The Simplex Method Of Linear Programming Using Lu Decomposition (With R. H. Bartels)

Abstract

Abstract This paper is a model for the way that Gene likes to work: find someone interesting in another field to talk with, find out what the problems, approaches, and computational practices are in that field, and then come up with some possible way in which good numerical linear algebra might help. Then tell someone else how a thing should really be done and get them to do the deed. The interesting person in question that led to this work was George Dantzig and the field was linear programming. My first contact with this interaction between Gene and George Dantzig must have been after Gene had the tableau method of the simplex method presented to him. I recall an incredulous statement from Gene roughly like, “It’s crazy; they use Gauss–Jordan elimination without pivot selection!” Gene proposed that some method of updating the LU decomposition of the basis matrix be found rather than the Gauss–Jordan update of the basis matrix inverse, as was used in the tableau. I needed a PhD thesis topic, so I became the “someone else” who was to do the deed. The updating process was Gene’s, and I was asked to see what I could say about its numerical stability using Wilkinson round-off analysis.