical Gram-Schmidt Process (CGS). In this talk we present a survey of the research on Gram--Schmidt orthogonalization, its related QR factorization, and the algebraic least squares problem. We begin by reviewing the two main versions of the Gram-Schmidt process and the related QR factorization and we briefly discuss the application of these concepts to least squares problems. This is followed by a short survey of eighteenth and nineteenth century papers on overdetermined linear systems and least squares problems. We then examine the original orthogonality papers of both Gram and Schmidt. The second part of the talk focuses on such issues as the use of Gram-Schmidt orthogonalization for stably solving least squares problems, loss of orthogonality, and reorthogonalization. In particular, we focus on noteworthy work by Ake Bjorck and Heinz Rutishauser and discuss later results by a host of contemporary authors. *S. J. Leon, Ake Bjorck and Walter Gander are co-authors of the paper Gram-Schmidt Orthogonalization: 100 years and more, Numer. Linear Algebra Appl (2013) This talk is to a large part based on that paper

Tuesday, January 5, 2016
11:00AM AP&M 2402