Summary
We propose a new inertia‐revealing factorization for sparse symmetric matrices. The factorization scheme and the method for extracting the inertia from it were proposed in the 1960s for ...dense, banded, or tridiagonal matrices, but they have been abandoned in favor of faster methods. We show that this scheme can be applied to any sparse symmetric matrix and that the fill in the factorization is bounded by the fill in the sparse QR factorization of the same matrix (but is usually much smaller). We describe our serial proof‐of‐concept implementation and present experimental results, studying the method's numerical stability and performance.
Any symmetric matrix can be reduced to antitriangular form in finitely many steps by orthogonal similarity transformations. This form reveals the inertia of the matrix and has found applications in, ...e.g., model predictive control and constraint preconditioning. Originally proposed by Mastronardi and Van Dooren, the existing algorithm for performing the reduction to antitriangular form is primarily based on Householder reflectors and Givens rotations. The poor memory access pattern of these operations implies that the performance of the algorithm is bound by the memory bandwidth. In this work, we develop a block algorithm that performs all operations almost entirely in terms of level 3 BLAS operations, which feature a more favorable memory access pattern and lead to better performance. These performance gains are confirmed by numerical experiments that cover a wide range of different inertia.
This note deals with the inertia of the solutions to Generalised Stein's Inequalities (GSI) i.e. Stein's inequalities related to discrete descriptor models described through linear matrix pencils. ...Only strict inequalities are considered. It is shown that, under some very slight assumptions, there always exists a solution to the GSI and some properties on the inertia of this solution can be derived very easily. As an application, the finite root‐clustering of the pencil associated with a descriptor system in a union of separate discs is considered.
We consider the generalized eigenvalue problem
A
x
=
λB
x
, where
A
and
B
are real symmetric matrices and
B
is also positive definite. All the eigenvalues of this problem are real, and it is often ...necessary to compute only a few eigenvalues which are important for applications. In electronic structure calculations of materials, specific interior eigenvalues are of fundamental interest, since they play crucial roles in various industrial applications. In this paper, we propose an approach based on the inertia of the linear matrix pencil of
A
and
B
. The eigenvalue problem is restated, and a class of algorithms is presented for separating the target eigenvalues from the others.
The Bunch-Kaufman factorization is widely accepted as the algorithm of choice for the direct solution of symmetric indefinite linear equations; it is the algorithm employed in both LINPACK and ...LAPACK. It has also been adapted to sparse symmetric indefinite linear systems. While the Bunch--Kaufman factorization is normwise backward stable, its factors can have unusual scaling, with entries bounded by terms depending both on |A| and on $\kappa(A)$. This scaling, combined with the block nature of the algorithm, may degrade the accuracy of computed solutions unnecessarily. Overlooking the lack of a triangular factor bound leads to a further complication in LAPACK such that the LAPACK Bunch--Kaufman factorization can be unstable. We present two alternative algorithms, close cousins of the Bunch-Kaufman factorization, for solving dense symmetric indefinite systems. Both share the positive attributes of the Bunch-Kaufman algorithm but provide better accuracy by bounding the triangular factors. The price of higher accuracy can be kept low by choosing between our two algorithms. One is appropriate as the replacement for the blocked LAPACK Bunch-Kaufman factorization; the other would replace the LINPACK-like unblocked factorization in LAPACK. Solving sparse symmetric indefinite systems is more problematic. We conclude that the Bunch-Kaufman algorithm cannot be rescued effectively in the sparse case. Imposing the constraint of bounding the triangular factors leads naturally to one particular version of the Duff-Reid algorithm, which we show gives better accuracy than Liu's sparse variant of the Bunch-Kaufman algorithm. We extend the work of Duff and Reid in two respects that often provide higher efficiency: a more effective procedure for finding pivot blocks and a stable extension to pivot blocks of size larger than two.
Integral quadratic constraints (IQCs) can be used for proving stability of systems with uncertainties and nonlinearities. Similarly, IQCs can also be used for controller synthesis. Necessary and ...sufficient conditions for the existence of such a controller is derived. These conditions include linear matrix inequalities (LMIs) and matrix inertia specifying the number of negative eigenvalues of a matrix. In general, these conditions are non-convex. Connections to bilinear matrix inequalities and LMIs with rank constraints are also given.
PDF
