1. Setting the Stage
Society for Industrial and Applied Mathematics eBooks · 1998
A very large condition number of the coefficient matrix A in a linear system of equations implies that some (or all) of the equations are numerically linearly dependent. Hence, the standard advice to avoid solving such systems numerically is not bad. Indeed, it is sometimes the case that the large condition number is caused by an incorrect mathematical model which should be modified before one attempts to compute a numerical solution. Numerical “tools,” such as the singular value decomposition (SVD) (see §2.1), can identify the linear dependencies and thus help to improve the model and lead to a modified system with a better-conditioned matrix. This modified system can then be solved by standard numerical techniques [36], [154], [230]. However, there are classes of problems for which the coefficient matrix is correctly very ill conditioned, i.e., where this property is part of the formulation of the problem. Then the standard linear algebra techniques no longer apply, and the numerical treatment often becomes more difficult. This book gives a survey of advanced numerical methods for solving such problems with ill-conditioned matrices. 1.1. Problems with Ill-Conditioned Matrices The numerical treatment of very ill conditioned linear systems of equations is more complicated than the treatment of well-conditioned systems, for the following two reasons. • The user should know what kind of ill conditioning to expect and how to deal with it. Is the problem rank deficient or ill posed? Is it possible to regularize, i.e., include additional information to stabilize the solution? What additional information is available and is it suited for stabilization purposes? • The user should also know which numerical regularization method should be used to treat the problem efficiently and reliably on a computer. Are both the analysis and the solution phases important? Should one prefer a direct or an iterative method? How much stabilization should be added?