Stability of Methods for Matrix Inversion

Abstract

Inversion of a triangular matrix can be accomplished in several ways. The standard methods are characterised by the loop ordering, whether matrix-vector multiplication, solution of a triangular system, or a rank-1 update is done inside the outer loop, and whether the method is blocked or unblocked. The numerical stability properties of these methods are investigated. It is shown that unblocked methods satisfy pleasing bounds on the left or right residual. However, for one of the block methods it is necessary to convert a matrix multiplication into the solution of a multiple right-hand side triangular system in order to have an acceptable residual bound. The inversion of a full matrix given a factorization PA=LU is also considered, including the special cases of symmetric inde nite and symmetric positive de nite matrices. Three popular methods are shown to possess satisfactory residual bounds, subject to a certain requirement on the implementation, and an attractive new method is described. This work was motivated by the question of what inversion methods should be used in LAPACK.

Algorithm for the construction of self-energies for electronic transport calculations based on singularity elimination and singular value decomposition

“…The relative error introduced by the inversion of Q is proportional to ͑Q ͒. 17,18,28,35,36 Close to a surface state the smallest singular value is of the order of ␦, so that ͑Q ͒ ϰ ␦ −1 . As this is the dominant source of error in the calculation of the SE close to a surface state, we can approximate the relative error as…”

The Fast Generalized Parker–Traub Algorithm for Inversion of Vandermonde and Related Matrices

“…Therefore the Parker variant also involves the subtraction of like-signed numbers. 3 Thus, the Parker variant of the Parker-Traub algorithm is also subject to the above remark, saying that a result of the form (4.2) will not hold for the Parker algorithm as well. Summarizing, the arguments of [15] suggest that the slow Björck-Pereyra algorithm is accurate, whereas the fast Parker-Traub inversion algorithm is the other way around.…”

“…This would be useful, for example, in the case of multiple right-hand sides, but numerous sources predict a numerical breakdown, thus ruling out such a possibility. For example [3], motivated by the question of what inversion methods should be used in LAPACK, concludes with the remark: "we wish to stress that all the analysis here pertains to matrix inversion alone. It is usually the case that when a computed inverse is used as a part of a larger computation, the stability properties are less favorable, and this is one reason why matrix inversion is generally discouraged."…”

Testing linear algebra software

“…that is so bounded, as explained in Du Croz & Higham (1992) and Higham (1996, §13.3). This error apparently has not lead to the test being failed in any of the instances when the test software was run, indicating that for the random matrices used the ratio (9) has always been smaller than the threshhold.…”