Mathematicians of Gaussian Elimination

G aussian elimination is universally known as “the” method for solving simultaneous linear equations. As Leonhard Euler remarked, it is the most natural way of proceeding (“der natürlichste Weg” [Euler, 1771, part 2, sec. 1, chap. 4, art. 45]). Because Gaussian elimination solves linear problems directly, it is an important technique in computational science and engineering, through which it makes continuing, albeit indirect, contributions to advancing knowledge and to human welfare. What is natural depends on the context, so the algorithm has changed many times with the problems to be solved and with computing technology. Gaussian elimination illustrates a phenomenon not often explained in histories of mathematics. Mathematicians are usually portrayed as “discoverers”, as though progress is a curtain that gradually rises to reveal a static edifice which has been there all along awaiting discovery. Rather, Gaussian elimination is living mathematics. It has mutated successfully for the last two hundred years to meet changing social needs. Many people have contributed to Gaussian elimination, including Carl Friedrich Gauss. His method for calculating a special case was adopted by professional hand computers in the nineteenth century. Confusion about the history eventually made Gauss not only the namesake but also the originator of the subject. We may write Gaussian elimination to honor him without intending an attribution. This article summarizes the evolution of Gaussian elimination through the middle of the twentieth century [Grcar, 2011a,b]. The sole development in ancient times was in China. An independent origin in modern Europe has had three phases. First came the schoolbook lesson, beginning with Isaac Newton. Next were methods for professional hand computers, which began with Gauss, who apparently was inspired by work of JosephLouis Lagrange. Last was the interpretation in matrix algebra by several authors, including John