Gaussian elimination: when is scaling beneficial?

Abstract For the linear system Ax = b , the ordered pair ( D, F ) of nonsingular diagonal matrices determine a scaling of the system through the two equations D ( AF ) y = Db , y = F −1 x . When scaling is implemented along with partial pivoting (PP) to solve Ax = b by Gaussian elimination (GE), it is well known that certain ordered pairs ( D, F ) produce better computed solutions than those obtained in the absence of scaling, while others produce worse solutions. The two most common explanations of this fact are (1) ( D, F ) modifies (magnifies or reduces) the classical condition number of A , and (2) ( D, F ) modifies the magnitudes of the elements of A . In case (2), if a scaling yields entries of approximately the same magnitude, it is called an equilibration. Here, the underlying hyperplane geometry of both the sweepout phase and the back-substitution phase of GE is used to achieve a new level of understanding. We present what we believe to be a better explanation of how scaling or equilibration influences PP in the selection of pivot equations, a process critical to both phases of GE.