How ordinary elimination became Gaussian elimination

How Ordinary Elimination Became Gaussian Elimination

Newton, in an unauthorized textbook, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method — that Newton did not want to publish, that Euler did not recommend, that Legendre called “ordinary,” and that Gauss called “common” — is now named after Gauss: “Gaussian” elimination. (One suspects, he would not be amused.) Gauss’s...

How Accurate is Gaussian Elimination?∗

J.H. Wilkinson put Gaussian elimination (GE) on a sound numerical footing in the 1960s when he showed that with partial pivoting the method is stable in the sense of yielding a small backward error. He also derived bounds proportional to the condition number κ(A) for the forward error ‖x − x̂‖, where x̂ is the computed solution to Ax = b. More recent work has furthered our understanding of GE, la...

Gaussian elimination

As the standard method for solving systems of linear equations, Gaussian elimination (GE) is one of the most important and ubiquitous numerical algorithms. However, its successful use relies on understanding its numerical stability properties and how to organize its computations for efficient execution on modern computers. We give an overview of GE, ranging from theory to computation. We explai...

Gaussian elimination without rounding

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 ma...